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A Simple 3D Immersed InterfaceMethod for Stokes flow with Singular Forces on Staggered Grids

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Communications in Computational Physics 2021年第6期30卷 227-254页

作者： Weiyi Wang Zhijun Tan School of Computer Science and Engineering Sun Yat-sen UniversityGuangzhou 510275China Guangdong Province Key Laboratory of Computational Science Sun Yat-sen Uni-versityGuangzhou 510275China

In this paper,a fairly simple 3D immersed interface method based on the CG-Uzawa type method and the level set representation of the interface is employed for solving three-dimensional Stokes flow with singular forces along the interface.The method is to apply the Taylor’s expansions only along the normal direction and incorporate the jump conditions up to the second normal derivatives into the finite difference schemes.A second order geometric iteration algorithm is employed for computing orthogonal projections on the surface with third-order accuracy.The stokes equations are discretized involving the correction terms on staggered grids and then solved by the conjugate gradient Uzawa type method.The major advantages of the present method are the special simplicity,the ability in handling the Dirichlet boundary conditions,and no need of the pressure boundary condition.The method can also preserve the volume conservation and the discrete divergence free condition very well.The numerical results show that the proposed method is second order accurate and efficient.

关键词： 3D immersed interface method CG-Uzawa method Stokes flow level set method staggered grids singular forces

A general solution for Stokes flow and its application to the problem of a rigid plate translating in a fluid

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Acta Mechanica Sinica 2015年第1期31卷 32-44页

作者： Xiang-Yu Li Si-Cong Ren Qi-Chang He School of Mechanics and Engineering Southwest Jiaotong University School of Mechanical Engineering Southwest Jiaotong University Universit'e Paris-Est Laboratoire Mod'elisation et Simulation Multi Echelle UMR 8208 CNRS 5 Boulevard Descartes77454 Marne-la-Vall'ee France

A general solution for 3D Stokes flow is given which is different from, and more compact than the exist ing ones and more compact than them in that it involves only two scalar harmonic functions. The general solution deduced is combined with the potential theory method to study the Stokes flow induced by a rigid plate of arbitrary shape trans lating along the direction normal to it in an unbounded fluid. The boundary integral equation governing this problem is derived. When the plate is elliptic, exact analytical results are obtained not only for the drag force but also for the ve locity distributions. These results include and complete the ones available for a circular plate. Numerical examples are provided to illustrate the main results for circular and ellip tic plates. In particular, the elliptic eccentricity of a plate is shown to exhibit significant influences.

关键词： Stokes flow 3D general solutions Potentialtheory method Circular plate Elliptic plate

Reduction of the Regularization Error of the Method of Regularized stokeslets for a Rigid Object Immersed in a Three-Dimensional Stokes flow

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Communications in Computational Physics 2014年第1期15卷 126-152页

作者： Hoang-Ngan Nguyen Ricardo Cortez Center for Computational Science Tulane University6823 St.Charles AveNew OrleansLouisiana70118USA.

We focus on the problem of evaluating the velocity field outside a solid object moving in an incompressible Stokes flow using the boundary integral formulation.For points near the boundary,the integral is nearly singular,and accurate computation of the velocity is not routine.One way to overcome this problem is to regularize the integral kernel.The method of regularized stokeslet(MRS)is a systematic way to regularize the kernel in this situation.For a specific blob function which is widely used,the error of the MRS is only of first order with respect to the blob parameter.We prove that this is the case for radial blob functions with decay propertyφ(r)=O(r−3−α)when r→∞for some constantα>1.We then find a class of blob functions for which the leading local error term can be removed to get second and third order errors with respect to blob parameter.Since the addition of these terms might give a flow field that is not divergence free,we introduce a modification of these terms to make the divergence of the corrected flow field close to zero while keeping the desired accuracy.Furthermore,these dominant terms are explicitly expressed in terms of blob function and so the computation time is negligible.

关键词： Stokes flow regularized stokeslet boundary integral equation nearly singular integral.

Solution of Two-Dimensional Stokes flow Problems Using Improved Singular BoundaryMethod

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Advances in Applied Mathematics and Mechanics 2015年第1期7卷 13-30页

作者： Wenzhen Qu Wen Chen Department of Engineering Mechanics College of Mechanics and MaterialsHohai UniversityNanjing 210098China

In this paper,an improved singular boundarymethod(SBM),viewed as one kind of modified method of fundamental solution(MFS),is firstly applied for the numerical analysis of two-dimensional(2D)Stokes flow problems.The key issue of the SBM is the determination of the origin intensity factor used to remove the singularity of the fundamental solution and its derivatives.The new contribution of this study is that the origin intensity factors for the velocity,traction and pressure are derived,and based on that,the SBM formulations for 2D Stokes flow problems are presented.Several examples are provided to verify the correctness and robustness of the presented method.The numerical results clearly demonstrate the potentials of the present SBM for solving 2D Stokes flow problems.

关键词： Singular boundary method origin intensity factor Stokes flow fundamental solution.

