The original Leray’s problem concerns the well-posedness of weak solutions to the steady incompressible Navier-Stokes equations in a distorted pipe,which approach the Poiseuille flow subject to the no-slip boundary c...The original Leray’s problem concerns the well-posedness of weak solutions to the steady incompressible Navier-Stokes equations in a distorted pipe,which approach the Poiseuille flow subject to the no-slip boundary condition at spatial infinity.In this paper,the same problem with the Navier-slip boundary condition instead of the no-slip boundary condition,is addressed.Due to the complexity of the boundary condition,some new ideas,presented as follows,are introduced to handle the extra difficulties caused by boundary terms.First,the Poiseuille flow in the semi-infinite straight pipe with the Navier-slip boundary condition will be introduced,which will serve as the asymptotic profile of the solution to the generalized Leray’s problem at spatial infinity.Second,a solenoidal vector function defined in the whole pipe,satisfying the Navierslip boundary condition,having the designated flux and equalling the Poiseuille flow at a large distance,will be carefully constructed.This plays an important role in reformulating our problem.Third,the energy estimates depend on a combined L2-estimate of the gradient and the stress tensor of the velocity.展开更多
We study the initial-boundary value problem of the Navier-Stokes equations for incompressible fluids in a general domain in R^n with compact and smooth boundary, subject to the kinematic and vorticity boundary conditi...We study the initial-boundary value problem of the Navier-Stokes equations for incompressible fluids in a general domain in R^n with compact and smooth boundary, subject to the kinematic and vorticity boundary conditions on the non-flat boundary. We observe that, under the nonhomogeneous boundary conditions, the pressure p can be still recovered by solving the Neumann problem for the Poisson equation. Then we establish the well-posedness of the unsteady Stokes equations and employ the solution to reduce our initial-boundary value problem into an initial-boundary value problem with absolute boundary conditions. Based on this, we first establish the well-posedness for an appropriate local linearized problem with the absolute boundary conditions and the initial condition （without the incompressibility condition）, which establishes a velocity mapping. Then we develop apriori estimates for the velocity mapping, especially involving the Sobolev norm for the time-derivative of the mapping to deal with the complicated boundary conditions, which leads to the existence of the fixed point of the mapping and the existence of solutions to our initial-boundary value problem. Finally, we establish that, when the viscosity coefficient tends zero, the strong solutions of the initial-boundary value problem in R^n（n ≥ 3） with nonhomogeneous vorticity boundary condition converge in L^2 to the corresponding Euler equations satisfying the kinematic condition.展开更多
An analytical approximation for the similarity solutions of the two-and three-dimensional stagnation slip flow and heat transfer is obtained by using the homotopy analysis method. This method is a series expansion met...An analytical approximation for the similarity solutions of the two-and three-dimensional stagnation slip flow and heat transfer is obtained by using the homotopy analysis method. This method is a series expansion method, but it is different from the perturbation technique, because it is independent of small physical parameters at all. Instead, it is based on a continuous mapping in topology so that it is applicable for not only weakly but also strongly nonlinear flow phenomena. Convergent [m,m] homotopy Padé approximants are obtained and compared with the numerical results and the asymptotic approximations. It is found that the homotopy Padé approximants agree well with the numerical results. The effects of the slip length and the thermal slip constant β on the heat transfer characteristics are investigated and discussed.展开更多
In this paper,we investigate the vanishing viscosity limit problem for the 3-dimensional(3D)incompressible Navier-Stokes equations in a general bounded smooth domain of R^3 with the generalized Navier-slip boundary co...In this paper,we investigate the vanishing viscosity limit problem for the 3-dimensional(3D)incompressible Navier-Stokes equations in a general bounded smooth domain of R^3 with the generalized Navier-slip boundary conditions u^ε·n=0,n×(ω^ε)=[Bu^ε]τon∂Ω.Some uniform estimates on rates of convergence in C([0,T],L2(Ω))and C([0,T],H^1(Ω))of the solutions to the corresponding solutions of the ideal Euler equations with the standard slip boundary condition are obtained.展开更多
The main purpose of this paper is to extend the result obtained by Beirao da Veiga(2000)from the whole-space case to slip boundary cases.Denote by a two components of the velocity u.To fix ideas setū=(u_(1),u_(2),0)(...The main purpose of this paper is to extend the result obtained by Beirao da Veiga(2000)from the whole-space case to slip boundary cases.Denote by a two components of the velocity u.To fix ideas setū=(u_(1),u_(2),0)(the half-space)orū=?_(1)ê_(1)+?_(2)ê_(2)(the general boundary case(see(7.1))).We show that there exists a constant K,which enjoys very simple and significant expressions such that if at some timeτ∈(0,T)one has lim sup_(t→^(τ)-0)‖ū(t)‖_(L^(3)(Ω))^(3)<‖ū(τ)‖_(L^(3)(Ω))^(3)+K,then u is continuous atτwith values in L^(3)(Ω).Roughly speaking,the above norm-discontinuity of merely two components of the velocity cannot occur for steps'amplitudes smaller than K.In particular,if the above condition holds at eachτ∈(0,T),the solution is smooth in(0,T)×Ω.Note that here there is no limitation on the width of the norms‖ū(t)‖_(L^(3)(Ω))^(3)·So K is independent of these quantities.Many other related results are discussed and compared among them.展开更多
