1. Introduction
Recently, pressure-driven fluid-infused microchannels have gained importance, as they facilitate fluid transport by allowing the bulk liquid to slide over the infused phase, thereby reducing hydrodynamic resistance (Rothstein Reference Ruckenstein and Churaev2010; Villegas et al. Reference Vinogradova2019; Pakzad et al. Reference Pakzad, Nouri-Borujerdi and Moosavi2022). These surfaces feature pillars or ribs and grooves or cavities that trap liquid or gas within their microstructures, forming a low-friction layer. Consequently, flow resistance in fluid-infused surface (FIS) microchannels is reduced, as the liquid slides over the trapped phase (Liravi et al. Reference Liu, Yu, Zhao, Hao, You and Yue2020; Park et al. Reference Patlazhan and Vagner2021; Li et al. Reference Liravi, Pakzad, Moosavi and Nouri-Borujerdi2024). For improving the efficiency of passive systems in microfluidic applications such as lab-on-a-chip devices (Dittrich & Manz Reference Dittrich and Manz2006; Mark et al. Reference Nair, Adarsh, Nandakumar and Ranjith2010), biochemistry (Duffy et al. Reference Duffy, McDonald, Schueller and Whitesides1998; Thorsen et al. Reference Tolstoi2002) and microelectronics (Van Erp et al. Reference Vega-Sánchez and Neto2020; Gong et al. Reference Gong, Shen, Dai, Li and Gong2021), these FISs are employed to reduce the high flow resistance associated with their large surface-area-to-volume ratio. The simplest configuration of an FIS channel is surfaces with longitudinally or transversely arranged ribs and cavities. Early studies on gas-infused surfaces (GISs), also known as superhydrophobic surfaces (Ou et al. Reference Ou and Rothstein2004; Ou & Rothstein Reference Pakzad, Nouri-Borujerdi and Moosavi2005), demonstrated that interface exposure dictates slip, achieving 20 %–40 % pressure drag reduction (PDR) for laminar flows (Reynolds number,
$Re{\lt } 1000$
). However, GIS channels suffer from high lubricant diffusivity and compressibility, leading to drainage, reduced lifespan and diminished drag-reducing efficiency (Dilip et al. Reference Dilip, Jha, Govardhan and Bobji2014; Zhang et al. Reference Zhang and Kwok2023; Jouin et al. Reference Khomami2024).
Alternatively, liquid-infused surface (LIS) microchannels (see figure 1 a) were introduced, where a lubricating liquid layer within surface textures replaces the entrapped gas layer in GIS, offering enhanced flow, reduced friction and greater stability under pressure variations (Asmolov et al. Reference Asmolov, Nizkaya and Vinogradova2018; Pakzad et al. Reference Park, Choi and Kim2023; Ahuja et al. Reference Ahuja, Joshi and Agrawal2024). Studies on Newtonian fluids slipping over Newtonian infused liquids show greater stability than GIS, achieving 16 %–20 % drag reduction at low Reynolds numbers (Solomon et al. Reference Song, Song, Hu, Du, Du, Choi and Rothstein2014; Chen et al. Reference Chen, Gao, Stone and Li2016). Given the demand for non-Newtonian flows in microfluidic and biological applications, the effect of various infused lubricants and working fluid combinations was subject to investigations (Rahimi et al. Reference Rahmani, Larachi and Taghavi2023; Ray et al. Reference Reiner2023). Indeed, many studies show that shear-thinning fluids enhance drag reduction in both super-hydrophobic channels and air-impregnated ribbed microchannels (Crowdy Reference Crowdy2017; Patlazhan & Vagner Reference Philip2017; Javaherchian & Moosavi Reference Joseph, Mathew, Krishnaraj, Dilip and Ranjith2019; Gaddam et al. Reference Gaddam, Sharma, Ahuja, Dimov, Joshi and Agrawal2021; Schnitzer & Ray Reference Schönecker, Baier and Hardt2024). While most LIS surfaces use Newtonian infused fluids, shear-thinning infused fluids, with viscosity decreasing under higher shear rates, achieve significant drag reduction (34.9 %–63.6 %) in confined LIS channels under power-law modelling (Nair et al. Reference Nair, Nandakumar Chandran and Kumar Ranjith2024). A practical motivation for considering such lubricants is the hypothesis that their apparent viscosity decreases near the interface under high shear, which may promote slip, while remaining higher than that of the base solvent in lower-shear regions, which could aid lubricant retention.
To evaluate the drag reduction performance of FIS channels with microscopic features, a macroscopic representation is crucial, serving as a foundational tool for both experimental analysis and theoretical modelling. On a lubricant-infused surface characterised by locally dependent boundary conditions, the concept of effective slip is most appropriate to describe the overall slip measured on these surfaces (Andrienko et al. Reference Andrienko, Dünweg and Vinogradova2003; Xia Reference Zhang, Hu, Wen, Ren and Xie2024). Indeed, Navier initially proposed that slip length – a hypothetical projected distance at which the tangential velocity vanishes – could be used to assess drag reduction. Aligned with this concept, various analytical and experimental studies have explored fluid flow in microchannels with superhydrophobic grooved surfaces. First, Philip (Reference Rahimi, Shahsavari, Pakzad, Moosavi and Nouri-Borujerdi1972) provided an analytical solution for flow over ribbed surfaces and microgrooves. Later, Lauga & Stone (Reference Li, Zhou, Jiao, Guo and Fu2003) analytically examined pressure-driven Stokes flow through fluid-infused surfaces, assuming a combination of shear-free (interface) and no-slip (rib) boundary conditions. Teo & Khoo (Reference Thorsen, Maerkl and Quake2009) and Cheng et al. (Reference Cheng, Teo and Khoo2009) also analysed Stokes flow in microchannels with transverse or longitudinal grooves, assuming a shear-free gas–liquid interface. Ng & Wang (Reference Ng and Wang2009) derived a semi-analytical solution for flow over square microgrooves, finding longitudinal grooves reduce drag more effectively than transverse ones. Recently, Hatte & Pitchumani (Reference Hatte and Pitchumani2021a , Reference Hatte and Pitchumanib ) made attempts to predict flow and heat transfer characteristics analytically for channels with longitudinal, transverse and square post microgrooves, considering arbitrary known slip lengths at the interface, which also indicated that longitudinal grooves are beneficial for higher drag reduction values (PDR values of 10 %–30 %). Further, numerous researchers studied how interface shape affects slip characteristics, noting that protrusion angle impacts slip behaviour (Davis & Lauga Reference Davis and Lauga2009; Joseph et al. Reference Jouin, Robinet and Cherubini2020; Arun et al. Reference Arun, Dilip and Ranjith2021; Nguyen et al. Reference Ou, Perot and Rothstein2024). Often mathematically, such effective slip models are restricted to Stokes flow regimes and with the elliptical slip velocity profile assumptions at the interface (Schönecker & Hardt Reference Sochi2013; Schönecker et al. Reference Schönecker and Hardt2014).
Previous investigations indicate that accurate modelling of fluid behaviour in FIS microchannels relies on appropriate boundary conditions, particularly at the solid–liquid/gas interface (Gluzdov & Gatapova Reference Gluzdov and Gatapova2024; Rahmani, Larachi & Taghavi Reference Ranjith2024). The simplest approach in modelling ribs and cavities is to assume a mathematically smooth surface and apply appropriate heterogeneous boundary conditions (Lauga & Stone Reference Li, Zhou, Jiao, Guo and Fu2003; Davies et al. Reference Davies, Maynes, Webb and Woolford2006; Ranjith et al. Reference Ray, Bouvier and Papageorgiou2015; Ranjith Reference Ranjith, Vedantam and Patnaik2015). However, it cannot capture the cavity flow patterns and their concurrent effect on the hydrodynamic drag, which are actually present in the microchannel (Joseph et al. Reference Jouin, Robinet and Cherubini2020; Arun et al. Reference Arun, Dilip and Ranjith2021; Nair et al. Reference Ng and Wang2022). Indeed, Chakraborty et al. (Reference Chakraborty, Das and Chattoraj2007) and Chakraborty & Anand (Reference Chakraborty and Anand2008) developed a generalised formulation to account for surface roughness effects on fluid friction and implied that microgroove geometry dictates the slip characteristics (Dey et al. Reference Dey, Saha and Chakraborty2018). Although theoretical studies assume a flat interface, local pressure variations create convex or concave menisci, slightly enhancing or reducing effective slip, respectively (Asmolov et al. Reference Asmolov, Nizkaya and Vinogradova2018; Song et al. Reference Tang, Lu, Zhang, Wang and Tao2018). Heterogeneous surface models explicitly consider solid, liquid and gas regions, simulating composite surfaces like superhydrophobic or liquid-infused ones, often requiring two-phase flow conditions. Finding an explicit formulation for effective slip length, which reduces the problem to a single-phase problem (Chen, Han & Hwang Reference Chen, Han and Hwang2020; Schönecker et al. Reference Schönecker and Hardt2014), has been a challenge and also neglects the velocity changes at the boundary due to the presence of ribs and cavities. Several researchers (Rahimi et al. Reference Rahmani, Larachi and Taghavi2023; Nair et al. Reference Nair, Nandakumar Chandran and Kumar Ranjith2024) have considered these complex microgroove geometries for assessing microchannel slip flow hydrodynamics, but this is a cumbersome numerical/experimental process.
The concept of apparent slip simplified the problem into a widely accepted two-phase flow approach, incorporating a hypothetical lubricant film that creates a slippery interface. Prior to apparent slip models, the concept of molecular slip, where the molecules of liquid slip directly over the solid surface, was taken into consideration for capillary flows (Tolstoi Reference Tretheway and Meinhart1952; Blake Reference Blake1990; Barrat & Bocquet Reference Barrat and Bocquet1999). However, later experimental studies imply anomalously high surface diffusion, which conflicts with the theory, leading to the adoption of continuum-level apparent slip models. One approach treats apparent slip as a continuum consequence of a viscosity-depleted near-wall layer – for instance, a hydrodynamic boundary layer with lower viscosity near hydrophobic substrates (Vinogradova Reference Xia1995; Sochi Reference Solomon, Khalil and Varanasi2011; Patlazhan & Vagner Reference Philip2017). Considering a main assumption as bulk fluid flow exhibits a linear shear profile, but within a thin region, the effective viscosity is reduced; extrapolating the bulk profile to the wall produces a finite slip velocity and defines a slip length. However, another framework proposes a tenuous interfacial fluid layer separating liquid and solid (Ruckenstein & Churaev Reference Schnitzer and Ray1991). In this limit, the boundary behaves akin to a fluid–fluid interface with weak tangential resistance; however, the postulate of a stable nanometric gap remains an idealisation and has not been conclusively verified. Unlike the viscosity-depletion approach, wherein the assumption of a bulk fluid linear profile with linear extrapolation (thereby giving slip length directly proportional to hydrodynamic boundary layer thickness), and lack of knowledge of the rheology of the near-wall layer and its thickness, make it a less suitable option for characterising the LIS channels. Conversely, the co-flowing formulation (treating the lubricating phase as a physical secondary fluid) becomes an effective method to develop generalised equations that aid the experiments of pressure drop measurement methods generally used in microfluidic devices (Vega-Sánchez & Neto Reference Villegas, Zhang, Jarad, Noor and Didar2022; Liu et al. Reference Mark, Haeberle, Roth, Von Stetten and Zengerle2024). Apparent slip studies for Newtonian liquid-infused channels are typically conducted under the assumption that the trapped fluid in the infused surfaces forms a hypothetical slip layer at the interface. In this framework, the fluid-infused surface is taken as the reference interface, while a no-slip boundary condition is applied at the bottom of the hypothetically extended slip layer, see figure 1(b). Using this modelling approach with predefined interface heights – often choosing the mean surface roughness as the slip-layer thickness and taking the interface as the reference boundary – Dey et al. (Reference Dey, Saha and Chakraborty2018) derived simplified solutions. However, discrepancies with experimental observations indicated that microgroove geometry strongly influences the slippery hydrodynamics of fluid-infused surfaces. This highlights the need for improved modelling of the assumed slip-layer thickness, which should depend on both the rheology of the entrapped fluid and the underlying surface geometry (Schönecker & Hardt Reference Sochi2013; Schönecker et al. Reference Schönecker and Hardt2014).
(a) Three-dimensional schematic of a ribbed microchannel with non-Newtonian infused lubricant, (b) simplified non-Newtonian LIS (NLIS) flow domain with immiscible fluids having primary fluid (PF) and secondary fluid (SF) modelled as power law fluid.
