Evolution of the joint motion of two viscous heat-conducting fluids in a plane layer under the action of an unsteady pressure gradient

A study is made of an invariant solution of the equations of a viscous heat-conducting fluid, which is treated as unidirectional motion of two such fluids in a plane layer with a common boundary under the action of an unsteady pressure gradient. A priori estimates of the velocity and temperature are obtained. The steady state is determined, and it is shown (under some conditions on the pressure gradient) that, at larger times, this state is the limiting one. For semiinfinite layers, a solution in closed form is obtained using the Laplace transform. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 4, pp. 94–107, July–August, 2008.     

On the non-persistence of irrotational motion in a viscous heat-conducting fluid

We consider the possibility of irrotational flow in a fluid exterior to a moving rigid obstacle, or interior to a moving rigid shell. Observations show that when a rigid body is impulsively set into motion an irrotational flow may exist initially but does not persist. The breakup of this irrotational flow and the associated phenomenon of generation of vorticity at the wall are generally attributed to the condition of adherence at the fluid-solid interface. Since this condition itself is derived from observation, one can ask whether there is another explanation for the phenomenon. The purpose of this paper is to show that a persistent irrotational flow is incompatible with the second law of thermodynamics.     

Spatial motion of interface of two heavy viscous fluids in a porous medium

The integrodifferential equation for the spatial motion of the interface of two fluids of differing weights and differing viscosities in a porous medium is presented. Results of its numerical solution are given. The general solution is given for the problem of the dynamics of a gently sloping surface.The author wishes to thank V. L. Danilov for his interest in the study and valuable advice.     

Nonstationary motion of a viscous fluid in a long tube

A number of problems are solved for the nonstationary motion of a viscous compressible fluid in a tube with elastic walls. It is assumed that the tube is semi-infinite, its axis horizontal, and that at one of its ends the flow rate of the fluid can change. The solution of each of the problems is reduced to the finding a generalized solution to a nonlinear system of partial differential equations for two functions — the mean values of the velocity and pressure in the tube section — with certain constant or null initial conditions and with a boundary condition specifying the time dependence of some function of the velocity and the pressure at the end of the tube. It is noted that the same problems can be solved by successive approximation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 35–43, November–December, 1980.     

Vortex flows of a viscous heat-conducting gas in a slightly divergent tube with volume energy supply

The results of a numerical investigation of viscous vortex flow in a slightly divergent tube with thermal energy supplied to the flow are presented. The initial stage of vortex flow development is considered for two different longitudinal velocity distributions simulating the velocity profiles in jet-like and wake-like vortex flows in the vicinity of the vortex axis. The first type of flow can be considered as a model for the near-axis region of the vortex formed in the flow around a delta wing at incidence. The second type can serve as a model for the near-axis region of the trailing vortex downstream of a high-aspect-ratio wing. The development of the two flows is studied for a constant area tube, a slightly divergent tube, and in the case of thermal energy supply from a volume energy source at a constant wall temperature.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 90–97, September–October, 1996.     

Motion of two viscous fluids in a precessing vessel

An approximate solution of the problem of the motion of two viscous fluids in a cylindrical vessel executing slow regular precession with an arbitrary angle of nutation is constructed. The axial component of the moment of the forces exerted by the fluid on the lateral surface of the vessel is determined.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 27–34, September–October, 1995.     

Three-dimensional motion of a viscous incompressible fluid in a narrow tube

An approximate solution of the problem of unsteady motion of a viscous incompressible fluid in a long narrow deformable tube at low Reynolds numbers is obtained. Pressure oscillations and tube deformation are shown to be related by an integrodifferential equation. The solution obtained extends the Poiseuille solution in elliptic tubes to the case of comparatively arbitrary small deformations in terms of the tube length and angle. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 4, pp. 28–32, July–August, 2009.     

Unsteady motion of viscous fluid in an expanding tube

The problem of unsteady motion of a viscous compressible fluid in a semiinfinite tube with horizontal axis is solved by successive approximation. The circular cross section of the tube depends exponentially on the coordinate measured along the tube axis. At the end of the tube there is a unit such as a sliding valve, compressor, reciprocating pump, or turbine that changes the flow rate. The process is assumed to be barotropic.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 79–82, May–June, 1983.     

