Investigation of Boundary-Value Problem for Stationary System of Equations of Viscous Non-Isothermal Fluid

In the Stokes approximation at small Reynolds and Peclet numbers, we obtain a solution to the boundary-value problem of flow around of particles of spherical shape for stationary system of equations of a viscous non-isothermal fluid comprising a linearized by speed Navier–Stokes equation system and the equation of heat transfer given an exponential-power law of dependence of viscosity of fluid on temperature.     

Integral equations for the generalized stokes operator: Applications to high reynolds number flows

This paper deals with a boundary integral equation for the generalized Stokes problem and its approximation by simpler integral equations when the Reynolds number tends to infinity. The two-dimensional case has been treated in [1]. This paper addresses the three-dimensional case. © 1994 John Wiley & Sons, Inc.     

Modeling of flows over inclined wavy bottoms

We study the flow of an incompressible liquid film down a wavy incline. Starting from the Navier–Stokes equations we derive an integral boundary layer equation (IBLe) by applying a Galerkin method with a single test and ansatz function. In comparison with older models this approach has several advantages. First, a linear stability analysis for stationary solutions yields a critical Reynolds number which corresponds in the limit of a flat incline with results directly obtained by the Navier–Stokes equations. Second, the IBLe is consistent with the pertinent Benney equation. Finally, the velocity profile is not assumed to be parabolic, so that also parameter regimes with recirculations can be modelled. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modulation equations and Reynolds averaging for finite-amplitude non-linear waves in an incompressible fluid

A formal perturbation scheme is developed to determine originalmodulation equations for laminar finite-amplitude non-linearwaves in an incompressible fluid. Three idealized problems areanalysed. The modulation equations comprise conservation ofwaves, averaged conditions for conservation of mass, momentum,kinetic energy and angular momentum and the averaged projectionof the Navier–Stokes equations onto the vorticity vector.The last of these modulation equations, which is related tovortex stretching, only appears in 3D problems. The techniqueof Reynolds averaging is also employed to obtain equations forthe mean velocities and pressure. The Reynolds-averaged Navier–Stokesequations correspond to the modulation equations for conservationof mass and momentum. However, the Reynolds stress transportequations are shown to be inconsistent with the other necessarymodulation equations. In two further idealized problems, exactsolutions of the Navier–Stokes equations are obtainedby employing the modulation equations.     

Homotopy analysis method for the heat transfer in a asymmetric porous channel with an expanding or contracting wall

The flow of a viscous incompressible fluid between two parallel plates due to the normal motion of the porous upper plate is investigated and an analysis is made to determine the heat and mass transfer. The unsteady Navier–Stokes equations are reduced to a generalization of the Proudman–Johnson equation retaining the effect of wall motion using a suitable similarity transformation. The analytical solution for stream function and heat transfer characteristics are obtained by employing homotopy analysis method. The effects of various physical parameters like expansion ratio, Prandtl number, Reynolds number on various momentum and heat transfer characteristics are discussed in detail.     

Comparison of computations of asymptotic flow models in a constricted channel

We aim at comparing computations with asymptotic models issued from incompressible Navier–Stokes at high Reynolds number: the Reduced Navier–Stokes/Prandtl (RNS/P) equations and the Double Deck (DD) equations. We treat the case of the steady two dimensional flow in a constricted pipe. In particular, finite differences and finite element solvers are compared for the RNS/P equations. It results from this study that the two codes compare well. Numerical examples also illustrate the interest of these asymptotic models as well as the flexibility of the finite element solver.     

Decoupled modified characteristics finite element method for the time dependent Navier–Stokes/Darcy problem

In this paper, a modified characteristics finite element method for the time dependent Navier–Stokes/Darcy problem with the Beavers–Joseph–Saffman interface condition is presented. In this method, the Navier–Stokes/Darcy equation is decoupled into two equations, one is the Navier–Stokes equation, the other is the Darcy equation, and the Navier–Stokes equation is solved by the modified characteristics finite element method. The theory analysis shows that this method has a good convergence property. In order to show the effect of our method, some numerical results was presented. The numerical results show that this method is highly efficient. Copyright © 2013 John Wiley & Sons, Ltd.     

