Stokes Flow Problems Research Articles

Effect of membrane permeability on capsule substrate adhesion: Computation using immersed interface method

A two-dimensional thin-walled capsule of a flexible permeable membrane is adhered onto a rigid planar substrate under adhesive forces (derived from a potential function). The capsule and the surrounding environment are filled with a diluted binary solution of different solute concentrations. The Stokes flow problem is solved by the immersed interface method (IIM) with equal viscosities for enclosed and surrounding fluid of the capsule. The numerical results obtained are verified against two simplified theoretical solutions with reasonable concurrence. The osmotic inflation of the adhered capsule is studied systematically as a function of solute concentration field and the membrane permeability properties. Our findings indicate that contact length shrinks in dimension and deformation decreases as capsule inflates. Also the equilibrium contact length does not depend on the hydraulic conductivity of the membrane as theoretically expected. Further numerical investigations show that the inflation and detachment of the initially adhered capsule depend significantly on the solute diffusive permeability and the reflection coefficient of capsule membrane.

A study of an arbitrary Stokes flow past a fluid coated sphere in a fluid of a different viscosity

A general method to discuss the problem of an arbitrary Stokes flow (both axisymmetric and non-axisymmetric flows) of a viscous, incompressible fluid past a sphere with a thin coating of a fluid of a different viscosity is considered. We derive the expressions for the drag and torque experienced by the fluid coated sphere and also discuss the conditions for the reduction of the drag on the fluid coated sphere. In fact, we show that the drag reduces compared to the drag on a rigid sphere of the same radius when the unperturbed velocity is either harmonic or purely biharmonic, i.e., of the form \({r^2\vec{\textbf{v}}}\), where \({\vec{\textbf{v}}}\) is a harmonic function. Previously Johnson (J Fluid Mech 110:217–238, 1981), who considered a uniform flow showed that the drag on the fluid coated sphere reduces compared to the drag on the uncoated sphere when the ratio of the surrounding fluid viscosity to the fluid-film viscosity is greater than 4. We show that this result is true when the undisturbed velocity is harmonic or purely biharmonic, uniform flow being a special case of the former. However, we illustrate by an example that the drag may increase in a general Stokes flow even if this ratio is greater than 4. Moreover, when the unperturbed velocity is harmonic or purely biharmonic, and the ratio of the surrounding fluid viscosity to the fluid-film viscosity is greater than 4 for a fixed value of the viscosity of the ambient fluid, we determine the thickness of the coating for which the drag is minimum.

Three-Dimensional Shape Optimization in Stokes Flow Problems

An integral equation constrained optimization approach to finding minimum-drag shapes for solid bodies of revolution in Stokes flows subject to constraints on body's volume and shape has been developed. An axially symmetric Stokes flow problem has been formulated in the form of an integral equation (state equation), and finding an optimal shape has been reduced to an integral equation constrained optimization problem. The total variation of the Lagrangian functional of the problem has been reduced to the variation only with respect to the shape by the adjoint equation-based method, and the steepest descent direction for the coefficients in the function series representing the shape has been found analytically. It has been shown that solutions to the state and adjoint integral equations are related by a simple algebraic formula, which eliminates the need for solving the adjoint equation. The approach has been illustrated in finding the minimum-drag shape for a solid unit volume particle and in finding the ...

Consistent approximations and boundary conditions for ice-sheet dynamics from a principle of least action

AbstractThe formulation of a physical problem in terms of a variational (or action) principle conveys significant advantages for the analytical formulation and numerical solution of that problem. One such problem is ice-sheet dynamics as described by non-Newtonian Stokes flow, for which the variational principle can be interpreted as stating that a measure of heat dissipation, due to internal deformation and boundary friction, plus the rate of loss of total potential energy is minimized under the constraint of incompressible flow. By carrying out low-aspect-ratio approximations to the Stokes flow problem within this variational principle, we obtain approximate dynamical equations and boundary conditions that are internally consistent and preserve the analytical structure of the full Stokes system. This also allows us to define an action principle for the popular first-order or ‘Blatter–Pattyn’ shallow-ice approximation that is distinct from the action principle for the Stokes problem yet preserves its most important properties and elucidates various details about this approximation. Further approximations within this new action functional yield the standard zero-order shallow-ice and shallow-shelf approximations, with their own action principles and boundary conditions. We emphasize the specification of boundary conditions, which are problematic to derive and implement consistently in approximate models but whose formulation is greatly simplified in a variational setting.

