The yield conditions for the gravitational displacement of three-dimensional fluid droplets from inclined solid surfaces are studied through a series of numerical computations. The study considers both sessile and pendant droplets and includes interfacial forces with constant surface tension. An extensive study is conducted, covering a wide range of Bond numbers Bd, angles of inclination β and advancing and receding contact angles, θA and θR. This study seeks the optimal shape of the contact line which yields the maximum displacing force (or BT ≡ Bd sin β) for which a droplet can adhere to the surface. The yield conditions BT are presented as functions of (Bd or β, θA, Δθ) where Δθ = θA − θR is the contact angle hysteresis. The solution of the optimization problem provides an upper bound for the yield condition for droplets on inclined solid surfaces. Additional contraints based on experimental observations are considered, and their effect on the yield condition is determined. The numerical solutions are based on the spectral boundary element method, incorporating a novel implementation of Newton's method for the determination of equilibrium free surfaces and an optimization algorithm which is combined with the Newton iteration to solve the nonlinear optimization problem. The numerical results are compared with asymptotic theories (Dussan V. & Chow 1983; Dussan V. 1985) and the useful range of these theories is identified. The normal component of the gravitational force BN ≡ Bd cos β was found to have a weak effect on the displacement of sessile droplets and a strong effect on the displacement of pendant droplets, with qualitatively different results for sessile and pendant droplets.

The yield conditions for the displacement of fluid droplets from solid boundaries are studied through a series of numerical computations. The study includes gravitational and interfacial forces, but is restricted to two-dimensional droplets and low-Reynoldsnumber flow. A comprehensive study is conducted, covering a wide range of viscosity ratio λ, Bond number B d , capillary number Ca and contact angles θ A and θ R . The yield conditions for drop displacement are calculated and the critical shear rates are presented as functions Ca(λ, B d , θ A , ∆θ) where ∆θ = θ A − θ R is the contact angle hysteresis. The numerical solutions are based on the spectral boundary element method, incorporating a novel implementation of Newton's method for the determination of equilibrium free surface profiles. The numerical results are compared with asymptotic theories (Dussan 1987) based on the lubrication approximation. While excellent agreement is found in the joint asymptotic limits ∆θ θ A 1, the useful range of the lubrication models proves to be extremely limited. The critical shear rate is found to be sensitive to viscosity ratio with qualitatively different results for viscous and inviscid droplets. Gravitational forces normal to the solid boundary have a significant effect on the displacement process, reducing the critical shear rate for viscous drops and increasing the rate for inviscid droplets. The low-viscosity limit λ → 0 is shown to be a singular limit in the lubrication theory, and the proper scaling for Ca at small λ is identified.

In the present study we investigate the dynamics of initially spherical capsules (made from elastic membranes obeying the strain-hardening Skalak or the strain-softening neo-Hookean law) in strong planar extensional flows via numerical computations. To achieve this, we develop a three-dimensional spectral boundary element algorithm for membranes with shearing and area-dilatation tensions in Stokes flow. The main attraction of this approach is that it exploits all the benefits of the spectral methods (i.e. high accuracy and numerical stability) but without creating denser systems. To achieve continuity of the interfacial geometry and its derivatives at the edges of the spectral elements during the interfacial deformation, a membrane-based interfacial smoothing is developed, via a Hermitian-like interpolation, for both the interfacial shape and the membrane elastic forces. Our numerical results show that no critical flow rate exists for both Skalak and neo-Hookean capsules in the moderate and strong planar extension flows considered in the present study. As the flow rate increases, both capsules reach elongated ellipsoidal steady-state configurations; the cross-section of the Skalak capsule preserves its elliptical shape, while the neo-Hookean capsule becomes more and more lamellar. The curvature at the pointed edges of these elongated steady-state shapes shows a very fast increase with the flow rate. The large interfacial deformations are accompanied with the development of strong membrane tensions especially for the strain-hardening Skalak capsule; the computed increase of the membrane tensions with the flow rate or the shape extension can be used to predict rupture of a specific membrane (with known lytic tension) due to excessive tensions. The type of the experiment imposed on the capsule as well as the applied flow rate affect dramatically the time evolution of the capsule edges owing to the interaction of the hydrodynamic forces with the membrane tensions; when a spherical Skalak capsule is let to deform in a strong flow, very large edge curvatures (with respect to the steady-state value) are developed during the transient evolution.

We develop a computationally efficient cytoskeleton-based continuum erythrocyte algorithm. The cytoskeleton is modeled as a two-dimensional elastic solid with comparable shearing and area-dilatation resistance that follows a material law (Skalak, R., A. Tozeren, R. P. Zarda, and S. Chien. 1973. Strain energy function of red blood cell membranes. Biophys. J. 13:245-264). Our modeling enforces the global area-incompressibility of the spectrin skeleton (being enclosed beneath the lipid bilayer in the erythrocyte membrane) via a nonstiff, and thus efficient, adaptive prestress procedure which accounts for the (locally) isotropic stress imposed by the lipid bilayer on the cytoskeleton. In addition, we investigate the dynamics of healthy human erythrocytes in strong shear flows with capillary number Ca =O(1) and small-to-moderate viscosity ratios 0.001 ≤ λ ≤ 1.5. These conditions correspond to a wide range of surrounding medium viscosities (4-600 mPa s) and shear flow rates (0.02-440 s(-1)), and match those used in ektacytometry systems. Our computational results on the cell deformability and tank-treading frequency are compared with ektacytometry findings. The tank-treading period is shown to be inversely proportional to the shear rate and to increase linearly with the ratio of the cytoplasm viscosity to that of the suspending medium. Our modeling also predicts that the cytoskeleton undergoes measurable local area dilatation and compression during the tank-treading of the cells.

