A body-forcing strategy is proposed to model the effects of surface roughness on heat transfer in a turbulent flow. The model introduces a source term into the temperature transport equation that depends on a characteristic velocity scale, the local wall-relative temperature and the roughness density. We test it in direct numerical simulation (DNS) of a turbulent open channel at a constant Prandtl number of 0.71, and through extensive comparison with roughness-resolved DNS data, we show that the instantaneous local friction velocity is the appropriate velocity scale. Despite its simple formulation, it accurately captures the salient physics of rough-wall heat transfer, such as the breakdown of the Reynolds analogy, the characteristic plateau of the temperature roughness function and the fully rough behaviour of the ratio of roughness lengths of momentum and heat transfer. An appropriate decomposition of the total heat flux reveals that the scaling of the conductive and body-forcing Stanton numbers is similar to that of the exposed and sheltered Stanton numbers from fully resolved roughness simulations, respectively. This suggests that the primary effect of roughness on heat transfer can be effectively captured so long as the surface heat flux is modelled correctly in a statistical sense, even if the precise contribution from individual roughness elements is not accurately resolved. Finally, the body-forcing model can be used over regular no-slip smooth walls, bypassing the need to resolve the exact details of the roughness geometry.

This study investigates how the impact of an internal heat source and anisotropic thermal diffusivity influences convective motion in a horizontally inclined porous medium saturated with fluid, assuming an adiabatic bottom boundary and an isothermal top boundary. Here, the momentum equation is formulated using the Darcy–Forchheimer model and assumes that the viscosity is a linear function of the temperature. Linear and nonlinear stability analyses have been conducted to determine the stability by calculating the critical Rayleigh number and the associated wavenumber. For nonlinear stability, conditional stability has been established for all initial data with higher-order norms, showing that the $L^2$ theory is insufficient to achieve similar stability, where $L^2$  represents Hilbert space over the domain  $\omega$. It has also been shown that the conditional stability bound can be attained by restricting the internal heat source parameter Q using higher-order norms. The Chebyshev tau-QZ algorithm is used to solve the generalised eigenvalue problem, which is obtained by employing the normal mode technique on the disturbance equations. The effect of various dimensionless parameters, such as anisotropic thermal diffusivity parameters $(\xi )$, Rayleigh number (Ra), angle of inclination ($\phi$) and viscosity parameter $(\varGamma )$, on the stability of the medium is investigated, and streamlines are also shown by contour plots to depict the flow pattern. The results show that the system is stable when $\xi \gt 1$ and unstable when $\xi \lt 1$, favouring convection. It is also observed that the convection is delayed with rising values of $\phi$, and the region of subcritical instability increases uniformly with rising values of $\phi$.

We investigate the transport and adsorption of solutes within graded porous filters characterised by a spatially varying microstructure. While classical homogenisation theory typically assumes periodic media, we employ the method of multiple scales to derive an effective macroscopic model for ‘near-periodic’ geometries where the porosity varies slowly over the longitudinal coordinate. A key novelty of this work is the departure from the standard solenoidal constraint; instead, we introduce a modified incompressibility condition derived from non-equilibrium thermodynamics that accounts for the coupling between the solute concentration and the solvent velocity. This leads to a generalised Darcy-scale description where the fluid velocity field is non-solenoidal within the porous domain. Through asymptotic analysis, we determine the leading-order concentration profiles and quantify first-order corrections that capture the interplay between the porosity gradient and the mixture composition. We evaluate filter performance across several metrics, including outflux concentration and total adsorption rate, under both fixed-flow and fixed-pressure-drop operating conditions. Our results demonstrate that the porosity gradient and the coupling parameter significantly influence the filtration efficiency, particularly as the medium approaches the clogging limit. The analysis reveals that the optimal filter design is highly sensitive to the chosen performance metric, highlighting the necessity of physically consistent boundary conditions and mixture dynamics in the design of high-efficiency graded filters.