An easy-to-implement multi-scale computation of permeability coefficient for porous materials

In this paper, the multi-scale analysis based on the homogenization method is applied to the Stokes flow. This formulation allows computing permeability by solving a special Stokes flow problem in a unit cell. In particular, a simplified unit cell with characteristic radius is proposed to account for the complexity of the pore network. The determination of this unit cell is an inverse homogenization optimization process: to find the characteristic radius that yields a single laboratory permeability test data. Using the determined characteristic radius, the permeability coefficients can be computed for various void radios. Such an approach is applied to sands and clays. An agreement between the computed and measured values of the permeability coefficient is found.

Influence of inlet radius on Stokes flow in a circular tube via the Hamiltonian systematic method

This paper presents a new semianalytical method, Hamiltonian systematic method, for solving axisymmetric problems of Stokes flow. In the system, nonzero-eigenvalue solutions can describe local effect near the boundary and therefore the influence of inlet radius on the flow can be investigated. A rule of minimal entrance length is discussed on the basis of the criteria which are defined by axial flow deviating from the full developed (Hagen–Poseuille) flow. Numerical results show that the entrance length is related to the inlet radius, and there is one minimal point on the relationship curve, namely, there is one minimal entrance length. Besides, pressures have the characteristic too and the minimal point is same. The method can also be generalized to other fields.

Fully Lagrangian Modeling of Dynamics of MEMS With Thin Beams—Part II: Damped Vibrations

Abstract Micro-electro-mechanical systems (MEMS) often use beam or plate shaped conductors that are very thin with h/L≈O(10−2–10−3) (in terms of the thickness h and length L of a beam or side of a square plate). A companion paper (Ghosh and Mukherjee, 2009, “Fully Lagrangian Modeling of Dynamics of MEMS With Thin Beams—Part I: Undamped Vibrations,” ASME J. Appl. Mech., 76, p. 051007) addresses the coupled electromechanical problem of MEMS devices composed of thin beams. A new boundary element method (BEM) is coupled with the finite element method (FEM) by Ghosh and Mukherjee, and undamped vibrations are addressed there. The effect of damping due to the surrounding fluid modeled as Stokes flow is included in the present paper. Here, the elastic field modeled by the FEM is coupled with the applied electric field and the fluid field, both modeled by the BEM. As for the electric field, the BEM is adapted to efficiently handle narrow gaps between thin beams for the Stokes flow problem. The coupling of the various fields is carried out using a Newton scheme based on a Lagrangian description of the various domains. Numerical results are presented for damped vibrations of MEMS beams.

Parallel scalable adjoint-based adaptive solution of variable-viscosity Stokes flow problems

We present a framework for parallel adaptive solution of variable-viscosity Stokes flow problems. We focus on data structures, algorithms, and solvers that can scale to thousands of processor cores. The problem is discretized by octree-based finite elements with explicit enforcement of continuity constraints at hanging nodes. The parallel octree structure allows for fast neighbor-finding and facilitates local coarsening and refinement of the mesh. Mesh adaptivity is driven by a posteriori error indicators, including adjoint-based goal-oriented techniques. Dynamic load-balancing is achieved by dynamically partitioning a Morton-ordered space-filling curve. The Stokes system is solved iteratively using the minimum residual method (MINRES), preconditioned by a Schur-complement-based approximate inverse that employs algebraic multigrid V-cycle approximations of the inverses of the Poisson-like operators. We demonstrate the effectiveness of this framework on several testbed problems with up to 6 orders of magnitude variation in viscosity and up to 1.7 billion unknowns, on up to 4096 cores. The results indicate that the overhead due to all AMR components is less than 3% of the overall solve time, the solver exhibits very good algorithmic and parallel implementation scalability, the solver is insensitive to the magnitude of viscosity variation, and adjoint-based adaptivity results in over two orders of magnitude reduction in number of unknowns and up to an order of magnitude improvement in runtime relative to a uniform mesh, for the same level of error.

Stability analysis of Trefftz methods for the stick-slip problem

The stick-slip problem is a two-dimensional Stokes flow problem, and is classified into biharmonic equation with crack singularities. The collocation Trefftz method (CTM) is used to provide the very accurate solutions and leading coefficients. In this paper, the error analysis is made, to show the exponential convergence rates, and the new stability analysis is explored more in detail. We derive the bounds of effective and traditional condition numbers, to have the polynomial and the exponential growth rates, respectively. The moderate effective condition number is a suitable criterion of stability for the CTM solution of the stick-slip problem, while the huge condition number is misleading. Besides, numerical experiments are carried out to support the stability analysis. Hence the effective condition number becomes a new trend of stability for numerical partial differential equations.

