Slightly deformable Darcy drop in linear flows

A two-phase flow model is developed to study the small deformation of a poroelastic drop under linear flows. Inside the drop a deformable porous network characterized by an elastic modulus is fully immersed in a viscous fluid. When the viscous dissipation of the interior fluid phase is negligible (compared to the friction between the fluid and the skeleton), the two-phase flow is reduced to a poroelastic Darcy flow with a deformable porous network. At the interface between the poroelastic drop and the exterior viscous Stokes flow, a set of boundary conditions are derived by the free-energy-dissipation principle. Both interfacial slip and permeability are taken into account and the permeating flow induces dissipation that depends on the elastic stress of the interior solid. Assuming that the porous network has a large elastic modulus, a small-deformation analysis is conducted. A steady equilibrium is computed for two linear applied flows: a uniaxial extensional flow and a planar shear flow. By exploring the interfacial slip, permeability, and network elasticity, various flow patterns about these slightly deformed poroelastic drops are found at equilibrium. The linear dynamics of the small-amplitude deviation of the poroelastic drop from the spherical shape is governed by a nonlinear eigenvalue problem, and the eigenvalues are determined.

Figure 1

Shown on the left is a sketch of a poroelastic drop (in ${\mathrm{\Omega }}_I$) immersed in a Stokesian fluid (in ${\mathrm{\Omega }}_E$). Inside the drop there is a fluid phase (subscript $f$) and a skeleton phase (subscript $s$) which is bounded by the deformable interface ${\mathrm{\Gamma }}_t$. Here $\widehat{\mathbf{n}}$ is the outward normal, $\beta$ is the interfacial slip coefficient, and $\eta$ is the interfacial permeability. On the right is a uniaxial extensional flow in the far field (top) and a planar shear flow in the far field (bottom).

Figure 2

Shown on the left is an illustration of a slightly deformed poroelastic drop under a uniaxial extensional flow, with ${\varphi }_0=0.5, \mathrm{\Lambda }=1/3$, and ${\alpha }_e=0.005$. The black dashed line is the original spherical drop shape. The thick red solid line is the equilibrium drop shape. Here $(\beta ,\eta )=(0,0)$. On the right are streamlines in the first quadrant for two sets of $(\beta ,\eta )$ as labeled.

Figure 3

Radial displacement (${\widehat{\mathbf{u}}}_s·\widehat{\mathbf{r}}$) of a poroelastic drop in a uniaxial extensional flow, with $\mathrm{\Lambda }=1/3, {\varphi }_0=0.5$, and ${\alpha }_e=0.005$. (a) Radial displacement versus $\beta$ for three values of $\eta$ as labeled. (b) Radial displacement versus $\eta$ for three values of $\beta$ as labeled.

Figure 4

Inflow into a poroelastic drop in a uniaxial extensional flow as a function of (a) $\eta$ and (b) $\beta$, with $\mathrm{\Lambda }=1/3, {\alpha }_e=0.005$, and ${\varphi }_0=0.5$.

Figure 5

Variation of (a) radial displacement and (b) inflow with respect to porosity ${\varphi }_0$, with $\mathrm{\Lambda }=1/3$ and ${\alpha }_e=0.005$. The interfacial slip $\beta =0$ for the results in (b).

Figure 6

Shown on the left is an illustration of a poroelastic drop in a planar shear flow $\mathbf{V}=(\dot{\gamma }y,0,0)$ and on the right decomposition of a planar shear flow into a straining component and a rotational component.

Figure 7

Flow around a slightly deformable poroelastic drop under a simple shear flow with ${\varphi }_0=0.5, \mathrm{\Lambda }=1/3, {\alpha }_e=0.01$, and $(\beta ,\eta )=(0,0)$. (a) Cross section on the $x-y$ plane where the shear flow lies. The black dashed curve is the initial spherical drop shape and the thick red solid curve is the equilibrium deformation. The diagonal dash-dotted curve is the elongation axis in the shear flow (see Fig. 6). (b) Three-dimensional rendition of ${\widehat{\mathbf{u}}}_s·\widehat{\mathbf{r}}$. The color bar represents the magnitude of the radial displacement.

Figure 8

Flow around a slightly deformable poroelastic drop under a simple shear flow with ${\varphi }_0=0.5, \mathrm{\Lambda }=1/3, {\alpha }_e=0.01$, and $(\beta ,\eta )=(0,0)$. (a) Three-dimensional rendition of ${\widehat{\mathbf{u}}}_s·\widehat{\mathit{\theta }}$. (b) Three-dimensional rendition of ${\widehat{\mathbf{u}}}_s·\widehat{\mathit{\varphi }}$. The color bar represents the magnitude of the displacement.

Figure 9

Illustration of a slightly deformed poroelastic drop under a uniaxial extensional flow. The black dashed line is the original spherical drop shape. The red solid line is the equilibrium drop shape. Arrows between the two shapes indicate the equilibrium displacement field ${\widehat{\mathbf{u}}}_s$ evaluated at $r=1$ with (a) $(\beta ,\eta )=({10}^2,0)$ and (b) $(\beta ,\eta )=({10}^3,10)$.

Figure 10

Radial displacement (${\widehat{\mathbf{u}}}_s·\widehat{\mathbf{r}}$) evaluated at $r=1$ with $\mathrm{\Lambda }=1/3, {\alpha }_e=0.005$, and ${\varphi }_0=0.5$. (a) Radial displacement versus $\beta$ for three values of $\eta$ as labeled. (b) Radial displacement versus $\eta$ for three values of $\beta$ as labeled.

Figure 11

Inflow into the first quarter of a poroelastic drop in a planar shear flow plotted against (a) interfacial permeability $\eta$ and (b) interfacial slip $\beta$, with $\mathrm{\Lambda }=1/3, {\varphi }_0=0.5$, and ${\alpha }_e=0.005$.

Figure 12

Variation of (a) radial displacement and (b) inflow with respect to porosity ${\varphi }_0$, with $\mathrm{\Lambda }=1/3$ and ${\alpha }_e=0.005$. The interfacial slip $\beta =0$ for the results in (b).

Figure 13

Eigenvalue $\omega$ as a function of ${\varphi }_0$ with $\mathrm{\Lambda }=1/3, {\alpha }_e={10}^{-2}$, and (a) $(\beta ,\eta )=(0,0)$, (b) $(\beta ,\eta )=({10}^2,0)$, and (c) $(\beta ,\eta )=(0,10)$.

Figure 14

Cross section ($x=0$ plane) of a viscous drop in a uniaxial extension flow. The black dashed curve is the original spherical shape and the thick red solid curve is the equilibrium drop shape with $\text{Ca}=0.1$. The viscosity ratio (interior to exterior) is (a) ${10}^{-2}$, (b) 1, and (c) ${10}^3$.

Figure 15

Cross section ($z=0$ plane) of a viscous drop in a simple shear flow. The black dashed curve is the original spherical shape and the thick red solid curve is the equilibrium drop shape with $\text{Ca}=0.1$. The viscosity ratio (interior to exterior) is (a) ${10}^{-2}$, (b) 1, and (c) ${10}^3$.