We consider the dynamics of two equal, spherical liquid drops of Newtonian fluid, brought into contact at a negligible speed, and merging driven by surface tension. The effects of an outer fluid or that of gravity are disregarded, so that the entire motion is determined uniquely by the Ohnesorge number $\textit{Oh }= \eta /\sqrt {\rho \gamma R}$, where $\eta$ is the viscosity, $\rho$ the density, $\gamma$ the surface tension and $R$ the drop radius. Using methods of matched asymptotics, in the limit that the minimum radius $r_0$ of the bridge connecting the drops is much smaller than the initial drop radius, we compute the dynamics for $r_0(t)$, as well as the shape of the gap between the drops. For very early times, the translational motion of the drops along the axis of symmetry is arrested by inertia, which affects the shape of the scaling function describing the gap on the scale of $r_0$. For $r_0/R \approx \sqrt {\ln \textit{Oh}}/\textit{Oh}$, the motion crosses over to an entirely viscous similarity solution, passing through a sequence of time-dependent scaling functions, which we compute. Our results agree well with full numerical simulations of the Navier–Stokes equation, as well as experiment.

Large-eddy simulations of open-channel flow are conducted to investigate how the channel aspect ratio (), defined as the width ($B$) to depth ($h$) ratio (), affects secondary currents (SCs), large-scale motions (LSMs) and very-large-scale motions (VLSMs). Four cases spanning are considered. For wide-channel flows (), certain flow features dominate depending on the distance from the sidewall, which are: (i) secondary current instability (SCI) within ${\sim} 0.6h$ of the sidewalls; (ii) VLSMs and LSMs from distance $2.4h$ onwards from the sidewalls; and (iii) prevalence of LSMs between (i) and (ii). In the SCI-dominated region (i), two time-averaged SC cells appear, one near the bottom wall–bed corner (BV) and another near the surface (SV). The size and position of the BV and SV oscillate quasi-periodically with a wavelength ${\sim} 6h$, reflecting the instability of the secondary currents. In the VLSM/LSM-dominated region (ii), pre-multiplied spectra of the streamwise velocity fluctuations peak at ${\sim} 3h$ and ${\sim} 20h$, consistent with the presence of LSMs and VLSMs. In the LSM-dominated region (iii), velocity spectra exhibit a LSM prevalent peak at ${\sim} 3h$. The near-sidewall SCI region (i) and central VLSM/LSM region (ii) are phase independent in their oscillations, likely due to a barrier effect of LSM-dominated region (iii) between them. For , the VLSM/LSM region (ii) disappears, while the other two regions exhibit distinct alterations. At , the SCI-dominated region resembles that for , while VLSMs in the LSM-dominated region have a remarkably shorter wavelength (${\sim} 10h$). At a lower (), VLSMs are completely absent in the LSM-dominated region. Based on these results, a schematic framework of the interrelations between SCs and VLSMs across different channel is proposed.

We investigate the dynamics of a rigid spherical particle in plane Couette flow with a viscosity that varies linearly across the channel, from the Stokes limit to non-zero Reynolds numbers, using analytical theory and immersed-boundary lattice Boltzmann simulations. In the Stokes limit, we find that the particle develops a constant, viscosity-gradient-induced translational slip, defined as the difference between the particle velocity and the undisturbed fluid velocity at its centre. The particle leads the flow when the viscosity increases towards the moving wall and lags when it decreases. Its angular velocity equals half the local, viscosity-gradient-modified fluid vorticity, experiencing no additional rotational slip. Furthermore, we show that cross-stream particle migration is strictly prohibited at any confinement ratio or gradient magnitude. At non-zero Reynolds numbers in a constant-viscosity fluid, increasing the channel Reynolds number beyond a critical value induces a supercritical pitchfork bifurcation that splits the stable centreline equilibrium into two symmetric off-centre equilibria. With a viscosity gradient, we find that for Reynolds numbers below this bifurcation point, the particle consistently settles at a single off-centre equilibrium in the less-viscous half. For Reynolds numbers exceeding this point, however, we identify a critical viscosity gradient threshold: below this limit, the particle can stabilise in either channel half depending on its initial placement; above it, the particle focuses exclusively in the less-viscous half regardless of its initial position. These findings enable predictive particle manipulation in viscosity-stratified microflows.