Stokes flow inside an evaporating liquid line for any contact angle

Evaporation of droplets or liquid films lying on a substrate induces internal viscous flow, which affects the transport of suspended particles and, thus, the final deposit profile in numerous applications. In this work, the problem of Stokes flow inside a two-dimensional droplet, representing the cross section of an evaporating liquid line lying on a flat surface, is considered. The stream function formulation is adopted, leading to the biharmonic equation in bipolar coordinates. A solution in closed form is obtained for any contact angle in (0,pi) and is, thus, valid for both hydrophilic and hydrophobic substrates. The solution can be used with any type of evaporation mechanism, including diffusion, convection, or kinetically controlled modes. Both pinned and depinned contact lines are considered. For the boundary conditions to be compatible at the contact lines, the Navier slip boundary condition is applied on the substrate. Numerical results are presented for kinetically and diffusion controlled evaporation. For pinned contact lines, the flow inside the evaporating liquid line is directed towards the edges, thus, promoting the coffee stain phenomenon. In the case of depinned contact lines and contact angle less than pi/2 , the flow is directed towards the center of the droplet, whereas, for strongly hydrophobic substrates it is directed outwards.

Effective permeability and thermal conductivity of open-cell metallic foams via homogenization on a microstructure model

Abstract To predict effective thermal conductivities and permeabilities of open-cell foams, produced by a SlipReactionFoamSintering (SRFS)-process, a multiscale approach based on the homogenization method is applied. This formulation allows to calculate effective aerothermal properties by solving either special thermal or Stokes flow problems on a unit cell. Starting from a detailed foam description by tomographic images, 3D microstructure models of the SlipReaction (SR)-foam are generated by combining spectral analyses with a specific mesh generation. Effective thermal conductivities of an Inconel foam sample are predicted with different FE discretizations by the homogenization procedure and compared with the experimental results, measured by the transient plane source technique. Effective Darcy permeabilities are evaluated in the foam center and compared with experimental pressure drop measurements. Moreover, permeability predictions of the unit cells with extremal porosity allow to measure the permeability scatter of the Inconel SR-foam sample.

Preconditioned iterative methods for Stokes flow problems arising in computational geodynamics

The numerical solution of a discrete Stokes flow problem is a common component of many computational geodynamic models. We compare using the Schur complement reduction (SCR) and a fully coupled (FC) approach to solve geodynamic problems which are characterized by incompressible fluids containing large variations in viscosity. The scalability and robustness of these methods is assessed by examining their sensitivity to the discretization parameter h and the viscosity contrast Δ η , respectively. We demonstrate that the scaled BFBt preconditioner [Elman, H.C., Howle, V.E., Shadid, J., Shuttleworth, R., Tuminaro, R., 2006. Block preconditioners based on approximate commutators. SIAM J. Sci. Comput. 27 (5), 1651–1668] can be an effective preconditioner for S = G T K − 1 G , even when the fluid viscosity varies in space, provided an appropriate scaling is used. The performance of this preconditioner is strongly linked to the scaling employed. Through a number of numerical experiments, we demonstrate that our new scaling strategy ensures that the scaled BFBt preconditioner remains effective in scenarios where the viscosity variations are either locally smooth or discontinuous. Of the solvers considered, applying FGMRES to the pressure Schur complement in combination with BFBt and our choice of scaling matrices was the most scalable and robust approach.

A justification for the thin film approximation of Stokes flow with surface tension

In the free boundary problem of Stokes flow driven by surface tension, we pass to the limit of small layer thickness. It is rigorously shown that in this limit the evolution is given by the well-known thin film equation. The main techniques are appropriate scaling and uniform energy estimates in Sobolev spaces of sufficiently high order, based on parabolicity.

Asymmetric Channel Divider in Stokes Flow

This article examines the classic problem of Stokes flow into a divided channel with, in contrast to previous literature, the divider barrier asymmetrically placed with respect to the moving, parallel channel walls. The boundary value problem is reduced to a Wiener–Hopf equation that is of matrix form and of a class for which no exact solution is known. An explicit approximate solution, in general accurate to any specified degree, is obtained by a recent method which employs Pade approximants. Numerical results exhibit the flows due to moving walls or various combinations of downstream pressure gradients.

