On a boundary layer problem related to the gas flow in shales

The development of gas deposits in shales has become a significant energy resource. Despite the already active exploitation of such deposits, a mathematical model for gas flow in shales does not exist. Such a model is crucial for optimizing the technology of gas recovery. In the present article, a boundary layer problem is formulated and investigated with respect to gas recovery from porous low-permeability inclusions in shales, which are the basic source of gas. Milton Van Dyke was a great master in the field of boundary layer problems. Dedicating this work to his memory, we want to express our belief that Van Dyke’s profound ideas and fundamental book Perturbation Methods in Fluid Mechanics (Parabolic Press, 1975) will live on—also in fields very far from the subjects for which they were originally invented.     

Solution of a nonlinear heat conduction problem with boundary conditions of the fourth kind

We present a method for electrically modelling nonlinear contact heat-transfer problems both with and without taking into account the thermal conductivity of the contact layer.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 18, No. 2, pp. 328–332, February, 1970.     

Study concerning the laminar sublayer region of a turbulent boundary layer with injection

Results are shown of an experimental study concerning the laminar sublayer region of a turbulent boundary layer with injection.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 22, No. 3, pp.460–466, March 1972.     

Conjugate problem of heat transfer in a laminar boundary layer with injection

The author examines two conjugate problems of heat transfer in the laminar boundary layer at the boundary of a semi-infinite porous medium on the assumption that fluid filters continuously through the porous surface and that the injection velocity varies as x^–1/2, where x is the distance in the direction of flow.     

Diffusion problem of a creeping gas. I. Calculation of distribution functions of a molecular gas in the knudsen layer

A modified method of half range expansions has been used to find distribution functions of a molecular gas mixture inhomogeneous in concentration in the Knudsen layer.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 40, No. 5, pp. 807–817, May, 1981.     

Solution of the elastic-viscoplastic boundary value problem

A variational method of solution of the elastic-viscoplastic boundary value problem is established. The geometrical construction of the generalized stress field in the space of admissible stress histories is presented. The necessary and sufficient condition for the existence of solution is given. The minimum principle for the generalized stress history is stated. The uniqueness of the stress field is proved. The regular elastic-viscoplastic material model is proposed.     

Solution of a plane steady heat conduction problem with boundary conditions of the third kind for regions of special type

It is shown that the steady problem of heat conduction theory for regions bounded by cochleas of order 4m + 2 (m=1, 2, 3, ..., N), which emit heat from their surfaces according to Newton's law, is reduced by conformai mapping to the solution of certain equations in finite differences. For the case m=1 the solution of the equations is expressed in terms of Bessel functions, and formulas for the temperature distribution are obtained.     

Solutions to the equations of a compressible laminar boundary layer at a plate with suddenly changing boundary conditions

A method is proposed for constructing the solutions to the equations of a compressible boundary layer at a plate with suddenly changing boundary conditions, when u T and Pr=1.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 24, No. 3, pp. 445–452, March, 1973.     

The problem of the boundary layer at a thermally isolated plate

New solutions are given for the energy equation of a compressible boundary layer at a plate in absence of heat transfer.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 18, No. 1, pp. 39–44, January, 1970.     

Approximate solution of the internal problem for a turbulent boundary layer

The development of a turbulent boundary layer in an axisymmetric channel is analyzed with an an account of the mutual effects of the boundary layer and the core of the flow. The coordinates of the separation point, the coordinates of the junction, and other characteristics are found as functions of the Reynolds number and the channel geometry. A simple approximate method is proposed for calculating the boundary layer in a channel with an arbitrary curvilinear generatrix.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 18, No. 2, pp. 224–232, February, 1970.     

Effect of boundary conditions on the heat transfer law for a turbulent boundary layer

The heat transfer at an impermeable plate has been experimentally investigated under various boundary conditions. The conservativeness of the heat transfer law St₀=f(Re _T ^** ) is demonstrated for a monotonic increase of temperature and heat flux along the surface.     

Solution of a viscoelastic boundary layer equation by orthogonal collocation

Summary The applicability of the method of orthogonal collocation to the solution of Newtonian and viscoelastic boundary layer problems is investigated. To this end, the method is applied to the boundary layer equation describing the flow of a second-order fluid near a two-dimensional stagnation point. The efficiency of a number of different approximating bases is investigated. It is shown that the method is applicable to both Newtonian and weakly viscoelastic fluids, and that it compares very favorably with other weighted residual methods.     