In the present study we investigate computationally the steady-state motion of an elastic capsule along the centerline of a square microfluidic channel and compare it with that in a cylindrical tube. In particular, we consider a slightly over-inflated elastic capsule made of a strain-hardening membrane with comparable shearing and area-dilatation resistance. Under the conditions studied in this paper (i.e. small, moderate and large capsules at low and moderate flow rates), the capsule motion in a square channel is similar to, and thus governed by the same scaling laws with the capsule motion in a cylindrical tube, even though in the channel the cross-section in the upstream portion of large capsules is non-axisymmetric (i.e. square-like with rounded corners). When the hydrodynamic forces on the membrane increase, the capsule develops a pointed downstream edge and a flattened rear (possibly with a negative curvature) so that the restoring tension forces are increased as also happens with droplets. Membrane tensions increase significantly with the capsule size while the area near the downstream tip is the most probable to rupture when a capsule flows in a microchannel. Because the membrane tensions increase with the interfacial deformation, a suitable Landau-Levich-Derjaguin-Bretherton analysis reveals that the lubrication film thickness h for large capsules depends on both the capillary number Ca and the capsule size a; our computations determine the latter dependence to be (in dimensionless form) h ~ a−2 for the large capsules studied in this work. For small and moderate capsule sizes a, the capsule velocity Ux and additional pressure drop ΔP+ are governed by the same scaling laws as for high-viscosity droplets. The velocity and additional pressure drop of large thick capsules also follow the dynamics of high-viscosity droplets, and are affected by the lubrication film thickness. The motion of our large thick capsules is characterized by a Ux−u~h~a−2 approach to the undisturbed average duct velocity and an additional pressure drop ΔP+ ~ a3/h ~ a5. By combining basic physical principles and geometric properties, we develop a theoretical analysis that explains the power laws we found for large capsules.

The yield conditions for the displacement of three-dimensional fluid droplets from solid boundaries are studied through a series of numerical computations. The study considers low-Reynolds-number shear flows over plane boundaries and includes interfacial forces with constant surface tension. A comprehensive study is conducted, covering a wide range of viscosity ratio γ, capillary number Ca and advancing and receding contact angles, θA and θR. This study seeks the optimal shape of the contact line which yields the maximum flow rate (or Ca) for which a droplet can adhere to the surface. The critical shear rates are presented as functions Ca(γ, θA, Δθ) where Δθ=θA−θR is the contact angle hysteresis. The solution of the optimization problem provides an upper bound for the yield condition for droplets on solid surfaces. Additional constraints based on experimental observations are considered, and their effect on the yield condition is determined. The numerical solutions are based on the spectral boundary element method, incorporating a novel implementation of Newton's method for the determination of equilibrium free surfaces and an optimization algorithm which is combined with the Newton iteration to solve the nonlinear optimization problem. The numerical results are compared with asymptotic theories (Dussan 1987) based on the lubrication approximation. While good agreement is found in the joint asymptotic limits Δθ[Lt ]θA[Lt ]1, the useful range of the lubrication models proves to be extremely limited. The critical shear rate is found to be sensitive to viscosity ratio with qualitatively different results for viscous and inviscid droplets.

Interfacial dynamics of membrane-enclosed fluid volumes in viscous flows is complicated due to the coupling of the fluid dynamics with the membrane properties. Based on computational investigation via our interfacial spectral boundary element algorithm, our study shows that a (strain-hardening) Skalak-type capsule in a planar extensional Stokes flow develops steady-state shapes whose edges from spindled become cusped with increasing flow rate owing to a transition of the edge tensions from tensile to compressive. A bifurcation in the steady-state shapes is also found (i.e., existence of both spindled and cusped edges for a range of high flow rates) by implementing different transient processes, owing to the different evolution of the membrane tensions.

In the present study we describe a novel three-dimensional spectral boundary element algorithm for interfacial dynamics in Stokes flow and/or gravity. The main attraction of this approach is that it exploits all the benefits of the spectral methods ͑i.e., exponential convergence and numerical stability͒ with the versatility of the finite element method. In addition, it is not affected by the disadvantage of the spectral methods used in volume discretization to create denser systems. Our algorithm also exploits all the benefits of the boundary element techniques, i.e., a reduction of the problem dimensionality and great parallel scalability. To achieve continuity of the interfacial geometry and its derivatives at the edges of the spectral elements during the droplet deformation, a suitable interfacial smoothing is developed based on a Hermitian-like interpolation. An adaptive mesh reconstructing procedure based on the relevant lengths of the spectral elements is also described. In addition, we consider the inertialess motion of a buoyant droplet left to rise ͑or sediment͒ near a vertical solid wall and compare our numerical results with analytical predictions. In our study we emphasize the need for computational studies for the accurate determination of droplet migration near solid walls.