Stress state of a saturated porous body near a cylindrical cavity with an exponential pressure change at its boundary

Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya Mekhanika, Vol. 24, No. 5, pp. 93–98, May, 1988.     

Slow Motion of a Porous Cylindrical Shell in a concentric cylindrical cavity

This paper concerns the Slow Motion of a Porous Cylindrical Shell in a concentric cylindrical cavity using particle-in-cell method. The Brinkman’s equation in the porous region and the Stokes equation for clear fluid in their stream function formulations are used. The hydrodynamic drag force acting on each porous cylindrical particle in a cell and permeability of membrane built up by cylindrical particles with a porous shell are evaluated. Four known boundary conditions on the hypothetical surface are considered and compared: Happel’s, Kuwabara’s, Kvashnin’s and Cunningham’s (Mehta-Morse’s condition). Some previous results for hydrodynamic drag force and dimensionless hydrodynamic permeability have been verified. Variation of the drag coefficient and dimensionless hydrodynamic permeability with permeability parameter σ, particle volume fraction γ has been studied and some new results are reported. The flow patterns through the regions have been analyzed by stream lines. Effect of particle volume fraction γ and permeability parameter σ on flow pattern is also discussed. In our opinion, these results will have significant contributions in studying, Stokes flow through cylindrical swarms.     

Motion of a free axisymmetric viscous liquid film

Axisymmetric free-film flows are encountered in connection with the atomization of liquids and the collision of jets [1, 2]. In [3] steady motion with transverse symmetry is examined and its inviscid instability is studied. Here, steady flow with an arbitrary velocity profile is investigated numerically by the collocation method. The study of the stability of the steady flow under the assumption of local plane-parallelism leads to the formulation of a sixth-order eigenvalue problem which is solved numerically. The existence of unstable disturbances of two types is demonstrated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 23–29, July–August, 1990.     

Motion of a vortex near the boundary between two heavy fluids

The motion of a vortex beneath the surface of a heavy fluid has been discussed in both linear [1, 2] and nonlinear [3–5] formulation. The density of the upper medium is neglected, which makes it possible to replace the continuity of pressure during transition through the boundary between the media by constancy of the pressure at the boundary of the heavy fluid. In this paper, the problem is solved in a general nonlinear formulation, including the mutual effects of media motion, and the vortex can be in either the upper or lower medium. Steady-state motion of a vortex of given intensity near the boundary between two heavy fluids is discussed in terms of a model of an ideal and incompressible medium. Approximate expressions are obtained for the boundary.     

Dynamics of gas bubbles in a cylindrical cavity filled with liquid

S. P. Timoshenko Institute of Mechanics, Academy of Sciences of the Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 30, No. 8, pp. 88–92, August, 1994.     

Stability of a fiber in an elastoplastic matrix near a free cylindrical surface

Institute of Mechanics, Ukrainian Academy of Sciences, Kiev. Translated from Prikladnaya Mekhanika, Vol. 28, No. 1, pp. 40–47, January, 1992.     

Continuation of the solution of a free boundary problem in cylindrical symmetry

Summary We perform an a priori analysis of the behavior of the solution to a Stefan type free boundary problem in cylindrical symmetry, in arbitrarily large time intervals; a nonlinear flux condition is prescribed on the fixed boundary, for t>0.Next we find some relations between the occurrence of each possible case and the behavior of the initial datum, assuming that the flux is null on the fixed boundary for t>0.

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Sommario Si esegue un'analisi a priori del comportamento, in intervalli di tempo arbitrariamente grandi, della soluzione di un problema a frontiera libera del tipo di Stefan in simmetria cilindrica; sulla frontiera fissa, per t>0, è stabilita una condizione di flusso non lineare.Si trovano quindi relazioni tra i vari casi possibili e il comportamento del dato iniziale, sotto l'ulteriore ipotesi che il flusso sulla frontiera fissa, per t>0, sia nullo.