In this paper,the authors study the 1 D steady Boltzmann flow in a channel.The walls of the channel are assumed to have vanishing velocity and given temperaturesθ0andθ1.This problem was studied by Esposito-Lebowitz-...In this paper,the authors study the 1 D steady Boltzmann flow in a channel.The walls of the channel are assumed to have vanishing velocity and given temperaturesθ0andθ1.This problem was studied by Esposito-Lebowitz-Marra(1994,1995)where they showed that the solution tends to a local Maxwellian with parameters satisfying the compressible Navier-Stokes equation with no-slip boundary condition.However,a lot of numerical experiments reveal that the fluid layer does not entirely stick to the boundary.In the regime where the Knudsen number is reasonably small,the slip phenomenon is significant near the boundary.Thus,they revisit this problem by taking into account the slip boundary conditions.Following the lines of[Coron,F.,Derivation of slip boundary conditions for the Navier-Stokes system from the Boltzmann equation,J.Stat.Phys.,54(3-4),1989,829-857],the authors will first give a formal asymptotic analysis to see that the flow governed by the Boltzmann equation is accurately approximated by a superposition of a steady CNS equation with a temperature jump condition and two Knudsen layers located at end points.Then they will establish a uniform L∞estimate on the remainder and derive the slip boundary condition for compressible Navier-Stokes equations rigorously.展开更多
This paper deals with the two-level Newton iteration method based on the pressure projection stabilized finite element approximation to solve the numerical solution of the Navier-Stokes type variational inequality pro...This paper deals with the two-level Newton iteration method based on the pressure projection stabilized finite element approximation to solve the numerical solution of the Navier-Stokes type variational inequality problem.We solve a small Navier-Stokes problem on the coarse mesh with mesh size H and solve a large linearized Navier-Stokes problem on the fine mesh with mesh size h.The error estimates derived show that if we choose h=O(|logh|^(1/2)H^(3)),then the two-level method we provide has the same H1 and L^(2) convergence orders of the velocity and the pressure as the one-level stabilized method.However,the L^(2) convergence order of the velocity is not consistent with that of one-level stabilized method.Finally,we give the numerical results to support the theoretical analysis.展开更多
The problem of the steady migration of an axially symmetric prolate particle along its axis of revolution coinciding with the centerline of a circular capillary is investigated semi-analytically in the limit of low Re...The problem of the steady migration of an axially symmetric prolate particle along its axis of revolution coinciding with the centerline of a circular capillary is investigated semi-analytically in the limit of low Reynolds number,where the viscous fluid may slip at the solid surfaces.A method of distribution of spherical singularities along the axis inside the particle is employed to establish the general solution of the fluid velocity satisfying the boundary conditions at the capillary wall and infinity.The slip condition at the particle surface is then satisfied by using a boundary collocation method to determine the unknown constants in this solution.The hydrodynamic drag force acting on the particle is obtained with good convergence for the cases of a prolate spheroid and a prolate Cassini oval with various values of the slip parameter of the particle,slip parameter of the capillary wall,aspect ratio or shape parameter of the particle,and spacing parameter between the particle and the wall.For the axially symmetric migrations of a spheroid and a Cassini oval in a capillary with no-slip surfaces and of a sphere in a capillary with slip surfaces,our results agree excellently with the numerical solutions obtained earlier.The capillary wall affects the particle migration significantly when the solid surfaces get close to each other.For a specified particle-in-capillary configuration,the normalized drag force exerted on the particle in general decreases with increasing slippage at the solid surfaces,except when the fluid slips little at the capillary wall and the particle-wall spacing parameter is relatively large.For fixed spacing parameter and slip parameters,the drag force increases with an increase in the axial-to-radial aspect ratio(or surface area effective for viscous interaction with the capillary wall)of the particle,but this tendency can be reversed when the particle is highly slippery.展开更多
In this article, we prove the existence and uniqueness of solutions of the NavierStokes equations with Navier slip boundary condition for incompressible fluid in a bounded domain of R^3. The results are established by...In this article, we prove the existence and uniqueness of solutions of the NavierStokes equations with Navier slip boundary condition for incompressible fluid in a bounded domain of R^3. The results are established by the Galerkin approximation method and improved the existing results.展开更多
