Flow of electrolytes in a porous medium

A two-scale model of ion transfer in a porous medium is obtained for one-dimensional horizontal flows under the action of a pressure gradient and an external electric field by the method of homogenization. Steady equations of electroosmotic flows in flat horizontal nano-sized slits separated by thin dielectric partitions are averaged over a small-scale variable. The resultant macroequations include Poisson’s equation for the vertical component of the electric field and Onsager’s relations between flows and forces. The total horizontal flow rate of the fluid is found to depend linearly on the pressure gradient and external electric field, and the coefficients in this linear relation are calculated with the use of microequations. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 4, pp. 162–173, July–August, 2008.     

Pulsatile pipe flow of pseudoplastic fluids

This note is concerned with a laminar pipe flow of a non-Newtonian fluid under the action of a small pulsating pressure gradient superposed to a steady one. The constitutive law describing the rheological behaviour of the fluid is the so-called power law (Ostwald–de Waele). An approximated analytical solution is found for the velocity, as power series of the amplitude of the periodic disturbance. The analytic solution is compared with a direct numerical solution and the perfect accord of the values obtained is underscored.     

Pulsatile flow with heat transfer of dusty magnetohydrodynamic Ree-Eyring fluid through a channel

The flow due to the pulsatile pressure gradient of dusty non-Newtonian fluid with heat transfer in a channel is considered. The system is stressed by an external magnetic field. The non-Newtonian fluid under consideration is obeying the rheological equation of state due to Ree-Eyring’s stress–strain relation. The equations of momentum and energy have been solved by using Lightill method. The velocity and temperature distributions of the two phase of the dusty fluid are obtained. The effects of various physical parameters of distributions the problem on these distributions are discussed and illustrated graphically through a set of figure.     

Effect of Relaxation and Retardation Time on Peristaltic Transport of the Oldroydian Viscoelastic Fluid

The influence of relaxation and retardation time on peristaltic transport of an incompressible Oldroydian viscoelastic fluid by means of an infinite train of sinusoidal waves traveling along the walls of a two-dimensional flexible channel is investigated. A perturbation solution is obtained for the case in which the amplitude ratio (wave amplitude to channel half-width) is small. The results show that the values of the mean axial velocity of an Oldroydian viscoelastic fluid is smaller than that for a Newtonian fluid. The reflux phenomena are discussed. It is found that the critical reflux pressure gradient decreases with increasing retardation time and increases with increasing relaxation time. Numerical results are reported for different values of the physical parameters of interest. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 6, pp. 86–95, November–December, 2005.     

On the stress-strain state of a transversally isotropic body with a paraboloidal inclusion

A closed solution is presented for the three-dimensional problem of the stress-strain state of an unbounded elastic body with a soldered-in transversally isotropic inclusion in the form of a paraboloid of revolution. Here, it is assumed that the body is under axisymmetric tension (compression). A solution of the corresponding problem for a paraboloidal recess is obtained as a special case. Podil’chuk [2, 3] has investigated similar problems for isotropic bodies with an inclusion assuming the form of a paraboloid of revolution or an elliptical paraboloid. Translated from Prikladnaya Mekhanika, Vol. 34, No. 11, pp. 16–22, November, 1998.     

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.     

A gas pendulum

It is shown that a periodic two-dimensional isentropic motion of a gas exists and it is described by an exact solution of the equations of gas dynamics. A polytropic gas that fills a circular cylinder rotates and oscillates (in the radial direction) simultaneously under the action of periodically changing external pressure. The solution obtained belongs to the class of solutions with a velocity field that is linear in the coordinates (with homogeneous deformation). Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 5, pp. 115–119, September–October, 2000.     

Peristaltic flow of a Johnson-Segalman fluid through a deformable tube

To understand theoretically the flow properties of physiological fluids we have considered as a model the peristaltic motion of a Johnson–Segalman fluid in a tube with a sinusoidal wave traveling down its wall. The perturbation solution for the stream function is obtained for large wavelength and small Weissenberg number. The expressions for the axial velocity, pressure gradient, and pressure rise per wavelength are also constructed. The general solution of the governing nonlinear partial differential equation is given using a transformation method. The numerical solution is also obtained and is compared with the perturbation solution. Numerical results are demonstrated for various values of the physical parameters of interest.      

