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 ...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.展开更多
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 t...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.展开更多
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 e...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.展开更多
A new three-dimensional fundamental solution to the Stokes flow was proposed by transforming the solid harmonic functions in Lamb's solution into expressions in terms Of the oblate spheroidal coordinates. These fu...A new three-dimensional fundamental solution to the Stokes flow was proposed by transforming the solid harmonic functions in Lamb's solution into expressions in terms Of the oblate spheroidal coordinates. These fundamental solutions are advantageous in treating flows past an arbitrary number of arbitrarily positioned and oriented oblate spheroids. The least squares technique was adopted herein so that the convergence difficulties often encountered in solving three-dimensional problems were completely avoided. The examples demonstrate that present approach is highly accurate, consistently stable and computationally efficient. The oblate spheroid may be used to model a variety of particle shapes between a circular disk and a sphere. For the first time, the effect of various geometric factors on the forces and torques exerted on two oblate spheroids were systematically studied by using the proposed fundamental solutions. The generality of this approach was illustrated by two problems of three spheroids.展开更多
A boundary integral method was developed for simulating the motion and deformation of a viscous drop in an axisymmetric ambient Stokes flow near a rigid wall and for direct calculating the stress on the wall. Numerica...A boundary integral method was developed for simulating the motion and deformation of a viscous drop in an axisymmetric ambient Stokes flow near a rigid wall and for direct calculating the stress on the wall. Numerical experiments by the method were performed for different initial stand-off distances of the drop to the wall, viscosity ratios, combined surface tension and buoyancy parameters and ambient flow parameters. Numerical results show that due to the action of ambient flow and buoyancy the drop is compressed and stretched respectively in axial and radial directions when time goes. When the ambient flow action is weaker than that of the buoyancy the drop raises and bends upward and the stress on the wall induced by drop motion decreases when time advances. When the ambient flow action is stronger than that of the buoyancy the drop descends and becomes flatter and flatter as time goes. In this case when the initial stand-off distance is large the stress on the wall increases as the drop evolutes but when the stand-off distance is small the stress on the wall decreases as a result of combined effects of ambient flow, buoyancy and the stronger wall action to the flow. The action of the stress on the wall induced by drop motion is restricted in an area near the symmetric axis, which increases when the initial stand-off distance increases. When the initial stand-off distance increases the stress induced by drop motion decreases substantially. The surface tension effects resist the deformation and smooth the profile of the drop surfaces. The drop viscosity will reduce the deformation and migration of the drop.展开更多
A representation for the velocity and pressure fields in three-dimensional Stokes flow was presented in terms of a biharmonic function A and a harmonic function B.This representation was used to establish a general th...A representation for the velocity and pressure fields in three-dimensional Stokes flow was presented in terms of a biharmonic function A and a harmonic function B.This representation was used to establish a general theorem for the calculation of Stokes flow due to fundamental singularities in a region bounded by a stationary no-slip plane boundary.Collins's theorem for axisymmetric Stokes flow before a rigid plane follows as a special case of the theorem.A few illustrative examples are given to show its usefulness.展开更多
Flow profiles are frequently engineered in microfluidic channels for enhanced mixing,reaction control,and material synthesis.Conventionally,flow profiles are engineered by inducing inertial secondary flow to redistrib...Flow profiles are frequently engineered in microfluidic channels for enhanced mixing,reaction control,and material synthesis.Conventionally,flow profiles are engineered by inducing inertial secondary flow to redistribute the streams,which can hardly be reproduced in microfluidic environments with negligible inertial flow.The employed symmetric channel structures also limit the variety of achievable flow profiles.Moreover,each of the flow profiles specifically corresponds to a strictly defined flow condition and cannot be generalized to other flow environments.To address these issues,we present a systematic method to engineer the flow profile using inertialess secondary flow.The flow is manipulated in the Stokes regime by deploying a cascaded series of microsteps with various morphologies inside a microchannel to shape the flow profile.By tuning the shapes of the microsteps,arbitrary outflow profiles can be customized.A numerical profile-transformation program is developed for rapid prediction of the output profiles of arbitrary sequences of predefined microsteps.The proposed method allows the engineering of stable flow profiles,including asymmetric ones,over a wide range of flow conditions for complex microfluidic environmental prediction and design.展开更多