Stokes flow before plane boundary with mixed stick-slip boundary conditions

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Applied Mathematics and Mechanics(English Edition) 2011年第6期32卷 795-804页

作者： N.AKHTAR G.A.H.CHOWDHURY S.K.SEN Department of Mathematics Shahjalal University of Science and Technology Research Centre for Mathematical and Physical Sciences University of Chittagong

A general theorem for the Stokes flow over a plane boundary with mixed stick-slip boundary conditions is established. This is done by using a representation for the velocity and pressure fields in the three-dimensional Stokes flow in terms of a biharmonic function and a harmonic function. The earlier theorem for the Stokes flow due to fundamental singularities before a no-slip plane boundary is shown to be a special case of the present theorem. Furthermore, in terms of the stokes stream function, a corollary of the theorem is also derived, providing a solution to the problem of the axisymmetric Stokes flow along a rigid plane with stick-slip boundary conditions. The formulae for the drag and torque exerted by the fluid on the boundary are established. An illustrative example is given.

关键词： Stokes flow stokes stream function stick-slip boundary conditions,harmonic function biharmonic function

Stokes flow of micro-polar fluids by peristaltic pumping through tube with slip boundary condition

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Applied Mathematics and Mechanics(English Edition) 2011年第12期32卷 1587-1598页

作者： D.TRIPATHI M.K.CHAUBE P.K.GUPTA Department of Mathematics Indian Institute of Technology Ropar Punjab 140001 India Department of Mathematics Raj Kumar Goel Institute of Technology Ghaziabad 201001 India Department of Applied Mathematics Institute of Technology Banaras Hindu University Varanasi 221005 India

This paper studies the Stokes flow of micro-polar fluids by peristaltic pumping through the cylindrical tube under the effect of the slip boundary condition. The motion of the wall is governed by the sinusoidal wave equation. The analytical and numerical solutions for the axial velocity, the micro-polar vector, the stream function, the pressure gradient, the friction force, and the mechanical efficiency are obtained by using the lu- brication theory under the low Reynolds number and long wavelength approximations. The impacts of the emerging parameters, such as the coupling number, the micro-polar parameter, the slip parameter on pumping characteristics, the friction force, the velocity profile, the mechanical efficiency, and the trapping phenomenon are depicted graphically. The numerical results infer that large pressure is required for peristaltic pumping when the coupling number is large, while opposite behaviors are found for the micro-polar parameter and the slip parameter. The size of the trapped bolus reduces with the increase in the coupling number and the micro-polar parameter, whereas it blows up with the increase in the slip parameter.

关键词： Stokes flow slip boundary condition mechanical efficiency micro-polarfluid peristaltic pumping trapping

Phragmén-Lindelf and continuous dependence type results in a Stokes flow

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Applied Mathematics and Mechanics(English Edition) 2010年第7期31卷 875-882页

作者： J.C.SONG Department of Applied Mathematics Hanyang University

This paper investigates the asymptotic behavior of end effects for a Stokes flow defined on a three-dimensional semi-infinite cylinder. With homogeneous Dirichlet conditions of the velocity on the lateral surface of the cylinder, solutions either grow or decay exponentially in the distance from the finite end of the cylinder. In the case of decay, the effect of perturbing the equation parameters is also investigated.

关键词： Phragmen-Lindelof continuous dependence decay bounds Stokes flow

A Hybrid Immersed Interface Method for Driven Stokes flow in an Elastic Tube

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Numerical Mathematics(Theory,Methods and Applications) 2013年第4期6卷 600-616页

作者： Yi Li Sarah A.Williams Anita T.Layton Department of Mathematics Duke UniversityDurhamNC 27708-0320USA

We present a hybrid numerical method for simulating fluid flow through a compliant,closed tube,driven by an internal source and sink.Fluid is assumed to be highly viscous with its motion described by Stokes flow.Model geometry is assumed to be axisymmetric,and the governing equations are implemented in axisymmetric cylindrical coordinates,which capture 3D flow dynamics with only 2D computations.We solve the model equations using a hybrid approach:we decompose the pressure and velocity fields into parts due to the surface forcings and due to the source and sink,with each part handled separately by means of an appropriate method.Because the singularly-supported surface forcings yield an unsmooth solution,that part of the solution is computed using the immersed interface method.Jump conditions are derived for the axisymmetric cylindrical coordinates.The velocity due to the source and sink is calculated along the tubular surface using boundary integrals.Numerical results are presented that indicate second-order accuracy of the method.

关键词： Stokes flow interface tracking immersed interface methods axisymmetric cylindrical coordinates boundary integrals

The Pressure-Streamfunction MFS Formulation for the Detection of an Obstacle Immersed in a Two-Dimensional Stokes flow

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Advances in Applied Mathematics and Mechanics 2010年第2期2卷 183-199页

作者： A.Karageorghis D.Lesnic Department of Mathematics and Statistics University of CyprusP.O.Box 205371678 NicosiaCyprus Department of Applied Mathematics University of LeedsLeeds LS29JTUK

In this paper we consider a geometric inverse problem which requires detecting an unknown obstacle such as a submarine or an aquatic mine immersed in a stokes slow viscous stationary flow of an incompressible fluid,from a single set of Cauchy(fluid velocity and stress force)boundary measurements.The numerical reconstruction is based on the method of fundamental solutions(MFS)for the pressure and streamfunction in two dimensions combined with regularization.Numerical results are presented and discussed.

关键词： Stokes flow method of fundamental solutions regularization streamfunction stress force