Regular, partially invariant solutions of rank 1 and defect 1 of equations of plane motion of a viscous heat-conducting gas

A system of the Navier-Stokes equations of two-dimensional motion of a viscous heat-conducting perfect gas with a polytropic equation of state is considered. Regular, partially invariant solutions of rank 1 and defect 1 are studied. A sufficient condition of their reducibility to invariant solutions of rank 1 is proved. All solutions of this class with a linear dependence of the velocity-vector components on spatial coordinates are examined. New examples of solutions that are not reducible to invariant solutions are obtained. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 6, pp. 23–33, November–December, 2006.     

Joint unidirectional motion of two viscous heat-conducting fluids in a tube

This paper studies an invariant solution of the problem of joint motion of two heat-conducting viscous immiscible fluids which have a common interface in a cylindrical tube under an unsteady pressure gradient. The problem reduces to a coupled initial-boundary-value problem for parabolic equations. A priori estimates of velocity and temperature perturbations are obtained. The steady state of the system is determined, and it is proved that if, in one of the fluids, the pressure gradient rapidly approaches zero, the perturbations of all quantities tend to zero. It is shown that if the pressure gradient has a nonzero limit, the solution reaches a steady state. In this case, the velocity field in the limit is the same as in conjugate Poiseuille flow, and the temperature is represented as a polynomial of the fourth order on the radial coordinate.     

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.     

On the unsteady unidirectional flows generated by impulsive motion of a boundary or sudden application of a pressure gradient

Some exact solutions of the time-dependent Navier-Stokes equations are discussed for flows impulsively started from rest by the motion of a boundary or two boundaries or by sudden application of a pressure gradient. It is shown that the expressions of the quantities such as velocity, flux and skin friction obtained in a series form for large times can also be used to find their values for small times or vice versa. Furthermore, conditions for which the values of the quantities at large times are nearly the same with those at small times are derived.     

Controlling the disturbances in a hypersonic flat-plate shock layer by unsteady action from the surface

The results of a wind-tunnel experiment on the joint action of periodic acoustic fast-mode disturbances of the outer flow and disturbances generated at the leading edge of a plate on the hypersonic (M_∞ = 21) viscous shock layer on the plate are presented. The possibility of positively controlling the intensity of density fluctuations in the plate shock layer by means of disturbances introduced from the leading edge is shown. Direct numerical simulation of the suppression (enhancement) of disturbances under the simultaneous action on the shock layer of the two-dimensional fast-mode acoustic waves in the outer flow and the source of two-dimensional suction/injection disturbances near the leading edge of the plate is performed under the experimental conditions. The experimental and calculated results are shown to be in good agreement.     

Evolution equation for unsteady shock waves of moderate intensity in a viscous heat-conducting gas

A simple model equation that takes into account the nonisentropicity of the flow is. obtained from the equations of a viscous heat-conducting gas. It differs from the Burgers equation in possessing an additional term with a clear physical significance. This equation is suitable for one-dimensional traveling waves on the Mach number interval 1M1.3. The equation obtained gives the asymptotic laws of damping of weak shock waves correct to small terms of the leading and next order for plane [2], cylindrical [3] and spherical [4] symmetry.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 187–190, September–October, 1989.     

Numerical study of the axially symmetric motion of an incompressible viscous fluid in an annulus between two concentric rotating spheres

The research reported herein involved the study of the transient motion of a system consisting of an incompressible Newtonian fluid in an annulus between two concentric, rotating, rigid spheres. The primary purpose of the research was to study the use of a numerical method for analysing the transient motion that results from the interaction between the fluid in the annulus and the spheres which are started suddenly by the action of prescribed torques. The problems considered in this research included cases where: (a) one or both spheres rotate with prescribed constant angular velocities and (b) one sphere rotates due to the action of an applied constant or impulsive t?orque. In this research the coupled solid and fluid equations were solved numerically by employing the finite difference technique. With the approach adopted in this research, only the derivatives with respect to spatial variables were approximated with the use of the finite difference formulae. The steady state problem was also solved as a separate problem (for verification purposes), and the results were compared with those obtained from the solution of the transient problem. Newton's algorithm was employed to solve the algebraic equations which resulted from the steady state problem, and the Adams fourth-order predictor–corrector method was employed to solve the ordinary differential equations for the transient problem. Results were obtained for the streamfunction, circumferential function, angular velocity of the spheres and viscous torques acting on the spheres as a function of time for various values of the system dimensionless parameters.     

Evolution of long nonlinear waves on the interface of a stratified viscous fluid flow in a channel

The dynamics of disturbances of the interface between two layers of incompressible immiscible fluids of different densities in the presence of a steady flow between the horizontal bottom and lid is studied analytically and numerically. A model integrodifferential equation is derived, which takes into account long-wave contributions of inertial layers and surface tension of the fluids, small but finite amplitude of disturbances, and unsteady shear stresses on all boundaries. Numerical solutions of this equation are given for the most typical nonlinear problems of transformation of both plane waves of different lengths and solitary waves. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 4, pp. 49–61, July–August, 2007.     

The linear stability of plane Poiseuille flow under unsteady distortion

This paper investigates the linear stability behaviour of plane Poiseuille flow underunsteady distortion by multiscale perturbation method and discusses further the problemproposed by paper[1].The results show that in the initial period of disturbancedevelopment,the distortion profiles presented by paper[1]will make the disturbances growup,thus augmenting the possibility of instability.     

Variational principles and generalized variational principles in hydrodynamics of viscous fluids

In this paper, the variational principles of hydrodynamic problems for the incompressible and compressible viscous fluids are established. These principles are principles of maximum power losses. Their generalized variational principles are also discussed on the basis of Lagrangian multiplier methods.     