In this paper, a new analytical method of symplectic system, Hamiltonian system, is introduced for solving the problem of the Stokes flow in a two-dimensional rectangular domain. In the system, the fundamental problem...In this paper, a new analytical method of symplectic system, Hamiltonian system, is introduced for solving the problem of the Stokes flow in a two-dimensional rectangular domain. In the system, the fundamental problem is reduced to an eigenvalue and eigensolution problem. The solution and boundary conditions can be expanded by eigensolutions using adjoint relationships of the symplectic ortho-normalization between the eigensolutions. A closed method of the symplectic eigensolution is presented based on completeness of the symplectic eigensolution space. The results show that fundamental flows can be described by zero eigenvalue eigensolutions, and local effects by nonzero eigenvalue eigensolutions. Numerical examples give various flows in a rectangular domain and show effectiveness of the method for solving a variety of problems. Meanwhile, the method can be used in solving other problems.展开更多
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-dimensiona...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.展开更多
A type of penalty-hybrid variational principle is suggested for the analysis of Stokesian flow. On such a basis, a finite element model is formulated featuring, among others, a priori satisfaction of the deviatoric st...A type of penalty-hybrid variational principle is suggested for the analysis of Stokesian flow. On such a basis, a finite element model is formulated featuring, among others, a priori satisfaction of the deviatoric stress and hydrostatic pressure on linear momentum balance equations. Also in the present scheme the hydrostatic pressure is successfully eliminated at the element level, leaving only nodal velocities as solution unknowns. A series of 4-node and 8-node quadrilateral elements are derived and examined. Numerical examples demonstrating their characteristic behaviors are also included.展开更多
This paper defines new kinds of functions——the conjugate axisymmetric poteptial functions. With the aid of them, we can prove the completeness of the solutions of the generalized axisymmetric Stokes flow equation wi...This paper defines new kinds of functions——the conjugate axisymmetric poteptial functions. With the aid of them, we can prove the completeness of the solutions of the generalized axisymmetric Stokes flow equation without the condition on the convexity of the domain.展开更多
For any study ofa suspension entering a pore, the knowledge of the force and moment exerted on a solute particle in an arbitrary position outside the pore is essential, 'This paper for the first lime presents appr...For any study ofa suspension entering a pore, the knowledge of the force and moment exerted on a solute particle in an arbitrary position outside the pore is essential, 'This paper for the first lime presents approximate analytical expressions (in closed form) of all the twelve force and moment coefficienis for a sphere outsied a circular orifice, on the basis of a number of discrete data computed by Yan et al(1987).These coefficients are then applied to calculate the trajectory and angular velocity of a spherical particle approaching the pore at zero Reynolds number. The trajectory is in excellent agreement with the available experimental results. An analysis of the relative importance of the coefficients shows that the rotation effect cannot be neglected near the pore opening or near the wall, and that the lateral force effect must be taken into account in the neighborhood of the edge of the pore opening. It is due to neglecting these factors that previous theoretical results deviate from the experimental ones near the pore opening. The effects of the ratio of the particle to pore radii as well as the influences of the graritytbuoyance on the particle trajectory, velocity distribution and rotation are discnssed in detail. It is pointed out that in the experiments of neutrally-buoyant suspensions, the restriction on the density of the particle is most demanding for a large particle size.The expressions of forces and moments presenled herein are complete, relatively accurate and convenient, thus providing a good prerequisite for further studies of any problems involving the entrance of particles to a pare.展开更多
We study the time-decay properties of weighted norms of solutions to the Stokes equations and the Navier-Stokes equations in the half-space Rn+ (n 2). Three kinds of the weighted Lp-Lr estimates are established for th...We study the time-decay properties of weighted norms of solutions to the Stokes equations and the Navier-Stokes equations in the half-space Rn+ (n 2). Three kinds of the weighted Lp-Lr estimates are established for the Stokes semigroup generated by the Stokes operator in the half-space R+n (n 2). As an application of the weighted estimates of the Stokes semigroup, a class of local and global strong solutions in weighted Lp (R+n) are constructed, following the approach given by Kato.展开更多