Figure 1. Long description
The diagram consists of two parts labeled (a) and (b). Part (a) shows a three-dimensional schematic of a ribbed microchannel with non-Newtonian infused lubricant. The channel wall is depicted in yellow, the infused lubricant in red, and the working fluid in blue. The flow direction is indicated by arrows labeled ‘In-flow’ and ‘Out-flow’. Part (b) shows a simplified NLIS flow domain with immiscible fluids, where the channel wall is in yellow, the hypothetical slip layer in red, and the working fluid in blue. The flow direction is similarly indicated by arrows labeled ‘In-flow’ and ‘Out-flow’.
Newtonian fluid-infused microchannels modelled with apparent slip have been investigated by several researchers using appropriately chosen fluid properties (Tretheway & Meinhart Reference Tretheway and Meinhart2002; Zhang & Kwok Reference Zhang and Kwok2004; Ng & Wang Reference Nguyen, Lee, Ryu, Kim, Yoon and Chang2010; Alinovi & Bottaro Reference Alinovi and Bottaro2018), channel dimensions, functional requirements (Busse et al. Reference Busse, Sandham, McHale and Newton2013) and slip-generating mechanisms (Tretheway & Meinhart Reference Van, Remco, Reza, Luca and Matioli2004). Frequently, the working fluids used in real-life applications involving microchannels are non-Newtonian in nature. The enhancement of slip during flow over superhydrophobic surfaces is strongly influenced by the fluid’s rheology and has been extensively studied by researchers seeking to elucidate the mechanisms responsible for slip generation (Tang et al. Reference Teo and Khoo2012; Ray et al. Reference Reiner2023; Huang et al. Reference Huang, Ye, Yin, Gao, Tao, Li and Li2025). To capture the hydrodynamic behaviour relevant to such applications, various slip models have been employed in the literature, with non-Newtonian fluid models (Huang et al. Reference Huang, Ye, Yin, Gao, Tao, Li and Li2025; Javaherchian & Moosavi Reference Joseph, Mathew, Krishnaraj, Dilip and Ranjith2019; Krishnaram et al. Reference Lauga and Stone2024; Schnitzer & Ray Reference Schönecker, Baier and Hardt2024). Non-Newtonian channel flow with slip boundary condition (Ferrás et al. Reference Ferrás, Nóbrega and Pinho2012) has been analytically studied to support the observations of non-Newtonian working fluid channel flow, stating that a shear-thinning working fluid is ideal for drag-reduction enhancement for a given surface textured with microgrooves.
With all this information, concerns have been raised about identifying the main experimental parameter affecting the accuracy and reproducibility of microfluidic experiments of slip measured from the pressure drop method (Vega-Sánchez & Neto Reference Villegas, Zhang, Jarad, Noor and Didar2022). The lack of a generalised theory for slip measurements for NLIS channel experiments necessitates investigating the aforementioned alternative simplified approach, the apparent slip model. Upon simplification, the complex ribbed microchannel with infused lubricant could be modelled as two adjacent immiscible power-law fluids co-flowing between parallel plates under a prescribed pressure gradient (see figure 1
b for schematics). In this reduced formulation, the cavity-filled lubricant is modelled as a hypothetical secondary layer of thickness
$\delta$
, while the bulk working fluid forms the primary layer above it. The purpose of this representation is not to resolve the detailed groove geometry, meniscus shape or cavity recirculation, but to obtain an analytically tractable framework in which the coupled roles of interfacial velocity continuity, tangential shear-stress continuity, viscosity contrast, slip layer thickness and non-Newtonian rheology can be examined explicitly. The model should therefore be interpreted as an idealised apparent slip representation for studying parametric trends in NLIS flows, rather than as a complete homogenised substitute for all texture-resolved liquid-infused microchannel geometries. Accordingly, geometric effects such as discrete groove topology, meniscus curvature and cavity-scale recirculation are outside the scope of the present model. This simplified co-flowing model does not capture the essential hydrodynamics of rib–cavity LIS flows, but rather represents the trapped lubricant as a low-viscosity sublayer that enables continuum-level prediction of apparent slip and drag reduction consistent with prior studies of Newtonian LIS channels (Dey et al. Reference Dey, Saha and Chakraborty2018; Liu et al. Reference Mark, Haeberle, Roth, Von Stetten and Zengerle2024). Thus, this work involves analytical modelling of NLIS channel slip quantification by simplifying the problem for a pressure-driven flow of two adjacent immiscible non-Newtonian fluids. The primary research objectives can be outlined as follows.
-
(i) To formulate generalised expressions for the velocity profile, apparent slip characteristics and drag reduction behaviour associated with slip flow in NLIS channels.
-
(ii) To develop a simplified analytical apparent slip model to replicate the superhydrophobic behaviour of NLIS microchannels under pressure-driven flow with a non-Newtonian core fluid.
-
(iii) To investigate the influence of the rheological properties of power-law-modelled non-Newtonian fluids and the interface viscosity ratio on pressure drag reduction, wall friction and slip flow rate.
The outline of the manuscript is as follows. Section 2 details the mathematical modelling of the apparent-slip model for an NLIS microchannel. In addition, the factors affecting the flow characteristics of the primary fluid flow, the apparent slip characteristics governing the phenomenon and the associated slip flow hydrodynamics are elaborated in § 3, followed by the conclusions in § 4.
2. Mathematical modelling
To simplify the complex geometry and fluid–fluid interface of an NLIS microchannel, a simplified apparent slip model is proposed. The specifics of model development, derivations of velocity profiles for pressure-driven NLIS microchannel flow, are discussed in this section.
2.1. Problem definition
(a) A two-dimensional symmetric schematic of pressure-driven flow in an NLIS ribbed microchannel illustrating the velocity profile of the PF over the no-slip rib and slip cavity regions. The cavity dimensions are
$w \times d$
with periodicity
$L_p$
and shear-free fraction. (b) A representation of a simplified two-phase co-flowing, pressure-driven Poiseuille flow of power-law fluids mimicking an NLIS channel, wherein the core (PF) fluid slips over the infused (SF) fluid with a slip-layer thickness of
$\delta$
. Here, the legend for the image can be taken as light blue area 
represents primary fluid (
$\mu _p = k_p \dot {\gamma }^{n_p-1}$
), 
represents secondary fluid (
$\mu _s ={} k_s \dot {\gamma }^{n_s-1}$
), black solid line 
denotes no-slip wall and blue dashed line 
denotes a fluid–fluid interface.
Figure 2. Long description
The diagram consists of two parts labeled (a) and (b). Part (a) shows a two-dimensional symmetric schematic of pressure-driven flow in a ribbed microchannel. It illustrates the velocity profile of the primary fluid over the no-slip rib and slip cavity regions. The cavity dimensions are depicted with periodicity and shear-free fraction. Part (b) represents a simplified two-phase co-flowing, pressure-driven Poiseuille flow of power-law fluids mimicking a fluid-infused channel. The core fluid slips over the infused fluid with a slip-layer thickness. The black solid line denotes a no-slip wall, and the dashed line represents a fluid-fluid interface.
When pressure is applied along the streamwise direction on the working fluid, the NLIS channel can be modelled in two dimensions with transverse ribs and periodically placed cavities (size
$w \times d$
and periodicity
$L_p$
) infused with lubricants, as depicted in figure 2(a). For simplicity, the complex ribbed surfaces can be replaced with a mathematically smooth surface and a hypothetical slip layer of finite thickness (
$\delta$
) at the wall, over which the core working fluid flows, as shown in figure 2(b). Specifically, the core flow is considered the primary fluid (PF) and the lubricant in the hypothetical slip layer is treated as the secondary fluid (SF), allowing the use of an apparent slip model to characterise the system. Further, it is assumed that the secondary and primary fluids are non-Newtonian, preserving the rheological properties of the infused lubricant and working fluid, respectively. Thereby, the NLIS channel with a slip layer is reduced to a problem of two immiscible non-Newtonian fluids sliding over one another (Bird, Armstrong & Hassager Reference Bird, Armstrong and Hassager1987; Bird Reference Bird2002). This is effectively equivalent to plane Poiseuille flow through an NLIS microchannel. Here, the symmetrical behaviour of fluid flow enables us to consider a symmetry plane
$AA'$
and of length
$L$
. The two parallel surfaces are separated by a distance
$H=2h$
, where
$h$
is the half-height of the channel, and the slip layer thickness is
$\delta$
, which is considered. The parameter
$\delta$
is of significance in drag reduction estimation, as it emerges naturally from the groove geometry and flow conditions governed by the Reynolds number and has the upper hand over the common practice of assuming a predefined interface shape, which may be inadequate for analysing slip flows. Here, the lower boundary of the secondary layer is treated as no-slip, representing the solid–lubricant interface inside the surface cavities, while interfacial mobility (
$u_{\textit{slip}}\gt 0$
) occurs at the liquid–liquid interface through shear continuity. The model fixes the reference plane at the channel surface and hypothetically shifts the lubricating layer downward by
$\delta$
, keeping the channel height constant. In this reduced formulation, apparent slip is attributed to the motion of the upper fluid over a lubricating film rather than to direct wall slip (Dey et al. Reference Dey, Saha and Chakraborty2018; Liu et al. Reference Mark, Haeberle, Roth, Von Stetten and Zengerle2024; Rahmani et al. Reference Ranjith2024). Unlike viscosity-depletion models such as that of Patlazhan & Vagner (Reference Philip2017), the reduced formulation directly represents momentum exchange through continuity of velocity and tangential shear stress at the fluid–fluid interface, making slip a consequence of layered fluid interaction rather than near-wall viscosity reduction. Moreover, to establish a connection with the conventional Maxwell–Navier effective slip length
$\beta$
, the interface slip velocity is obtained from the linearly extrapolated velocity profile as
$u_{\textit{slip}}=\beta ({\text{d}u}/{\text{d}y})$
, as depicted in figure 2(b). The present formulation builds on multilayer co-flow analyses used for immiscible and rheologically distinct fluids, where interfacial shear continuity dictates velocity distribution (Han & Shetty Reference Han and Shetty1976; Khomami Reference Krishnaram, Chandran, Ha and Kumar1990). It extends this framework to microchannel LIS flows, where apparent slip arises from the interfacial boundary conditions between the working and lubricating fluids.
The non-Newtonian fluids are modelled as power-law liquids whose viscosity depends on the shear force exerted by the counterpart fluid. The power-law model (Reiner Reference Rothstein1926; Chhabra & Richardson Reference Chhabra and Richardson2011) describes the rheological behaviour of the fluid and is identified by the viscosity index
$n$
, given by
$\mu _i = k_i\dot {\gamma }^{n_i-1},$
where
$k$
is the fluid-specific flow consistency index,
$\dot {\gamma }$
is the strain rate,
${\text{d}u}/{\text{d}y}$
, and the suffix i can denote
$p$
and
$s$
that represent primary and secondary fluids. Note that the PF and SF can be shear-thinning (
$n_i\lt 1$
), Newtonian (
$n_i=1$
) or shear-thickening (
$n_i\gt 1$
), depending on the viscosity index. Although more general rheological models, such as Carreau–Yasuda, Cross, Herschel–Bulkley and Oldroyd-type models are available, the power-law model is adopted here as a reduced constitutive approximation that permits closed-form analytical development of the present two-layer problem using minimal fitting parameters, namely the flow consistency (
$k_i$
) and flow behaviour (
$n_i$
) indices. This choice is made primarily for analytical tractability and for examining how shear-dependent viscosity variation influences the predicted apparent slip behaviour. At the same time, the limitations of the power-law model should be recognised clearly. In particular, it does not reproduce the finite zero-shear and infinite-shear viscosity plateaus captured by more general rheological models, and it may therefore over- or under-predict viscosity outside the intermediate shear-rate regime for which it is typically used. Accordingly, the present results should be interpreted within the scope of this constitutive idealisation rather than as a generally validated constitutive description for all NLIS flow conditions. Our aim is to develop a general expression for the velocity profile in this geometry. Since the flow is pressure-driven, the velocity profiles for both fluids will be nonlinear, as shown in figure 2(b). At the hypothetical wall boundary, a no-slip condition for the SF (
$u_{wall=0}$
at
$y = -\delta$
) is defined, so the interface experiences a finite velocity
$u_{\textit{slip}}$
at
$y = 0$
. Moreover, the stress at the interface (
$y = 0$
) is the same for both the PF and SF.
