This article gives a general model using specific periodic special functions, that is, degenerate elliptic Weierstrass P functions composed with the LambertW function, whose presence in the governing equations through...This article gives a general model using specific periodic special functions, that is, degenerate elliptic Weierstrass P functions composed with the LambertW function, whose presence in the governing equations through the forcing terms simplify the periodic Navier Stokes equations (PNS) at the centers of arbitrary r balls of the 3-Torus. The continuity equation is satisfied together with spatially periodic boundary conditions. The yicomponent forcing terms consist of a function F as part of its expression that is arbitrarily small in an r ball where it is associated with a singular forcing expression both for inviscid and viscous cases. As a result, a significant simplification occurs with a v3(vifor all velocity components) only governing PDE resulting. The extension of three restricted subspaces in each of the principal directions in the Cartesian plane is shown as the Cartesian product ℋ=Jx,t×Jy,t×Jz,t. On each of these subspaces vi,i=1,2,3is continuous and there exists a linear independent subspace associated with the argument of the W function. Here the 3-Torus is built up from each compact segment of length 2R on each of the axes on the 3 principal directions x, y, and z. The form of the scaled velocities for non zero scaled δis related to the definition of the W function such that e−W(ξ)=W(ξ)ξwhere ξdepends on t and proportional to δ→0for infinite time t. The ratio Wξis equal to 1, making the limit δ→0finite and well defined. Considering r balls where the function F=(x−ai)2+(y−bi)2+(z−ci)2−ηset equal to −1e+rwhere r>0. is such that the forcing is singular at every distance r of centres of cubes each containing an r-ball. At the centre of the balls, the forcing is infinite. The main idea is that a system of singular initial value problems with infinite forcing is to be solved for where the velocities are shown to be locally Hölder continuous. It is proven that the limit of these singular problems shifts the finite time blowup time ti∗for first and higher derivatives to t=∞thereby indicating that there is no finite time blowup. Results in the literature can provide a systematic approach to study both large space and time behaviour for singular solutions to the Navier Stokes equations. Among the references, it has been shown that mathematical tools can be applied to study the asymptotic properties of solutions.展开更多
This paper is concerned with the global well-posedness of the solution to the compressible Navier-Stokes/Allen-Cahn system and its sharp interface limit in one-dimensional space.For the perturbations with small energy...This paper is concerned with the global well-posedness of the solution to the compressible Navier-Stokes/Allen-Cahn system and its sharp interface limit in one-dimensional space.For the perturbations with small energy but possibly large oscillations of rarefaction wave solutions near phase separation,and where the strength of the initial phase field could be arbitrarily large,we prove that the solution of the Cauchy problem exists for all time,and converges to the centered rarefaction wave solution of the corresponding standard two-phase Euler equation as the viscosity and the thickness of the interface tend to zero.The proof is mainly based on a scaling argument and a basic energy method.展开更多
A time semi-discrete Crank-Nicolson (CN) formulation with second-order time accuracy for the non-stationary parabolized Navier-Stokes equations is firstly established. And then, a fully discrete stabilized CN mixed ...A time semi-discrete Crank-Nicolson (CN) formulation with second-order time accuracy for the non-stationary parabolized Navier-Stokes equations is firstly established. And then, a fully discrete stabilized CN mixed finite volume element (SCNMFVE) formu- lation based on two local Gaussian integrals and parameter-free with the second-order time accuracy is established directly from the time semi-discrete CN formulation so that it could avoid the discussion for semi-discrete SCNMFVE formulation with respect to spatial wriables and its theoretical analysis becomes very simple. Finally, the error estimates of SCNMFVE solutions are provided.展开更多
In this paper,we study the controllability of compressible Navier-Stokes equations with density dependent viscosities.For when the shear viscosityμis a positive constant and the bulk viscosityλis a function of the d...In this paper,we study the controllability of compressible Navier-Stokes equations with density dependent viscosities.For when the shear viscosityμis a positive constant and the bulk viscosityλis a function of the density,it is proven that the system is exactly locally controllable to a constant target trajectory by using boundary control functions.展开更多
