We derive a closed form expression for the regularized Stokeslet in two space dimensions with periodic boundary conditions in the x-direction and a solid plane wall at y=0.To accommodate the no-slip condition on the w...We derive a closed form expression for the regularized Stokeslet in two space dimensions with periodic boundary conditions in the x-direction and a solid plane wall at y=0.To accommodate the no-slip condition on the wall,a system of images for the regularized Stokeslets was used.The periodicity is enforced by writing all elements of the image system in terms of a Green’s function whose periodic extension is known.Although the formulation is derived in the context of regularized Stokeslets,the expression for the traditional(singular)Stokeslet is easily found by taking the limit as the regularization parameter approaches zero.The new formulation is validated by comparing results of two test problems:the Taylor infinite waving sheet and the motion of a cylinder moving near a wall.As an example of an application,we use our formulation to compute the motion and flow generated by cilia using a model that does not prescribe the motion so that the beat period and synchronization of neighboring cilia are a result of the forces developed along the cilia.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
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
The present work is concerned with a two-dimensional(2D)Stokes flow through a channel bounded by two parallel solid walls.The distance between the walls may be arbitrary,and the surface of one of the walls can be arbi...The present work is concerned with a two-dimensional(2D)Stokes flow through a channel bounded by two parallel solid walls.The distance between the walls may be arbitrary,and the surface of one of the walls can be arbitrarily rough.The main objective of this work consists in homogenizing the heterogeneous interface between the rough wall and fluid so as to obtain an equivalent smooth slippery fluid/solid interface characterized by an effective slip length.To solve the corresponding problem,two efficient numerical approaches are elaborated on the basis of the method of fundamental solution(MFS)and the boundary element methods(BEMs).They are applied to different cases where the fluid/solid interface is periodically or randomly rough.The results obtained by the proposed two methods are compared with those given by the finite element method and some relevant ones reported in the literature.This comparison shows that the two proposed methods are particularly efficient and accurate.展开更多
The fundamental solutions of the Stokes/Oseen equations due to a point force in an unbounded viscous fluid are referred to as the Stokeslet/Oseenlet,for which a systematic derivation are analytically presented here in...The fundamental solutions of the Stokes/Oseen equations due to a point force in an unbounded viscous fluid are referred to as the Stokeslet/Oseenlet,for which a systematic derivation are analytically presented here in terms of a uniform expression.By means of integral transforms,the closed-form solutions are explicitly deduced in a formula which involves the Hamiltonian,Hessian,and Laplacian operators,and elementary functions.Secondly,interfacial viscous capillary-gravity waves between two semi-infinite fluids due to oscillating singularities,including a simple source in the upper inviscid fluid and a Stokeslet in the low viscous fluid,were analytically studied by the Laplace-Fourier integral transform and asymptotic analysis.The dynamics responses consist of the transient and steady-state components,which are dealt with by the method of stationary phase and the Cauchy residue theorem,respectively.The transient response is made up of one short capillarity・dominated and one long gravity-dominated wave with the former riding on the latter.The steady-state wave has the same frequency as that of oscillating singularities.Asymptotic solutions for the wave profiles and the exact solution for the wave number are analytically derived,which show the combined effects of fluid viscosity,surface capillarity and an upper layer fluid.展开更多
The interaction of laminar flows with free surface waves generated by submerged bodies in an incompressible viscous fluid of infinite depth is investigated analytically. The analysis is based on the linearized Navier-...The interaction of laminar flows with free surface waves generated by submerged bodies in an incompressible viscous fluid of infinite depth is investigated analytically. The analysis is based on the linearized Navier-Stokes equations for disturbed flows. The kinematic and dynamic boundary conditions are linearized for the small-amplitude free-surface waves, and the initial values of the flow are taken to be those of the steady state cases. The submerged bodies are mathematically represented by fundamental singularities of viscous flows. The asymptotic representations for unsteady free-surface waves produced by the Stokeslets and Oseenlets are derived analytically. It is found that the unsteady waves generated by a body consist of steady-state and transient responses. As time tends to infinity, the transient waves vanish due to the presence of a viscous decay factor. Thus, an ultimate steady state can be attained.展开更多
基金Thework of the authorswas supported in part by theNational Science Foundation(NSF)Grant No.DMS-1043626.