2.2. Analytical model of velocity profile
To develop an apparent slip model for the NLIS microchannel flow, the flow is assumed to be fully developed, laminar in nature, and having steady and uniform characteristics. In addition, both the lubricant and the working fluids are also considered non-Newtonian, immiscible to each other, and incompressible. With regards to channel dimensions, a high aspect ratio channel, i.e. the width of the channel is much larger than its height so that the effect of side walls can be neglected, is investigated here. Considering the above-mentioned assumptions for the primary and secondary fluids flowing one over the other at a low Reynolds number with
$u_p$
and
$u_s$
velocities, respectively, under a constant pressure gradient of
${\text{d} P}/{\text{d} x}$
in the stream-wise direction. The governing momentum balance equations for primary and secondary non-Newtonian fluids, as shown in figure 2(b), can be written as follows:
\begin{align} k_p \frac{\text{d}}{\text{d}y} \left [ \left ( \frac{\text{d}u_{p}}{\text{d}y} \right )^{n_p} \right ]-\frac{\text{d} P}{{\rm d} x}&=0 \quad \text{for}\ 0 \leqslant y \leqslant h, \quad \text{and}\nonumber\\ k_s \frac{\text{d}}{\text{d}y} \left [ \left ( \frac{\text{d}u_{s}}{\text{d}y} \right )^{n_s} \right ] -\frac{\text{d} P}{{\rm d} x}&=0 \quad \text{for} \ -\delta \leqslant y \leqslant 0. \end{align}
These equations are integrated twice and the constants are evaluated using the following boundary conditions (no-slip at the lower wall, continuity of velocity and tangential shear stress at the interface, and symmetry at the channel centreline) to obtain velocity profiles:
\begin{align} &\qquad\qquad\qquad\,\,\,\text{at } y = -\delta , \quad u_{s} = 0; \end{align}
\begin{align} &\text{at } y = 0, \quad u_{p} = u_{s}\quad \text{and}\quad k_p \left (\frac{\text{d}u_{p}}{\text{d}y}\right )^{n_p} = k_s \left (\frac{\text{d}u_{s}}{\text{d}y}\right )^{n_s}; \end{align}
\begin{align} &\qquad\qquad\qquad\text{and at } y = h , \quad \frac{\text{d}u_{p}}{\text{d}y} = 0. \end{align}
In fact, the governing equations presented in (2.1) are obtained by reducing the general x-momentum equation (Gerhart et al. Reference Gerhart, Hochstein and Gerhart2020) under the appropriate flow assumptions. Although the resulting form resembles the Stokes equation in its mathematical structure, it remains valid for moderate Reynolds numbers as long as the flow is steady, laminar and fully developed (Ou et al. Reference Ou and Rothstein2004; Dey et al. Reference Dey, Saha and Chakraborty2018). For convenience, a dummy variable for the pressure gradient,
$ G = -({\text{d} P}/{\text{d}x})$
, is introduced. Solving the pressure-driven Poiseuille flow in an NLIS microchannel yields the final expression for the fully developed velocity profiles of both the primary and secondary fluids, given by
\begin{align} \nonumber u_{p} &= \left ( \frac {G}{k_p} \right )^{\tfrac{1}{n_p}} \left ( \frac {n_p}{n_p + 1} \right ) \left ( h \right )^{\tfrac{n_p +1}{n_p}} \left [ 1 - \left ( 1 -\frac {y}{h} \right ) ^{\tfrac{n_p +1}{n_p}}\right ] \nonumber\\ &\quad + \left ( \frac {G}{k_s} \right )^{\tfrac{1}{n_s}} \left ( \frac {n_s}{n_s + 1} \right ) (h)^{\tfrac{n_s + 1}{n_s}} \left [ \left (\frac {\delta }{h} + 1\right )^{\tfrac{n_s + 1}{n_s}} - 1 \right ] \end{align}
\begin{align} \text{and}\quad u_{s} &= \left ( \frac {G}{k_s} \right )^{\tfrac{1}{n_s}} \left ( \frac {n_s}{n_s + 1} \right ) (h)^{\tfrac{n_s + 1}{n_s}} \left [ \left (\frac {\delta }{h} + 1\right )^{\tfrac{n_s + 1}{n_s}} - (1 - \frac {y}{h})^{\tfrac{n_s + 1}{n_s}} \right ]\!. \end{align}
Detailed steps of the solution method employed to solve the governing equations are provided in Appendix A.
Note that at
$ y = 0$
, the primary fluid slips over the secondary fluid and it experiences a finite velocity at the interface, which could be modelled as Navier’s slip condition. This phenomenon could be mathematically formulated similarly to the Navier slip boundary condition experienced by the primary fluid, i.e. an apparent slip mimics the real scenario of effective slip offered by a transverse rib surface, a no-slip-slip surface in the case of an NLIS microchannel. The Navier slip model in general form is given by
$u_{wall}=\beta (\text{d} u/\text{d} y)_{wall}$
, where
$\beta$
is the apparent slip length of the NLIS channel, and is the expression for the hypothetical space-averaged velocity at the smooth wall. However, for the current setting, the apparent slip length
$\beta$
is assumed to be linked to the PF velocity gradient at the interface, such that
$u_{\textit{slip}}=\beta (\text{d} u_p/\text{d} y)_{\textit{interface}}$
. The notable difference between effective and apparent slip lies in the origin of the slip velocity: in the former, slip is explicitly prescribed, whereas in the latter, it arises implicitly from the lubricant’s velocity field. The factors influencing flow phenomenon, slip velocity and the associated apparent-slip characteristics shall be discussed in § 3. Additionally, limited two-dimensional two-phase computations for selected steady laminar cases are carried out for illustrative comparisons with the analytical velocity profiles of the layered flow; the numerical model details for the same are provided in the supplementary material S1 available at https://doi.org/10.1017/jfm.2026.11719. As a remark, since no dedicated linear stability analysis is performed here, these numerical results should not be interpreted as establishing the stability of the stratified fluid flow.
3. Results and discussion
This section details the flow characteristics of the layered pressure-driven flow of non-Newtonian fluids mimicking primary fluid flow with apparent slip in an NLIS microchannel. Particularly, the interaction of shear forces, slip velocity at the interface and their combined effects on apparent slip characteristics such as slip length, flow rate and pressure drag reduction are discussed. Furthermore, the slip flow hydrodynamics, including shear stress distributions and the Poiseuille number, are examined to offer deeper insights into the behaviour of ribbed microchannels with non-Newtonian lubricants.
3.1. Non-dimensional velocity distribution
It is important to understand the velocity profile associated with the non-Newtonian immiscible fluid flow over one another in a channel to model the apparent slip length formulations. To non-dimensionalise the velocity profiles in (2.5) and (2.6), the average velocity of the primary fluid is used. This choice is appropriate, as it aligns well with the apparent slip model for the primary fluid and is also a measurable quantity in experiments. Here, the average velocity of the primary fluid
$\langle \overline {u_p}\rangle$
is defined as
\begin{align} \langle \overline {u_p}\rangle = \frac {1}{h} \int _{0}^{h} u_p(y) \, \text{d}y,\end{align}
and using (2.5), the average velocity is obtained as
\begin{align} \langle \overline{u_p} \rangle = \left( \frac{G}{k_p} \right)^{\tfrac{1}{n_p}} \!\left( \frac{n_p}{2n_p + 1} \right) \!\left( h \right)^{\tfrac{n_p +1}{n_p}} \!+ \left( \frac{G}{k_s} \right)^{\tfrac{1}{n_s}}\! \left( \frac{n_s}{n_s + 1} \right)\! (h)^{\tfrac{n_s + 1}{n_s}} \left[ \!\left(\frac{\delta}{h} + 1\!\right)^{\tfrac{n_s + 1}{n_s}} - 1 \!\right].\\[-6pt]\nonumber \end{align}
Further, the velocity profiles for both primary and secondary fluid are non-dimensionalised as
\begin{align} \frac{u_p}{\langle \overline{u_p}\rangle} &= \frac{\frac{1}{\Gamma} \left( \frac{n_p}{n_p + 1} \right) \left( \frac{n_s + 1}{n_s} \right) \left[ \frac{ 1 - \left( 1 -\frac{y}{h} \right) ^{\tfrac{n_p +1}{n_p}}}{\left(\frac{\delta}{h} + 1\right)^{\tfrac{n_s + 1}{n_s}} - 1} \right] + 1} {\frac{1}{\Gamma} \left( \frac{n_p}{2n_p + 1} \right) \left( \frac{n_s + 1}{n_s} \right) \left[ \frac{ 1}{\left(\frac{\delta}{h} + 1\right)^{\tfrac{n_s + 1}{n_s}} - 1} \right] + 1}, \end{align}
\begin{align} \frac{u_s}{\langle\overline{u_p}\rangle} &= \frac{ \left[ \frac{ \left(\frac{\delta}{h} + 1\right)^{\tfrac{n_s + 1}{n_s}} - \left(1 - \frac{y}{h}\right)^{\tfrac{n_s + 1}{n_s}}}{\left(\frac{\delta}{h} + 1\right)^{\tfrac{n_s + 1}{n_s}} - 1} \right] } {\frac{1}{\Gamma} \left( \frac{n_p}{2n_p + 1} \right) \left( \frac{n_s + 1}{n_s} \right) \left[ \frac{ 1}{\left(\frac{\delta}{h} + 1\right)^{\tfrac{n_s + 1}{n_s}} - 1} \right] + 1}, \end{align}
where the interface apparent viscosity ratio
$\varGamma = (Gh )^{({n_{p} - n_{s}})/({n_{p} n_{s}}) } ( {k_{p}^{1/n_{p}}}/{k_{s}^{1/n_{s}}} )$
. Note that
$Gh$
corresponds to the interfacial shear stress, and its significance and influence on the hydrodynamics are discussed in Appendix B. This indicates that the governing parameters for flow behaviour are viscosity indices (
$n_s$
and
$n_p$
), interface apparent viscosity ratio of primary and secondary fluids (
$\varGamma$
), and the slip layer thickness (
$\delta /h$
). Further, a generalised velocity profile for the entire domain is represented as a piecewise linear function:
\begin{equation} \frac {u}{\langle \overline {u_p}\rangle } = \gamma \frac {u_s}{\langle \overline {u_p}\rangle } + (1- \gamma ) \frac {u_p}{\langle \overline {u_p}\rangle } \hspace {0.5cm} \begin{cases} \gamma = 1 & -\delta \leqslant y \leqslant 0, \\ \gamma = 0 & 0 \lt y \leqslant h. \end{cases} \end{equation}
Perhaps the slip velocity at the interface is the most important parameter dictating the performance of an NLIS channel – the greater the
$u_{\textit{slip}}$
, the higher the efficiency. The extent of sliding showcased by the primary fluid over the secondary fluid can be quantified by the interface velocity, commonly termed the primary fluid’s slip velocity. This non-dimensional slip velocity at interface (
$y=0$
) is defined as
$({u_{\textit{slip}}}/{\langle \overline {u_p}\rangle })=({u}/{\langle \overline {u_p}\rangle }_{|y=0})$
, which is obtained from (3.5) as
\begin{equation} \frac{u_{\textit{slip}}}{\langle\overline{u_p}\rangle} = \frac{ 1 } {\frac{1}{\Gamma} \left( \frac{n_p}{2n_p + 1} \right) \left( \frac{n_s + 1}{n_s} \right) \left[ \frac{ 1}{\left(\frac{\delta}{h} + 1\right)^{\tfrac{n_s + 1}{n_s}} - 1} \right] + 1}. \end{equation}
Alternatively, the interfacial slip velocity, implicitly obtained from two-phase simulations, is quantitatively compared with analytical predictions, demonstrating good agreement, and the comparison is provided in the supplementary material S1.
3.2. Effect of influencing parameters
The effects of individual model parameters are evaluated with respect to slip behaviour to further investigate the drag reduction mechanism associated with apparent-slip behaviour. It is evident from figure 3(a) that, among the parameters, one that influences the slip nature of the primary fluid is the defined interface apparent viscosity ratio,
$\varGamma$
. For brevity, the expression for
$\varGamma$
can be further reduced to
\begin{equation} \Gamma = \frac{k_{p}\left(\frac{{\rm d}u_{p}}{{\rm d}y}\right)_{|y=0}^{n_p - 1}} {k_{s}\left(\frac{{\rm d}u_{s}}{{\rm d}y}\right)_{|y=0}^{n_s - 1}} = \frac{\mu_{p |y=0}}{\mu_{s |y=0}} . \end{equation}
Developed velocity profiles for non-Newtonian viscous double layer with varying (a) interface apparent viscosity ratio,
$\varGamma$
(
$n_s = 0.7$
,
$n_p = 1.3$
,
$\delta /h = 0.5$
), (b) secondary fluid power-law index,
$n_s$
(
$\varGamma = 1$
,
$n_p = 1$
,
$\delta /h = 0.5$
), (c) primary fluid power-law index,
$n_p$
(
$\varGamma = 1$
,
$n_s = 1$
,
$\delta /h = 0.5$
) and (d) slip layer thickness,
$\delta /h$
(
$\varGamma = 1$
,
$n_s = n_p = 1$
).
