2.1. Governing equations and 2-D formulation

In formulating the dimensionless equations governing film thinning and rupture and the 2-D algorithm used to solve them, the problem variables are non-dimensionalized by using the undisturbed film half-thickness as the characteristic length, $l_c \equiv h_0$, the visco-capillary time as the characteristic time, $t_c \equiv \mu _0 h_0/\sigma$, the ratio of these two scales as the characteristic velocity, $v_c \equiv l_c/t_c$, the zero-deformation-rate viscosity as the characteristic viscosity $\mu _c \equiv \mu _0$ and the capillary pressure as the characteristic pressure $p_c \equiv \sigma /h_0$. As a result of choosing these characteristic scales, the dynamics is governed by four dimensionless groups (see below): the Ohnesorge number, $Oh \equiv {\mu _0}/(\rho \sigma h_0)^{1/2}$, which represents the ratio of the viscous force to the square root of the product of the inertial and capillary forces; the van der Waals number, $A \equiv A_H/48 {\rm \pi}\sigma {h_0}^2$, which represents the relative importance of intermolecular forces to capillary forces; the power-law exponent or index $n$; and the characteristic deformation rate, $m^{-1} = t_c {\tilde {m}}^{-1}$. For the remainder of this paper, variables without tildes $(\sim )$ over them represent the dimensionless counterparts of the corresponding dimensional variables with tildes over them, e.g. $\tilde {z}$ is the dimensional lateral coordinate whereas $z \equiv \tilde {z}/l_c$ is its dimensionless counterpart.

The dynamics of the liquid sheet is governed by the continuity and Cauchy momentum equations which are given in dimensionless form by

(2.1)\begin{equation} \boldsymbol{\nabla} \boldsymbol{\cdot} {\boldsymbol v} = 0, \end{equation}

(2.2)\begin{equation} \frac{1}{Oh^2} \left ( \frac {\partial {\boldsymbol v}}{\partial t} + {\boldsymbol v} \boldsymbol{\cdot} \boldsymbol{\nabla} {\boldsymbol v} \right ) = \boldsymbol{\nabla} \boldsymbol{\cdot} {\boldsymbol T}. \end{equation}

Here, $\boldsymbol {\nabla } \equiv h_0 \tilde {\boldsymbol {\nabla }}$ is the dimensionless gradient operator, $\boldsymbol v \equiv \tilde {\boldsymbol {v}} /v_c = v_y \boldsymbol {e_y} + v_z \boldsymbol {e_z}$ is the dimensionless liquid velocity, where $v_y$ and $v_z$ are the $y$- and $z$-components of $\boldsymbol v$ and $\boldsymbol {e_y}$ and $\boldsymbol {e_z}$ are the unit vectors in those directions, $t \equiv \tilde {t}/ t_c$ is the dimensionless time, and ${\boldsymbol T} = - p {\boldsymbol I} + \mu [ \boldsymbol {\nabla } \boldsymbol v + (\boldsymbol {\nabla } \boldsymbol v)^{\rm T} ]$ is the dimensionless stress tensor, where $p \equiv \tilde {p}/p_c$ is the dimensionless pressure and $\mu = |2m \dot \gamma |^{n-1}$ is the dimensionless viscosity with $\dot \gamma \equiv \tilde {\dot \gamma } t_c$ denoting the dimensionless deformation rate.

The kinematic and traction boundary conditions are applied at the liquid–gas interface $S(t)$ to enforce mass conservation and to account for the discontinuity or jump in stress due to surface tension and van der Waals forces:

(2.3)\begin{gather} \boldsymbol n \boldsymbol{\cdot} \left ( {\boldsymbol v} - {\boldsymbol v}_s \right ) = 0, \end{gather}

(2.4)\begin{gather} \boldsymbol n \boldsymbol{\cdot} {\boldsymbol T} = 2{H}\boldsymbol n - \frac {A}{h^3}\, \boldsymbol{n}, \end{gather}

where ${\boldsymbol v}_s$ is the velocity of points on the interface $S( t)$, $\boldsymbol {n}$ is the unit normal vector to $S(t)$ and $2{H}$ is twice the mean curvature of $S(t)$. Symmetry boundary conditions are enforced at the two symmetry planes located at $z = 0$ and $z = {\lambda }/2$ by requiring that the lateral velocity and tangential stress equal zero. Finally, along the midplane located at $y = 0$, the vertical velocity and tangential stress both vanish on account of symmetry.

