Displacement of a Newtonian fluid by a non-Newtonian fluid in a porous medium

This paper presents an analytical Buckley-Leverett-type solution for one-dimensibnal immiscible displacement of a Newtonian fluid by a non-Newtonian fluid in porous media. The non-Newtonian fluid viscosity is assumed to be a function of the flow potential gradient and the non-Newtonian phase saturation. To apply this method to field problems a practical procedure has been developed which is based on the analytical solution and is similar to the graphic technique of Welge. Our solution can be regarded as an extension of the Buckley-Leverett method to Non-Newtonian fluids. The analytical result reveals how the saturation profile and the displacement efficiency are controlled not only by the relative permeabilities, as in the Buckley-Leverett solution, but also by the inherent complexities of the non-Newtonian fluid. Two examples of the application of the solution are given. One application is the verification of a numerical model, which has been developed for simulation of flow of immiscible non-Newtonian and Newtonian fluids in porous media. Excellent agreement between the numerical and analytical results has been obtained using a power-law non-Newtonian fluid. Another application is to examine the effects of non-Newtonian behavior on immiscible displacement of a Newtonian fluid by a power-law non-Newtonian fluid.     

Impact of a strip of compressible liquid on a barrier

The exact solution of the plane problem of the impact of a finite liquid strip on a rigid barrier is obtained in the linearized formulation. The velocity components, the pressure and other elements of the flow are determined by means of a velocity potential that satisfies a two-dimensional wave equation. The final expressions for them are given in terms of elementary functions that clearly reflect the wave nature of the motion. The exact solution has been thoroughly analyzed in numerous particular cases. It is shown directly that in the limit the solution of the wave problem tends to the solution of the analogous problem of the impact of an incompressible strip obtained in [1]. A logarithmic singularity of the velocity parallel to the barrier in the corner of the strip is identified. A one-dimensional model of the motion, which describes the behavior of the compressible liquid in a thin layer on impact and makes it possible to obtain a simple solution averaging the exact wave solution, is proposed. Inefficient series solutions are refined and certain numerical data on the impact characteristics for a semi-infinite compressible liquid strip, previously considered in [2–4] in connection with the study of the earthquake resistance of a dam retaining water in a semi-infinite basin, are improved. The solution obtained can be used to estimate the forces involved in the collision of solids and liquids. It would appear to be useful for developing correct and reliable numerical methods of solving the nonlinear problems of fluid impact on solids often examined in the literature [5].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 138–145, November–December, 1990.The results were obtained by the author under the scientific supervision of B. M. Malyshev (deceased).     

Nonlinear resonance of a subsatellite on a short constant tether

The nonlinear resonant behavior of a subsatellite on a short constant tether during station-keeping phase is investigated in this paper. The nonlinear dynamic equations of in-plane motion of the system are derived based on Kane’s method first. Then an approach of multiple scales expressed in matrix form is employed in solving the simplified nonlinear system of cubic nonlinearity near its local equilibrium position. Analysis shows that there exists a three-to-one resonance in such a nonlinear system with two degrees of freedom. Afterward, the approximate solution up to third order determined analytically by the Weierstrass elliptic function is obtained and the comparison between the approximate and numerical solutions presented as well. The results show that the approximate solution is coincide well with the numerical solution of original system. The nonlinear resonance of the subsatellite on short tether exhibits coexistent quasiperiodic motions or a quasiperiodic oscillation near local equilibrium position.     

An investigation into the spreading of a thin liquid drop under gravity on a slowly rotating disk

The axisymmetric spreading of a thin liquid drop under the influence of gravity and rotation is investigated. The effects of the Coriolis force and surface tension are ignored. The Lie group method is used to analyse the non-linear diffusion-convection equation modelling the spreading of the liquid drop under gravity and rotation. A stationary group invariant solution is obtained. The case when rotation is small is considered next. A straightforward perturbation approach is used to determine the effects of the small rotation on the solution given for spreading under gravity only. Over a short period of time no real difference is observed between the approximate solution and the solution for spreading under gravity only. After a long period of time, the approximate solution tends toward a dewetting solution. We find that the approximate solution is valid only in the interval t∈[0,t∗), where t∗ is the time when dewetting takes place. An approximation to t∗ is obtained.     