文摘We derive a closed form expression for the regularized Stokeslet in two space dimensions with periodic boundary conditions in the x-direction and a solid plane wall at y=0.To accommodate the no-slip condition on the wall,a system of images for the regularized Stokeslets was used.The periodicity is enforced by writing all elements of the image system in terms of a Green’s function whose periodic extension is known.Although the formulation is derived in the context of regularized Stokeslets,the expression for the traditional(singular)Stokeslet is easily found by taking the limit as the regularization parameter approaches zero.The new formulation is validated by comparing results of two test problems:the Taylor infinite waving sheet and the motion of a cylinder moving near a wall.As an example of an application,we use our formulation to compute the motion and flow generated by cilia using a model that does not prescribe the motion so that the beat period and synchronization of neighboring cilia are a result of the forces developed along the cilia.
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.
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supported by the Vietnam National Foundation for Science and Technology Development(NAFOSTED)(No.107.02-2017.310)。
文摘The present work is concerned with a two-dimensional(2D)Stokes flow through a channel bounded by two parallel solid walls.The distance between the walls may be arbitrary,and the surface of one of the walls can be arbitrarily rough.The main objective of this work consists in homogenizing the heterogeneous interface between the rough wall and fluid so as to obtain an equivalent smooth slippery fluid/solid interface characterized by an effective slip length.To solve the corresponding problem,two efficient numerical approaches are elaborated on the basis of the method of fundamental solution(MFS)and the boundary element methods(BEMs).They are applied to different cases where the fluid/solid interface is periodically or randomly rough.The results obtained by the proposed two methods are compared with those given by the finite element method and some relevant ones reported in the literature.This comparison shows that the two proposed methods are particularly efficient and accurate.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11872239,10602032).
文摘The fundamental solutions of the Stokes/Oseen equations due to a point force in an unbounded viscous fluid are referred to as the Stokeslet/Oseenlet,for which a systematic derivation are analytically presented here in terms of a uniform expression.By means of integral transforms,the closed-form solutions are explicitly deduced in a formula which involves the Hamiltonian,Hessian,and Laplacian operators,and elementary functions.Secondly,interfacial viscous capillary-gravity waves between two semi-infinite fluids due to oscillating singularities,including a simple source in the upper inviscid fluid and a Stokeslet in the low viscous fluid,were analytically studied by the Laplace-Fourier integral transform and asymptotic analysis.The dynamics responses consist of the transient and steady-state components,which are dealt with by the method of stationary phase and the Cauchy residue theorem,respectively.The transient response is made up of one short capillarity・dominated and one long gravity-dominated wave with the former riding on the latter.The steady-state wave has the same frequency as that of oscillating singularities.Asymptotic solutions for the wave profiles and the exact solution for the wave number are analytically derived,which show the combined effects of fluid viscosity,surface capillarity and an upper layer fluid.
文摘The interaction of laminar flows with free surface waves generated by submerged bodies in an incompressible viscous fluid of infinite depth is investigated analytically. The analysis is based on the linearized Navier-Stokes equations for disturbed flows. The kinematic and dynamic boundary conditions are linearized for the small-amplitude free-surface waves, and the initial values of the flow are taken to be those of the steady state cases. The submerged bodies are mathematically represented by fundamental singularities of viscous flows. The asymptotic representations for unsteady free-surface waves produced by the Stokeslets and Oseenlets are derived analytically. It is found that the unsteady waves generated by a body consist of steady-state and transient responses. As time tends to infinity, the transient waves vanish due to the presence of a viscous decay factor. Thus, an ultimate steady state can be attained.