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 ke...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.展开更多
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...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.展开更多
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...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.展开更多
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,fr...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.展开更多
We present a new numerical method for solving two-dimensional Stokes flow with deformable interfaces such as dynamics of suspended drops or bubbles.The method is based on a boundary integral formulation for the interf...We present a new numerical method for solving two-dimensional Stokes flow with deformable interfaces such as dynamics of suspended drops or bubbles.The method is based on a boundary integral formulation for the interfacial velocity and is spectrally accurate in space.We analyze the singular behavior of the integrals(single-layer and double-layer integrals)appearing in the equations.The interfaces are formulated in the tangent angle and arc-length coordinates and,to reduce the stiffness of the evolution equation,the marker points are evenly distributed in arc-length by choosing a proper tangential velocity along the interfaces.Examples of Stokes flow with bubbles are provided to demonstrate the accuracy and effectiveness of the numerical method.展开更多
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 singu...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.展开更多
In a recent paper (Li et al., Acta Mech. Sin. 31, 32-44, 2015), the authors claimed that the general solution of steady Stokes flows can be compactly expressed using only two harmonic functions. They present two cases...In a recent paper (Li et al., Acta Mech. Sin. 31, 32-44, 2015), the authors claimed that the general solution of steady Stokes flows can be compactly expressed using only two harmonic functions. They present two cases of a flat plate translating through a viscous fluid. The present paper shows that such a two-harmonic solution does not describe the rotation of a circular plate in an unbounded fluid and thus confirms that at least three independent harmonics are required to express the general solution of Stokes equations.展开更多
基金supported by the National Natural Science Foundation of China(11102171)the Program for New Century Excellent Talents in University of Ministry of Education of China(NCET-13-0973)
文摘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.
基金supported by the National Research Foundation of Korea (NRF) (No.2010-0012215)
文摘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.
文摘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.
文摘A new three-dimensional fundamental solution to the Stokes flow was proposed by transforming the solid harmonic functions in Lamb's solution into expressions in terms Of the oblate spheroidal coordinates. These fundamental solutions are advantageous in treating flows past an arbitrary number of arbitrarily positioned and oriented oblate spheroids. The least squares technique was adopted herein so that the convergence difficulties often encountered in solving three-dimensional problems were completely avoided. The examples demonstrate that present approach is highly accurate, consistently stable and computationally efficient. The oblate spheroid may be used to model a variety of particle shapes between a circular disk and a sphere. For the first time, the effect of various geometric factors on the forces and torques exerted on two oblate spheroids were systematically studied by using the proposed fundamental solutions. The generality of this approach was illustrated by two problems of three spheroids.
基金Project supported by the National Natural Science Foundation of China (No. 10272032)
文摘A boundary integral method was developed for simulating the motion and deformation of a viscous drop in an axisymmetric ambient Stokes flow near a rigid wall and for direct calculating the stress on the wall. Numerical experiments by the method were performed for different initial stand-off distances of the drop to the wall, viscosity ratios, combined surface tension and buoyancy parameters and ambient flow parameters. Numerical results show that due to the action of ambient flow and buoyancy the drop is compressed and stretched respectively in axial and radial directions when time goes. When the ambient flow action is weaker than that of the buoyancy the drop raises and bends upward and the stress on the wall induced by drop motion decreases when time advances. When the ambient flow action is stronger than that of the buoyancy the drop descends and becomes flatter and flatter as time goes. In this case when the initial stand-off distance is large the stress on the wall increases as the drop evolutes but when the stand-off distance is small the stress on the wall decreases as a result of combined effects of ambient flow, buoyancy and the stronger wall action to the flow. The action of the stress on the wall induced by drop motion is restricted in an area near the symmetric axis, which increases when the initial stand-off distance increases. When the initial stand-off distance increases the stress induced by drop motion decreases substantially. The surface tension effects resist the deformation and smooth the profile of the drop surfaces. The drop viscosity will reduce the deformation and migration of the drop.