Influence of distributed addition of a polymer solution on the characteristics of a turbulent boundary layer

The motion of a plate in the presence of distributed addition of a polymer solution is studied on the basis of a system of partial differential equations. The calculations take into account the influence of the additives on the coefficients of molecular and turbulent viscosity and diffusion. The influence of the concentration and rate of injection of the polymer solution on the profiles of the tangential frictional stress, the averaged velocity, and the local and total frictional drag coefficients is analyzed. The results of calculations are compared with data on the drag of a plate when a polymer solution is injected near its leading edge.     

Harmonic vibrations of a half-space with a surface-breaking crack under conditions of out-of-plane deformation

The paper presents a solution of the problem of determining the stress state in an elastic isotropic half-space with a crack intersecting its boundary under harmonic longitudinal shear vibrations. The vibrations are excited by a regular action of a harmonic shear load on the crack shores. The solution method is based on the use of the discontinuous solution of the Helmholtz equation, which allows one to reduce the original problem to a singular integro-differential equation for the unknown jump of the displacement on the crack surface. The solution of this equation is complicated by the existence of a fixed singularity of its kernel. Therefore, one of the main results is the development of an efficient approximate method for solving such equations, which takes into account the true asymptotics of the unknown function. The latter allows one to obtain a high-precision approximate formula for calculating the stress intensity factor.     

Interaction of a diaphragm pressure gage with a viscoelastic halfspace

The results of an analysis to determine the interaction between a diaphragm pressure transducer and a solid propellant grain are presented. The solutions to a clamped circular plate and a halfspace are superposed to yield the desired solution. The boundary conditions on the halfspace are shown to be such that the solution to an internally pressurized Sneddon “penny-shaped” crack is applicable for an incompressible material. The problem is first solved elastically, in terms of a material-stiffness parameter which relates the diaphragm stiffness to the propellant stiffness. The solution is then extended to viscoelastic behavior through parameterization of the stiffness parameter. The electrical output of the diaphragm gage is determined and compared with the output from hydrostatic calibration, in order to determine the error or loss in gage sensitivity based on hydrostatic calibration, due to the interaction between the gage and the propellant.     

Diffraction of a plane stress wave by a semi-infinite crack in a general anisotropic elastic material

The problem of a semi-infinite crack subjected to an incident stress wave in a general anisotropic elastic solid is considered. The plane wave impinges the crack at a general oblique angle and is of any of the three types propagating in that direction. A related problem of a semi-infinite crack loaded by a pair of concentrated forces moving along the crack surfaces is also considered. In contrast to the conventional approach by Laplace transforms, a Stroh-like formalism is employed to construct the solution directly in the time domain. The solution is shown to depend on a Wiener–Hopf factorization of a symmetric matrix. Closed-form solution of the stress intensity factors is derived. A remarkably simple expression for the energy release rate is obtained for normal incidence.     

Generalized solution of the problem on a circular membrane loaded by a lumped force

The solution of the problem on a circular membrane loaded at the center by a lumped force is a classical example of a singular solution of equations of mathematical physics. In this paper, the problem is solved by using relations of the generalized theory of elasticity, which contain a structural parameter and permit obtaining a regular solution. An experiment for determining the structural parameter in the problem of bending of a membrane is described.     

Free surface deformation due to a source or a sink in Stokes flow

Free surface shape and cusp formation are analyzed by considering two-dimensional viscous flow due to a line source or a line sink below the free surface where the strength of source/sink is given arbitrarily. In the analysis, the Stokes' approximation is used and surface tension effects are included, but gravity is neglected. The solution is obtained analytically by using conformal mapping and complex function theory. From the solution, shapes of the free surface are shown and the formation of a cusp on the free surface is discussed. As the capillary number decreases in negative, the free surface shape becomes singular and in a real fluid a cusp should form on the free surface below some negative critical capillary number. Typically, streamline patterns for some capillary numbers are also shown. As the small capillary number vanishes, the solution is reduced to a linearized potential flow solution.     