The film is initially quiescent, and the wavelength of the perturbations ${\lambda }$ imposed on its two free surfaces are taken to be larger than the dimensionless critical wavelength obtained from linear stability analysis and given by ${\lambda }_c = {4 \sqrt {2}{\rm \pi} ^{3/2}}{h_0}/{d}$. Thus, the film in its initial configuration is unstable and thereafter thins continuously until it ruptures. Because the dynamics in the rupture zone is insensitive to the values of the amplitude $\epsilon$ and wavelength $\lambda$ of the interface perturbation when $\lambda > \lambda _c$ and $\epsilon \ll 1$, these two parameters are not included in the list of dimensionless groups governing the dynamics in the body of the paper. However, we address briefly the effects of perturbation amplitude and wavelength on the dynamics in Appendix A. Following earlier studies of pinch-off singularities discussed in the introduction, we adopt in the next few sections the universally practiced usage of reporting results as a function of dimensionless time $\tau \equiv (\tilde {t}_R - \tilde {t})/t_c$ remaining until rupture (but also see below).

2.2. Numerical methods for solving the 2-D equations

The 2-D free surface flow governed by the PDEs (2.1)–(2.2) subject to the aforementioned boundary and initial conditions is solved using a fully implicit, method of lines (MOL), arbitrary Langrangian–Eulerian (ALE) algorithm. Here, the Galerkin/finite element method (G/FEM) is used for spatial discretization (Feng & Basaran Reference Feng and Basaran1994) and a predictor–corrector method with an adaptive, implicit finite difference scheme is employed for time integration (Wilkes, Phillips & Basaran Reference Wilkes, Phillips and Basaran1999).

Two features of the flow make the spatial discretization challenging. First, the film's free surfaces undergo large deformations as the sheet approaches rupture. Second, because $h_0 \gg d$, the lateral extent of the domain is three to five orders of magnitude larger than its vertical extent. Therefore, to capture the large deformations that the film's surface $S(t)$ undergoes and to accurately capture the dynamics that occurs over highly disparate length scales in the lateral and vertical directions, the elliptic mesh generation method (Christodoulou & Scriven Reference Christodoulou and Scriven1992; Notz & Basaran Reference Notz and Basaran2004) is used to discretize the domain and determine the vertical and lateral coordinates of each grid point in the moving, adaptive mesh simultaneously with the velocity and pressure unknowns within the liquid sheet. Here, the velocity and pressure unknowns are solved in a mixed interpolation scheme where the nodal values of the velocity are represented in terms of biquadratic basis functions while the pressure unknowns are represented by means of bilinear basis functions. The unknown coordinates of the grid or mesh points are also represented by biquadratic basis functions.

The PDEs are converted to ODEs by the G/FEM formulation while time integration reduces the system of ODEs to a system of nonlinear algebraic equations. These are solved using Newton's method with an analytically computed Jacobian. The linearized system of equations at each iteration is solved using a multi-frontal algorithm (Anthony et al. Reference Anthony, Kamat, Harris and Basaran2019). Variants of this algorithm have been employed by our research group to successfully study hydrodynamic singularities that arise in thread pinchoff of both Newtonian and non-Newtonian fluids (Yildirim & Basaran Reference Yildirim and Basaran2001; Notz & Basaran Reference Notz and Basaran2004; Suryo & Basaran Reference Suryo and Basaran2006; Bhat et al. Reference Bhat, Appathurai, Harris, Pasquali, McKinley and Basaran2010; Castrejón-Pita et al. Reference Castrejón-Pita, Castrejón-Pita, Thete, Sambath, Hutchings, Hinch, Lister and Basaran2015; Kamat et al. Reference Kamat, Wagoner, Thete and Basaran2018), drop and bubble coalescence (Paulsen et al. Reference Paulsen, Burton, Nagel, Appathurai, Harris and Basaran2012; Munro et al. Reference Munro, Anthony, Basaran and Lister2015; Anthony et al. Reference Anthony, Kamat, Thete, Munro, Lister, Harris and Basaran2017), and rupture of supported thin films (Garg et al. Reference Garg, Kamat, Anthony, Thete and Basaran2017). The reader is thus referred to these works for a more complete description of the solution method and numerical implementation.

2.3. 1-D equations for asymptotic analysis

To study sheet dynamics analytically, the long-wavelength approximation (Deen Reference Deen1998; Leal Reference Leal2007) can be invoked to reduce the system of 2-D PDEs (2.1)–(2.2) to a set of 1-D PDEs for the film thickness and lateral velocity. These 1-D PDEs governing thinning and rupture of power-law films have already been derived by Thete et al. (Reference Thete, Anthony, Basaran and Doshi2015). In this paper, we are interested in studying the dynamics of sheets in both the Stokes and the inviscid limits in which the governing equations can be obtained by formally setting the density $\rho = 0$ (or equivalently the reciprocal of the Ohnesorge number $1/Oh = 0$) in the former case and the viscosity $\mu _0=0$ (or equivalently $Oh=0$) in the latter case. Thus, it turns out to be necessary to non-dimensionalize the 1-D evolution equations in two different ways such that (a) one formulation is suitable for studying the dynamics when $Oh \gg 1$ and, in particular, in the limit of $1/Oh = 0$ and (b) the other formulation is appropriate for studying the dynamics when $Oh \ll 1$ and, in particular, in the limit of $Oh = 0$.