A study on the effects of Navier slip, in conjunction with other flow parameters, on unsteady flow of reactive variable viscosity third-grade fluid through a porous saturated medium with asymmetric convective boundary...A study on the effects of Navier slip, in conjunction with other flow parameters, on unsteady flow of reactive variable viscosity third-grade fluid through a porous saturated medium with asymmetric convective boundary conditions is presented. The channel walls are assumed to be subjected to asymmetric convective heat exchange with the ambient, and exothermic chemical reactions take place within the flow system. The heat exchange with the ambient obeys Newton＇s law of cooling. The coupled equations, arising from the law of conservation of momentum and the first law of thermodynamics, then the derived system are non- dimensionalised and solved using a semi-implicit finite difference scheme. The lower wall slip parameter is observed to increase the fluid velocity profiles, whereas the upper wall slip parameter retards them because of backflow at the upper channel wall. Heat pro- duction in the fluid is seen to increase with the slip parameters. The wall shear stress increases with the slip parameters while the wall heat transfer rate is largely unaltered by the lower wall slip parameter but marginally increased by the upper wall slip parameter.展开更多
The effects of the Born repulsive force on the stability and dynamics of ultra-thin slipping films under the influences of intermolecular forces are investigated with bifurcation theory and numerical simulation. Resul...The effects of the Born repulsive force on the stability and dynamics of ultra-thin slipping films under the influences of intermolecular forces are investigated with bifurcation theory and numerical simulation. Results show that the repulsive force has a stabilizing effect on the development of perturbations, and can suppress the rupture process induced by the van der Waals attractive force. Although slippage will enhance the growth of disturbances, it does not have influence on the linear cutoff wave number and the final shape of the film thickness as time approaches to infinity.展开更多
基金supported by Natural Science Foundation of Jiangsu Province(Grant No.BK20200803)National Natural Science Foundation of China(Grant No.12001285)+1 种基金supported by National Natural Science Foundation of China(Grant Nos.11801268 and 12031006)supported by National Natural Science Foundation of China(Grant No.12001429)。
文摘The original Leray’s problem concerns the well-posedness of weak solutions to the steady incompressible Navier-Stokes equations in a distorted pipe,which approach the Poiseuille flow subject to the no-slip boundary condition at spatial infinity.In this paper,the same problem with the Navier-slip boundary condition instead of the no-slip boundary condition,is addressed.Due to the complexity of the boundary condition,some new ideas,presented as follows,are introduced to handle the extra difficulties caused by boundary terms.First,the Poiseuille flow in the semi-infinite straight pipe with the Navier-slip boundary condition will be introduced,which will serve as the asymptotic profile of the solution to the generalized Leray’s problem at spatial infinity.Second,a solenoidal vector function defined in the whole pipe,satisfying the Navierslip boundary condition,having the designated flux and equalling the Poiseuille flow at a large distance,will be carefully constructed.This plays an important role in reformulating our problem.Third,the energy estimates depend on a combined L2-estimate of the gradient and the stress tensor of the velocity.
基金supported in part by the National Science Foundation under Grants DMS-0807551, DMS-0720925, and DMS-0505473the Natural Science Foundationof China (10728101)supported in part by EPSRC grant EP/F029578/1
文摘We study the initial-boundary value problem of the Navier-Stokes equations for incompressible fluids in a general domain in R^n with compact and smooth boundary, subject to the kinematic and vorticity boundary conditions on the non-flat boundary. We observe that, under the nonhomogeneous boundary conditions, the pressure p can be still recovered by solving the Neumann problem for the Poisson equation. Then we establish the well-posedness of the unsteady Stokes equations and employ the solution to reduce our initial-boundary value problem into an initial-boundary value problem with absolute boundary conditions. Based on this, we first establish the well-posedness for an appropriate local linearized problem with the absolute boundary conditions and the initial condition （without the incompressibility condition）, which establishes a velocity mapping. Then we develop apriori estimates for the velocity mapping, especially involving the Sobolev norm for the time-derivative of the mapping to deal with the complicated boundary conditions, which leads to the existence of the fixed point of the mapping and the existence of solutions to our initial-boundary value problem. Finally, we establish that, when the viscosity coefficient tends zero, the strong solutions of the initial-boundary value problem in R^n（n ≥ 3） with nonhomogeneous vorticity boundary condition converge in L^2 to the corresponding Euler equations satisfying the kinematic condition.