Solution of the boundary layer equation for a power law fluid in magneto-hydrodynamics

Summary The problem of a non-Newtonian power law conducting fluid past a semi-finite plate, in the presence of an external electromagnetic field, is studied. The electric conductivity is taken as a function of the velocity. The method of expansion for a small parameter is used to obtain the solution and for a special form of the free stream velocity, the method of similarity solutions is used to obtain the exact solutions of the boundary layer equations associated with this problem in a closed form. It was found that the presence of the electromagnetic field reduces the velocity of the fluid.     

Effect of the boundary conditions on convection in a horizontal fluid layer with the Soret contribution

Summary.  In this paper, the phenomenon of natural convection in a horizontal binary fluid layer is studied. The four walls of the cavity are subjected to uniform fluxes of heat. The Soret effect is taken into consideration. The different hydrodynamic boundary conditions applied on the horizontal walls are rigid-rigid, rigid-free and free-free. Based on the parallel flow approach, an analytical solution for the stream function, the temperature and the concentration is established. The numerical results obtained from a finite volume method show a good agreement with those predicted analytically. The system is governed by five control parameters; the aspect ratio of the cavity, A, the Rayleigh number, R _T , the Lewis number, Le, the buoyancy ratio, ϕ, and the lateral heating intensity, α. For given sets of these parameters, the existence of natural and anti-natural flows is discussed. Received March 14, 2002; revised August 5, 2002 Published online: February 10, 2003     

Velocity profile in a turbulent boundary layer of a supersonic gas stream

A semiempirical equation is proposed for relating the coefficients of molecular and turbulent viscosity in the laminar sublayer and in the transition region. Simple formulas are derived for determining the velocity throughout the boundary layer and the frictional stress at the surface of a plate at various values of the Mach number. The results of calculations are compared with experimental data.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 23, No. 1, pp. 112–120, July, 1972.     

A boundary layer theory for the end problem of a circular cylinder

Summary This paper discusses the nature of an approximate solution for the hollow circular cylinder whose fixed ends are given a uniform relative axial displacement and whose cylindrical surfaces are free from traction. We shall take the solution of this problem to be given by a super-position of the following two problems: problem I considers a finite length cylinder whose ends are given a relative axial displacement, but are no longer fixed; problem II removes the radial displacement at the end of the cylinder obtained in problem I.Nomenclature a mid-surface radius of cylinder - c half-height of cylinder - E, in-plane elastic moduli - E_t, _t, G_t transverse elastic moduli - _z, , _r axial, circumferential, and normal strain - _rz transverse shear strain - h cylinder thickness - _z, , _r axial, circumferential, and normal stress - _rz transverse shear stress - z, r axial and radial coordinates - u_z, u_r axial and normal displacements     

Compressible turbulent boundary layer on a permeable plate with injection of foreign gas

A numerical solution is obtained for equations of motion, diffusion, and energy using a three-parameter model of turbulence for boundary layer with different thermal boundary conditions on a permeable plate with injection of foreign gas. For an ideal gas with constant values of Prandtl and Schmidt numbers, the numerical solution is compared with limiting dependences for dimensionless temperature and enthalpy of gas on the wall. For helium/xenon mixtures with a low value of Prandtl number, the effect of intensity of injection and of the values of Reynolds and Mach numbers on the integral and local characteristics of flow and heat and mass transfer is investigated.     

Conical crack problem in semi-infinite media with stress-free boundary conditions

In this study an axially symmetric conical crack problem in semi-infinite media is considered. Stress-free boundary conditions are satisfied at the boundary of the half-space. By using Papkovich-Neuber functions and Hankel transform techniques the problem is reduced to a system of two singular integral equations which are then solved numerically. The numerical examples are given for a constant pressure and constant shear stress on the crack surface separately. The stress intensity factors are evaluated and presented for various crack geometries and Poisson's ratios.

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Résumé Dans cette étude, on considère le problème d'une fissure en forme de conique à symétrie axiale dans un milieu semi-infini. Les conditions d'absence de contraintes aux limites sont satisfaites à l'extrémité du demi-espace. En utilisant les fonctions de Papkovich-Neuber et une transformée de Hankel, on réduit le problème à un système de deux intégrales singulières, que l'on résout ensuite par voie numique.A titre d'exemples numériques, on considère une pression constante et une tension de cisaillement constante, agissant séparément sur la surface de la fissure.On évalue les facteurs d'intensité de contraintes et on les présente pour diverses géométries de fissures et pour diverses valeurs du module de Poisson.

     

Solution of a drying problem for a colloidal porous material with a moving evaporation boundary

Closed solutions are derived for linear differential equations for one-dimensional drying with an evaporation boundary moving in accordance witht.Translated from Inzhenerno-Fizieheskii Zhurnal, Vol. 25, No. 5, pp. 871–876, November, 1973.