2.3.1. 1-D evolution equations: highly viscous sheets

For highly viscous sheets, the 1-D PDEs governing the dynamics are non-dimensionalized by using $l_c \equiv h_0$ as the characteristic length in the vertical direction and $l_z \equiv (48 {\rm \pi}{h_0}^4 \sigma /A_H )^{1/2}$ as the characteristic length in the lateral direction, which incorporates the disparity of the vertical and the lateral length scales into the formulation from the outset. Also in this limit, we use $t_V \equiv 48 {\rm \pi}{h_0}^3 \mu _0/A_H$ as the characteristic time and $v_V \equiv l_z/t_V$ as the characteristic velocity. The characteristic time used here is related to the one used earlier in the 2-D formulation as $t_V \equiv t_c/A$. The dimensionless 1-D evolution equations are then given by

(2.5)\begin{gather} \frac{\partial h}{\partial t} + \frac{\partial \left( h v \right)}{\partial z} = 0, \end{gather}

(2.6)\begin{gather} \frac{1}{Oh^2}\underbrace{\left(\frac{\partial v}{\partial t} + v \frac {\partial v} {\partial z} \right)}_{\text{Inertial (I)}} = \underbrace{\frac{\partial^3 h}{\partial { z}^3}}_{\text{Capillary (C)}} + \underbrace{\frac{\partial \left( { h}^{{-}3} \right)}{\partial { z}}}_{\text{van der Waals (vdW)}} + \underbrace{\frac{4}{ h} \frac {\partial}{\partial z} \left({\mu_V} {h} \frac {\partial v}{\partial z} \right)}_{\text{Viscous (V)}}. \end{gather}

Here, $z \equiv \tilde {z} / l_z$ is the dimensionless lateral length, $t \equiv \tilde {t}/t_V$ is the dimensionless time, $h ( z, t )$ denotes the dimensionless film half-thickness, $v ( z, t )$ denotes the dimensionless fluid velocity in the lateral or $z$ direction and $\mu _V = {|2 m_1 {\partial v}/{\partial z} |}^{n-1}$, where $m_1$ is related to $m$ in the 2-D formulation as $m_1 = mA$. The different forces that influence van der Waals-driven rupture of sheets, namely inertial (I), capillary (C), van der Waals (vdW) and viscous (V) forces, are identified by labels placed under each of the corresponding terms in (2.6).

2.3.2. 1-D evolution equations: slightly viscous sheets

For slightly viscous sheets, the 1-D PDEs governing the dynamics are non-dimensionalized by using $l_c \equiv h_0$ as the characteristic length in the vertical direction and $l_z \equiv (48 {\rm \pi}{h_0}^4 \sigma /A_H )^{1/2}$ as the characteristic length in the lateral direction which, once again, recognizes the disparity of length scales in the two directions from the beginning. The characteristic length scales in the two directions used in this limit are thus the same as those used for highly viscous sheets. In this limit, however, since viscosity should not be involved in defining characteristic scales, we use $t_I \equiv ({\rho l_z^4}/{\sigma h_0})^{1/2}$ as the characteristic time and $v_I \equiv l_z/t_I$ as the characteristic velocity. The characteristic time used here is related to the one used in the 2-D formulation as $t_I = t_c/(Oh \, A)$. The dimensionless 1-D evolution equations are then given by

(2.7)\begin{gather} \frac{\partial h}{\partial t} + \frac{\partial \left( h v \right)}{\partial z} = 0, \end{gather}

(2.8)\begin{gather} \underbrace{\left(\frac{\partial v}{\partial t} + v \frac {\partial v} {\partial z} \right)}_{\text{Inertial (I)}} = \underbrace{\frac{\partial^3 h}{\partial { z}^3}}_{\text{Capillary (C)}} + \underbrace{\frac{\partial \left( { h}^{{-}3} \right)}{\partial { z}}}_{\text{van der Waals (vdW)}} + \underbrace{\frac{4 Oh}{ h} \frac {\partial}{\partial z} \left({\mu_I} {h} \frac {\partial v}{\partial z} \right)}_{\text{Viscous (V)}}. \end{gather}

Here, $z \equiv \tilde {z} / l_z$ is the dimensionless lateral length, $t \equiv \tilde {t}/t_I$ is the dimensionless time, $h ( z, t )$ denotes the dimensionless film half-thickness, $v ( z, t )$ denotes the dimensionless fluid velocity in the lateral or $z$ direction and $\mu _I = {|2 m_2 {\partial v}/{\partial z} |}^{n-1}$, where $m_2$ is related to $m$ in the 2-D formulation as $m_2 = m \, Oh \, A$. The different forces that influence van der Waals-driven rupture of sheets have once again been identified by labels placed under each of the corresponding terms in (2.8).

