Fluid flow beneath a semipermeable membrane during drying processes

The dynamic interactions between a semipermeable membrane and a long, thin layer of liquid beneath it are investigated in the context of drying processes. The membrane separates two aqueous solutions of sugar, and the transport of water across the membrane is driven by concentration and pressure gradients across it. A model is formulated using a long-wave approximation that includes the effects of volume loss due to water transport across the membrane, the incompressibility and bending stiffness of the membrane, and the dynamical effects that arise owing to the viscous stresses generated by the fluid flow. This model is first applied to study the desiccation of a sessile vesicle that is clamped to a rigid substrate and then also to study the behavior of blisters on laminated substrates. For each problem, equilibrium membrane shapes are obtained and their bifurcation structures are described as the sugar concentration above the membrane is varied. It is demonstrated that a wrinkled membrane coarsens to lessen the frequency of wrinkles and that if the membrane is clamped symmetrically so that it meets the substrate at a nonzero angle, then the membrane favors an asymmetric shape as water is drawn out through it.

Figure 1

Diagram of the problem.

Figure 2

Schematic diagrams of the three problems considered in this paper. (a) Problem I: a vesicle that is adhered to a substrate. (b) Problem II: an isolated blister in a membrane that coats a substrate. (c) Problem III: the buckling of a long film that lies on a cushion of liquid.

Figure 3

Equilibrium solutions for Problem I with $P_0=0$. The other parameter values used are $\ell =16/3$, $\overline{S}=2/300$ (so that $c_{\mathrm{init}}={10}^{-2}$), $k_1={10}^2$, and $k_2={10}^{-2}$. (a) The variation of internal sugar concentration $c_{\mathrm{in}}$ as the external sugar concentration $c_{\mathrm{out}}$ is varied. Circles denote bifurcation points, and arrows denote the variation of the membrane shapes [shown in (b)] as the branch is traversed (see text). The chain-dashed branch (labeled I) represents the parabolic-cap solution, the solid branches (two of which are labeled II and III) represent asymmetric solutions, and the dashed branch (three portions of which are labeled IV-VI) represents symmetric solutions. (b) Equilibrium membrane shapes. Arrows show the direction in which the branches are traversed in (a), and solutions are plotted for $c_{\mathrm{in}}=0.015$ and multiples of $0.01$ thereafter.

Figure 4

Equilibrium solutions for Problem I, with $P_0={10}^8$ and $\lambda =5\times {10}^{-3}$. The other parameter values are the same as in Fig. 3. (a) The variation of internal sugar concentration $c_{\mathrm{in}}$ as the external sugar concentration $c_{\mathrm{out}}$ is varied. Circles denote bifurcation points, and arrows denote the variation of the membrane shapes [shown in (b)] as the branch is traversed (see text). (b) Equilibrium membrane shapes. Arrows show the direction in which the branches are traversed in (a) and solutions are plotted for $c_{\mathrm{in}}=0.015$ and multiples of $0.005$ thereafter.

Figure 5

Equilibrium solutions for Problem II, in the absence (dash-dotted, labeled I, and dotted, labeled II) and presence (dashed, labeled III, and solid, labeled IV) of hydration stresses. Parameter values used are the same as for Fig. 4. (a) The variation of internal sugar concentration $c_{\mathrm{in}}$ as the external sugar concentration $c_{\mathrm{out}}$ is varied. The symmetric solution branches are labeled I and III, and the asymmetric solution branches are labeled II and IV. (b) The variation of the membrane's compression $\Gamma$ as $c_{\mathrm{out}}$ is varied. (c) The same plot as (a) except with a narrower range of values for $c_{\mathrm{in}}$ and $c_{\mathrm{out}}$ so that the bifurcation points may be seen more clearly. (d) Equilibrium membrane shapes in the absence (I and II) and presence (III and IV) of hydration stresses. Arrows show the direction in which the branches are traversed, and solutions are shown for $c_{\mathrm{out}}=0.08$ and multiples of $0.02$ thereafter.

Figure 6

Equilibrium solutions for Problem III, in the absence (thin lines) and presence (bold lines) of hydration stresses. Parameter values used are the same as for Fig. 4. (a) The variation of internal sugar concentration $c_{\mathrm{in}}$ as the external sugar concentration $c_{\mathrm{out}}$ is varied. (b) The variation of membrane compression $\Gamma$ as the external sugar concentration $c_{\mathrm{out}}$ is varied. (c) Equilibrium membrane shapes, in the absence (thin lines, and offset in the $y$ direction) and presence (bold lines) of hydration stresses, for each of the four modes shown in (a) and (b). Arrows show the direction in which the solution branches are traversed, and the solutions are shown at $c_{\mathrm{out}}=0.05$ and multiples of $0.05$ thereafter. For these parameter values, the membrane's shape for mode IV are almost indistinguishable from each other.

Figure 7

The stability of equilibrium solutions and the height profiles of disturbance eigenmodes for Problem I. Parameter values used are the same as for Fig. 4, and the height profiles are plotted for $c_{\mathrm{out}}=0.1$. (a) The bifurcation diagram shown in Fig. 4 with the stable solutions shown using bold, continuous lines. Unstable solutions are shown using dotted lines. (b) The amplitude $\widehat{h}$ of the disturbance (above) and the perturbed membrane shape (beneath). The amplitudes have been normalized to have a maximum magnitude of 0.5, and the perturbed membrane shapes are a superposition of the shape of the steady state (shown by a solid line) and $\pm 0.4$ times the disturbance mode.

Figure 8

The stability of equilibrium solutions and the height profiles of disturbance eigenmodes for Problem II. Parameter values used are the same as for Figs. 4 and 5, and the height profiles in (b) are plotted for $c_{\mathrm{out}}=0.15$. (a) The bifurcation diagram shown in Fig. 5 with the stable solutions shown using bold lines. Unstable solutions are shown using thin dotted lines and can be seen for $0.055\le c_{\mathrm{out}}\le 0.075$. (b) The amplitude $\widehat{h}$ of the disturbance (above) and the perturbed membrane shape (beneath). The amplitudes have been normalized to have a maximum magnitude of 0.5, and the perturbed membrane shapes are a superposition of the shape of the steady state (shown by a solid line) and $\pm 0.4$ times the disturbance mode.

Figure 9

Growth rates and height profiles of disturbance eigenmodes for Problem III. Parameter values used are the same as for Fig. 4, and the height profiles are plotted for $c_{\mathrm{out}}=0.1$. (a) The growth rates of the most unstable disturbance mode for each of the four steady states given by $n=\mathrm{I},...,\mathrm{IV}$. Two growth rates are shown for mode II because the stability properties of this mode depend on where the symmetry boundary conditions are prescribed (see text). (b) The amplitude $\widehat{h}$ of the disturbance (above) and the perturbed membrane shape (beneath). The amplitudes have been normalized to have a maximum magnitude of 0.5, and the perturbed membrane shapes are a superposition of the shape of the steady state (shown by a solid line) and $\pm 0.4$ times the disturbance mode.