We consider the global well-posedness of strong solutions to the Cauchy problem of compressible barotropic Navier-Stokes equations in R^(2). By exploiting the global-in-time estimate to the two-dimensional(2D for shor...We consider the global well-posedness of strong solutions to the Cauchy problem of compressible barotropic Navier-Stokes equations in R^(2). By exploiting the global-in-time estimate to the two-dimensional(2D for short) classical incompressible Navier-Stokes equations and using techniques developed in(SIAM J Math Anal, 2020, 52(2): 1806–1843), we derive the global existence of solutions provided that the initial data satisfies some smallness condition. In particular, the initial velocity with some arbitrary Besov norm of potential part Pu_0 and large high oscillation are allowed in our results. Moreover, we also construct an example with the initial data involving such a smallness condition, but with a norm that is arbitrarily large.展开更多
We investigate the low Mach number limit for the isentropic compressible NavierStokes equations with a revised Maxwell's law(with Galilean invariance) in R^(3). By applying the uniform estimates of the error syste...We investigate the low Mach number limit for the isentropic compressible NavierStokes equations with a revised Maxwell's law(with Galilean invariance) in R^(3). By applying the uniform estimates of the error system, it is proven that the solutions of the isentropic Navier-Stokes equations with a revised Maxwell's law converge to that of the incompressible Navier-Stokes equations as the Mach number tends to zero. Moreover, the convergence rates are also obtained.展开更多
Many applications in fluid mechanics require the numerical solution of sequences of linear systems typically issued from finite element discretization of the Navier-Stokes equations. The resulting matrices then exhibi...Many applications in fluid mechanics require the numerical solution of sequences of linear systems typically issued from finite element discretization of the Navier-Stokes equations. The resulting matrices then exhibit a saddle point structure. To achieve this task, a Newton-based root-finding algorithm is usually employed which in turn necessitates to solve a saddle point system at every Newton iteration. The involved linear systems being large scale and ill-conditioned, effective linear solvers must be implemented. Here, we develop and test several methods for solving the saddle point systems, considering in particular the LU factorization, as direct approach, and the preconditioned generalized minimal residual (ΡGMRES) solver, an iterative approach. We apply the various solvers within the root-finding algorithm for Flow over backward facing step systems. The particularity of Flow over backward facing step system is an interesting case for studying the performance and solution strategy of a turbulence model. In this case, the flow is subjected to a sudden increase of cross-sectional area, resulting in a separation of flow starting at the point of expansion, making the system of differential equations particularly stiff. We assess the performance of the direct and iterative solvers in terms of computational time, numbers of Newton iterations and time steps.展开更多
The aims of this paper are to discuss global existence and uniqueness of strong solution for a class of isentropic compressible navier-Stokes equations with non-Newtonian in one-dimensional bounded intervals. We prove...The aims of this paper are to discuss global existence and uniqueness of strong solution for a class of isentropic compressible navier-Stokes equations with non-Newtonian in one-dimensional bounded intervals. We prove two global existence results on strong solutions of isentropic compressible Navier-Stokes equations. The first result shows only the existence. And the second one shows the existence and uniqueness result based on the first result, but the uniqueness requires some compatibility condition.展开更多
A new seven-modes truncation of Fourier series of Navier-Stokes equations for a two-dimensional incompressible fluid on a torus is obtained.And its stationary solutions,the existence of attractor and the global stabil...A new seven-modes truncation of Fourier series of Navier-Stokes equations for a two-dimensional incompressible fluid on a torus is obtained.And its stationary solutions,the existence of attractor and the global stability of the equations are firmly proved.At the same time,several issues such as some basic dynamical behaviors and routs to chaos are shown numerically by changing Reynolds number.The system exhibits a stochastic behavior approached through an involved sequence of bifurcations.展开更多