The Framework ofk-Harmonically Analytic Functions for Three-Dimensional Stokes Flow Problems, Part I

A solution form representing the velocity field and pressure for asymmetric three-dimensional (3D) Stokes flows has been constructed in terms of three k-harmonically analytic functions. It has also been shown that it uniquely determines an external velocity field vanishing at infinity. With the obtained solution form, problems of 3D Stokes flows due to asymmetric motions of solid bodies of revolution have been reduced to boundary-value problems for the three k-harmonically analytic functions, and the resisting force and torque, exerted on bodies in corresponding motions, have been expressed in terms of the k-harmonically analytic functions entering the solution form. For regions, in which Laplace's equation admits separation of variables, the boundary-value problems can be solved in closed form via series or integral representations of k-harmonically analytic functions in corresponding curvilinear coordinates. This approach has been demonstrated for asymmetric translation and rotation of solid sphere and ...

The Framework ofk-Harmonically Analytic Functions for Three-Dimensional Stokes Flow Problems, Part II

The framework of generalized analytic functions arising from the related potentials (so-called k-harmonically analytic functions) has been developed in application to three-dimensional (3D) axially symmetric Stokes flow problems. Cauchy's integral formula for the class of k-harmonically analytic functions has been obtained, and series representations for k-harmonically analytic functions for the regions exterior to sphere and prolate and oblate spheroids have been derived. As the central result in the developed framework, a solution form representing the velocity field and pressure for 3D axially symmetric Stokes flows has been constructed in terms of two 0-harmonically analytic functions. It has also been shown that it uniquely determines an external velocity field vanishing at infinity. With the obtained solution form, the problem of 3D Stokes flows due to the axially symmetric translation of a solid body of revolution has been reduced to a boundary-value problem for two 0-harmonically analytic function...

A new fast multipole boundary element method for solving 2-D Stokes flow problems based on a dual BIE formulation

A fast multipole boundary element method (BEM) is presented in this paper for large-scale analysis of two-dimensional (2-D) Stokes flow problems based on a dual boundary integral equation (BIE) formulation. In this dual BIE formulation, a linear combination of the conventional BIE for velocity and the hypersingular BIE for traction is employed to achieve better conditioning for the BEM systems of equations. In both the velocity and traction BIEs, the direct formulations are used, that is, the boundary variables involved are the velocity and traction directly. The fast multipole formulations for both the velocity BIE and traction BIE for 2-D Stokes flow problems are presented in this paper based on the complex variable representations of the fundamental solutions. Several numerical examples are presented to study the accuracy and efficiency of the proposed approach. The numerical results clearly demonstrate the potentials of the developed fast multipole BEM for solving large-scale 2-D Stokes flow problems.

Asymmetric three-dimensional Stokes flows about two fused equal spheres

Exact solutions to three-dimensional Stokes flow problems for asymmetric translation and rotation of two fused rigid spheres of equal size have been obtained in toroidal coordinates. The problems have been reduced to three-contour equations for meromorphic functions from a certain class, and then the latter have been reduced to Fredholm integral equations of the second kind by the Mehler–Fock transform of order 1. For the specified class of functions, the equivalence of the corresponding three-contour and Fredholm equations has been established in the framework of Riemann boundary-value problems for analytic functions. As an illustration for the obtained solutions, the pressure has been calculated at the surface of the body for both problems, and resisting force and torque, experienced by the body in asymmetric translation and rotation, have been computed as functions of a geometrical parameter of the body.

Topology optimization of large scale stokes flow problems

This note considers topology optimization of large scale 2D and 3D Stokes flow problems using parallel computations. We solve problems with up to 1,125,000 elements in 2D and 128,000 elements in 3D on a shared memory computer consisting of Sun UltraSparc IV CPUs.

Microstructure based model for permeability predictions of open-cell metallic foams via homogenization

To predict effective permeabilities of open-cell foams, produced by a SlipReactionFoamSintering (SRFS)-process, a multi-scale approach based on the homogenization method is applied. This formulation allows to calculate effective Darcy permeabilities by solving special Stokes flow problems on a unit cell. Starting from a detailed foam description by tomographic images, a new procedure based on a spectral analysis of the foam microstructure is presented in order to derive symmetric and non-symmetric unit cell models of the foam. Effective permeabilities are determined by using these unit cell models for a representative iron-based SlipReaction (SR)-foam sample. The analysis of these predictions leads to define one, rather simple, unit cell model suitable for further stochastic homogenization simulations.