基金Supported by the National Natural Science Foundation of China (Grant No.10872129)
文摘An analytical approximation for the similarity solutions of the two-and three-dimensional stagnation slip flow and heat transfer is obtained by using the homotopy analysis method. This method is a series expansion method, but it is different from the perturbation technique, because it is independent of small physical parameters at all. Instead, it is based on a continuous mapping in topology so that it is applicable for not only weakly but also strongly nonlinear flow phenomena. Convergent [m,m] homotopy Padé approximants are obtained and compared with the numerical results and the asymptotic approximations. It is found that the homotopy Padé approximants agree well with the numerical results. The effects of the slip length and the thermal slip constant β on the heat transfer characteristics are investigated and discussed.
基金This research is supported in part by NSFC 10971174,and Zheng Ge Ru Foundation,and Hong Kong RGC Earmarked Research Grants CUHK-4041/11P,CUHK-4042/08P,a Focus Area Grant from the Chinese University of Hong Kong,and a grant from Croucher Foundation.
文摘In this paper,we investigate the vanishing viscosity limit problem for the 3-dimensional(3D)incompressible Navier-Stokes equations in a general bounded smooth domain of R^3 with the generalized Navier-slip boundary conditions u^ε·n=0,n×(ω^ε)=[Bu^ε]τon∂Ω.Some uniform estimates on rates of convergence in C([0,T],L2(Ω))and C([0,T],H^1(Ω))of the solutions to the corresponding solutions of the ideal Euler equations with the standard slip boundary condition are obtained.
基金supported by Portuguese Foundation for Science and Technology(Portugal)(Grant No.UIDB/MAT/04561/2020)supported by National Natural Science Foundation of China(Grant No.12001429)。
文摘The main purpose of this paper is to extend the result obtained by Beirao da Veiga(2000)from the whole-space case to slip boundary cases.Denote by a two components of the velocity u.To fix ideas setū=(u_(1),u_(2),0)(the half-space)orū=?_(1)ê_(1)+?_(2)ê_(2)(the general boundary case(see(7.1))).We show that there exists a constant K,which enjoys very simple and significant expressions such that if at some timeτ∈(0,T)one has lim sup_(t→^(τ)-0)‖ū(t)‖_(L^(3)(Ω))^(3)<‖ū(τ)‖_(L^(3)(Ω))^(3)+K,then u is continuous atτwith values in L^(3)(Ω).Roughly speaking,the above norm-discontinuity of merely two components of the velocity cannot occur for steps'amplitudes smaller than K.In particular,if the above condition holds at eachτ∈(0,T),the solution is smooth in(0,T)×Ω.Note that here there is no limitation on the width of the norms‖ū(t)‖_(L^(3)(Ω))^(3)·So K is independent of these quantities.Many other related results are discussed and compared among them.
基金supported by the National Natural Science Foundation of China(Nos.11971201,11731008)the General Research Fund from RGC of Hong Kong(No.14301719)+1 种基金a Direct Grant from CUHK(No.4053397)the Fundamental Research Funds for the Central Universities and a fellowship award from the Research Grants Council of the Hong Kong Special Administrative Region,China(No.SRF2021-1S01)。
文摘In this paper,the authors study the 1 D steady Boltzmann flow in a channel.The walls of the channel are assumed to have vanishing velocity and given temperaturesθ0andθ1.This problem was studied by Esposito-Lebowitz-Marra(1994,1995)where they showed that the solution tends to a local Maxwellian with parameters satisfying the compressible Navier-Stokes equation with no-slip boundary condition.However,a lot of numerical experiments reveal that the fluid layer does not entirely stick to the boundary.In the regime where the Knudsen number is reasonably small,the slip phenomenon is significant near the boundary.Thus,they revisit this problem by taking into account the slip boundary conditions.Following the lines of[Coron,F.,Derivation of slip boundary conditions for the Navier-Stokes system from the Boltzmann equation,J.Stat.Phys.,54(3-4),1989,829-857],the authors will first give a formal asymptotic analysis to see that the flow governed by the Boltzmann equation is accurately approximated by a superposition of a steady CNS equation with a temperature jump condition and two Knudsen layers located at end points.Then they will establish a uniform L∞estimate on the remainder and derive the slip boundary condition for compressible Navier-Stokes equations rigorously.