Figure 3. Long description
The image contains four line graphs labeled (a), (b), (c), and (d), each depicting developed velocity profiles for a non-Newtonian viscous double layer under different conditions. Graph (a) shows the effect of varying interface apparent viscosity ratio with three different values represented by red dashed, black dashed, and blue solid lines. Graph (b) illustrates the impact of the secondary fluid power-law index with five different values shown by red dashed, red dashed-dotted, black solid, blue dotted, and blue dashed lines. Graph (c) presents the influence of the primary fluid power-law index with five different values indicated by red dashed, red dashed-dotted, black solid, blue dotted, and blue dashed lines. Graph (d) demonstrates the effect of slip layer thickness with five different values represented by red dashed, red dashed-dotted, black solid, blue dotted, and blue dashed lines. The x-axis represents the normalized velocity (u/up), and the y-axis represents the normalized height (y/h). All values are approximated.
First, the effect of the interface apparent viscosity ratio on the velocity distribution is analysed. For that, flow parameters are fixed (
$n_s = 0.7$
,
$n_p = 1.3$
,
$\delta /h = 0.5$
) and viscosity ratios are changed
$\varGamma =0.1, 1 \text{ and } 10$
. Subsequently, the results obtained for shear-thinning SF and shear-thickening PF are presented in figure 3(a). Three different cases are discussed in detail here,
$\varGamma =1$
,
$\varGamma \lt 1$
and
$\varGamma \gt 1$
. Indeed,
$\varGamma =1$
corresponds to
$\mu _s=\mu _p$
at the interface, implying equal interfacial apparent viscosities of the two fluids, the slope mismatch across the interface vanishes and the composite velocity profile reduces to the classical parabolic Poiseuille profile. In contrast, when
$\varGamma \lt 1$
, the interface viscosity
$\mu _s\gt \mu _p$
(since
$n_s=0.7$
and
$n_p=1.3$
) and thus the viscous drag in the slip layer is greater in the PF due to a rapid change in viscosity. Consequently, the piecewise velocity profile develops a kink at the interface, reflecting the mismatch in velocity gradients across the two layers. Likewise, for
$\varGamma \gt 1$
, a pronounced change in slope is observed at the interface because the two layers sustain different velocity gradients. Importantly,
$\varGamma$
is a flow-governed parameter rather than an intrinsic fluid property. Since it explicitly depends on the imposed pressure gradient
$G$
and the channel height
$h$
, any change in the rheological index of the primary fluid (
$n_p$
) or secondary fluid (
$n_s$
) alters the exponent of
$(Gh)$
and, consequently, the magnitude of
$\varGamma$
. Therefore, to maintain the same
$\varGamma$
for different
$n_p$
and
$n_s$
, the driving pressure gradient
$G$
must change accordingly, because the interfacial shear stress is set by the pressure gradient scale
$Gh$
. Notably, the consistency parameter
$k$
and the power-law index
$n$
are fluid parameters that are explicitly prescribed, whereas
$\varGamma$
is an implicit property that evolves from the flow-induced velocity gradients at the interface, as evident from (3.7).
Generally, the non-dimensional parameter
$\varGamma$
governs the velocity field and its gradients in both fluids. Smaller
$\varGamma$
implies weaker interfacial slip, whereas larger
$\varGamma$
suggests stronger slip, as shown in figure 3(a). For Newtonian fluids,
$\varGamma =( {k_p}/{k_s})$
, equivalent to the conventional LIS parameter, viscosity ratio ’
$N$
’ (Schönecker & Hardt Reference Sochi2013; Nair et al. Reference Nair, Nandakumar Chandran and Kumar Ranjith2024). Among the cases shown, the maximum interfacial velocity occurs at
$\varGamma =10$
and the minimum at
$\varGamma =0.1$
. This behaviour is governed by interfacial shear, with fluid rheology described by the power-law indices
$n_s$
and
$n_p$
. Previous studies showed that a shear-thinning infused fluid promotes drag reduction; accordingly, the primary fluid develops larger velocity gradients and greater interfacial slip (Nair et al. Reference Nair, Nandakumar Chandran and Kumar Ranjith2024).
It is critical from a design perspective to understand the combination of primary and secondary fluids to enhance slip characteristics within the geometrical constraints provided. Moreover, the PF and SF can be Newtonian, shear-thinning or shear-thickening, depending on the viscosity index
$n_i$
, which significantly affects the hydrodynamics. First, the PF properties are fixed (
$\varGamma = 1$
,
$n_p = 1$
,
$\delta /h = 0.5$
), the SF viscosity index is varied over the range,
$ n_s = 0.25{-} 1.75$
, and the velocity profiles obtained are shown in figure 3(b). In contrast, figure 3(c) depicts the velocity distribution when the SF properties are fixed (
$\varGamma = 1$
,
$n_s = 1$
,
$\delta /h = 0.5$
) and the primary fluid index is varied in the range
$ n_p = 0.25{-} 1.75$
. From figure 3(b), it is understandable that for a constant
$\varGamma$
, a stronger shear-thinning behaviour of the secondary fluid enhances the slip behaviour of the PF. Similarly, as the PF viscosity index decreases, the fluid becomes more shear-thinning and the velocity profile becomes flatter, as shown in figure 3(c). Furthermore, a shear-thinning primary–secondary fluid combination would be ideal for enhanced drag reduction; however, the nature of the working fluid (PF) is constrained by practical applications.
Another influencing parameter is the non-dimensionalised slip layer thickness (
$\delta /h$
), which indirectly indicates the volume flow rate of secondary fluid flow for a given pressure gradient. The importance of slip layer thickness lies in the fact that it dictates the slip velocity required for the PF to achieve a targeted flow rate in an NLIS channel. Therefore, an increase in
$\delta /h$
elevates the interface slip velocity of the primary fluid, as shown in figure 3(d). Most importantly, as slip layer thickness vanishes,
$\delta /h = 0$
, the velocity profile traces the benchmark no-slip Poiseuille flow solution. Another major factor is that, as
$\delta /h$
increases, the interface slip velocity (at
$y=0$
) also rises rapidly. This is due to the movement of the hypothetical no-slip wall away from the interface; consequently,
$u_{\textit{slip}}$
enhances. From a practical point of view, higher
$\delta /h$
values, more than 0.5, are unrealistic for a LIS microchannel; therefore, a lower slip layer thickness (
$ \lt \lt \delta /h$
) is sufficient to achieve similar behaviour. In real scenarios with varied microgroove geometries (
$w$
,
$d$
and
$L_p$
), the defined slip layer thickness
$\delta$
relates to infused liquid exposure and core flow conditions; while this is mathematically established for Newtonian flow in the Stokes regime (Dey et al. Reference Dey, Saha and Chakraborty2018; Schönecker et al. Reference Schönecker and Hardt2014), it remains unexplored at higher Reynolds numbers. Similarly, for NLIS microchannel flow, such relations need to be derived to connect the hydrodynamic parameters to the slip layer thickness, which is introduced in the present study.
3.3. Apparent slip flow characteristics
For an NLIS channel, the apparent slip length formulation is proposed to simplify the multiphase problem to a single-phase flow problem with wall slip boundary conditions at the interface. Indeed, the characteristics of such a simplification are expressed quantitatively using the apparent slip length (
$\beta$
), slip flow rates and pressure drag reduction experienced.
3.3.1. Apparent slip length
The simplest expression for a slip length from known slip velocity could be derived using the Navier slip boundary condition, which relates the slip velocity to the velocity gradients, having a proportionality constant slip length,
$\beta$
, and the mathematical expression of such a condition can be written as
\begin{equation} u_{\textit{slip}} = \beta \frac{\text{d}u_{p}}{\text{d}y}, \quad \text{at } y = 0, \end{equation}
where
$ \beta$
is the apparent slip length and
$u_{\textit{slip}}$
is obtained from (3.6). A non-dimensionalised expression of the derived apparent slip length with respect to the half-height of the microchannel,
$\beta ^{\prime } = ({\beta }/{h})$
, gives the following expression:
\begin{equation} \beta^{\prime} = \Gamma \left( \frac{n_s}{n_s + 1} \right) \left[ \left(\frac{\delta}{h} + 1\right)^{\tfrac{n_s + 1}{n_s}} - 1 \right]. \end{equation}
Thus,
$\beta ^{\prime }$
is a function of SF viscosity index
$n_s$
and slip layer thickness
$\delta /h$
, so that
$u_{\textit{slip}}$
is decided by the properties of the interface and hypothetical layer. Thereby, the hydrodynamic resistance and the performance of NLIS channels are governed by these factors. Note that for a channel with a Newtonian lubricant (
$n_s =1$
) and a core fluid that is also Newtonian (
$n_p =1$
), the relation reduces to
\begin{equation} \beta ^{\prime } = \frac {k_{p}}{2 k_{s}} \left [ \left (\frac {\delta }{h} + 1\right )^2 - 1 \right ]\!. \end{equation}
This relation matches that of Dey et al. (Reference Dey, Saha and Chakraborty2018), except that they defined the channel half-height as
$h-\delta$
(where the mean surface roughness
$\delta$
is subtracted), later showing that
$\delta$
depends on cavity geometry and lubricant rheology.
To delineate the thin- and thick-layer regimes, the variation of
$\beta ^{\prime }$
with
$\delta /h$
(on a logarithmic horizontal axis) is illustrated in figures 4(a) and 4(b) as predicted by (3.9). Here, a nonlinear increase in
$ \beta ^{\prime }$
with
$ \delta /h$
is observed, underscoring the important role that slip layer thickness plays in amplifying slip effects. Both figures show
$\beta ^{\prime }$
rising monotonically with
$\delta /h$
: weak sensitivity for
$\delta /h \ll 1$
, then rapid growth as the slip layer thickness nears the channel height. This marks a transition from a near no-slip regime to an interface-dominated regime, where interfacial mobility governs as
$\delta \gt h$
. Consequently, the lubricating layer sustains the imposed shear with a larger velocity variation, thereby reducing hydrodynamic resistance and amplifying the apparent slip. Figure 4(a) illustrates that at a constant
$\varGamma$
, variations in
$n_s$
alter the balance between shear rate and apparent viscosity within the secondary layer. From the definition of
$\varGamma$
(refer to (3.7)), maintaining a fixed interface apparent viscosity ratio as
$n_s$
increases requires a lower interfacial shear rate
$(\text{d}u_s/\text{d}y)|_{y=0}$
, since the term
$k_s(\text{d}u_s/\text{d}y)^{n_s-1}$
remains constant. Physically, this means that the secondary fluid, which is now less shear-thinning, must sustain a larger local shear rate to achieve the same apparent interfacial viscosity. This implies that the same interfacial stress,
$\tau _{int} = k_s ({\text{d}u_s}/{\text{d}y} )^{n_s} = k_p ({\text{d}u_p}/{\text{d}y} )^{n_p}$
, is sustained with a different distribution of velocity gradients across the two layers. However, when the secondary fluid becomes shear-thickening (
$n_s\gt 1$
), the slip layer responds less readily to the imposed shear, so that the same interfacial stress is sustained with a lower shear rate within the lubricating layer. As a result, the velocity increases across the lubricating layer and, hence, the apparent slip length
$\beta '$
decreases with increasing
$n_s$
. Conversely, when
$n_s \ll 1$
(strongly shear-thinning behaviour), the decrease in the effective viscosity of the secondary layer is much more pronounced under shear, thereby reducing the viscous resistance to tangential momentum transfer at the interface and promoting greater interfacial mobility.
Non-dimensionalised slip length (
$\beta ^{\prime }$
) variations with respect to non-dimensionalised slip layer thickness
$({\delta }/{h})$
(a) at
$\varGamma = 0.5$
for varying
$n_s$
values; (b) at
$n_s = 1$
for varying
$\varGamma$
values. Here, (
) symbols denote the results of Schönecker et al. (Reference Schönecker and Hardt2014) and 
symbols denote the results of Nair et al. (Reference Nair, Nandakumar Chandran and Kumar Ranjith2024) (both having
$n_p =n_s = 1$
).
Figure 4. Long description
Two line graphs compare non-dimensionalized slip length variations with non-dimensionalized slip layer thickness. The first graph (a) shows variations at different values of n sub s, with lines representing n sub s values of 0.5, 0.7, 1, 1.3, and 1.5. The second graph (b) shows variations at different values of Gamma, with lines representing Gamma values of 0.1, 0.2, 0.5, 0.7, and 1.0. Both graphs include insets highlighting specific data ranges. The symbols denote results from Schnecker et al. (2014) and Nair et al. (2024), both with specific conditions. The x-axis represents non-dimensionalized slip layer thickness, and the y-axis represents non-dimensionalized slip length. The graphs illustrate how these parameters influence slip length under varying conditions.
