Viscous incompressible flow past a finite plate

We consider the problem of viscous incompressible flow past a finite plate. The apparatus of hydrodynamic potential reduces the problem to Volterra— Fredholm boundary-value equations. Numerical results are reported.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 64, pp. 103–107, 1988.     

Predominantly Unidirectional Flow of a Viscous Fluid

Journal of Applied and Industrial Mathematics - We consider the problem of the flow of a viscous fluid in the presence of solid bodies (namely, the two walls and a permeable plate) under...     

Viscous Shock Motion for Advection-Diffusion Equations

This paper studies the asymptotic solution of the initial-boundary value problem for scalar convection-dominated evolution equations on a bounded spatial domain when initial and boundary conditions are such that the solution develops a single thin shock layer of steep change. The exponentially slow motion of the shock is determined for exponentially long times using an ansatz based on the solution for the special case of Burgers' equation, obtained through the Cole-Hopf transformation. Results obtained analytically are confirmed by numerical experiments.     

Viscous incompressible free-surface flow down an inclined perturbed plane

The plane stationary free boundary value problem for the Navier-Stokes equations is studied. This problem models the viscous fluid free-surface flow down a perturbed inclined plane. For sufficiently small data the solvability and uniqueness results are proved in Hölder spaces. The asymptotic behavior of the solution is investigated.     

Monotone Mappings and Flows of Viscous Media

We establish solvability of the boundary-value problems describing stationary and periodic flows of viscoplastic media. In the case of stationary flows we study the question of convergence of the Galerkin method. For the problem of periodic flows we prove a version of the second Bogolyubov theorem.     

Global Existence of Heat-Conductive Incompressible Viscous Fluids

In this paper, we consider the Cauchy problem of non-stationary motion of heat-conducting incompressible viscous fluids in \(\mathbb{R}^{2}\), where the viscosity and heat-conductivity coefficient vary with the temperature. It is shown that the Cauchy problem has a unique global-in-time strong solution \((u, \theta)(x,t)\) on \(\mathbb{R}^{2}\times(0,\infty)\), provided the initial norm \(\|\nabla u_{0}\|_{L^{2}}\) is suitably small, or the lower-bound of the coefficient of heat conductivity (i.e. \(\underline{\kappa}\)) is large enough, or the derivative of viscosity (i.e. \(|\mu'(\theta)|\)) is small enough.     

Finite-amplitude Rotational Waves in Viscous Shear Flows

Growing finite-amplitude initially spanwise-independent two-dimensional rotational waves and their nonlinear interaction with unidirectional viscous shear flows of various strengths are considered. Both primary and secondary instabilities are studied, but only secondary instabilities are permitted to vary in the spanwise direction. A generalized Lagrangian-mean formulation is employed to describe wave-mean interactions, and a separate theory is constructed to account for the back effect of the developing mean flow on the wave field. Viscosity is seen to significantly complicate calculation of the back effect. The primary instability is seen to act as a platform for, and catalyst to, secondary instabilities. The analysis leads to an eigenvalue problem for the initial growth of the secondary instability, this being a generalization of the eigenvalue problem constructed by Craik for inviscid neutral waves. Two inviscid secondary instability mechanisms to longitudinal vortex form are observed: the first has as its basis the Craik–Leibovich type 2 mechanism. The second, which is as yet unproven, requires that both the wave and flow field distort in concert at all levels of shear. Both mechanisms excite exponential growth on a convective rather than diffusive scale in the presence of neutral waves, but growing waves alter that growth rate.     

Blow-Up Solution of the 3D Viscous Incompressible MHD System

The 3D viscous incompressible magneto-hydrodynamic (MHD) system comprised by the 3D incompressible viscous Navier-Stokes equation couples with Maxwell equation. The global well-posedness of the coupled system is still an open problem. In this paper, we study the Cauchy problem of this coupled system and establish some logarithmical type of blow-up criterion for smooth solution in Lorentz spaces.     