Existence and zero‐electron‐mass limit of strong solutions to the stationary compressible Navier‐Stokes‐Poisson equation with large external force

In this study, we consider a viscous compressible model of plasma and semiconductors, which is expressed as a compressible Navier‐Stokes‐Poisson equation. We prove that there exists a strong solution to the boundary value problem of the steady compressible Navier‐Stokes‐Poisson equation with large external forces in bounded domain, provided that the ratio of the electron/ions mass is appropriately small. Moreover, the zero‐electron‐mass limit of the strong solutions is rigorously verified. The main idea in the proof is to split the original equation into 4 parts, a system of stationary incompressible Navier‐Stokes equations with large forces, a system of stationary compressible Navier‐Stokes equations with small forces, coupled with 2 Poisson equations. Based on the known results about linear incompressible Navier‐Stokes equation, linear compressible Navier‐Stokes, linear transport, and Poisson equations, we try to establish uniform in the ratio of the electron/ions mass a priori estimates. Further, using Schauder fixed point theorem, we can show the existence of a strong solution to the boundary value problem of the steady compressible Navier‐Stokes‐Poisson equation with large external forces. At the same time, from the uniform a priori estimates, we present the zero‐electron‐mass limit of the strong solutions, which converge to the solutions of the corresponding incompressible Navier‐Stokes‐Poisson equations.     

2D numerical simulation of submerged hydraulic jumps

Hydraulic jumps are usually used to dissipate energy in hydraulic engineering. In this paper, the turbulent submerged hydraulic jumps are simulated by solving the unsteady Reynolds averaged Navier–Stokes equations along with the continuity equation and the standard k–? equations for turbulence modeling. The Lagrangian moving grid method is employed for the simulation of the free surface. In the developed model, kinematic free-surface boundary condition is solved simultaneously with the momentum and continuity equations, so that the water elevation can be obtained along with velocity and pressure fields as part of the solution. Computational results are presented for Froude numbers ranging from 3.2 to 8.2 and submergence factors ranging from 0.24 to 0.85. Comparisons with experimental measurements show that numerical model can simulate the velocity field, variation of free surface, maximum velocity, Reynolds shear and normal stresses at various stations with reasonable accuracy.     

Global existence of weak solutions for the anisotropic compressible Stokes system

In this paper, we study the problem of global existence of weak solutions for the quasi-stationary compressible Stokes equations with an anisotropic viscous tensor. The key idea is a new identity that we obtain by comparing the limit of the equations of the energies associated to a sequence of weak-solutions with the energy equation associated to the system verified by the limit of the sequence of weak-solutions. In the context of stability of weak solutions, this allows us to construct a defect measure which is used to prove compactness for the density and therefore allowing us to identify the pressure in the limiting model. By doing so we avoid the use of the so-called effective flux. Using this new tool, we solve an open problem namely global existence of solutions à la Leray for such a system without assuming any restriction on the anisotropy amplitude. This provides a flexible and natural method to treat compressible quasilinear Stokes systems which are important for instance in biology, porous media, supra-conductivity or other applications in the low Reynolds number regime.     

Algorithm for solving the Navier–Stokes equations for the modeling of creeping flows

We study a coupled algorithm for solving the two-dimensional Navier–Stokes equations in the stream function–vorticity variables. The algorithm is based on a finite-difference scheme in which the inertial terms in the vortex transport equation are taken from the lower time layer and the dissipative terms, from the upper time layer. In the linear approximation, we study the stability of this algorithm and use test computations to show its advantages when modeling flows at moderate Reynolds numbers.     

A new more consistent Reynolds model for piezoviscous hydrodynamic lubrication problems in line contact devices

Hydrodynamic lubrication problems in piezoviscous regime are usually modeled by the classical Reynolds equation combined with a suitable law for the pressure dependence of viscosity. For the case of pressure–viscosity dependence in the Stokes equation, a new Reynolds equation in the thin film limit has been proposed by Rajagopal and Szeri. However, these authors consider some additional simplifications. In the present work, avoiding these simplifications and starting from a Stokes equation with pressure dependence of viscosity through Barus law, a new Reynolds model for line contact lubrication problems is deduced, in which the cavitation phenomenon is also taken into account. Thus, the new complete model consists of a nonlinear free boundary problem associated to the proposed new Reynolds equation.Moreover, the classical model, the one proposed by Rajagopal and Szeri and the here proposed one are simulated through the development of some numerical algorithms involving finite elements method, projected relaxation techniques, duality type numerical strategies and fixed point iteration techniques. Finally, several numerical tests are performed to carry out a comparative analysis among the different models.     

Lagrangian Description of Three-Dimensional Viscous Flows at Large Reynolds Numbers

Computational Mathematics and Mathematical Physics - Boundary layer theory is used to show that, at large Reynolds numbers, the three-dimensional Navier–Stokes equations can be rewritten in a...     

