Stability of two-layer fluid flows with evaporation at the interface

The problem of stability of two-layer (fluid-gas) flows with account of evaporation at the thermocapillary interface is studied under the condition of a fixed gas flow rate. In the upper gas-vapor layer, the Dufour effect is taken into account. A novel exact solution of the Navier–Stokes equations in the Boussinesq approximation is constructed. The effects of longitudinal temperature gradients, gravity, thicknesses of the gas and fluid layers, and the gas flow rate on the flow structure, the onset of recirculated flows near the interface, the evaporation rate, and the properties of characteristic disturbances are investigated.     

Instability of the interface in two-layer flows with large viscosity contrast at small Reynolds numbers

The Kelvin–Helmholtz instability is believed to be the dominant instability mechanism for free shear flows at large Reynolds numbers. At small Reynolds numbers, a new instability mode is identified when the temporal instability of parallel viscous two fluid mixing layers is extended to current-fluid mud systems by considering a composite error function velocity profile. The new mode is caused by the large viscosity difference between the two fluids. This interfacial mode exists when the fluid mud boundary layer is sufficiently thin. Its performance is different from that of the Kelvin–Helmholtz mode. This mode has not yet been reported for interface instability problems with large viscosity contrasts.These results are essential for further stability analysis of flows relevant to the breaking up of this type of interface.     

Dynamic problem for a two-layer medium with a vibrational source at the boundary interface

The present study is concerned with an analysis of gravitational and acoustic waves which are excited by a vibrational source deeply placed in a liquid covered by ice. An analysis of the rigidity characteristics of ice modeled by an elastic layer or by a Kirchhoff plate is done by factorization of the solution to the integral equation equivalent to an initially combined boundary value problem. The uncombined boundary condition is used to solve problems for unrestricted ice fields in [1–3], whereas combined conditions with vibrational sources positioned at the boundary of the medium are used in [4].Translated from Zhurnal Prikladnoi Mekhaniki, No. 3, pp. 125–129, May–June, 1986.     

Example of an exact solution of the problem of the distribution of an ionized impurity in the surface region of a semiconductor

An exact solution of the nonlinear problem of determining the electrostatic potential and the distribution profile of an ionized impurity in the surface region of a semiconductor is constructed. Altai State University, Barnaul 656099. Institute of Semiconductor Physics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 4, pp. 17–24, July–August, 1998.     

Example of exact solution of a problem of gas compression by a spherical shell

A study is made of the problem of isentropic compression of gas by a spherical shell of finite thickness on the exterior of which there is a vacuum. The complete solution to the problem with different boundary conditions and different equations of state for the shell and the compressible medium is possible only numerically. However, there exists a class of exact solutions to the equations of gas dynamics [1, 2] with linear radial distribution of the velocities of the particles in which contact discontinuities are allowed. For this it is necessary that both the shell and the compressible medium be described by the same equation of state p = ( – 1) E with the same specific heat ratio = c_p/c_v. There can be arbitrarily many such discontinuities in the solution, i.e., this class of solutions can describe the compression of matter by multilayer shells. In the present paper, a restriction is made to a single-layer shell with specific heat ratio = 5/3.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 176–179, November–December, 1982.     

Optimization of two-phase flows using the solution of an inverse problem

The results of optimizing two- phase flows on the basis of the solution of an inverse problem with the pressure distribution given by a two-parameter function are presented. The efficiency of the developed approach is illustrated with reference to nozzle and ejector flows with large liquid phase flow rates (the liquid droplet flow rate being greater than that of the gas by a factor of several tens).     

从Stokes第一问题的精确解谈起

以Stokes第一问题为例,阐述在Navier-Stokes方程精确解的教学过程中,如何引导学生从不同数学方法的角度展开积极思维. 研究发现,除了在国内外流体力学教材中普遍使用的量纲分析法,还可以使用分离变量法、拉普拉斯变换和运算微积法等多种方法求解Stokes第一问题. 在此基础上,可不断提出新问题,鼓励学生探索创新,给出了几个例子.     

Solitary waves at the interface of a two-layer fluid

In this paper, we discuss the solitary waves at the interface of a two-layer incompressible inviscid fluid confined by two horizontal rigid walls, taking the effect of surface tension into account. First of all, we establish the basic equations suitable for the model considered, and hence derive the Korteweg-de Vries (KdV) equation satisfied by the first-order elevation of the interface with the aid of the reductive perturbation method under the approximation of weak dispersion. It is found that the KdV solitary waves may be convex upward or downward. It depends on whether the signs of the coefficients and of the KdV equation are the same or not. Then we examine in detail two critical cases, in which the nonlinear effect and the dispersion effect cannot balance under the original approximation. Applying other appropriate approximations, we obtain the modified KdV equation for the critical case of first kind (=0), and conclude that solitary waves cannot exist in the case considered as >0, but may still occur as <0, being in the form other than that of the KdV solitary wave.As for the critical case of second kind (=0), we deduce the generalized KdV equation, for which a kind of oscillatory solitary waves may occur. In addition, we discuss briefly the near-critical cases. The conclusions in this paper are in good agreement with some classical results which are extended considerably.     

Some exact solutions of stationary gas flows in a perpendicular magnetic field

The plane flow of a perfectly conducting gas subject to a perpendicular magnetic field is considered. Ladikov [1] has solved this problem as a special case of the more general solution for a specific condition. It is shown that, following the method of [2], this problem may be reduced to the determination of the unknown function which satisfies a partial differential equation. Since the equation for is very complex, we have considered only two interesting cases, namely: 1) when the motion is irrotational, and 2) when the pressure is constant along the streamlines.     

General exact solution of incompressible potential flows around two circles

Three exact solutions are obtained for 2-D incompressible potential flows around two moving circles in three cases: (i) expansion (or contraction) of themselves, (ii) approaching (or departing from) each other, (iii) moving perpendicularly to the line connecting the centres in opposite directions. Meanwhile, another set of two exact solutions is obtained for 2-D incompressible potential flows between two moving eccentric circles in two cases: moving parallely or perpendicularly to the line connecting the centres.     

Analytical solution of the strong evaporation (condensation) problem

An exact solution of the model Boltzmann equation with BGK (Bhatnagar-Gross-Krook) collision operator is obtained in the problem of strong evaporation (condensation) from a plane surface. The generalized eigenvectors of the corresponding characteristic equation are found. The existence and uniqueness of the expansion of the solution in eigenvectors of the continuous and discrete spectra are demonstrated. This expansion reduces to a vector Riemann-Hilbert boundary-value problem with matrix coefficient. An apparatus for the diagonalization and factorization of the boundary-value problem coefficient is developed. The matrix diagonalizing the problem coefficient has branch points in the complex plane which depend parametrically on the evaporation (condensation) rateu. The solution of the problem is investigated in terms ofu and the physical characteristics of the evaporation process are described.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.6, pp. 143–155, November–December, 1993.     

New exact solution of the problem of rotationally symmetric Couette-Poiseuille flow

An exact solution is obtained for the problem of steady-state viscous incompressible flow under a pressure difference in the gap between coaxial cylinders for the case where the inner cylinder rotates at a constant angular velocity. The solution differs from the classical Couette-Poiseuille result by the presence of radial mass transfer, which provides for interaction between the poloidal and azimuthal circulations. The flow rate is found to depend linearly on the angular velocity of rotation of the inner cylinder. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 5, pp. 71–77, September–October, 2007.     

Onset of thermocapillary convection in a two-layer system with the release of heat at the interface

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 69–73, January–February, 1990.