2.3.3. Local analyses and similarity variables

The film thickness and velocity profiles are expected to be self-similar in the vicinity of the location where the film thickness is minimum as the space–time singularity is approached. In § 3, we explore the dynamical behaviour of the film thickness and lateral velocity in this so-called ‘rupture zone’, which is shown in figure 1(b), by numerical solution of the 2-D system of (2.1)–(2.2) and theoretical analysis of the 1-D system of equations that have just been presented (see §§ 2.3.1 and 2.3.2).

In the 1-D analysis, we presume that the film half-thickness $h(z,t)$ and the lateral velocity $v(z,t)$ can be represented by adopting the similarity ansatz

(2.9a–c)\begin{equation} h \equiv \tau_*^{\alpha}H(\xi), \quad v \equiv \tau_*^{\gamma}V(\xi),\quad \xi \equiv \zeta / \tau_*^{\beta}, \end{equation}

where $\tau _* \equiv (\tilde {t}_R - \tilde {t})/t_*$ is the dimensionless time remaining until rupture and $t_*$ is a characteristic time, with both $\tau _*$ and $t_*$ being defined differently depending on whether the sheet is highly or slightly viscous (see below). Here, $\xi$ is the similarity variable, $\zeta \equiv (\tilde {z} - \tilde {z}_R)/l_z$ is the dimensionless lateral coordinate relative to the rupture point, $\alpha$, $\beta$ and $\gamma$ are the scaling exponents, and $H(\xi )$ and $V(\xi )$ are the scaling functions for the film thickness profile and lateral velocity in similarity space. When using the non-dimensionalization adopted with the 1-D analysis for highly viscous sheets and, in particular, when analysing the Stokes limit ($1/Oh^2=0$ in (2.6)), we denote $\tau _*$ by $\tau _V$ and note that $t_* \equiv t_V$, viz. $\tau _v \equiv (\tilde {t}_R - \tilde {t})/t_V$ (the use of subscripts ‘$V$’ is intended to highlight that we are in a highly viscous regime). Similarly, when using the non-dimensionalization adopted with the 1-D analysis for slightly viscous sheets and, in particular, when analysing the inviscid limit ($Oh=0$ in (2.8)), we denote $\tau _*$ by $\tau _I$ and note that $t_* \equiv t_I$ (the use of subscripts ‘$I$’ is intended to highlight that we are in a slightly viscous or nearly inviscid regime). We note the three dimensionless times remaining until rupture are related as $\tau _V = A \tau$ and $\tau _I = Oh \, A\tau$.

5.1. Thinning of sheets of power-law fluids when $Oh \gg 1$

For sheets of $Oh \gg 1$, depending on the value of the power-law index $n$, the initial dynamics is expected to lie in one of the two scaling regimes that arise in Stokes flow, as described in § 3. To determine when a transition from these creeping flow regimes may occur as the sheet continues to thin, it proves indispensable to examine how the inertial terms vary as time advances or the time remaining to rupture $\tau _v$ tends to zero. For a power-law fluid of $0.58 < n \le 1$, we expect from the results of § 3 that the initial dynamics will lie in the PLV regime. Thus, the variation with $\tau _v$ of the inertial terms (I) in the momentum equation (2.2) is given by

(5.1)\begin{equation} I \sim \frac{v}{t} \sim \frac{\tau_v^{\beta - 1}}{\tau_v}\sim \tau_v^{\beta - 2}, \end{equation}

where $\tau _v$ is defined in § 3. Given the functional dependence of $\beta$ on $n$ for $0.58 < n \le 1$ shown in figure 4 and the variation of the dominant viscous (V) and van der Waals (vdW) forces with $\tau _v$ in the PLV regime (cf. (3.2a,b)), it is clear that the inertial terms are blowing up at a faster rate than the two dominant forces in this regime. Thus, while the inertial (I) terms initially might be negligible on account of the large value of $Oh$, they can catch up to the V and vdW forces as time advances because of the larger rate of growth of the I terms compared to the V and vdW forces. Hence, while the initial dynamics lies in the PLV regime, a transition to a new regime is expected where inertial force features in the dominant balance of forces for real fluids of $Oh \gg 1$. To predict when this transition occurs, it proves useful to calculate the instantaneous or local Reynolds number $Re \equiv Re(\tau _v)$ in the rupture zone (this local Reynolds number can also be thought of as being equivalent to the reciprocal of the square of the local Ohnesorge number):