Similarly, figure 4(b) presents the variation of the normalised apparent slip length,
$\beta ^{\prime }$
, with the relative secondary-layer thickness,
$\delta /h$
, for different interfacial apparent viscosity ratios (
$\varGamma$
), at a fixed SF power-law index
$n_s$
. Note that
$\varGamma \gt 1$
indicates the primary fluid is locally more viscous than the secondary layer, while
$\varGamma \lt 1$
implies that the secondary (lubricating) fluid is relatively more viscous near the interface under shear-flow conditions. For very thin films (
$\delta /h \ll 1$
),
$\beta ^{\prime }$
remains close to zero for all
$\varGamma$
, indicating only marginal slip increase, as the interface is too close to the wall for appreciable slip to develop. As
$\delta /h$
increases, however,
$\beta ^{\prime }$
grows nonlinearly, with larger
$\varGamma$
values producing substantially higher slip lengths. Physically, as
$\varGamma$
increases, the apparent viscosity of the secondary fluid at the interface becomes smaller relative to that of the primary layer. Since the interfacial shear stress is governed by the imposed pressure gradient and remains continuous across the interface, the increase in apparent slip does not result from a reduction in interfacial shear stress, but rather from the redistribution of velocity gradients across the two layers under the same stress. In particular, the less viscous lubricating layer sustains a higher shear rate under the same interfacial stress, which increases the interfacial velocity and, consequently, the apparent slip length. As a remark, a log plot is provided here to showcase the scaling nature of apparent slip length with respect to slip layer thickness; however, in practice, we may not observe such large values of
$ \delta /h$
for a fluid-infused microchannel surface. This trend suggests that shear-thinning lubricants may be attractive for apparent-slip applications because their apparent viscosity decreases under high shear near the interface, which can enhance slip, while their viscosity remains higher in lower-shear regions than that of the solvent alone. This spatially adaptive rheology alters how the imposed interfacial shear stress is distributed across the two layers, thereby enhancing the apparent slip response; any corresponding benefit for lubricant retention or stability, however, should be regarded here as a plausible hypothesis rather than a demonstrated conclusion of the present model.
Note that when evaluating the apparent slip characteristics associated with NLIS microchannel flow, from a practical perspective, the geometrical features like periodicity (
$L_p$
) and shear-free fraction (
$F_s$
) associated with actual microchannels should be mapped with the hypothetical lubricant layer thickness (
${\delta }/{h}$
) introduced in the present modelling. Establishing such a theoretical connection,
${\delta }= f(h, w, d, L_p, \varGamma )$
, requires a detailed analysis and is therefore reserved for future investigations. However, we could illustrate the validity of the formulation for the slip characteristic associated with Newtonian lubricant-infused microchannels by defining its secondary fluid layer thickness,
$ ( {\delta }/{h} )_{\textit{Newt}}$
. For Newtonian LIS microchannels, this can be achieved under Stokes flow conditions by equating the effective slip length (
$\beta ^{\prime }_{\textit{Newt}}$
) from previous Newtonian models (see table 1) to the apparent slip length (
$\beta ^{\prime }$
) obtained from the current approach. This thereby implicitly relates the liquid fraction or cavity dimensions to fluid slip at the interface. Equating the LIS effective slip models,
$\beta ^{\prime }_{\textit{Newt}}$
, to the apparent slip model provides the relation for the lubricant layer thickness,
${\delta }/{h}$
, as
\begin{align} \left (\frac {\delta }{h}\right )_{\textit{Newt}} = \sqrt {2\left (\frac {k_{s}}{k_{p}}\right ) \beta ^{\prime }_{\textit{Newt}} + 1 } - 1 . \end{align}
Using this relation, the estimation of
$ ({\delta }/{h} )_{\textit{Newt}}$
for various slip conditions reported in the existing literature with Newtonian LIS microchannels (
$n_s = 1$
) is overplotted in figures 4(a) and 4(b).
Details of effective slip models with Newtonian fluid,
$\beta ^{\prime }_{\textit{Newt}}$
, under Stokes flow conditions (
$Re\approx 0$
) in LIS microchannels considering the transversely oriented ribs and cavities.
Table 1. Long description
The table presents a comparison of slip length models for Newtonian fluids under Stokes flow conditions in LIS microchannels. It includes three rows and three columns. The columns are labeled ‘Previous works’, ‘Slip length (β′Newt model)’, and ‘Remark’. The rows list different models by Lauga & Stone (2003), Hatte & Pitchumani (2021b), and Schönecker et al. (2014), along with their respective equations and remarks. The remarks describe the models as shear-free, interface fluid slip as a prerequisite, and elliptical velocity profile assumption at the interface.
where F s, liquid fraction; L p , length of a period; b f , interface fluid slip; h, half-height of channel; D, Schronecker parameter for maximum local slip length; N, viscosity ratio.
The slip length and the apparent viscosity ratio at the interface are linearly related (see (3.9)); however, in a log scale,
$ \beta ^{\prime }$
increases rapidly with
$ \varGamma$
, see figure 5. For all
$n_s$
values,
$\beta '$
increases monotonically with
$\varGamma$
, indicating that a larger interfacial apparent viscosity ratio – i.e. a smaller apparent viscosity of the secondary fluid relative to that of the primary fluid at the interface – promotes stronger interfacial mobility and higher apparent slip. However, the sensitivity of
$\beta ^{\prime }$
to
$\varGamma$
clearly diminishes as
$n_s$
increases. Also, for smaller
$ n_s$
, the slope of this relationship is steeper, supporting the finding that the slip is enhanced with increasing interfacial apparent viscosity ratio
$\varGamma$
, i.e. when the apparent viscosity of the secondary fluid at the interface becomes smaller relative to that of the primary fluid. When
$n_s$
is small (strongly shear-thinning secondary fluid), the apparent viscosity
$\mu _s = k_s (\text{d}u_s/\text{d}y)^{\,n_s-1}$
decreases sharply with increasing shear, so that the same interfacial shear stress can be sustained with a larger shear rate within the lubricating layer. As
$\varGamma$
increases, even modest viscosity contrasts can produce a rapid growth in
$\beta ^{\prime }$
, because the secondary layer becomes less viscous relative to the primary fluid at the interface. In contrast, when
$n_s$
increases towards unity or above (weakly shear-thinning or shear-thickening behaviour), the viscosity of the secondary layer varies less with shear rate. Under these conditions, the lubricating film exhibits a more uniform shear response and the local shear rate adjusts so that the interfacial shear stress remains continuous across the interface. This reduces the sensitivity of the interfacial velocity response to changes in
$\varGamma$
, resulting in lower
$\beta '$
and a weaker dependence on
$\varGamma$
. Geometric factors (
$ \delta /h$
) and viscosity contrasts (
$ n_s$
and
$ \varGamma$
) have complementary effects on slip length, as demonstrated. These findings highlight the need to take interface viscosity gradients and slip layer thickness into account when designing microchannels for applications including lubrication, drag reduction and microfluidic devices. To ascertain the slip predictions through the apparent slip model suggested, the results are compared with the effective slip length results of Schronecker et al. (Reference Schönecker and Hardt2014) (primary and secondary fluids are Newtonian) and Nair et al. (Reference Nair, Nandakumar Chandran and Kumar Ranjith2024) (primary fluid is Newtonian and secondary fluid is non-Newtonian); the results overlayed in figures 4(a), 4(b) and 5 showcase the capability of present model as an alternative to compare and study microchannel flows with non-Newtonian fluid-infused surfaces.
Non-dimensionalised slip length (
$\beta ^{\prime }$
) variations with respect to interface apparent viscosity ratio (
$\varGamma$
) in
$\log$
scale for
${\delta }/{h} = 0.2$
. Here, 
symbol denote the results of Schönecker et al. (Reference Schönecker and Hardt2014) (
$n_s = n_p = 1$
) and diamond symbol denotes the results of Nair et al. (Reference Nair, Nandakumar Chandran and Kumar Ranjith2024) (
$n_p =1$
having
$n_s = 0.9$
, 
;
$n_s = 1$
, black 
and
$n_s = 1.3$
, 
).
Figure 5. Long description
A line graph showing the variation of non-dimensionalized slip length with respect to interface apparent viscosity ratio for different values of n subscript s. The x-axis represents the interface apparent viscosity ratio on a logarithmic scale ranging from 10 to the power of negative 2 to 10 to the power of 2. The y-axis represents the non-dimensionalized slip length ranging from 0 to 20. The graph includes five data lines corresponding to different values of n subscript s: 0.2, 0.5, 1, 1.3, and 1.5. Each data line is represented by a different line style: red dashed, red dash-dotted, black solid, blue dotted, and blue dash-dotted, respectively. The graph also includes two insets labeled A and B, which zoom in on specific regions of the main graph. Inset A focuses on the lower left region, showing detailed variations for small values of the interface apparent viscosity ratio. Inset B focuses on the upper right region, showing detailed variations for larger values of the interface apparent viscosity ratio. The graph includes data points marked with different symbols, such as diamonds and crosses, representing results from different studies. All values are approximated.
To understand the influence of dependent parameters (
$\theta$
) such as
$\delta /h$
,
$\varGamma$
and
$n_{s}$
on slip characteristics, a comprehensive sensitivity analysis of the apparent slip length is carried out (see supplementary material S2 for details). For that, normalised sensitivity coefficients
$E_\theta = (\theta /\beta ^{\prime })(\partial \beta ^{\prime }/\partial \theta )$
, which measure the percentage change in
$\beta ^{\prime }$
produced by a percentage change in the parameter
$\theta$
, are used to quantify the sensitivity. The results show that
$E_\varGamma = 1$
identically, confirming that
$\varGamma$
acts purely as a proportional scaling factor. In contrast, the sensitivities associated with
$\delta /h$
and
$n_s$
exhibit strong interdependence in behaviour:
$E_{\delta /h}$
approaches unity in the thin-film limit and converges to
$(n_s+1)/n_s$
for large
$\delta /h$
indicating that any percentage increase in
$\delta /h$
will positively aid the slip length; while
$E_{n_s}$
diverges negatively as
$n_s \to 0$
and decays monotonically towards zero as
$n_s \to \infty$
, indicating that any percentage increase in
$n_s$
will inhibit the slip length growth. These trends, and their continuous variation across the parameter space, are illustrated in supplementary material S2, together with the limiting-behaviour analysis.
(a) Slip flow rate ratio
$({Q_{\textit{slip}}}/{Q_{\textit{no}\text{-}\textit{slip}}})$
variations with respect to non-dimensionalised slip length (
$\beta ^{\prime }$
) for varying primary fluid conditions [Overplotting the previous results (Choi et al. Reference Choi, Ulmanella, Kim, Ho and Kim2006) obtained using the relation
$({Q_{\textit{slip}}}/{Q_{\textit{no}\text{-}\textit{slip}}})_{\Delta P} = 1 + 3\beta ^{\prime }$
(black
$\blacktriangle$
)]. (b) Contour plot of slip flow rate ratio (
${Q_{\textit{slip}}/}{Q_{\textit{no}\text{-}\textit{slip}}}$
) for varying
$n_s$
and
$n_p$
values at
$\varGamma = 0.5,({\delta }/{h}) = 0.5$
.
Figure 6. Long description
The image contains two graphs. The first graph is a line graph showing slip flow rate ratio variations with respect to non-dimensionalized slip length for varying primary fluid conditions. Different lines represent different values of n_p, with a legend indicating n_p values of 0.5, 0.8, 1, 1.2, and 1.5. The second graph is a contour plot of slip flow rate ratio for varying n_p and n_s values at a specific condition. The contour plot uses a color gradient to represent different slip flow rate ratios, with a red diamond marker indicating a specific value. The graphs illustrate how slip flow rate ratios change under different conditions, providing insights into fluid transport efficiency in microchannels.
3.3.2. Slip flow rate ratio
Often, to experimentally estimate the slip length, we require an expression in terms of pressure ratio or flow rate ratio, which can be measured. Slip flow rate is one of the metrics to compare two microchannels under the condition of equal pressure drop and geometric features. The channel with a higher flow rate corresponds to lower hydrodynamic friction and greater drag reduction. To evaluate this mathematically, the slip flow rate of PF is estimated as
$Q_{\textit{slip}} = 2W \int ^h_0 u_p(y)\, \text{d}y$
, where
$W$
is the width of the channel and
$u_p(y)$
is taken from (3.2). Typically, the performance of an LIS channel is evaluated by comparing its slip flow rate (
$Q_{\textit{slip}}$
) with that of a benchmark no-slip channel (
$Q_{\textit{no}\text{-}\textit{slip}}$
) having the same dimensions and pressure gradient. Here, the volumetric flow rate of a no-slip channel is obtained by setting
${\delta }/{h} = 0$
for
$Q_{\textit{slip}}$
. Thus, the volumetric flow ratio is obtained as
\begin{equation} \left(\frac{Q_{\textit{slip}}}{Q_{no-slip}}\right)_{\Delta P} = 1 + \left(\frac{2n_p +1}{n_p} \right)\ \Gamma \left( \frac{n_s}{n_s + 1} \right) \left[ \left(\frac{\delta}{h} + 1\right)^{\tfrac{n_s + 1}{n_s}} - 1 \right], \end{equation}
which further reduces to
\begin{equation} \left (\frac {Q_{\textit{slip}}}{Q_{\textit{no}\text{-}\textit{slip}}}\right )_{\Delta P} = 1 + \left (\frac {2n_p +1}{n_p} \right )\ \beta ^{\prime } . \end{equation}
It is noteworthy that (3.13) reduces to the classical slip flow rate expression (Blevins Reference Blevins1984; Choi et al. Reference Choi, Ulmanella, Kim, Ho and Kim2006) when both the primary and secondary fluids have Newtonian viscosity indices, i.e.