The pressureless Navier-Stokes equations for non-Newtonian fluid are studied. The analytical solutions with arbitrary time blowup, in radial symmetry, are constructed in this paper. With the previous results for the a...The pressureless Navier-Stokes equations for non-Newtonian fluid are studied. The analytical solutions with arbitrary time blowup, in radial symmetry, are constructed in this paper. With the previous results for the analytical blowup solutions of the N-dimensional (N ≥ 2) Navier-Stokes equations, we extend the similar structure to construct an analytical family of solutions for the pressureless Navier-Stokes equations with a normal viscosity term (μ(ρ)| u|^α u).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
We consider the Cauchy problem, free boundary problem and piston problem for one-dimensional compressible Navier-Stokes equations with density-dependent viscosity. Using the reduction to absurdity method, we prove tha...We consider the Cauchy problem, free boundary problem and piston problem for one-dimensional compressible Navier-Stokes equations with density-dependent viscosity. Using the reduction to absurdity method, we prove that the weak solutions to these systems do not exhibit vacuum states, provided that no vacuum states are present initially. The essential re- quirements on the solutions are that the mass and energy of the fluid are locally integrable at each time, and the Lloc1-norm of the velocity gradient is locally integrable in time.展开更多
A seven-mode truncation system of the Navier-Stokes equations for a two-dimensional incompressible fluid on a torus is considered. Its stationary solutions and stability are presented;the existence of the attractor an...A seven-mode truncation system of the Navier-Stokes equations for a two-dimensional incompressible fluid on a torus is considered. Its stationary solutions and stability are presented;the existence of the attractor and the global stability of the system are discussed. The whole process, which shows a chaos behavior approached through instability of invariant tori, is simulated numerically by computers with the changing of Reynolds number. Based on numerical simulation results of bifurcation diagram, Lyapunov exponent spectrum, Poincare section, power spectrum and return map of the system, some basic dynamical behaviors of the new chaos system are revealed.展开更多
In this paper,we study the one-dimensional motion of viscous gas near a vacuum,with the gas connecting to a vacuum state with a jump in density.The interface behavior,the pointwise decay rates of the density function ...In this paper,we study the one-dimensional motion of viscous gas near a vacuum,with the gas connecting to a vacuum state with a jump in density.The interface behavior,the pointwise decay rates of the density function and the expanding rates of the interface are obtained with the viscosity coefficientμ(ρ)=ρ^(α)for any 0<α<1;this includes the timeweighted boundedness from below and above.The smoothness of the solution is discussed.Moreover,we construct a class of self-similar classical solutions which exhibit some interesting properties,such as optimal estimates.The present paper extends the results in[Luo T,Xin Z P,Yang T.SIAM J Math Anal,2000,31(6):1175-1191]to the jump boundary conditions case with density-dependent viscosity.展开更多
The preconditioning method is used to solve the low Mach number flow. The space discritisation scheme is the Roe scheme and the DES turbulence model is used. Then, the low Mach number turbulence flow around the NACA00...The preconditioning method is used to solve the low Mach number flow. The space discritisation scheme is the Roe scheme and the DES turbulence model is used. Then, the low Mach number turbulence flow around the NACA0012 airfoil is used to verify the efficiency of the proposed method. Two cases of the low Mach number flows around the multi-element airfoil and the circular cylinder are also used to test the proposed method. Numerical results show that the methods combined the preconditioning method and compressible Navier-Stokes equations are efficient to solve low Mach number flows.展开更多
A global weak solution to the isentropic Navier-Stokes equation with initial data around a constant state in the L^(1)∩BV class was constructed in[1].In the current paper,we will continue to study the uniqueness and ...A global weak solution to the isentropic Navier-Stokes equation with initial data around a constant state in the L^(1)∩BV class was constructed in[1].In the current paper,we will continue to study the uniqueness and regularity of the constructed solution.The key ingredients are the Holder continuity estimates of the heat kernel in both spatial and time variables.With these finer estimates,we obtain higher order regularity of the constructed solution to Navier-Stokes equation,so that all of the derivatives in the equation of conservative form are in the strong sense.Moreover,this regularity also allows us to identify a function space such that the stability of the solutions can be established there,which eventually implies the uniqueness.展开更多