文摘A representation for the velocity and pressure fields in three-dimensional Stokes flow was presented in terms of a biharmonic function A and a harmonic function B.This representation was used to establish a general theorem for the calculation of Stokes flow due to fundamental singularities in a region bounded by a stationary no-slip plane boundary.Collins's theorem for axisymmetric Stokes flow before a rigid plane follows as a special case of the theorem.A few illustrative examples are given to show its usefulness.
基金This work was supported by the General Research Fund(17306315,17304017,and 17305518)and Research Impact Fund(R7072-18)from the Research Grants Council(RGC)of Hong Kong,Chinathe Excellent Young Scientists Fund(Hong Kong and Macao)(21922816)from the National Natural Science Foundation of China(NSFC)+1 种基金the Seed Funding for Strategic Interdisciplinary Research Scheme 2017/18 from the University of Hong Kongas well as the Sichuan Science and Technology Program(2018JZ0026).
文摘Flow profiles are frequently engineered in microfluidic channels for enhanced mixing,reaction control,and material synthesis.Conventionally,flow profiles are engineered by inducing inertial secondary flow to redistribute the streams,which can hardly be reproduced in microfluidic environments with negligible inertial flow.The employed symmetric channel structures also limit the variety of achievable flow profiles.Moreover,each of the flow profiles specifically corresponds to a strictly defined flow condition and cannot be generalized to other flow environments.To address these issues,we present a systematic method to engineer the flow profile using inertialess secondary flow.The flow is manipulated in the Stokes regime by deploying a cascaded series of microsteps with various morphologies inside a microchannel to shape the flow profile.By tuning the shapes of the microsteps,arbitrary outflow profiles can be customized.A numerical profile-transformation program is developed for rapid prediction of the output profiles of arbitrary sequences of predefined microsteps.The proposed method allows the engineering of stable flow profiles,including asymmetric ones,over a wide range of flow conditions for complex microfluidic environmental prediction and design.
文摘In this paper, a new analytical method of symplectic system, Hamiltonian system, is introduced for solving the problem of the Stokes flow in a two-dimensional rectangular domain. In the system, the fundamental problem is reduced to an eigenvalue and eigensolution problem. The solution and boundary conditions can be expanded by eigensolutions using adjoint relationships of the symplectic ortho-normalization between the eigensolutions. A closed method of the symplectic eigensolution is presented based on completeness of the symplectic eigensolution space. The results show that fundamental flows can be described by zero eigenvalue eigensolutions, and local effects by nonzero eigenvalue eigensolutions. Numerical examples give various flows in a rectangular domain and show effectiveness of the method for solving a variety of problems. Meanwhile, the method can be used in solving other problems.
文摘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.
文摘A type of penalty-hybrid variational principle is suggested for the analysis of Stokesian flow. On such a basis, a finite element model is formulated featuring, among others, a priori satisfaction of the deviatoric stress and hydrostatic pressure on linear momentum balance equations. Also in the present scheme the hydrostatic pressure is successfully eliminated at the element level, leaving only nodal velocities as solution unknowns. A series of 4-node and 8-node quadrilateral elements are derived and examined. Numerical examples demonstrating their characteristic behaviors are also included.
文摘This paper defines new kinds of functions——the conjugate axisymmetric poteptial functions. With the aid of them, we can prove the completeness of the solutions of the generalized axisymmetric Stokes flow equation without the condition on the convexity of the domain.