Small-scale shear layers in temporally developing turbulent boundary layers are investigated using direct numerical simulations at momentum-thickness Reynolds numbers from 2000 to 13 000. Flow statistics are evaluated in a local coordinate system aligned with each shear layer identified by the shear intensity. Conditional mean fields show that shear layers are embedded in a biaxial strain. Near the wall, these conditional fields further reveal two distinct regions of intense rigid-body rotation beneath the shear layer, leading to momentum transfer through a mechanism consistent with counter-rotating vortex pairs. Two distinct scalings are found for the shear layers. In the logarithmic and outer layers, the mean shear-related velocity jump $\Delta u_3$ and its width $\delta _3$ scale with the Kolmogorov velocity $u_{\eta T}$ and length $\eta _T$ evaluated in the turbulent region. In the buffer layer, $\delta _3$ scales with the viscous length $\delta _\nu =\nu /u_\tau$ ($\nu$ is the kinematic viscosity and $u_\tau$ the friction velocity), and $\Delta u_3$ scales with $\delta _\nu /\tau _{\eta T}$, consistent with the predominance of shear in energy dissipation, where $\tau _{\eta T}$ is the Kolmogorov time scale. Velocity jumps associated with the biaxial strain exhibit Kolmogorov-based scaling in the logarithmic and outer layers. In the buffer layer, their widths scale with $\delta _\nu$, but their velocity jumps do not scale with $\delta _\nu /\tau _{\eta T}$ because strain contributes little to dissipation. Comparisons with turbulent jets indicate weak wall influence in the logarithmic and outer layers. These scalings provide a basis for predicting shear-layer properties, which is crucial for developing flow-control strategies that leverage small-scale shear instability.

We extend the linear stability analysis of electroconvective instability in a quiescent Oldroyd-B viscoelastic fluid subjected to an electric field into the strongly elastic regime, up to Weissenberg numbers ($\textit{Wi}$) of order $10^{3}$. The oscillatory onset is governed by the interplay among destabilising Coulomb forcing, viscous dissipation and elastic stress relaxation. For the quiescent base state, the linearised constitutive relation allows the polymer stress to be represented through a complex effective viscosity. We then apply a single-mode Galerkin projection based on the Newtonian critical eigenfunction, obtaining a reduced dispersion relation and a closed-form prediction for the neutral oscillation frequency. In the limit of large $\textit{Wi}$, this analytical solution yields two asymptotic scaling laws, namely a frequency scaling law $\omega _{c} \propto \sqrt {(1-\beta )/\textit{Wi}}$ ($\omega _{c}$ is the critical frequency and $\beta$ is the solvent to total viscosity ratio) and a threshold scaling law $T_{c} = \beta T_{c}^{N} + a(\beta )/\textit{Wi}$ ($T_{c}$ is the critical electric Rayleigh number, $T_{c}^{N}$ is the Newtonian critical electric Rayleigh number and $a(\beta )$ is the coefficient related to $\beta$), which satisfy a self-consistency relation. Here, $T_{c}$ exhibits a three-stage evolution with $\textit{Wi}$, and a frozen-elasticity limit is identified at large $\textit{Wi}$. With a single geometric constant as input, the values of $\omega _{c}$ and $T_{c}$ obtained from the analytical solution and from high-resolution spectral computations agree to within 0.3 %, providing mutual corroboration.

We present a theory for the hydrodynamic forces in the drainage of an electrolyte film confined between charged curved surfaces, which applies to small gaps under local equilibrium across the film thickness. We derive a local Onsager formalism describing the coupled flow rate, charge and solute fluxes, which accounts for all degrees of overlap of the electrical double layers (EDLs) and can be adapted to a variety of electrical, chemical or hydrodynamic boundary conditions. We explore in detail the case of a monovalent electrolyte with constant surface charge and no-slip boundary conditions. A complete phase diagram is established for the electro–kinetic damping, including non-overlapping EDLs at both high and low Dukhin numbers, as well as all degrees of EDLs overlap including the Gouy–Chapman, Debye–Hückel and ideal-gas domains. We highlight a simple relation between the apparent hydrodynamic distance at high Dukhin number and the surface electrical conductance. Our theory also predicts a diffusio–kinetic contribution to the so-called electroviscous effect, which does not reduce to a damping but exhibits a conservative (stiffness) component. We show that this stiffness decays as an algebraic function of the gap between the surfaces and may dominate the stiffness due to the equilibrium Derjaguin–Landau–Verwey–Overbeek interaction force.