The analytical solution of the multi-dimensional,time-fractional model of Navier-Stokes equation using the triple and quadruple Elzaki transformdecompositionmethod is presented in this article.The aforesaidmodel is an...The analytical solution of the multi-dimensional,time-fractional model of Navier-Stokes equation using the triple and quadruple Elzaki transformdecompositionmethod is presented in this article.The aforesaidmodel is analyzed by employing Caputo fractional derivative.We deliberated three stimulating examples that correspond to the triple and quadruple Elzaki transform decomposition methods,respectively.The findings illustrate that the established approaches are extremely helpful in obtaining exact and approximate solutions to the problems.The exact and estimated solutions are delineated via numerical simulation.The proposed analysis indicates that the projected configuration is extremely meticulous,highly efficient,and precise in understanding the behavior of complex evolutionary problems of both fractional and integer order that classify affiliated scientific fields and technology.展开更多
In this paper,we study a one-dimensional motion of viscous gas near vacuum. We are interested in the case that the gas is in contact with the vacuum at a finite interval. This is a free boundary problem for the one-di...In this paper,we study a one-dimensional motion of viscous gas near vacuum. We are interested in the case that the gas is in contact with the vacuum at a finite interval. This is a free boundary problem for the one-dimensional isentropic Navier-Stokes equations, and the free boundaries are the interfaces separating the gas from vacuum,across which the density changes discontinuosly.Smoothness of the solutions and the uniqueness of the weak solutions are also discussed.The present paper extends results in Luo-Xin-Yang[12] to the jump boundary conditions case.展开更多
The zero dissipation limit of the compressible heat-conducting Navier–Stokes equations in the presence of the shock is investigated. It is shown that when the heat conduction coefficient κ and the viscosity coeffici...The zero dissipation limit of the compressible heat-conducting Navier–Stokes equations in the presence of the shock is investigated. It is shown that when the heat conduction coefficient κ and the viscosity coefficient ε satisfy κ = O(ε), κ/ε≥ c 〉 0, as ε→ 0 (see (1.3)), if the solution of the corresponding Euler equations is piecewise smooth with shock wave satisfying the Lax entropy condition, then there exists a smooth solution to the Navier–Stokes equations, which converges to the piecewise smooth shock solution of the Euler equations away from the shock discontinuity at a rate of ε. The proof is given by a combination of the energy estimates and the matched asymptotic analysis introduced in [3].展开更多
文摘This article gives a general model using specific periodic special functions, that is, degenerate elliptic Weierstrass P functions composed with the LambertW function, whose presence in the governing equations through the forcing terms simplify the periodic Navier Stokes equations (PNS) at the centers of arbitrary r balls of the 3-Torus. The continuity equation is satisfied together with spatially periodic boundary conditions. The yicomponent forcing terms consist of a function F as part of its expression that is arbitrarily small in an r ball where it is associated with a singular forcing expression both for inviscid and viscous cases. As a result, a significant simplification occurs with a v3(vifor all velocity components) only governing PDE resulting. The extension of three restricted subspaces in each of the principal directions in the Cartesian plane is shown as the Cartesian product ℋ=Jx,t×Jy,t×Jz,t. On each of these subspaces vi,i=1,2,3is continuous and there exists a linear independent subspace associated with the argument of the W function. Here the 3-Torus is built up from each compact segment of length 2R on each of the axes on the 3 principal directions x, y, and z. The form of the scaled velocities for non zero scaled δis related to the definition of the W function such that e−W(ξ)=W(ξ)ξwhere ξdepends on t and proportional to δ→0for infinite time t. The ratio Wξis equal to 1, making the limit δ→0finite and well defined. Considering r balls where the function F=(x−ai)2+(y−bi)2+(z−ci)2−ηset equal to −1e+rwhere r>0. is such that the forcing is singular at every distance r of centres of cubes each containing an r-ball. At the centre of the balls, the forcing is infinite. The main idea is that a system of singular initial value problems with infinite forcing is to be solved for where the velocities are shown to be locally Hölder continuous. It is proven that the limit of these singular problems shifts the finite time blowup time ti∗for first and higher derivatives to t=∞thereby indicating that there is no finite time blowup. Results in the literature can provide a systematic approach to study both large space and time behaviour for singular solutions to the Navier Stokes equations. Among the references, it has been shown that mathematical tools can be applied to study the asymptotic properties of solutions.