基金Project supported by the National Natural Science Foundation of China
文摘For any study ofa suspension entering a pore, the knowledge of the force and moment exerted on a solute particle in an arbitrary position outside the pore is essential, 'This paper for the first lime presents approximate analytical expressions (in closed form) of all the twelve force and moment coefficienis for a sphere outsied a circular orifice, on the basis of a number of discrete data computed by Yan et al(1987).These coefficients are then applied to calculate the trajectory and angular velocity of a spherical particle approaching the pore at zero Reynolds number. The trajectory is in excellent agreement with the available experimental results. An analysis of the relative importance of the coefficients shows that the rotation effect cannot be neglected near the pore opening or near the wall, and that the lateral force effect must be taken into account in the neighborhood of the edge of the pore opening. It is due to neglecting these factors that previous theoretical results deviate from the experimental ones near the pore opening. The effects of the ratio of the particle to pore radii as well as the influences of the graritytbuoyance on the particle trajectory, velocity distribution and rotation are discnssed in detail. It is pointed out that in the experiments of neutrally-buoyant suspensions, the restriction on the density of the particle is most demanding for a large particle size.The expressions of forces and moments presenled herein are complete, relatively accurate and convenient, thus providing a good prerequisite for further studies of any problems involving the entrance of particles to a pare.
基金supported in part by National Basic Research Program of China (Grant No. 2006CB805902)Knowledge Innovation Funds of Chinese Academy of Sciences(Grant No. KJCX3-SYW-S03)+1 种基金supported in part by Scientific Research Plan Projects of Shaanxi Education (Grant No. 09JK770)China Postdoctoral Science Foundation (Grant No. 20090461305)
文摘We study the time-decay properties of weighted norms of solutions to the Stokes equations and the Navier-Stokes equations in the half-space Rn+ (n 2). Three kinds of the weighted Lp-Lr estimates are established for the Stokes semigroup generated by the Stokes operator in the half-space R+n (n 2). As an application of the weighted estimates of the Stokes semigroup, a class of local and global strong solutions in weighted Lp (R+n) are constructed, following the approach given by Kato.
基金supported by the National Basic Research Program of China(2010CB832702)the National Science Funds for Distinguished Young Scholars of China(11125208)+1 种基金the R&D Special Fund for Public Welfare Industry(Hydrodynamics,201101014)Programme of Introducing Talents of Discipline to Universities(111 project,Grant No.B12032).
文摘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.
基金supported by the Special Project on High-performance Computing under the National Key R&D Program(No.2016YFB0200604)National Natural Science Foundation of China(11971502,11571385)Guangdong Natural Science Foundation(2017A030313017).
文摘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.
基金supported in part by the National Science Foundation under Grant DMS-0715021.
文摘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.
文摘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.
基金supported by the grants NSF-DMS 0511411,0914923 and 0923111.
文摘We present a new numerical method for solving two-dimensional Stokes flow with deformable interfaces such as dynamics of suspended drops or bubbles.The method is based on a boundary integral formulation for the interfacial velocity and is spectrally accurate in space.We analyze the singular behavior of the integrals(single-layer and double-layer integrals)appearing in the equations.The interfaces are formulated in the tangent angle and arc-length coordinates and,to reduce the stiffness of the evolution equation,the marker points are evenly distributed in arc-length by choosing a proper tangential velocity along the interfaces.Examples of Stokes flow with bubbles are provided to demonstrate the accuracy and effectiveness of the numerical method.
基金supported by LONI Institute Graduate Fellowship.
文摘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.
基金supported by the National Natural Science Foundation of China (Grant 11372186)the Specialized Research Fund for the Doctoral Program of Higher Education (Grant 20130073110059)
文摘In a recent paper (Li et al., Acta Mech. Sin. 31, 32-44, 2015), the authors claimed that the general solution of steady Stokes flows can be compactly expressed using only two harmonic functions. They present two cases of a flat plate translating through a viscous fluid. The present paper shows that such a two-harmonic solution does not describe the rotation of a circular plate in an unbounded fluid and thus confirms that at least three independent harmonics are required to express the general solution of Stokes equations.