On the Generation of Discontinuous Shearing Motions of a Non-Newtonian Fluid

The system under study models unsteady, one-dimensional shear flow of a highly elastic and viscous incompressible non-Newtonian fluid with fading memory under isothermal conditions. The flow, in a channel, is driven by a constant pressure gradient, is symmetric about the center line, and satisfies a no-slip boundary condition at the wall. The non-Newtonian contribution to the stress is assumed to obey a differential constitutive law (due to Oldroyd, Johnson & Segalman), the key feature of which is a non-monotone relation between the total steady shear stress and strain rate. In a regime in which the Reynolds number is much smaller than the Deborah (or Weissenberg) number, one obtains a degenerate, singularly perturbed system of nonlinear reaction-diffusion equations. It is shown that if the driving pressure gradient exceeds a critical value (the local shear stress maximum of the steady stress vs. strain rate relation), then the solution to the governing system, starting from rest at , tends as to a particular discontinuous steady state solution (the “top-jumping” steady state), except in a small neighborhood of the discontinuity. This discontinuous steady state is shown to be nonlinearly stable in a precise sense with respect to perturbations yielding smooth initial data. Such discontinuous steady states have been proposed to explain “spurting” flows, which exhibit a large increase in mean flow rate when the driving pressure is raised above a critical value. (Accepted April 22, 1996)     

On the Long-Time Behaviour of Solutions to the Navier–Stokes–Fourier System with a Time-Dependent Driving Force

In this paper, we give a complete characterization of the asymptotic behaviour of solutions to the Navier–Stokes–Fourier system. We show that either the driving force behaves asymptotically as a gradient of a scalar function, in which case any solution tends to a static state, or the total energy goes to infinity with growing time.     

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.     

Stability and initial postbuckling behavior of anisotropic cylindrical shells under external pressure

A solution is obtained to describe the stability and initial postbuckling behavior of cylindrical shells made of composites with one plane of symmetry. The solution is based on the Donnell-Mushtari-Vlasov nonlinear theory of anisotropic shells and Koiter’s theory of buckling and postbuckling behavior. Calculated results are presented for boron plastic shells with reinforcement of different types under external pressure. It is shown that the conventional model of a composite as an orthotropic material is erroneous for many types of reinforcement __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 3, pp. 86–103, March 2007.     

Gradient model in the problem of stress-concentration around a circular hole in two-component stochastic composites

The gradient model of stochastically inhomogeneous media is used to study the stress concentration around a circular hole in a two-component elastic composite. The study is based on a general solution of the system of equilibrium equations expressed in terms of harmonic functions and functions that satisfy the Helmholtz equation. This solution is used to solve problems for an infinite plane with a circular hole under uniform and uniaxial tension. The results obtained are compared with the solutions found using the theory of effective moduli, which is simpler __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 12, pp. 41–53, December 2007.     

The stress state of a laminated nonlinearly elastic thick-walled spherical shell

This paper deals with spatial axisymmetric boundary-value problems of the physically nonlinear theory of elasticity for piecewise-homogeneous spherical bodies. The passage to dimensionless characteristics of the stress-strain state allows us to extract a physical dimensionless small parameter in the nonlinear state equations. The solution of nonlinear equilibrium equations and boundary-value problems is searched for in the form of series in positive degrees of the small parameter. This approach allows reducing the stated physically nonlinear boundary-value problem to a sequence of corresponding linear nonhomogeneous problems. A specific analytical solution and numerical results are obtained for a two-layer nonlinearly elastic spherical shell under bilateral pressure. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 35, No. 12, pp. 26–32, December, 1999.     