On the Rayleigh-Taylor Instability for Two Uniform Viscous Incompressible Flows

The authors study the Rayleigh-Taylor instability for two incompressible immis- cible fluids with or without surface tension, evolving with a free interface in the presence of a uniform gravitational field in Eulerian coordinates. To deal with the free surface, instead of using the transformation to Lagrangian coordinates, the perturbed equations in Eule- rian coordinates are transformed to an integral form and the two-fluid flow is formulated as a single-fluid flow in a fixed domain, thus offering an alternative approach to deal with the jump conditions at the free interface. First, the linearized problem around the steady state which describes a denser immiscible fluid lying above a light one with a free interface separating the two fluids, both fluids being in （unstable） equilibrium is analyzed. By a general method of studying a family of modes, the smooth （when restricted to each fluid domain） solutions to the linearized problem that grow exponentially fast in time in Sobolev spaces are constructed, thus leading to a global instability result for the linearized problem. Then, by using these pathological solutions, the global instability for the corresponding nonlinear problem in an appropriate sense is demonstrated.     

Viscous limits for piecewise smooth solutions of the p-system

We study the zero-dissipation problem for a one-dimensional model system for the isentropic flow of a compressible viscous gas, the so-called p-system with viscosity. When the solution of the inviscid problem is piecewise smooth and having finitely many noninteracting shocks satisfying the entropy condition, there exists unique solution to the viscous problem which converges to the given inviscid solution away from shock discontinuities at a rate of order ε as the viscosity coefficient ε goes to zero. The proof is given by a matched asymptotic analysis and an elementary energy method. And we do not need the smallness condition on the shock strength.     

Viscous two‐fluid flows in perturbed unbounded domains

Two stationary plane free boundary value problems for the Navier‐Stokes equations are studied. The first problem models the viscous two‐fluid flow down a perturbed or slightly distorted inclined plane. The second one describes the viscous two‐fluid flow in a perturbed or slightly distorted channel. For sufficiently small data and under certain conditions on parameters the solvability and uniqueness results are proved for both problems. The asymptotic behaviour of the solutions is investigated. For the second problem an example of nonuniqueness is constructed. Computational results of flow problems that are very close to the above problems are presented. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Viscous and inviscid magneto-hydrodynamics equations

In this paper we study the classical problem in turbulence for the magneto-hydrodynamics (MHD) equations: whether the solutions (u ^(v),B ^(v)) of the viscous MHD equations tend to the solutions (u ⁽⁰⁾,B ^(v)) of the inviscid MHD equations as the Reynolds numbersRe, Rm → ∞. As a preparation we first derive bounds for ||(u ⁽⁰⁾,B ⁽⁰⁾(t)||H ^m) (m ≥3) in terms of deformation tensor related quantities (0.1) {ie251-1} We then show that asRe → ∞ andRm → ∞, the difference {ie-251-2} {ie-251-3} converges to zero uniformly int as long as the quantities in (0.1) remain finite. The convergence rates are explicit. Supported by the NSF grant DMS 9304580 at IAS.     

Regularity of Viscous Solutions for a Degenerate Non-linear Cauchy Problem

We consider the Cauchy problem for a class of nonlinear degenerate parabolic equation with forcing. By using the vanishing viscosity method it is possible to construct a generalized solution. Moreover, this solution is a Lipschitz function on the spatial variable and Hölder continuous with exponent 1/2 on the temporal variable.     

Stochastic Representation of Weak Solutions of Viscous Conservation Laws: A BSDE Approach

We consider the Cauchy problem for systems of viscous conservation laws. We obtain three different but related stochastic representations of weak solutions of the problem: in terms of solutions to systems of usual backward stochastic differential equations, in terms of solutions to some stochastic backward systems, and in terms of solutions to some forward-backward stochastic differential equations.     