     

Computation of Hele-Shaw free boundary problems near obstacles

The time-dependent evolution of source-driven Hele-Shaw free boundary flows in the presence of an obstacle is computed numerically. The Baiocchi transformation is used to convert the Hele-Shaw Laplacian growth problem into a free boundary problem for a streamfunction-like variable u(x, y, t) governed by Poisson’s equation (with constant right-hand side) with the source becoming a point vortex of strength linearly dependent on time. On the free boundary, both u and its normal derivative vanish, and on the obstacle, the normal derivative of u vanishes. Interpreting u as a streamfunction, at a given time, the problem becomes that of finding a steady patch of uniform vorticity enclosing a point vortex of given strength such that the velocity vanishes on the free boundary and the tangential velocity vanishes on the obstacle. A combination of contour dynamics and Newton’s method is used to compute such equilibria. By varying the strength of the point vortex, these equilibria represent a sequence of source-driven growing blobs of fluid in a Hele-Shaw cell. The practicality and accuracy of the method is demonstrated by computing the evolution of Hele-Shaw flow driven by a source near a plane wall; a case for which there is a known exact solution. Other obstacles for which there are no known exact solutions are also considered, including a source both inside and outside a circular boundary, a source near a finite-length plate and the interaction of an infinite free boundary impinging on a circular disc.     

Motion of a slender body near the free surface of a compressible fluid

An analytic solution to the problem of motion of a slender rigid body in a semi-infinite domain of a compressible fluid is obtained for the case when the body moves in parallel to the free surface at a constant velocity. This problem is similar to the problem of motion of a hydrofoil ship whose wing-like device allows it to lift its hull above the water surface and to decrease the friction and drag forces limiting the speed of usual ships. During its motion in water, a hydrofoil produces a lift force. The obtained analytic solution allows one to derive explicit expressions for the drag force and for the lift force in the limiting cases of relatively small and large depths. When depth is small, the drag force is greater than that in an infinite medium, since the wave drag is additionally evolved. When the velocity increases and approaches the sound velocity, the forces exerted on the body increase without limit, which is typical for a linear formulation of the problem.     

Convection development in a liquid layer with a free boundary

In certain calculations of the critical Rayleigh number for a liquid layer with free boundary which is heated from below, the linearization method has been used and it has been assumed that the temperature perturbations disappear at the undisturbed free boundary.Proper linearization shows that the temperature perturbation is proportional to the free surface perturbation, and the latter is proportional to the normal stress perturbation with the proportionality factor F=²/gh³ (g is the free-fall acceleration, is the kinematic viscosity, h is the liquid layer thickness). In §1 we present a formulation of the problem with account for the parameter F; in §2 we consider the linearized equations and the existence of a stability threshold is proved-a positive eigenvalue-and it is established that with an increase in the parameter F/P (P is the Prandtl number) the value of the critical Rayleigh number Ra_* decreases; §3 presents the results of a numerical calculation of Ra as a function of the parameter F/P.Convection development in a liquid layer with a free surface on which a given temperature is maintained was studied in [1, 2]. The value R_*=1100 found for the critical Rayleigh number agrees well with the experimental value. In the calculations made in [1, 2] the linearization method is used, and it is assumed that the temperature perturbations disappear at the undisturbed free boundary. Strictly speaking, this assumption is not correct.Correct linearization shows that the temperature perturbation is proportional to the perturbation of the free boundary, and the latter is proportional to the normal stress perturbation (see below (2.3)).The problem formulation is presented in §1; §2 deals with the linearized equations and the existence (Theorem 2.1) is demonstrated of a stability threshold—which is a simple positive eigenvalue; §3 presents the results of a numerical calculation of R_* as a function of the parameter =F/P.     

Motion of a gas shock layer with a free boundary under nonadiabatic conditions

This paper is a study of the effect of heat input (removal) on the characteristics of a shock layer produced by a gas at high supersonic velocity encountering a mobile boundary, which for generality is assumed to be free. We will use the Chernyi method, which was employed previously to solve the problem of a shock layer in an adiabatic flow [1, 2]. The results obtained can be useful for analysis of the effect of radiation (absorption) and processes involving the relaxation of internal degrees of freedom of molecules, condensation, chemical reactions, etc., whose effect on the gasdynamics of the flow in a shock layer may be similar to heat input or removal [3–5].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 152–154, May–June, 1976.The author thanks A. K. Rebrov for discussion of the results.