An indirect pressure-gradient technique for measuring instantaneous flow rates in unsteady duct flows

A new technique is presented for making measurements of the instantaneous flow rate in unsteady laminar pipe flows. It utilizes a relationship between flow rate and pressure-gradient history that is an exact solution to the Navier–Stokes equations for parallel, developed flow of constant-property Newtonian fluids undergoing arbitrary unsteadiness from an initially steady or stationary state. The method does not rely on any assumption about velocity profiles, and applies instantaneously in momentarily reversing flow. Experimental comparisons between direct measurements of the cumulative flow and the results of this technique indicate that it is capable of providing measurements of cumulative flow and flow rate which are accurate to within a few percent at any instant during a flow transient, provided the instantaneous pressure gradient can be measured with this accuracy.     

Elastic state of a transversely isotropic piezoelectric body with an arbitrarily oriented elliptic crack

The elastic stress state in a piezoelectric body with an arbitrarily oriented elliptic crack under mechanical and electric loads is analyzed. The solution is obtained using triple Fourier transform and the Fourier-transformed Green’s function for an unbounded piezoelastic body. Solving the problem for the case of a crack lying in the isotropy plane, for which there is an exact solution, demonstrates that the approach is highly efficient. The distribution of the stress intensity factors along the front of a crack in a piezoelectric body under uniform mechanical loading is analyzed numerically for different orientations of the crack __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 2, pp. 39–48, February 2008.     

Stress-strain state of a flexible spherical shell with an eccentric circular hole

The stress-strain state of thin flexible spherical shells weakened by an eccentric circular hole is analyzed. The shells are made of an isotropic homogeneous material and subjected to internal pressure. A problem formulation is given, and a method of numerical solution with allowance for geometrical nonlinearity is outlined. The distribution of displacements, strains, and stresses along the hole boundary and in the region of their concentration is examined. The data obtained are compared with numerical solutions of the linear problem. The stress-strain state around the eccentric circular hole is analyzed with allowance for geometrical nonlinearity __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 10, pp. 92–98, October 2007.     

Forced Convection with Laminar Pulsating Counterflow in a Saturated Porous Circular Tube

An analytical solution is obtained for forced convection in a circular tube occupied by a core–sheath-layered saturated porous medium with counterflow produced by pulsating pressure gradients. The case of the constant heat-flux boundary conditions is considered, and the Brinkman model is employed for the porous medium. A perturbation approach is used to obtain analytical expressions for the velocity, temperature distribution, and transient Nusselt number for convection produced by an applied pressure gradient that fluctuates with small amplitude harmonically in time about a non-zero mean. It is shown that the fluctuating part of the Nusselt number alters in magnitude and phase as the dimensionless frequency increases. The magnitude increases from zero, goes through a peak, and then decreases to zero. The height of the peak depends on the values of various parameters. The phase (relative to that of the steady component) decreases from π/2 to − π/2 as the frequency increases.     

Discrete-continuum model of symmetric separation of a material

A problem of the beginning of motion of a finite-width cut in a linearly elastic plane under the action of symmetric external loading is formulated. The material on the way of cut propagation forms a layer (interaction layer). The stress-strain state of the material is postulated to be homogeneous across this layer. A system of integral boundary equations is obtained for determining the stress-strain state. Based on this system of equations, a discrete model of separation of the layer material is constructed under the assumption of a constant stress-strain state in an element of the interaction layer. The stress distribution in the pre-fracture zone is determined. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 1, pp. 134–140, January–February, 2009.     

Single-Phase (Brine) and Two-Phase (DNAPL–Brine) Flows in Induced Fractures

This study reports an investigation on the characteristics of single-phase (brine) and two-phase (DNAPL–brine) flows in induced fractures. The fracture aperture and fluid phase distributions were determined using X-ray computer tomography. In the single-phase flow tests, the pressure gradient across the induced fractures increases linearly with increasing flow rate. However, models based on the measured aperture do not yield a consistent match with the experimental data because the effect of pressure losses due to aperture variation and undulation are not taken into account. In the two-phase flow tests, the measured phase distributions reveal that the flow pattern is dominated by a dispersed or mixed flow in which either DNAPL or brine phase is discontinuous. The channel flow pattern, in which DNAPL and brine phases are continuous in the fracture and well represented by the widely used Romm’s relative permeability relationship was not observed in this study. In contrast, a Lockhart–Martinelli-type correlation developed for gas–liquid flow in pipes was found to match the pressure gradient and phase saturation results obtained from the laboratory tests.