(5.2)\begin{equation} Re = \frac{\rho \tilde{z} \tilde{v}}{\tilde{ \mu}} = \frac{\rho l_z v_V}{\mu_0}\tau_v^{2\beta + n -2} = \frac{\tau_v^{2\beta + n -2}}{Oh^2}, \end{equation}

where $l_z$ and $v_V$ are defined in § 2.3. Figure 4 shows that $\beta < 0.39$ and since $n \le 1$, the exponent $2\beta + n -2 < 0$ in the PLV regime in which the power-law index ranges as $0.58 < n \le 1$. Therefore, when $Oh \gg 1$, while the Reynolds number $Re \ll 1$ initially, because the exponent of $\tau _v < 0$ in (5.2), $Re$ grows without bound as $\tau _v \rightarrow 0$. Indeed, the inertial terms would become comparable to the viscous and van der Waals ones when $Re \approx 1$. From (5.2), this occurs when

(5.3)\begin{equation} \tau_v \sim Oh^{2/(2\beta + n - 2)}. \end{equation}

When the local Reynolds number $Re = O(1)$, the dynamics in the rupture zone is expected to follow the power-law inertial–viscous (PLIV) scaling behaviour described by Thete et al. (Reference Thete, Anthony, Basaran and Doshi2015) for sheets of power-law fluids of $Oh = O(1)$ when the power-law index $n$ is restricted to lie in the range $6/7 < n \le 1$. Thus, in the present situation, a transition from the PLV regime to the PLIV regime occurs at the time remaining until rupture given by (5.3). This estimate of the time can then be substituted in (3.8a–c) to obtain estimates of the values of the minimum film half-thickness, $h_{min,t}$, and lateral length scale, $z_t^{\prime }$, when the transition should occur:

(5.4a,b)\begin{equation} h_{min,t} \sim Oh^{2n/3(2\beta + n - 2)}, \quad z'_t \sim \frac{l_z}{h_0}Oh^{2\beta/(2\beta + n - 2)}. \end{equation}

Figure 6 shows the variation with $\tau$ of several quantities of interest for a sheet of a power-law fluid of $Oh = 50$ and $n=0.9$ undergoing rupture. First, figure 6(a) shows that $h_{min}$ decreases with $\tau$ as $h_{min} \sim {\tau }^{0.9/3} \sim \tau ^{0.3}$ all the way until rupture. However, since the scaling exponent of $h$ is the same in both the PLV and PLIV regimes, the latter result is necessary but insufficient to demonstrate the occurrence of the expected transition between the two scaling regimes. Nevertheless, the pre-factor of the best-fit line to the data undergoes a change in value when $h_{min} \approx 10^{-2}$, which is in excellent agreement with the expected value of $h_{min,t} \sim 1.2 \times 10^{-2}$ calculated from (5.4a,b). Figure 6(b) shows the variation with $\tau$ of the lateral length scale $z'$. As the figure makes clear, the lateral length initially varies with time to rupture as $z' \sim \tau ^{0.28}$: thus, the dynamics initially is in the PLV regime as the scaling exponent of the lateral length $\beta = 0.28$ when $n = 0.9$, as shown earlier in § 3. However, the dynamics is clearly seen to transition to a final asymptotic PLIV regime in the later stages of thinning as the simulations show that $z' \sim \tau ^{0.55} \sim \tau ^{1-n/2}$ as $\tau \to 0$. From the simulation results, this transition is seen to occur at a value of $z' \approx 1 \times 10^1$, which is in good agreement with the expected value of $z'_{t} \sim (l_z/h_0)Oh^{2\beta /(2\beta + n - 2)} \sim 5.7 \times 10^{1}$ from (5.4a,b). This transition can also be seen clearly in figure 6(c), where the lateral velocity initially varies with $\tau$ as $v' \sim \tau ^{-0.72} \sim \tau ^{\beta -1}$ but at larger times (or smaller $\tau$) varies as $v' \sim \tau ^{-0.45} \sim \tau ^{-n/2}$. Thus, for sheets of power-law fluids of $Oh \gg 1$ and $6/7 < n \le 1$, the thinning dynamics initially lie in the PLV regime but eventually transition into the PLIV regime as rupture is approached.

Figure 6.

Computed scaling behaviour of key variables in the rupture zone during thinning of a moderately viscous sheet of a power-law fluid. Here, $Oh = 50$, $n=0.9$, $A = 9.21 \times 10^{-8}$ and $m = 1/A$. The variation with time remaining to rupture $\tau$ of (a) the minimum film half-thickness $h_{min}$, (b) lateral length scale $z'$ and (c) lateral velocity $v'$ evaluated at the same lateral location as $z'$. Results of 2-D simulations are shown by the data points while the straight lines represent best fits to the data. For this set of parameters, a transition from the PLV regime to the PLIV regime is observed.