$n_s = n_p =1$
, thus,
$ ({Q_{\textit{slip}}}/{Q_{\textit{no}\text{-}\textit{slip}}} )_{\Delta P} = 1 + 3\beta ^{\prime }$
.
As the value of
$n_p$
decreases (see (3.13)), the slip flow rate ratio increases due to the reduction in viscosity. The graphical representation of the slip flow rate versus slip length for varying power-law indices of the PF is shown in figure 6(a). As the PF becomes increasingly shear-thinning (
$n_p\lt 1$
), the slip length enhances significantly. Further, to understand how secondary fluid affects the slip flow rates simultaneously with primary fluid, a contour plot of
$ ({Q_{\textit{slip}}}/{Q_{\textit{no}\text{-}\textit{slip}}} )$
is presented in figure 6(b). It indicates that a higher slip flow rate ratio is possible for shear-thinning fluid combinations in both primary and secondary fluids. The contour trends indicate that slip enhancement is favoured when the lubricating layer exhibits stronger shear-thinning behaviour (
$n_s\ll 1$
), although the present analysis does not address lubricant stability directly. However, it is interesting that for a Newtonian fluid combination, i.e.
$n_s =n_p = 1$
, the slip flow rate ratio (3.13) becomes
\begin{equation} \left (\frac {Q_{\textit{slip}}}{Q_{\textit{no}\text{-}\textit{slip}}}\right )^{\textit{Newtonian}}_{\Delta P} = 1 + \left (\frac {3}{2} \right )\ \left (\frac {k_{p}}{k_{s}}\right ) \left [ \left (\frac {\delta ^2}{h^2} + 2\frac {\delta }{h}\right ) \right ] ,\end{equation}
which is dependent on slip layer thickness and viscosity ratio, as in the reported work (Dey et al. Reference Dey, Saha and Chakraborty2018). For instance, from the provided contour (see figure 6
b), when the value of
$\varGamma$
is 0.5 and the slip layer thickness (
${\delta }/{h}$
) is 0.5, the flow rate ratio is 1.9375, wherein
$n_p=n_s=1$
. Note that when the flow rate ratio is greater than unity, it indicates a favourable condition for NLIS channels, where
$Q_{\textit{slip}}\gt Q_{\textit{no}\text{-}\textit{slip}}$
. Hence, the contour plot serves as a guideline for selecting the viscosity indices of the PF and SF to ensure an advantageous flow rate.
(a) Pressure ratio
$({G_{\textit{no}\text{-}\textit{slip}}}/{G_{\textit{slip}}})$
variations against slip length (
$\beta ^{\prime }$
) for varying primary fluid conditions [overplotting the results from the relation,
$({G_{\textit{no}\text{-}\textit{slip}}}/{G_{\textit{slip}}})_{Q} = 1 + 3\beta ^{\prime }$
for
$n_p =1$
(black
$\triangle$
) (Choi et al. Reference Choi, Ulmanella, Kim, Ho and Kim2006), also numerical results (Javaherchian & Moosavi Reference Joseph, Mathew, Krishnaraj, Dilip and Ranjith2019) having
$n_p = 0.8$
(
),
$n_p = 1$
(black 
) and
$n_p = 1.2$
(
)]. (b) Effect of interface apparent viscosity ratio (
$\varGamma$
) and secondary fluid viscosity indices (
$n_s$
) on PDR (%) [here,
$n_p = 0.7$
,
$({\delta }/{h}) = 0.5$
].
Figure 7. Long description
The image contains two line graphs. The first graph (a) shows the pressure ratio variations against slip length for varying primary fluid conditions. It includes overplotted results from a relation for specific values, as well as numerical results for different viscosity indices. The second graph (b) illustrates the effect of interface apparent viscosity ratio and secondary fluid viscosity indices on PDR percentage. The x-axis of the first graph represents slip length, while the y-axis represents the pressure ratio. The x-axis of the second graph represents the interface apparent viscosity ratio, and the y-axis represents the PDR percentage. Different lines in each graph represent varying conditions and parameters.
3.3.3. Pressure drag reduction
Similar to flow rate ratios, another vital performance indicator is the pressure drag reduction ratio (PDR), which is evaluated at constant flow rates and is given by
$PDR= ( ({G_{\textit{no}\text{-}\textit{slip}}- G_{\textit{slip}} })/{G_{\textit{no}\text{-}\textit{slip}}})\times 100$
. Considering (3.13) for constant flow rates, the pressure ratio is expressed as follows:
\begin{eqnarray} \left (\frac {G_{\textit{no}\text{-}\textit{slip}}}{G_{\textit{slip}}}\right )_{Q} = &\left [ 1 + \left (\frac {2n_p +1}{n_p} \right )\ \beta ^{\prime }\right ]^{n_p}, \end{eqnarray}
note that this pressure ratio favours drag reduction in its inverse form. As the slip length
$\beta ^{\prime }$
increases for shear-thinning primary fluid flow, its value decreases with increasing power-law index
$n_p$
and the opposite trend is observed for shear-thickening fluids, as shown in figure 7(a). Here, the pressure ratio for different flow behaviour indices (
$n_p$
) is presented with overplotted data from previous studies for validation. The Newtonian limit (
$n_p = 1$
) agrees closely with results by Choi et al. (Reference Choi, Ulmanella, Kim, Ho and Kim2006) and numerical predictions of power-law working fluid over gas-infused surfaces by Javaherchian & Moosavi (Reference Joseph, Mathew, Krishnaraj, Dilip and Ranjith2019), confirming the accuracy of the present formulation in the linear regime. For shear-thinning working fluids (
$n_p \lt 1$
), the predicted pressure reduction is stronger; the opposite holds for shear-thickening cases (
$n_p \gt 1$
). This is consistent with the observed trend that lower effective viscosity enhances apparent slip and improves flow efficiency. These comparisons reinforce the model’s capability to recover known Newtonian fluids over gas/liquid-infused surface behaviours while extending predicatively to non-Newtonian regimes.
Next, the effect of the interface apparent viscosity ratio (
$\varGamma$
) on PDR (%) is examined to evaluate the performance of NLIS as the secondary fluid becomes more shear-thinning (
$n_s \lt 1$
). Figure 7(b) illustrates PDR (%) across a wide range of
$\varGamma$
values plotted on a logarithmic scale. The results show that PDR increases with
$\varGamma$
, highlighting a nonlinear relationship that saturates near 100 % for large
$\varGamma$
. Lower
$n_s$
values generally enhance performance, evident from the red and blue dashed curves outpacing their solid black line corresponding to Newtonian infused fluid. This underscores the significant role of viscosity ratios and viscosity indices in modulating the efficiency of interface-driven systems such as the NLIS channel.
In addition to the slip characteristics discussed previously, the present formulation also yields compact expressions for two resistance-based quantities, viz., the average wall shear stress ratio and the Poiseuille number, which may be useful for design-oriented interpretation of NLIS channel performance. Using the relation between the pressure gradient and the average primary fluid velocity, the non-dimensionalised average wall shear stress under slip conditions is obtained as
\begin{equation} \frac{\overline{\tau}^{\mathrm{\textit{slip}}}_{w}}{\overline{\tau}^{\mathrm{\textit{no}\text{-}\textit{slip}}}_{w}} = \left( \frac{\frac{n_p}{2n_p+1}} {\frac{n_p}{2n_p+1}+\beta^{\prime}} \right)^{n_p}, \end{equation}
which shows that the wall shear stress decreases monotonically with increasing apparent slip length (refer to (B4)). Likewise, the corresponding Poiseuille number is expressed as
\begin{equation} Po = 2^{(2n_p+3)} \left( \frac{1} {\frac{n_p}{2n_p+1}+\beta^{\prime}} \right)^{n_p}, \end{equation}
recovering the classical no-slip Newtonian limit
$Po=96$
when
$n_p=1$
and
$\beta ^{\prime }=0$
(refer to (B6)). These quantities follow directly from the analytical model and provide supplementary measures of hydrodynamic resistance. The detailed derivation steps, together with the extended discussion and corresponding figures, are provided in Appendix B.
3.4. Applicability and limitations of the proposed model
In practical experiments, it is often difficult to measure molecular-scale slip lengths directly because of the limited spatial resolution of flow visualisation techniques. Thus, experimentally characterising flow slip in parallel-plate microchannels is challenging, since the microscopic slip at the fluid–solid interface cannot be measured directly. Instead, it must be inferred from macroscopic hydrodynamic quantities such as the volumetric flow rate and pressure drop. In a typical experimental set-up, a microchannel containing a structured or liquid-infused surface is connected to a high-precision syringe pump or pressure controller. In this approach, either the flow rate is maintained at a constant value while the corresponding pressure drop is recorded using differential pressure sensors, or alternatively, the flow rate is measured while a fixed pressure drop is imposed across the channel. For Newtonian liquid-infused or superhydrophobic surfaces with Newtonian working fluid under investigation, a well-established relation,
\begin{align} \left (\frac {Q_{{\textit{slip}}}}{Q_{\textit{no}\text{-}\textit{slip}}}\right )_{\Delta P}^{\textit{Newt}} = \left (\frac {G_{\textit{no}\text{-}\textit{slip}}}{G_{{\textit{slip}}}}\right )_{Q}^{\textit{Newt}} = 1 + 3\beta ^{\prime }, \end{align}
serves as a practical bridge between measurable macroscopic quantities and the microscopic interfacial slip (Ou et al. Reference Ou and Rothstein2004; Choi et al. 2006). Experimental set-ups generally use microchannels (50–500 μm wide) made of glass, PDMS or silicon, with cavities infused with lubricants such as silicone oil, Krytox or fluorinated liquids. Assuming laminar flow, the apparent slip shifts the parabolic velocity profile in a manner consistent with the Navier-slip description, thereby reducing the complex interfacial effects to a single slip-length parameter. This formulation replaces complex interfacial effects – such as meniscus curvature and local recirculation within microgrooves – with a single slip length,
$\beta$
, thereby linking measurable flow data to key surface parameters – cavity depth, lubricant viscosity and surface fraction that govern drag reduction. If an effective slip length is already available from an experiment, a conventional single-phase Navier-slip description is sufficient for predicting the corresponding channel-flow response. The role of the present two-layer model is different: it provides an analytical interpretation of how the apparent slip length itself depends on the lubricating-layer thickness, interfacial viscosity contrast and non-Newtonian rheology. To illustrate its interpretive role, consider two identical channels operated under the same flow conditions, but containing different primary-secondary fluid combinations. If their effective slip lengths differ,
$\beta _1$
and
$\beta _2$
, the corresponding pressure-flow relations may be used to compare the extent to which the lubricating-layer thickness, interfacial viscosity contrast and fluid rheology influence the apparent slip response. In this way, the present formulation is not intended to replace a standard Navier-slip description when an effective slip length is already known, but rather to provide a reduced analytical framework for relating that apparent slip to underlying layer properties.
In the present two-layer model problem, the conventional Newtonian slip–flow relations become inadequate when both the working and lubricating fluids exhibit shear-dependent viscosities. Under these conditions, the linear scaling between flow rate and pressure drop no longer holds in its Newtonian form and the apparent slip depends nonlinearly on the rheological indices of the two fluids. Within the present two-layer model, the generalised formulation of flow rate and pressure ratio (refer to (3.13) and 3.15) obtained in this work,
\begin{align} \left (\!\frac {Q_{{\textit{slip}}}}{Q_{\textit{no}\text{-}\textit{slip}}}\!\right )_{\Delta P}^{non\text{-}Newt}\! = 1+\left (\frac {2n_p+1}{n_p}\right )\beta ^{\prime } ;\, \left (\!\frac {\Delta P_{\textit{no}\text{-}\textit{slip}}}{\Delta P_{{\textit{slip}}}}\!\right )_{Q}^{non\text{-}Newt} \!= \left [\!1+\left (\frac {2n_p+1}{n_p}\right )\beta ^{\prime }\right ]^{n_p}\!\!, \end{align}
extended the slip–flow correlation to the power-law regime. This nonlinear relation reveals that even moderate shear-thinning behaviour can amplify apparent slip far beyond Newtonian predictions, while shear-thickening tends to suppress it.