Evolution of the free surface of a plane channel during filling with a viscous fluid

The behavior of the free surface of a viscous incompressible fluid is studied in the process of filling of a plane channel inclined to the gravity direction. The problem is numerically solved within the framework of the Navier-Stokes equations under the assumption that the fluid is Newtonian and capillary effects can be neglected. The numerical technique is based on the joint use of the SIMPLE and invariant methods. Two filling regimes are found to exist, namely, that with a steady shape of the free surface without the formation of internal cavities and that with cavity formation in the initial filling stage characterized by spouting behavior of the free boundary.     

Stability of the equilibrium state in a convection model with nonlinear temperature and pressure dependences of density

The convection of a heat-conducting viscous liquid is considered. It is assumed that the liquid density depends quadratically on the temperature and pressure. The instability of the equilibrium state of a free-boundary horizontal layer with respect to small perturbations is studied using a linearization method. It is found that the state of mechanical equilibrium is unstable. Neutral curves are constructed and the critical Rayleigh numbers are found. The results are compared with the well-known solution of the same problem for the limiting case where the density is a quadratic function of temperature and does not depend on pressure. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 2, pp. 66–74, March–April, 2007.     

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.     

On the flows produced by sudden application of a constant pressure gradient or by impulsive motion of a boundary

Some properties of the time-dependent Navier-Stokes equations are discussed for flows impulsively started from rest by sudden application of a constant pressure gradient or by the impulsive motion of a boundary. Five illustrative examples are given. They are: unsteady flow in a circular cylinder moving parallel to its length, starting flow in a circular pipe, unsteady flow in a rotating cylinder, starting flow in a rectangular channel moving parallel to its length and unsteady flow in a channel of rectangular cross-section. It is found that the expressions of the quantities such as velocity, flux and skin friction are in series forms which may be rapidly convergent for large values of the time but slowly convergent for small values of the time or vice versa. It is shown that if their expressions can be found for one of large values of the time or small values of the time, these expressions can be used for the other.     

Slip effects and heat transfer analysis in a viscous fluid over an oscillatory stretching surface

In this study, we investigate the heat transfer problem in a viscous fluid over an oscillatory infinite sheet with slip condition. The sheet is moved back and forth in its own plane. The derived problem involves a dimensionless parameter indicating the relative magnitude of frequency to sheet stretching rate. A system of non‐linear partial differential equations is solved numerically using the finite‐difference scheme, in which a coordinate transformation is employed to transform the semi‐infinite physical space to a bounded computational domain. The physical features of interesting parameters on the velocity and temperature distributions are shown graphically and discussed. The values of the skin‐friction coefficient and the local Nusselt number are given in tabular form. Copyright © 2008 John Wiley & Sons, Ltd.     

Bifurcation of a Main Steady-State Viscous Fluid Flow in a Plane Divergent Channel

The evolution of steady-state viscous incompressible fluid flows in a plane divergent channel is investigated. For the classical formulation of the Jeffery-Hamel problem a complete solution is given as a function of the determining parameters. For a fixed angle of divergence the behavior of the main unimodal flow is determined as a function of the Reynolds number. Critical values at which the flow pattern bifurcates and the steady-state unimodal flow ceases to exist are found. The mechanism of bifurcation is established and its diagram is constructed. This mechanism and the diagram were not previously known in the scientific literature in connection with the investigation of the Jeffery-Hamel problem. The critical Reynolds number at which bifurcation occurs is given as a function of the channel divergence angle. The results may be of interest for hydromechanical, technological, and geophysical applications.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, 2005, pp. 25–36.Original Russian Text Copyright © 2005 by Akulenko and Kumakshev.     

Optimum shape of a cavity for the collection of a soil-saturating viscous fluid

The process of inflow of an oil-bearing fluid from an infinite region of porous soil into a cavity (reservoir) and its subsequent pumping into a well located at the center of the reservoir is investigated. The cavity is chosen from the class of flattened ellipsoids of revolution. Thus, a combined problem of the optimization of the shape of a cavity of given volume for which the maximum seepage flow is achieved and the approximate determination of the additional constraints on the cavity dimensions necessary to ensure the outflow of the oil-bearing fluid is considered.     

Unsteady boundary layer flow due to a stretching surface in a rotating fluid

The induced unsteady flow due to a stretching surface in a rotating fluid, where the unsteadiness is caused by the suddenly stretched surface is studied in this paper. After a similarity transformation, the unsteady Navier–Stokes equations have been solved numerically using the Keller-box method. Also, the perturbation solution for small times as well as the asymptotic solution for large times, when the flow becomes steady, has been obtained. It is found that there is a smooth transition from the small time solution to the large time or steady state solution.     

Simulation of interface and free surface flows in a viscous fluid using adapting quadtree grids

A new adaptive quadtree method for simulating laminar viscous fluid problems with free surfaces and interfaces is presented in this paper. The Navier–Stokes equations are solved with a SIMPLE‐type scheme coupled with the Compressive Interface Capturing Scheme for Arbitrary Meshes (CICSAM) (Numerical prediction of two fluid systems with sharp interfaces, Ph.D. Thesis, Imperial College of Science, Technology and Medicine, London, 1997) volume of fluid (VoF) method and PLIC reconstruction of the volume fraction field during refinement and derefinement processes. The method is demonstrated for interface advection cases in translating and shearing flow fields and found to provide high interface resolution at low computational cost. The new method is also applied to simulation of the collapse of a water column and the results are in excellent agreement with other published data. The quadtree grids adapt to follow the movement of the free surface, whilst maintaining a band of the smallest cells surrounding the surface. The calculation is made on uniform and adapting quadtree grids and the accuracy of the quadtree calculation is shown to be the same as that made on the equivalent uniform grid. Copyright © 2004 John Wiley & Sons, Ltd.