For sheets with smaller values of the power-law index in the range $0.58 < n \le 6/7$, the initial dynamics is still expected to lie in the PLV regime as made plain by the analysis in § 3. However, when the local Reynolds number becomes $O(1)$, a different transition is expected to occur than the one that is described in the previous paragraph based on the results of Thete et al. (Reference Thete, Anthony, Basaran and Doshi2015). These authors have shown that when $Oh = O(1)$ and $0 < n \le 6/7$, the local dynamics lies in the IC regime and the scaling exponents are given by (1.5a–c). Thus, in the present case where $Oh \gg 1$, once the local $Re = O(1)$, the dynamics of sheets of $0.58 < n \le 6/7$ is expected to transition from the initial PLV regime to a late-stage IC regime. The values of the minimum film half-thickness and lateral length scale at which this transition occurs can once again be estimated from (5.4a,b) as the initial regimes in both cases, viz. when $6/7 < n \le 1$ and $0.58 < n \le 6/7$, are identical. Figure 7 shows the variation with $\tau$ of several quantities of interest for a sheet of a power-law fluid of $Oh = 1000$ and $n=0.6$ undergoing rupture. In this situation, the minimum film half-thickness $h_{min}$ is seen to initially vary with $\tau$ (figure 7a) as $h_{min} \sim \tau ^{0.2} \sim \tau ^{0.6/3}$ but is later observed to vary as $h_{min} \sim \tau ^{2/7}$, thereby signifying a transition to the IC regime. Corresponding transitions are also observed for both the lateral length scale $z'$ and lateral velocity $v'$ in figures 7(b) and 7(c). Moreover, the transition for $h_{min}$ obtained from simulations is observed to occur at $h_{min} \approx 7 \times 10^{-3}$, which is in good agreement with $h_{min,t} \sim 1.33 \times 10^{-2}$ determined from (5.4a,b). Similarly, the transition for $z'$ is observed to occur at $z' \approx 6 \times 10^{-1}$, which is again in good agreement with $z'_t \sim 1.14 \times 10^{0}$ determined from (5.4a,b). Thus, for sheets of power-law fluids of $Oh \gg 1$ and $0.58 < n \le 6/7$, the thinning dynamics transitions from an initial PLV regime to a final IC regime where the film essentially behaves like an inviscid fluid in the rupture zone.

Figure 7.

Computed scaling behaviour of key variables in the rupture zone during thinning of a highly viscous sheet of a moderately deformation-rate-thinning power-law fluid. Here, $Oh = 1000$, $n = 0.6$, $A = 9.21 \times 10^{-8}$ and $m = 1/A$. The variation with time remaining to rupture $\tau$ of (a) the minimum film half-thickness $h_{min}$, (b) lateral length scale $z'$ and (c) lateral velocity $v'$ evaluated at the same lateral location as $z'$. Results of 2-D simulations are shown by the data points while the straight lines represent best fits to the data. For this set of parameters, a transition from the PLV regime to the IC regime is observed.

For sheets of power-law fluids with power-law exponents in the range $0 < n \le 0.58$, it is shown in § 3 that the dynamics lies in the PLCV regime in which the sheets are undergoing Stokes flow. Thus, for a sheet of $Oh \gg 1$ and $n$ values in this range, we expect the local dynamics to initially lie in the PLCV regime. In this regime, the variation of inertial terms (I) with time remaining to rupture is given by

(5.5)\begin{equation} I \sim \frac{v}{t} \sim \frac{\tau_v^{2n/3 - 1}}{\tau_v}\sim \tau_v^{2n/3- 2}. \end{equation}

Since $n \le 0.58$, it is clear from (3.7) and (5.5) that the inertial terms blow up faster than the capillary, van der Waals and viscous forces that are in balance in this regime. Once again, it proves advantageous to compute the instantaneous Reynolds number in the rupture zone which varies with $\tau _v$ as

(5.6)\begin{equation} Re = \frac{\rho \tilde{z} \tilde{v}}{\tilde{ \mu}} = \frac{\rho l_z v_V}{\mu_0}\tau^{7n/3 -2} = \frac{\tau_v^{(7n-6)/3}}{Oh^2}. \end{equation}