基金supported by the National Natural Science Foundation of China(12361044)supported by the National Natural Science Foundation of China(12171024,11971217,11971020)supported by the Academic and Technical Leaders Training Plan of Jiangxi Province(20212BCJ23027)。
文摘This paper is concerned with the global well-posedness of the solution to the compressible Navier-Stokes/Allen-Cahn system and its sharp interface limit in one-dimensional space.For the perturbations with small energy but possibly large oscillations of rarefaction wave solutions near phase separation,and where the strength of the initial phase field could be arbitrarily large,we prove that the solution of the Cauchy problem exists for all time,and converges to the centered rarefaction wave solution of the corresponding standard two-phase Euler equation as the viscosity and the thickness of the interface tend to zero.The proof is mainly based on a scaling argument and a basic energy method.
基金supported by National Science Foundation of China(11271127)Science Research Project of Guizhou Province Education Department(QJHKYZ[2013]207)
文摘A time semi-discrete Crank-Nicolson (CN) formulation with second-order time accuracy for the non-stationary parabolized Navier-Stokes equations is firstly established. And then, a fully discrete stabilized CN mixed finite volume element (SCNMFVE) formu- lation based on two local Gaussian integrals and parameter-free with the second-order time accuracy is established directly from the time semi-discrete CN formulation so that it could avoid the discussion for semi-discrete SCNMFVE formulation with respect to spatial wriables and its theoretical analysis becomes very simple. Finally, the error estimates of SCNMFVE solutions are provided.
基金partially supported by the National Science Foundation of China(11971320,11971496)the National Key R&D Program of China(2020YFA0712500)the Guangdong Basic and Applied Basic Research Foundation(2020A1515010530)。
文摘In this paper,we study the controllability of compressible Navier-Stokes equations with density dependent viscosities.For when the shear viscosityμis a positive constant and the bulk viscosityλis a function of the density,it is proven that the system is exactly locally controllable to a constant target trajectory by using boundary control functions.
基金Zhai was partially supported by the Guangdong Provincial Natural Science Foundation (2022A1515011977)the Science and Technology Program of Shenzhen(20200806104726001)+1 种基金Zhong was partially supported by the NNSF of China (11901474, 12071359)the Exceptional Young Talents Project of Chongqing Talent (cstc2021ycjh-bgzxm0153)。
文摘We consider the global well-posedness of strong solutions to the Cauchy problem of compressible barotropic Navier-Stokes equations in R^(2). By exploiting the global-in-time estimate to the two-dimensional(2D for short) classical incompressible Navier-Stokes equations and using techniques developed in(SIAM J Math Anal, 2020, 52(2): 1806–1843), we derive the global existence of solutions provided that the initial data satisfies some smallness condition. In particular, the initial velocity with some arbitrary Besov norm of potential part Pu_0 and large high oscillation are allowed in our results. Moreover, we also construct an example with the initial data involving such a smallness condition, but with a norm that is arbitrarily large.
基金Yuxi HU was supported by the NNSFC (11701556)the Yue Qi Young Scholar ProjectChina University of Mining and Technology (Beijing)。
文摘We investigate the low Mach number limit for the isentropic compressible NavierStokes equations with a revised Maxwell's law(with Galilean invariance) in R^(3). By applying the uniform estimates of the error system, it is proven that the solutions of the isentropic Navier-Stokes equations with a revised Maxwell's law converge to that of the incompressible Navier-Stokes equations as the Mach number tends to zero. Moreover, the convergence rates are also obtained.