A High-Order Kernel-Free Boundary Integral Method for Incompressible Flow Equations in Two Space Dimensions

This paper presents a fourth-order kernel-free boundary integral method for the time-dependent, incompressible Stokes and Navier-Stokes equations defined on irregular bounded domains. By the stream function-vorticity formulation, the incompressible flow equations are interpreted as vorticity evolution equations. Time discretization methods for the evolution equations lead to a modified Helmholtz equation for the vorticity, or alternatively, a modified biharmonic equation for the stream function with two clamped boundary conditions. The resulting fourth-order elliptic boundary value problem is solved by a fourth-order kernel-free boundary integral method, with which integrals in the reformulated boundary integral equation are evaluated by solving corresponding equivalent interface problems, regardless of the exact expression of the involved Green's function. To solve the unsteady Stokes equations, a four-stage composite backward differential formula of the same order accuracy is employed for time integration. For the Navier-Stokes equations, a three-stage third-order semi-implicit Runge-Kutta method is utilized to guarantee the global numerical solution has at least third-order convergence rate. Numerical results for the unsteady Stokes equations and the Navier-Stokes equations are presented to validate efficiency and accuracy of the proposed method.     

Elrod–Adams cavitation model for a new nonlinear Reynolds equation in piezoviscous hydrodynamic lubrication

In previous studies, different cavitation models have been incorporated into the classical Reynolds equation in piezoviscous regimes. The advantages of the Elrod–Adams cavitation model compared with the Reynolds model have been demonstrated in this classical framework. Recently, a new nonlinear Reynolds equation was rigorously justified [15] for lubricated line contact problems by introducing the piezoviscous Barus law into the departure Navier–Stokes equations before passing to the thin film limit. In addition, the corresponding nonlinear first order ordinary differential equation (ODE) has been proposed.In the present study, we incorporate the Elrod–Adams model for cavitation and we pose the free boundary problem associated with the nonlinear first order ODE, which involves a multivalued Heaviside operator for the relationship between the lubricant pressure and saturation. After analyzing the qualitative properties of the solution, we propose suitable numerical techniques for solving the problem as well as obtaining the lubricant pressure, saturation, and viscosity. Finally, we give some numerical results to illustrate the performance of the proposed numerical methods as well as comparisons with alternative models.     

Finite element error analysis of a variational multiscale method for the Navier-Stokes equations

The paper presents finite element error estimates of a variational multiscale method (VMS) for the incompressible Navier–Stokes equations. The constants in these estimates do not depend on the Reynolds number but on a reduced Reynolds number or on the mesh size of a coarse mesh. This work is partially supported by NSF grants DMS9972622, DMS20207627 and INT9814115.     

On a free boundary problem for the Reynolds equation derived from the Stokes system with Tresca boundary conditions

The asymptotic behaviour of a Stokes flow with Tresca free boundary friction conditions when one dimension of the fluid domain tends to zero is studied. A specific Reynolds equation associated with variational inequalities is obtained and uniqueness is proved.     

A stochastic Lagrangian proof of global existence of the Navier–Stokes equations for flows with small Reynolds number

We consider the incompressible Navier–Stokes equations with spatially periodic boundary conditions. If the Reynolds number is small enough we provide an elementary short proof of the existence of global in time Hölder continuous solutions. Our proof uses a stochastic representation formula to obtain a decay estimate for heat flows in Hölder spaces, and a stochastic Lagrangian formulation of the Navier–Stokes equations.     

求解定常不可压Stokes方程的两层罚函数方法

借助于两套有限元网格空间提出了一种求解定常不可压Stokes方程的两层罚函数方法.该方法只需要求解粗网格空间上的Stokes方程和细网格空间上的两个易于求解的罚参数方程（离散后的线性方程组具有相同的对称正定系数矩阵）.收敛性分析表明粗网格空间相对于细网格空间可以选择很小，并且罚参数的选取只与粗网格步长和问题的正则性有关.因此罚参数不必选择很小仍能够得到最优解.最后通过数值算例验证了上述理论结果，并且数值对比可知两层罚函数方法对于求解定常不可压Stokes方程具有很好的效果.     

Study of free convective boundary layer of isothermal lateral surface of axisymmetrical horizontal body

Approximate analytical solution of simplified Navier–Stokes and Fourier–Kirchhoff equations describing free convective heat transfer from isothermal surface has been presented. It is supposed that the surface has the horizontal axis of symmetry and its axial cross-section lateral boundary is a concave function. The equation for the boundary layer thickness is derived for typical for natural convection assumptions. The most important are that the convective fluid flow is stationary and the normal to the surface component of velocity is negligibly small in comparison with the tangential one. The theoretical results are verified by two characteristic cases of the revolution surfaces namely for horizontal conic and vertical round plate. Both limits of presented solution coincide with known formulas.