As $n \le 0.58$, no matter how large the Ohnesorge number is, the local Reynolds increases without bound as $\tau _v$ decreases. Thus, neglect of inertia is inconsistent asymptotically as $\tau _v \to 0$. Therefore, when the local $Re$ in the rupture zone becomes $O(1)$, a transition is expected to occur from the initial PLCV regime to a final stage IC regime, as has already been demonstrated by Thete et al. (Reference Thete, Anthony, Basaran and Doshi2015) for films of $Oh = O(1)$. Once again, we can determine the value of $\tau _v$ at which $Re = O(1)$ and make use of the scaling relations and exponents in the PLCV regime in (3.9a–c) to estimate the minimum film half-thickness and lateral length scale for which this transition occurs:

(5.7a,b)\begin{equation} h_{min,t} \sim Oh^{2n/(7n-6)}, \quad z'_t \sim \frac{l_z}{h_0}Oh^{4n/(7n-6)}. \end{equation}

Figure 8 shows the variation with $\tau$ of several quantities of interest for a sheet undergoing rupture for a fluid of $Oh = 1000$ and $n=0.5$. The minimum film half-thickness $h_{min}$ is seen to initially vary with $\tau$ (figure 8a) as $h_{min} \sim \tau ^{0.5/3}$ but later transitions and thereafter varies as $h_{min} \sim \tau ^{2/7}$, signifying a transition from the initial PLVC regime to the final IC regime. Similarly, the variation of the lateral length scale with $\tau$ (figure 8b) is seen to transition from $z' \sim \tau ^{1/3} \sim \tau ^{2n/3}$ at early times where the dynamics lies in the PLCV regime to $z' \sim \tau ^{4/7}$ at later times, once again signifying that the dynamics has transitioned to the IC regime. A corresponding transition is also observed for the lateral velocity in figure 8(c). Moreover, the computed transition between the two regimes is observed to occur at $h_{min} \approx 4 \times 10^{-2}$, which is in good agreement with the scaling estimate $h_{min,t} \sim 6.3 \times 10^{-2}$ determined from (5.7a,b). Similarly, the computed value of the lateral length scale at which the transition takes place is observed to occur at $z' \approx 7 \times 10^{0}$, which is also in good agreement with the scaling estimate $z'_t \sim 1.3 \times 10^{1}$ determined from (5.7a,b). Thus, for sheets of power-law fluids of $Oh \gg 1$ and $0 < n \le 0.58$, the thinning dynamics transitions from an initial PLCV regime to a final IC regime. Hence, in the rupture zone, the film for all practical purposes behaves like an inviscid fluid asymptotically as $\tau \to 0$.

Figure 8.

Computed scaling behaviour of key variables in the rupture zone during thinning of a highly viscous sheet of a highly deformation-rate-thinning power-law fluid. Here, $Oh = 1000$, $n = 0.5$, $A = 9.21 \times 10^{-8}$ and $m = 1/A$. The variation with time remaining to rupture $\tau$ of (a) the minimum film half-thickness $h_{min}$, (b) lateral length scale $z'$ and (c) lateral velocity $v'$ evaluated at the same lateral location as $z'$. Results of 2-D simulations are shown by the data points while the straight lines represent best fits to the data. For this set of parameters, a transition from the PLCV to the IC regime is observed.

5.2. Thinning of sheets of power-law fluids when $Oh \ll 1$

For sheets of power-law fluids of $Oh \ll 1$, the initial dynamics is expected to lie in the IC regime discussed in § 4 as viscous force is negligible on account of the small value of $Oh$. However, as the film thins and fluid velocity in the rupture zone increases, it is possible for viscous force to become significant. To test this possibility, it proves useful to determine how the viscous term (V) in the momentum (2.2) varies with time to rupture $\tau _I$ in the inertial regime:

(5.8)\begin{equation} V \sim \mu\frac{v}{z^2} \sim \frac{\tau_I^{4/7-n}}{\tau_I^{8/7}}\sim \tau_I^{{-}n -4/7}. \end{equation}

Here, $\tau _I$ is the dimensionless time to rupture where time is nondimensionalized using $t_I$ as the characteristic time scale. From (4.1) and (5.8), it is clear that viscous force blows up faster than inertial, capillary and van der Waals forces that are in balance when $6/7 < n \le 1$ as $\tau _I \rightarrow 0$. Thus, while viscous force might be initially negligible, it is expected to become significant and catch up to the other three forces as the sheet continues to thin. To quantify the importance of viscous force relative to others and to determine whether it can catch up to the other three forces, it proves useful once more to calculate the instantaneous Reynolds number which is given by

(5.9)\begin{equation} Re = \frac{\rho \tilde{z} \tilde{v}}{\tilde{ \mu}} = \frac{\rho l_z v_I}{\mu_0}\tau^{n-6/7} = \frac{\tau_I^{(7n-6)/7}}{Oh}. \end{equation}