文摘Many applications in fluid mechanics require the numerical solution of sequences of linear systems typically issued from finite element discretization of the Navier-Stokes equations. The resulting matrices then exhibit a saddle point structure. To achieve this task, a Newton-based root-finding algorithm is usually employed which in turn necessitates to solve a saddle point system at every Newton iteration. The involved linear systems being large scale and ill-conditioned, effective linear solvers must be implemented. Here, we develop and test several methods for solving the saddle point systems, considering in particular the LU factorization, as direct approach, and the preconditioned generalized minimal residual (ΡGMRES) solver, an iterative approach. We apply the various solvers within the root-finding algorithm for Flow over backward facing step systems. The particularity of Flow over backward facing step system is an interesting case for studying the performance and solution strategy of a turbulence model. In this case, the flow is subjected to a sudden increase of cross-sectional area, resulting in a separation of flow starting at the point of expansion, making the system of differential equations particularly stiff. We assess the performance of the direct and iterative solvers in terms of computational time, numbers of Newton iterations and time steps.
文摘The aims of this paper are to discuss global existence and uniqueness of strong solution for a class of isentropic compressible navier-Stokes equations with non-Newtonian in one-dimensional bounded intervals. We prove two global existence results on strong solutions of isentropic compressible Navier-Stokes equations. The first result shows only the existence. And the second one shows the existence and uniqueness result based on the first result, but the uniqueness requires some compatibility condition.
基金Supported by the Natural Science Foundation of China(41174090)
文摘A new seven-modes truncation of Fourier series of Navier-Stokes equations for a two-dimensional incompressible fluid on a torus is obtained.And its stationary solutions,the existence of attractor and the global stability of the equations are firmly proved.At the same time,several issues such as some basic dynamical behaviors and routs to chaos are shown numerically by changing Reynolds number.The system exhibits a stochastic behavior approached through an involved sequence of bifurcations.
基金Supported by the NSFC of China (1087117510931007+1 种基金10901137)supported by the Scientific Research Fund of Education Department of Zhejiang Province (Y200803203)
文摘The pressureless Navier-Stokes equations for non-Newtonian fluid are studied. The analytical solutions with arbitrary time blowup, in radial symmetry, are constructed in this paper. With the previous results for the analytical blowup solutions of the N-dimensional (N ≥ 2) Navier-Stokes equations, we extend the similar structure to construct an analytical family of solutions for the pressureless Navier-Stokes equations with a normal viscosity term (μ(ρ)| u|^α u).
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Project supported by the National Natural Science Foundation of China (No. 10571158) and the DFG
文摘We consider the Cauchy problem, free boundary problem and piston problem for one-dimensional compressible Navier-Stokes equations with density-dependent viscosity. Using the reduction to absurdity method, we prove that the weak solutions to these systems do not exhibit vacuum states, provided that no vacuum states are present initially. The essential re- quirements on the solutions are that the mass and energy of the fluid are locally integrable at each time, and the Lloc1-norm of the velocity gradient is locally integrable in time.
文摘A seven-mode truncation system of the Navier-Stokes equations for a two-dimensional incompressible fluid on a torus is considered. Its stationary solutions and stability are presented;the existence of the attractor and the global stability of the system are discussed. The whole process, which shows a chaos behavior approached through instability of invariant tori, is simulated numerically by computers with the changing of Reynolds number. Based on numerical simulation results of bifurcation diagram, Lyapunov exponent spectrum, Poincare section, power spectrum and return map of the system, some basic dynamical behaviors of the new chaos system are revealed.