Within this model, the hypothetical slip-layer thickness
$\delta /h$
may be interpreted as a reduced parameter associated with the experimentally inferred value of
$\beta '$
under the assumed constitutive description. This apparent slip value (
$\beta '$
) shall now become a parameter to characterise the channel for the prevailing specific flow condition. The proposed relation captures the flow behaviour of the present two-layer model problem and provides a reduced analytical framework for interpreting experimental pressure-flow data. Thus, the present generalised framework is intended to give researchers the flexibility to address both single-component primary-fluid flow with Navier slip, for which
$\beta ' = f\! (n_p,\; ( {Q_{{\textit{slip}}}}/{Q_{\textit{no}\text{-}\textit{slip}}} )_{\Delta P}^{non\text{-}Newt} )$
, and two-layer flow with apparent slip governed by infused fluid rheology and primary–secondary fluid interface mechanics, for which
$\beta ' = f(\varGamma , n_s, \delta /h)$
. In this sense, the model serves as an interpretive extension linking effective-slip descriptions to the underlying two-fluid physics, rather than replacing simpler Navier-slip formulations when single-phase descriptions suffice. The
$(n_s/(n_s+1))$
term in (3.9) reflects the lubricating film’s rheology: shear-thinning fluids (
$n_s\lt 1$
) reduce viscosity and amplify slip, while shear-thickening (
$n_s\gt 1$
) increases resistance. Moreover, if the channel geometry and interfacial properties are known, the relation can be inverted to estimate the effective rheological parameters (
$n_p$
,
$k_p$
) of the working fluid. Thus, the model links macroscopic measurements with microscopic interfacial physics, offering a practical means to characterise both surface slip and non-Newtonian rheology. Table 2 provides a concise summary illustrating how different primary–secondary fluid combinations and limiting wall conditions influence the apparent slip characteristics and hydrodynamic resistance of channel flow, thereby serving as a reference for interpreting the model’s parametric behaviour.
Summary of apparent slip characteristic – including the non-dimensionalised slip length (
$\beta ^{\prime }$
), non-dimensional average wall shear stress (
$( {\overline {\tau }^{\textit{slip}}_{w}}/{\overline {\tau }^{\textit{no}\text{-}\textit{slip}}_{w}})$
), Poiseuille number (
$Po$
) – for various primary-fluid/secondary-fluid combinations and channel-wall conditions, highlighting the limiting behaviour of the model parameters.
Table 2. Long description
The table presents a summary of apparent slip characteristics, including non-dimensionalized slip length, non-dimensional average wall shear stress, and Poiseuille number for different combinations of primary and secondary fluids and channel wall conditions. It highlights the limiting behavior of model parameters, showing how different fluid combinations and wall conditions influence slip characteristics and hydrodynamic resistance in channel flow.
From the perspective of numerical analysts and modellers, the derived relations may also be useful for interpreting simulations in which wall-slip or effective interfacial descriptions are imposed. Computational fluid dynamics simulations often incorporate modified wall boundary conditions such as Navier slip or effective viscosity layers to mimic real interfacial behaviour. The developed formulations enable direct comparison between such simulations and experimental data. For instance, if a NLIS channel exhibits a higher volumetric flow rate than the corresponding no-slip channel under the same pressure gradient, the ratio predicted by (3.13) and (3.15) directly links this enhancement to the flow behaviour index (
$n_p$
) and the apparent slip length (
$\beta ^{\prime }$
). For shear-thinning fluids (
$ n_p \lt 1$
), even a modest increase in flow rate corresponds to a comparatively larger effective (
$\beta ^{\prime }$
), as the nonlinear exponent (
$(2n_p + 1)/n_p$
) magnifies the slip contribution. Conversely, for shear-thickening fluids (
$n_p \gt 1$
), the same percentage increase in flow rate implies a larger apparent slip length induced by secondary fluid having a viscosity index less than the working fluid, i.e.
$n_s \lt n_p$
. This comparison allows numerical researchers to verify whether their imposed slip models, interfacial viscosities or surface patterns reproduce experimentally observed enhancements in mass flux or reductions in pressure loss. More broadly, these simple yet physically meaningful expressions provide a unified language for experimentalists and theorists to quantify interfacial mobility, correlate it with surface design, and systematically optimise microchannel performance for drag reduction, heat transfer enhancement or liquid-infused surface applications. In this regard, the derived generalised relations provide a reduced analytical framework for linking measurable macroscopic pressure-flow behaviour to the apparent slip response within the present model.
It should also be noted that the present two-layer formulation is an idealsed apparent-slip representation introduced for analytical tractability; it is not intended to resolve the detailed texture-scale hydrodynamics of actual rib-cavity LIS geometries, such as meniscus shape, cavity recirculation or discrete groove topology. The use of the power-law model for both fluids is a limitation of the present formulation because, although it is adopted for analytical tractability, it does not capture finite low- and high-shear viscosity plateaus, and the resulting predictions should therefore be interpreted as reduced analytical trends within the intermediate shear-rate regime. Likewise, the relation between the hypothetical slip-layer thickness and the actual surface geometry has not been derived here and remains a subject for future work. Nevertheless, the resulting generalised pressure-drop/flow-rate relations provide a useful reduced framework for interpreting apparent slip in non-Newtonian liquid-infused microchannels.
4. Conclusion
Fluid slip at solid–liquid interfaces has drawn increasing attention with the advent of micro- and nanoscale fluidic systems. Surface-trapped fluid acts as a localised slip region and consequently decreases the flow resistance, thereby enhancing drag reduction. Given the growing interest in employing shear-dependent primary and lubricating fluids in apparent-slip microchannel flows, there is a need for a general model capable of systematically examining how non-Newtonian rheology influences the predicted slip and drag-reduction behaviour. To this end, the authors adopt an analytical approach to solve the velocity profile of co-flowing immiscible non-Newtonian power-law fluids and extend it as an apparent slip model for non-Newtonian liquid-infused channel flow. Additionally, the slip hydrodynamics, including pressure gradient and shear stress, are thoroughly investigated, leading to a refined formulation of the Poiseuille number for NLIS parallel-plate microchannels. The key findings are summarised as follows.
-
(i) The classical relationship for apparent slip length for characterising an NLIS microchannel flow is derived, and the mathematical formulation indicates that slip length quantification depends upon interfacial apparent viscosity ratio, lubricating layer rheology and slip-layer thickness.
-
(ii) The apparent slip length increases nonlinearly with slip-layer thickness defined delineating the thin and thick layer regimes. Thin lubricating films keep the interface close to the wall, thereby limiting the development of appreciable slip, whereas thicker, less viscous layers sustain the imposed shear with a larger velocity variation, resulting in a higher interfacial velocity and an interface-dominated, high-slip regime. Shear-thinning lubricants amplify this by lowering their effective viscosity in high-shear regions and increasing the velocity rise across the lubricating layer, whereas weakly shear-thinning layers exhibit a weaker shear response and thus damp the apparent slip enhancement.
-
(iii) The apparent slip length is directly proportional to the non-dimensional interfacial apparent viscosity ratio, a flow-governed parameter set by the imposed pressure gradient and channel geometry, so that it reflects the interfacial shear distribution and overall flow state rather than the intrinsic material properties alone.
-
(iv) Quantitative estimations of slip flow rates and pressure drag reduction, relative to no-slip channels, indicate significant benefits for NLIS microchannel application under appropriate conditions. Within the present two-layer model, combinations involving shear-thinning primary and secondary fluids yield the largest predicted slip enhancement and drag-reduction trends at a constant interface apparent viscosity ratio.
-
(v) For Newtonian liquid-infused or superhydrophobic channels, the slip-induced enhancement in flow rate and the reduction in pressure drop follow the same linear dependence on the non-dimensional slip length, whereas in the present non-Newtonian formulation, this linear slope is generalised from the conventional value of 3 to
$(2n_p+1)/n_p$
. Additionally, the pressure-drop ratio is raised to the power
$n_p$
, thereby extending the classical slip law into a unified, rheology-dependent framework that connects macroscopic flow rate/pressure drop–flow measurements to interfacial slip and fluid rheology in NLIS channels.
Though the methodology discussed simplifies the multiphase problem associated with complex geometry, potentially overestimating slip by underestimating dissipation, the model remains valid since domain scales are not negligible compared with system size. The resulting pressure drop–flow rate relationship reveals how slip length depends on the surface defined via the Navier condition. These generalised relations provide a reduced analytical framework for interpreting apparent slip trends in non-Newtonian liquid-infused microchannels with potential applications spanning energy conversion to biomedical technologies and for guiding future model-based comparisons. Since the present analysis employs a power-law constitutive approximation, the results should be interpreted as rheology-dependent analytical trends within that model framework; extension to constitutive models with finite low- and high-shear viscosity plateaus remains for future work. Furthermore, the hypothetical slip layer thickness for non-Newtonian fluid combinations associated with the modelling should emerge naturally from the groove geometry, and Reynolds-number-dependent flow conditions have not been addressed here, but could be a potential topic for future research.
Supplementary material
Supplementary material is available at https://doi.org/10.1017/jfm.2026.11719.
Declaration of interests
The authors report no conflict of interest.
Appendix A. Derivation steps for analytical model
Here, we detail the steps of derivation that were involved in solving the coupled governing equations that represent the fully developed laminar steady and uniform flow of the immiscible lubricant and the working fluid (both non-Newtonian fluids) co-flow in a pressure-driven Poiseuille flow. Considering the above-mentioned governing equation (2.1) and boundary conditions (2.2)–(2.4), the derivation steps are as follows.
-
(i) Upon substitution of variable change in governing equations for
$ G = -({\text{d} P}/{\text{d}x})$
and proceeding with integration yields (A1)
\begin{align} \frac{\text{d}u_{p}}{\text{d}y} &= \left [ \frac {-Gy}{k_p} + C_1\right ] ^{1/n_p}, \quad 0 \leqslant y \leqslant h, \end{align}
(A2)
\begin{align} \frac{\text{d}u_{s}}{\text{d}y} &= \left [ \frac {-Gy}{k_s} + C_3\right ] ^{1/n_s}, \quad -\delta \leqslant y \leqslant 0. \end{align}
-
(ii) The resultant relations could be integrated again with respect to
$y$
, and (A1) and (A2) become (A3)
\begin{align} u_{p} &= C_2 - \frac {k_p \left ( C_1 - ( \dfrac{G y}{k_p} )\right )^{\left (\tfrac{1}{n_p} + 1\right )}}{G \left ( \frac{1}{n_p} + 1 \right )}, \end{align}
(A4)
\begin{align} u_{s} &= C_4 - \frac {k_s \left ( C_3 - \dfrac{G y}{k_s} \right )^{\left (\tfrac{1}{n_s} + 1\right )}}{G \left ( \dfrac{1}{n_s} + 1 \right )}, \end{align}
where
$C_1, C_2, C_3$
and
$C_4$
are constants of integration to be resolved with the help of known boundary conditions. -
(iii) Imposing the boundary conditions, as given by (2.2)–(2.4), the constants of integration are found to be
(A5)
\begin{align} C_{1} &= \frac {G h}{k_p},\end{align}
(A6)
\begin{align} C_{2} &= \frac{k_p \left( \frac{G h}{k_p} \right)^{\left( \tfrac{1}{n_p} + 1 \right)}}{G \left( \frac{1}{n_p} + 1 \right)} - \frac{k_s \left[\left( \frac{G h}{k_s} \right)^{\left( \tfrac{1}{n_s} + 1 \right)} + \left( \frac{G \delta}{k_s} + \frac{G h}{k_s} \right)^{\left( \tfrac{1}{n_s} + 1 \right)}\right]}{G \left( \frac{1}{n_s} + 1 \right)} ,\end{align}
(A7)
\begin{align} C_{3} &= \frac {G h}{k_s},\end{align}
(A8)
\begin{align} C_{4} &= \frac{k_s \left( \frac{G \delta}{k_s} + \frac{G h}{k_s} \right)^{\left( \tfrac{1}{n_s} + 1 \right)}}{G \left( \frac{1}{n_s} + 1 \right)}.\end{align}
-
(iv) Replacing these constants in (A4) and (A3) yields the final equation for a fully developed velocity profile for primary and secondary fluids, (2.5) and (2.6).
The solution represents the pressure-driven Poiseuille flow in an NLIS microchannel mimicking the slip phenomenon through the apparent slip model adopted.