Plainly, when $6/7 < n \le 1$, the local Reynolds number cannot remain large as $\tau \to 0$ no matter how small the Ohnesorge number. Therefore, it is asymptotically inconsistent to neglect viscous force as $\tau \to 0$. Viscous force is expected to become significant when $Re = O(1)$, which, from (5.9), can be estimate to occur when

(5.10)\begin{equation} \tau_I \sim Oh^{7/(7n-6)}. \end{equation}

At this time, since inertial and viscous forces are now comparable in the rupture zone, a transition is expected to occur from the initial IC regime to the PLIV regime. The transition between these two regimes can be estimated to occur for values of the minimum film half-thickness $h_{min}$ and lateral length scale $z'$ given by

(5.11a,b)\begin{equation} h_{min,t} \sim Oh^{2/(7n-6)}, \quad z'_t \sim \frac{l_z}{h_0} Oh^{4/(7n-6)}. \end{equation}

Figure 9 shows the variation with $\tau$ of several quantities of interest for a sheet of a power-law fluid of $Oh = 0.08$ and $n=0.97$ undergoing rupture. The results of the 2-D simulations shown in figure 9(a) make plain that the minimum film half-thickness $h_{min}$ initially varies with $\tau$ as $h_{min} \sim \tau ^{2/7}$, in excellent agreement with the theoretically predicted scaling law in the IC regime, but also that the dynamics transitions in the later stages of thinning so that $h_{min} \sim \tau ^{0.97/3} \sim \tau ^{n/3}$, in agreement with the theoretically predicted scaling in the PLIV regime. Furthermore, the lateral length scale $z'$ is seen to vary with $\tau$ in figure 9(b) as $z' \sim \tau ^{4/7}$ at early times, which is again in excellent agreement with theory in the IC regime, but transitions later to $z' \sim \tau ^{0.515} \sim \tau ^{1-n/2}$, which is also in excellent agreement with the expected scaling in the PLIV regime. Finally, figure 9(c) shows the corresponding transition between the IC and PLIV regimes for the lateral velocity $v'$. Moreover, the transition in $h_{min}$ is observed to occur at $h_{min} \approx 4 \times 10^{-3}$, which is in good agreement with the scaling estimate $h_{min,t} \sim 1.67 \times 10^{-3}$ determined from (5.11a). Similarly, the transition in $z'$ is observed to occur at $z' \approx 2 \times 10^{-2}$, which is again in good agreement with $z'_t \sim 9.21 \times 10^{-3}$ determined from the scaling estimate given in (5.11b). Thus, for sheets of power-law fluids of $Oh \ll 1$ and $6/7 < n \le 1$, the thinning dynamics transitions from the IC regime to the PLIV regime as $\tau \to 0$ and as viscous forces become significant on account of the inexorable increase in the fluid velocity in the rupture zone as the sheet tends toward breakup.

Figure 9.

Computed scaling behaviour of key variables in the rupture zone during thinning of a slightly viscous sheet of a power-law fluid. Here, $Oh = 0.08$, $n = 0.97$, $A = 9.21 \times 10^{-8}$ and $m = 1/A$. The variation with time remaining to rupture $\tau$ of (a) the minimum film half-thickness $h_{min}$, (b) lateral length scale $z'$ and (c) lateral velocity $v'$ evaluated at the same lateral location as $z'$. Results of 2-D simulations are shown by the data points while the straight lines represent best fits to the data. For this set of parameters, a transition from the IC to the PLIV regime is observed.

For sheets of power-law fluids of $Oh \ll 1$ and $0 < n \le 6/7$, the dynamics is expected to remain in the IC regime throughout the entire duration of thinning until the film ruptures. This expectation is clear from (4.1) and (5.8): viscous force blows up at a slower rate than and hence is subdominant to the inertial, capillary and van der Waals forces that remain in balance as the sheet thins. The correctness of this expectation is also confirmed from 2-D numerical simulations in figure 10 which shows computed predictions for the thinning of a sheet of a power-law fluid of $Oh = 0.08$ and $n=0.6$. The computed variation with $\tau$ of $h_{min}$, $z'$ and $v'$ is identical to that expected to occur from theory in the IC regime throughout the duration of thinning.

Figure 10.

Computed scaling behaviour of key variables in the rupture zone during thinning of a slightly viscous sheet of a highly deformation-rate-thinning power-law fluid. Here, $Oh = 0.08$, $n=0.6$, $A = 9.21 \times 10^{-8}$ and $m = 1/A$. The variation with time remaining to rupture $\tau$ of (a) the minimum film half-thickness $h_{min}$, (b) lateral length scale $z'$ and (c) lateral velocity $v'$ evaluated at the same lateral location as $z'$. Results of 2-D simulations are shown by the data points while the straight lines represent best fits to the data. For this set of parameters, the dynamics is observed to lie in the IC regime throughout the duration of thinning.