Formation of one-dimensional flows of a heat-conducting viscous gas

The article discusses the development of one-dimensional flows in a viscous heat-conducting gas using the example of two flows: 1) the flow arising with the decomposition of a discontinuity of the pressure in the quiescent gas (flow in a shock tube); 2) the flow arising with the application of a constant heat flow at a gassolid interface. For such flows, there has been very little study of the initial stage of the process, right up to the time when nonheat-conducting zones are separated out, described by the Euler equations, as well as dissipation zones of the type of a shock wave or a boundary layer, which can be treated using asymptotic methods [1–3]. With the investigation of the initial stage, the complete solution of the system of Navier—Stokes equations is required. The present article discusses the initial stage of the flows on the basis of a numerical solution of problems 1 and 2. A study is made of the effect of the Prandtl number and of the viscosity coefficient on the behavior of the gas.     

Unsteady motion of a viscous incompressible fluid in a tube with a deformable wall

A flow of a viscous incompressible fluid in a deformable tube is considered. Solutions of unsteady three-dimensional Navier-Stokes equations are obtained for low-Reynolds-number flows in the tube (under the condition of small deformations of the wall): generalized peristaltic flow and flow with elliptical deformations of the vessel walls. At small unsteady deformations of the tube walls, the solutions satisfy the equations and boundary conditions with an error smaller than the tube wall deformation level by an order of magnitude. In the case of elliptical deformations of the vessel, the solution agrees well with experimental data.     

Numerical modeling of impulsive jets of a viscous heat-conducting gas

Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, No. 4, pp. 93–99, July–August, 1992.     

Flow of a viscous heat-conducting gas and binary mixtures in a sink

The class of exact solutions of the one-dimensional Navier-Stokes equations corresponding to gas flows from a spherical source or sink has been investigated analytically and numerically on a number of occasions (see, for example, [1, 2]). Here, the solution for a sink is considered in the presence of heat transfer from the ambient medium. Apart from seeking the solution itself, the object of the investigation was to establish the conditions of transi tion from viscous to inviscid flow in the sink as the Reynolds number tends to infinity. As shown in [3], for zero heat flux at an infinitely remote point there is no such transition for flow in a sink. The sink flow characteristics of a binary gas mixture are investigated in detail. In the transonic flow region an asymptotic solution is obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 56–62, January–February, 1989.     

Variational principle for a viscous heat-conducting fluid with relaxation

In continuum mechanics a large number [1] of variational principles are known, but only some of them can be used for continua with energy dissipation and heat conduction. A classical example is the Helmholtz minimum energy dissipation principle for creeping motion. It is well known [1] that there is no holonomic variational principle from which the Navier-Stokes equations follow. Nevertheless, the local potential method [2], which represents a certain variational approach to obtaining these equations, has been developed. The disadvantages of this method include the dependence of the functional on the varied variables satisfying the Navier-Stokes equation, which, essentially, must also be obtained by minimizing the functional. Accordingly, the local potential method requires the use of a certain convergent iterative procedure that minimizes the functional. In this article an alternative approach is considered. The variational principle is holonomic; therefore its extremals are not solutions of the Navier-Stokes equations. However, it is possible to construct the unknown solutions by imposing simple constraints on the extremals obtained. This approach requires an extension of the space on which the varied variables are defined.     

The role of angular momentum in the laminar motion of viscous fluids

In laminar flow, viscous fluids must exert appropriate elastic shear stresses normal to the flow direction. This is a direct consequence of the balance of angular momentum. There is a limit, however, to the maximum elastic shear stress that a fluid can exert. This is the ultimate shear stress, \(\tau _\mathrm{y}\), of the fluid. If this limit is exceeded, laminar flow becomes dynamically incompatible. The ultimate shear stress of a fluid can be determined from experiments on plane Couette flow. For water at \(20\,^{\circ }\hbox {C}\), the data available in the literature indicate a value of \(\tau _\mathrm{y}\) of about \(14.4\times 10^{-3}\, \hbox {Pa}\). This study applies this value to determine the Reynolds numbers at which flowing water reaches its ultimate shear stress in the case of Taylor–Couette flow and circular pipe flow. The Reynolds numbers thus obtained turn out to be reasonably close to those corresponding to the onset of turbulence in the considered flows. This suggests a connection between the limit to laminar flow, on the one hand, and the occurrence of turbulence, on the other.     

Hyperbolic system for viscous fluids and simulation of shock tube flows

A system of field equations for viscous fluids with heat conduction is formulated. Unlike the theory of Navier-Stokes fluids, a hyperbolic system of field equations in conservative form is obtained. Numerical simulation of a viscous shock tube flow is presented as an application.