Optimal Boundary Control of Steady-State Flow of a Viscous Inhomogeneous Incompressible Fluid

We study the problem of optimal boundary control of two-dimensional steady-state flow of a viscous inhomogeneous incompressible fluid. The role of control is played by the values of the velocity on a part of the boundary of the domain considered. On the remaining part of the boundary, the vector of flow velocity and the fluid density are given. We seek the fluid density as a scalar function (determined by the initial data) of the stream function, study the solvability of the problem, and obtain necessary optimality conditions.     

Solvability of Control Problems for Stationary Equations of Magnetohydrodynamics of a Viscous Fluid

We consider the boundary-value problem for the stationary equations of magnetohydrodynamics of a viscous incompressible fluid with nonhomogeneous boundary conditions for the velocity and electromagnetic field. We study global solvability of this problem and establish some sufficient conditions for uniqueness of its solution. We state control problems for the model of magnetohydrodynamics under consideration, study their solvability, give and examine optimality systems for both arbitrary and particular quality functionals.     

Computing Viscous Flow in an Elastic Tube

We have developed a numerical method for simulating viscous flow through a compliant closed tube, driven by a pair of fluid source and sink. As is natural for tubular flow simulations, the problem is formulated in axisymmetric cylindrical coordinates, with fluid flow described by the Navier-Stokes equations. Because the tubular walls are assumed to be elastic, when stretched or compressed they exert forces on the fluid. Since these forces are singularly supported along the boundaries, the fluid velocity and pressure fields become unsmooth. To accurately compute the solution, we use the velocity decomposition approach, according to which pressure and velocity are decomposed into a singular part and a remainder part. The singular part satisfies the Stokes equations with singular boundary forces. Because the Stokes solution is unsmooth, it is computed to second-order accuracy using the immersed interface method, which incorporates known jump discontinuities in the solution and derivatives into the finite difference stencils. The remainder part, which satisfies the Navier-Stokes equations with a continuous body force, is regular. The equations describing the remainder part are discretized in time using the semi-Lagrangian approach, and then solved using a pressure-free projection method. Numerical results indicate that the computed overall solution is second-order accurate in space, and the velocity is second-order accurate in time.     

Local Artificial Boundary Conditions for the Incompressible Viscous Flow in a Slip Channel

1.IntroductionManyboundaxyvaJueproblemsofpartialdiffereotialequationsinvo1vingunboundeddomainoccurinmanyareasofapplications,e-g.lfluidflowaroundobstacles,couplingofstructureswithfoundationandsoon.Forgettingthenumericalsolutionsoftheproblemsonunboundeddomian,anaturalapproachistocutoffanunboundedpartofthedomainbyintroducinganartificialboundaryandsetupanaPpropriatear-tificialboundaryconditiononthearti%ialboundaryThentheoriginalproblemisapproximatedbyaproblemonbou.d.dfdomain.Inthelastteny6aJrs,b…     

A Theory of Exact Solutions for Annular Viscous Blobs

Summary. A new theory of exact solutions is presented for the problem of the slow viscous Stokes flow of a plane, doubly connected annular viscous blob driven by surface tension. The formulation reveals the existence of an infinite number of conserved quantities associated with the flow for a certain general class of initial conditions. These conserved quantities are associated with a class of exact solutions. This work is believed to provide the first exact solutions for the evolution of a doubly connected fluid region evolving under Stokes flow with surface tension. Received December 19, 1996; revised September 22, 1997, and accepted October 13, 1997     

一个与硅氧化有关的粘性可压缩流体的自由边界问题

本文研究一个描述硅的氧化过程的自由边界问题．它的数学模型是一个可压缩的Ｎａｖｉｏｒ－Ｓｔｏｋｅｓ方程与一个抛物方程以及一个双曲方程的耦合，其中在自由边界上存在表面张力并且密度方程是非齐次的．本文将证明只要已知数据满足相容性条件，则上述问题有唯一局部强解．