基金supported by the NSFC(11931013)the GXNSF(2022GXNSFDA035078)。
文摘In this paper,we study the one-dimensional motion of viscous gas near a vacuum,with the gas connecting to a vacuum state with a jump in density.The interface behavior,the pointwise decay rates of the density function and the expanding rates of the interface are obtained with the viscosity coefficientμ(ρ)=ρ^(α)for any 0<α<1;this includes the timeweighted boundedness from below and above.The smoothness of the solution is discussed.Moreover,we construct a class of self-similar classical solutions which exhibit some interesting properties,such as optimal estimates.The present paper extends the results in[Luo T,Xin Z P,Yang T.SIAM J Math Anal,2000,31(6):1175-1191]to the jump boundary conditions case with density-dependent viscosity.
文摘The preconditioning method is used to solve the low Mach number flow. The space discritisation scheme is the Roe scheme and the DES turbulence model is used. Then, the low Mach number turbulence flow around the NACA0012 airfoil is used to verify the efficiency of the proposed method. Two cases of the low Mach number flows around the multi-element airfoil and the circular cylinder are also used to test the proposed method. Numerical results show that the methods combined the preconditioning method and compressible Navier-Stokes equations are efficient to solve low Mach number flows.
基金partially the National Key R&D Program of China(2022YFA1007300)the NSFC(11901386,12031013)+2 种基金the Strategic Priority Research Program of the Chinese Academy of Sciences(XDA25010403)the NSFC(11801194,11971188)the Hubei Key Laboratory of Engineering Modeling and Scientific Computing。
文摘A global weak solution to the isentropic Navier-Stokes equation with initial data around a constant state in the L^(1)∩BV class was constructed in[1].In the current paper,we will continue to study the uniqueness and regularity of the constructed solution.The key ingredients are the Holder continuity estimates of the heat kernel in both spatial and time variables.With these finer estimates,we obtain higher order regularity of the constructed solution to Navier-Stokes equation,so that all of the derivatives in the equation of conservative form are in the strong sense.Moreover,this regularity also allows us to identify a function space such that the stability of the solutions can be established there,which eventually implies the uniqueness.
基金supported by the Natural Science Foundation of China(GrantNos.61673169,11301127,11701176,11626101,11601485).
文摘The analytical solution of the multi-dimensional,time-fractional model of Navier-Stokes equation using the triple and quadruple Elzaki transformdecompositionmethod is presented in this article.The aforesaidmodel is analyzed by employing Caputo fractional derivative.We deliberated three stimulating examples that correspond to the triple and quadruple Elzaki transform decomposition methods,respectively.The findings illustrate that the established approaches are extremely helpful in obtaining exact and approximate solutions to the problems.The exact and estimated solutions are delineated via numerical simulation.The proposed analysis indicates that the projected configuration is extremely meticulous,highly efficient,and precise in understanding the behavior of complex evolutionary problems of both fractional and integer order that classify affiliated scientific fields and technology.
文摘In this paper,we study a one-dimensional motion of viscous gas near vacuum. We are interested in the case that the gas is in contact with the vacuum at a finite interval. This is a free boundary problem for the one-dimensional isentropic Navier-Stokes equations, and the free boundaries are the interfaces separating the gas from vacuum,across which the density changes discontinuosly.Smoothness of the solutions and the uniqueness of the weak solutions are also discussed.The present paper extends results in Luo-Xin-Yang[12] to the jump boundary conditions case.
基金the Knowledge Innovation Program of the Chinese Academy of Sciences
文摘The zero dissipation limit of the compressible heat-conducting Navier–Stokes equations in the presence of the shock is investigated. It is shown that when the heat conduction coefficient κ and the viscosity coefficient ε satisfy κ = O(ε), κ/ε≥ c 〉 0, as ε→ 0 (see (1.3)), if the solution of the corresponding Euler equations is piecewise smooth with shock wave satisfying the Lax entropy condition, then there exists a smooth solution to the Navier–Stokes equations, which converges to the piecewise smooth shock solution of the Euler equations away from the shock discontinuity at a rate of ε. The proof is given by a combination of the energy estimates and the matched asymptotic analysis introduced in [3].