基金funded by the National Natural Science Foundation of China under Grant No.10901122 and No.11001205by Zhejiang Provincial Natural Science Foundation of China under Grant No.LY12A01015.
文摘This paper deals with the two-level Newton iteration method based on the pressure projection stabilized finite element approximation to solve the numerical solution of the Navier-Stokes type variational inequality problem.We solve a small Navier-Stokes problem on the coarse mesh with mesh size H and solve a large linearized Navier-Stokes problem on the fine mesh with mesh size h.The error estimates derived show that if we choose h=O(|logh|^(1/2)H^(3)),then the two-level method we provide has the same H1 and L^(2) convergence orders of the velocity and the pressure as the one-level stabilized method.However,the L^(2) convergence order of the velocity is not consistent with that of one-level stabilized method.Finally,we give the numerical results to support the theoretical analysis.
文摘The problem of the steady migration of an axially symmetric prolate particle along its axis of revolution coinciding with the centerline of a circular capillary is investigated semi-analytically in the limit of low Reynolds number,where the viscous fluid may slip at the solid surfaces.A method of distribution of spherical singularities along the axis inside the particle is employed to establish the general solution of the fluid velocity satisfying the boundary conditions at the capillary wall and infinity.The slip condition at the particle surface is then satisfied by using a boundary collocation method to determine the unknown constants in this solution.The hydrodynamic drag force acting on the particle is obtained with good convergence for the cases of a prolate spheroid and a prolate Cassini oval with various values of the slip parameter of the particle,slip parameter of the capillary wall,aspect ratio or shape parameter of the particle,and spacing parameter between the particle and the wall.For the axially symmetric migrations of a spheroid and a Cassini oval in a capillary with no-slip surfaces and of a sphere in a capillary with slip surfaces,our results agree excellently with the numerical solutions obtained earlier.The capillary wall affects the particle migration significantly when the solid surfaces get close to each other.For a specified particle-in-capillary configuration,the normalized drag force exerted on the particle in general decreases with increasing slippage at the solid surfaces,except when the fluid slips little at the capillary wall and the particle-wall spacing parameter is relatively large.For fixed spacing parameter and slip parameters,the drag force increases with an increase in the axial-to-radial aspect ratio(or surface area effective for viscous interaction with the capillary wall)of the particle,but this tendency can be reversed when the particle is highly slippery.
基金supported by National Board for Higher Mathematics(02011/9/2019NBHM(R.P.)/R and D Ⅱ/1324)
文摘In this article, we prove the existence and uniqueness of solutions of the NavierStokes equations with Navier slip boundary condition for incompressible fluid in a bounded domain of R^3. The results are established by the Galerkin approximation method and improved the existing results.
文摘A study on the effects of Navier slip, in conjunction with other flow parameters, on unsteady flow of reactive variable viscosity third-grade fluid through a porous saturated medium with asymmetric convective boundary conditions is presented. The channel walls are assumed to be subjected to asymmetric convective heat exchange with the ambient, and exothermic chemical reactions take place within the flow system. The heat exchange with the ambient obeys Newton＇s law of cooling. The coupled equations, arising from the law of conservation of momentum and the first law of thermodynamics, then the derived system are non- dimensionalised and solved using a semi-implicit finite difference scheme. The lower wall slip parameter is observed to increase the fluid velocity profiles, whereas the upper wall slip parameter retards them because of backflow at the upper channel wall. Heat pro- duction in the fluid is seen to increase with the slip parameters. The wall shear stress increases with the slip parameters while the wall heat transfer rate is largely unaltered by the lower wall slip parameter but marginally increased by the upper wall slip parameter.
基金Project supported by the National Natural Science Foundation of China (Grant No:10472062 ) and Shanghai Leading Academic Discipline Project (Grant No: Y0103 ).
文摘The effects of the Born repulsive force on the stability and dynamics of ultra-thin slipping films under the influences of intermolecular forces are investigated with bifurcation theory and numerical simulation. Results show that the repulsive force has a stabilizing effect on the development of perturbations, and can suppress the rupture process induced by the van der Waals attractive force. Although slippage will enhance the growth of disturbances, it does not have influence on the linear cutoff wave number and the final shape of the film thickness as time approaches to infinity.