Appendix B. Discussions on hydrodynamic resistance
From the perspective of hydraulic resistance, wall shear stress provides the direct measure of friction in an NLIS channel, while the corresponding Poiseuille number serves as a useful index for drag-reduction comparison. From an engineering perspective, these quantities are also important for the design and evaluation of liquid-infused microchannels. Accordingly, the wall-slip hydrodynamics and the associated Poiseuille-number formulation are derived here for the reader’s repeatability.
From the principles of fluid dynamics, shear stress for a non-Newtonian power-law fluid flow is defined as
$\tau (y) = k (({\text{d}u}/{\text{d}y}) )^{n}$
, where
$k$
and
$n$
are model coefficients. Accordingly, the shear stress distributions in the primary and secondary fluid layers are
\begin{align} \tau _p(y)=k_p\left (\frac{\text{d}u_p}{\text{d}y}\right )^{n_p}=G(h-y), \qquad \tau _s(y)=k_s\left (\frac{\text{d}u_s}{\text{d}y}\right )^{n_s}=G(h-y). \end{align}
The shear stress distribution is observed to have a linear relation and is governed by the pressure gradient and half-height of the microchannel. The interface shear stress, i.e at the slip interface, could be deduced,
$\tau (y=0) = G h$
, and using this to non-dimensionalise the shear stress distributions gives
\begin{equation} \begin{gathered} \frac {\tau (y)}{\tau (y=0)} = \frac {\left (h-y\right )}{h} = 1- \frac {y}{h} \end{gathered}. \end{equation}
An average wall shear stress is often defined as
$\overline {\tau }_w = ({1}/{p}) \int _0^ p \tau _w \, \text{d}s$
, where
$p$
is the perimeter of the channel and
${\rm d}s$
is the element of the arc length. In general, the average wall shear stress can be related to the required pressure gradient in the channel, and subsequently, force balance indicates that
$\text{d}z \int _0^p \tau _w \text{d}s = -A \, \text{d}P$
, where
$A$
is the cross-sectional area. When the average wall shear stress is substituted, that is,
$\int _0^p \tau _w \,\text{d}s = \overline {\tau }_w p$
, the average wall shear stress equation becomes simpler to
$\overline {\tau }_w = G({A}/{p})$
. The equivalent diameter for a non-circular channel is given by
$D_h = ({4A}/{p})$
, and for a parallel-plate microchannel, it can be expressed in terms of the aspect ratio
$\alpha = W/2h$
and channel height, thereby
$D_h = ({4\alpha h}/{\alpha +1})$
. Thus, the relationship between the wall shear stress and pressure gradient becomes
$\overline {\tau }_w = G( {D_h}/{4})$
.
Knowing the average velocity in terms of the pressure gradient for the NLIS microchannel simplifies the expression for wall shear stress, which becomes a function of the fluid’s transport properties and boundary slip. Considering (3.2) and (3.9), the relationship between the pressure gradient and average velocity is obtained as
$G = k_p (( {\langle \overline {u_p}\rangle })/({h^{({(n_p + 1)}/{n_p})} (( {n_p}/{(2n_p + 1)}) + \beta ^{\prime } )} ))^{n_p}$
. Substituting the value of
$G$
obtained for
$\overline {\tau }_w$
provides us with the wall shear stress as a function of transport properties of working/primary fluid,
\begin{equation} \overline {\tau }^{\textit{slip}}_{w} = k_p \left (\frac {\langle \overline {u_p}\rangle }{h^{\tfrac{n_p + 1}{n_p}} \left (\dfrac{n_p}{2n_p + 1}+ \beta ^{\prime } \right )}\right )^{n_p} \frac {\alpha h}{\alpha + 1} . \end{equation}
Now, consider a case where the primary fluid flows through a no-slip boundary channel (
$\beta ^{\prime } = 0$
), the wall shear stress expression changes to
$\overline {\tau }^{\textit{no}\text{-}\textit{slip}}_{w}$
indicating that as
$\beta ^{\prime }$
decreases and approaches zero, the wall shear stress maximises. This average wall shear stress at non-slip boundary,
$\overline {\tau }^{\textit{no}\text{-}\textit{slip}}_{w}$
, is a suitable candidate to non-dimensionalise the average shear stress under slip conditions, enabling proper comparison of microchannel performance. Therefore, the non-dimensionalised average wall shear stress is expressed as
\begin{equation} \frac{\overline{\tau}^{\textit{slip}}_{w}}{\overline{\tau}^{no-slip}_{w}} = \left(\frac{\frac{n_p}{2n_p + 1}}{\frac{n_p}{2n_p + 1} + \beta^{\prime}} \right)^{n_{p}}. \end{equation}
Conventionally, dimensionless shear stress is represented as the friction coefficient,
$C_f = ({\tau _w}/{({1}/{2}) \rho \langle \overline {u_p}\rangle ^2})$
, where dynamic pressure (
$( {1}/{2}) \rho \langle \overline {u_p}\rangle ^2$
) is used for the non-dimensionalisation. This could also be a choice for design comparisons based on the user’s preference. Figure 8(a) shows that the normalised wall shear stress decreases with increasing slip length
$\beta ^{\prime }$
and that stronger shear-thinning behaviour
$(n_p\lt 1)$
yields lower wall shear stress. As
$\beta ^{\prime }\to \infty$
, (B4) approaches the asymptotic limit
$( {\overline {\tau }^{\textit{slip}}_{w}}/{\overline {\tau }^{\textit{no}\text{-}\textit{slip}}_{w}}) \to 0.$
In addition, similar to the slip velocity comparison, the interface shear stress can also be estimated from the illustrative numerical cases used for velocity-profile comparison. A quantitative analytical to numerical comparison is provided in supplementary material S1.
(a) Non-dimensionalised wall shear stress ratio
$(\overline {\tau }^{{\textit{slip}}}_w / \overline {\tau }^{\textit{no}\text{-}\textit{slip}}_w)$
as a function of non-dimensionalised slip length (
$\beta ^{\prime }$
) for shear-thinning and shear-thickening primary fluid flow. (b) Variation of Poiseuille number
$Po$
for increasing non-dimensional slip length
$\beta ^{\prime }$
for different values of primary fluid power-law index
$n_p$
. [Overplotting the results obtained from Davies et al. (Reference Davies, Maynes, Webb and Woolford2006)
$Po =( {96}/{ (1 + 3 \beta ^{\prime } )})$
for
$n_p =1$
(
$+$
). Numerical results of Nair et al. (Reference Nair, Nandakumar Chandran and Kumar Ranjith2024) having
$n_p = 1$
with
$n_s = 0.7$
(
),
$n_s = 1$
(black
$\diamondsuit$
) and
$n_s = 1.3$
(
). Numerical results of Javaherchian & Moosavi (Reference Joseph, Mathew, Krishnaraj, Dilip and Ranjith2019) having
$n_s =1$
with
$n_p = 0.8$
(
$\circ$
),
$n_p = 0.9$
(
),
$n_p = 1$
(black
$\circ$
),
$n_p = 1.1$
(
$\square$
) and
$n_p = 1.2$
(
)].
Figure 8. Long description
The image contains two line graphs. The first graph on the left shows the non-dimensionalized wall shear stress ratio as a function of non-dimensionalized slip length for shear-thinning and shear-thickening primary fluid flow. Different lines represent various values of the primary fluid power-law index, with red dashed lines for lower indices, a solid black line for an index of 1, and blue dotted and dashed lines for higher indices. The second graph on the right illustrates the variation of the Poiseuille number for increasing non-dimensional slip length for different values of the primary fluid power-law index. The lines are color-coded similarly to the first graph. Both graphs include data points from previous studies, marked with different symbols and colors. The graphs show how the wall shear stress ratio and Poiseuille number change with varying slip lengths and fluid properties.
In the case of Newtonian fluid flow in plane-Poiseuille microchannel flow, it is important to define another dimensionless coefficient, Darcy friction factor,
$f = ({8\overline {\tau }_w}/{\rho U^2})$
, note that,
$f=4 C_f$
. Upon substitution of the obtained relationship between the average wall shear stress and the pressure gradient, the Darcy friction factor is rewritten as
$ f =( { (G ) D_h}/{({1}/{2}) \rho U^2})$
Bird (Reference Bird2002) and Choi et al. (Reference Choi, Westin and Breuer2003) use the equation
$\Delta P = f( {L}/{D_h}) ({\rho U^2}/{2})$
, which can be derived by substituting
$ (G ) = (-({\text{d}P}/{\text{d}x}) ) = ({\Delta P}/{L})$
. In this context, Reynolds number for power-law primary fluid flow based on the hydraulic diameter is defined as
$ Re = (({\rho \langle u_p \rangle ^{2-n_p} (D_h)^{n_p}})/{k_p})$
; thereby, the Darcy friction factor is modified as
$f = (G )( {D^{n_p +1}_h}/{({1}/{2}) k_p \langle \overline {u_p}\rangle ^{n_p}} )\times ({1}/{Re}).$
Furthermore, by substituting the expressions for average primary fluid velocity and hydraulic radius, the friction factor relation becomes
\begin{equation} f = \frac{2}{Re} \left(\frac{1}{ \left(\frac{n_p}{2n_p + 1} + \beta^{\prime} \right)}\right)^{n_p} \left(\frac{4\alpha}{\alpha+1}\right)^{n_p +1}. \end{equation}
For planar Poiseuille flow, where the fluid moves between two infinite parallel plates (
$W\xrightarrow {}\infty$
), the aspect ratio
$\alpha$
becomes very large and, thus,
$({\alpha }/{\alpha +1}) \approx 1$
. Finally, the Poiseuille number,
$Po=f.Re$
for the apparent slip flow is obtained as
\begin{equation} Po = 2^{(2n_p +3)} \left(\frac{1}{ \left(\frac{n_p}{2n_p + 1} + \beta^{\prime} \right)}\right)^{n_p}. \end{equation}
Similar to the normalised wall shear stress, the variation of the Poiseuille number
$Po$
with slip length
$\beta ^{\prime }$
for different primary-fluid power-law indices is shown in figure 8(b). In general,
$Po$
decreases with increasing
$\beta ^{\prime }$
, reflecting the reduction in hydrodynamic resistance caused by slip. Shear-thinning fluids exhibit lower
$Po$
owing to their higher interfacial slip, whereas the decrease is steeper for shear-thickening fluids. The overplotted results from earlier studies show that the present formulation recovers the Newtonian limit correctly, while comparisons in non-Newtonian slip-based cases should be interpreted with care because the rheological dependence and apparent-slip response are related differently across models. In particular, the Newtonian limit
$(n_p=1)$
agrees well with the correlation of Davies et al. (Reference Davies, Maynes, Webb and Woolford2006) and the simulations of Nair et al. (Reference Ng and Wang2022), confirming the consistency of the derived expression. Likewise, the results of Javaherchian & Moosavi (Reference Joseph, Mathew, Krishnaraj, Dilip and Ranjith2019) for power-law fluid flow over gas-infused surfaces also compare favourably, further supporting the validity of the model over a range of slip conditions. It should be noted that the friction factor expression in (B6) is directly applicable when the aspect ratio is sufficiently large. Although the aspect-ratio term does not appear explicitly in (B5) for parallel-plate channels, the factor
$ ({4\alpha }/{\alpha +1} )^{n_p+1}$
must be retained for rectangular microchannels with only moderately large aspect ratios.
w×d
Lp
δ
μp=kpγ˙np−1
μs=ksγ˙ns−1
Γ
ns=0.7
np=1.3
δ/h=0.5
ns
Γ=1
np=1
δ/h=0.5
np
Γ=1
ns=1
δ/h=0.5
δ/h
Γ=1
ns=np=1
β′
(δ/h)
Γ=0.5
ns
ns=1
Γ
np=ns=1
βNewt′
Re≈0
β′
Γ
log
δ/h=0.2
ns=np=1
np=1
ns=0.9
ns=1
ns=1.3
(Qslip/Qno-slip)
β′
(Qslip/Qno-slip)ΔP=1+3β′
▴
Qslip/Qno-slip
ns
np
Γ=0.5,(δ/h)=0.5
(Gno-slip/Gslip)
β′
(Gno-slip/Gslip)Q=1+3β′
np=1
△
np=0.8
np=1
np=1.2
Γ
ns
np=0.7
(δ/h)=0.5
β′
(τ¯wslip/τ¯wno-slip)
Po
(τ¯wslip/τ¯wno-slip)
β′
Po
β′
np
Po=(96/(1+3β′))
np=1
+
np=1
ns=0.7
ns=1
♢
ns=1.3
ns=1
np=0.8
∘
np=0.9
np=1
∘
np=1.1
◻
np=1.2