System with a non-linear negative self-excitation

A two-mass system is analyzed consisting of a self-excited basic system, which is mounted on a foundation subsystem consisting of a mass on a spring. The self-excitation is expressed in differential equations by a non-linear term of the second power. The efficiency of the self-excited vibration suppressing of different positive damping components in both the subsystems is investigated by means of analytical and numerical solution. Phase plane trajectories gained by numerical solution show the distortion of pure harmonic forms of oscillations presumed in analytical solution. Ranges of system parameters in which the approximate bifurcation diagrams coincide with numerical results are ascertained.     

Asymptotic Stability of a Stationary Solution to a Thermal Hydrodynamic Model for Semiconductors

The present paper concerns the existence and the asymptotic stability of a stationary solution to the initial boundary value problem for a one-dimensional heat-conductive hydrodynamic model for semiconductors. It is important to analyze thermal influence on the motion of electrons in semiconductor device to improve the reliability of devices by handling a hot carrier problem. We show the unique existence of the stationary solution satisfying a subsonic condition by using the Leray–Schauder and the Schauder fixed-point theorems. Then the asymptotic stability of the stationary solution is proved by deriving the a priori estimate uniformly in time. Here an energy form plays an essential role. We also prove that the solution converges to the stationary solution exponentially fast as time tends to infinity.     

In-plane bending of a beam resting on a rigid rough foundation

Summary The bending of a finite-length beam that lies on a rigid, rough, flat foundation and interacts with it in accordance to the dry friction law is considered. Loading by bending moments applied at the ends of the beam is studied in detail. The problem is found to be a self-similar one. For small moments, the central part of the beam remains undeflected, and the problem reduces to the solution of an infinite system of algebraic equations. Large moments deflect the entire length of the beam, and the problem partly loses its self-similarity. In this case, the problem reduces to the solution of a successively decreasing number of ordinary differential equations along with some algebraical equations. The solution for the latter case provides initial conditions for the former one. This permits to obtain a solution for any value of the moment. Received 5 November 1996; accepted for publication 27 January 1997     

Surface tension driven oscillations of a bubble in a viscoelastic liquid

Small amplitude surface tension driven oscillations of a spherical bubble in a dilute polymer solution are considered. The rheological properties of the liquid are modelled by using a 3-constant constitutive equation of the Oldroyd type. The Laplace transform of the solution of the initial value problem is inverted numerically. As in the Newtonian fluid case, both a discrete and a continuous spectrum occurs. In addition to the non-dimensional parameters in the corresponding problem for a Newtonian fluid, the results depend on two other parameters: the ratio of the relaxation time of the polymer solution and the time scale of the flow (the Deborah number) and the product of the polymer concentration and the intrinsic viscosity. For small bubbles in an aqueous solution having a small relaxation time, significant additional damping is found even for dilute solutions.     

Pressure distribution in the neighborhood of a growing fracture with a constant wedge force

Exact solutions of the problem of the pressure field in the neighborhood of a hydraulic fracture developing in accordance with a square root law in a permeable porous medium with a constant wedge force acting on the fracture edges are constructed. A particular case admitting a self-similar formulation and an exact solution and, as a result, the fairly complete investigation, is considered. The solution constructed holds for an arbitrary self-similar pressure distribution over the fracture edges. The problem considered reduces to the solution of a mixed boundary-value problem for the Helmholtz equation. The solution found can be useful both in itself and for testing more universal numerical algorithms.     

Effect of yield stress on the flow of a Casson fluid in a homogeneous porous medium bounded by a circular tube

The effect of yield stress on the flow characteristics of a Casson fluid in a homogeneous porous medium bounded by a circular tube is investigated by employing the Brinkman model to account for the Darcy resistance offered by the porous medium. The non-linear coupled implicit system of differential equations governing the flow is first transformed into suitable integral equations and are solved numerically. Analytical solution is obtained for a Newtonian fluid in the case of constant permeability, and the numerical solution is verified with that of the analytic solution. The effect of yield stress of the fluid and permeability of the porous medium on shear stress and velocity distributions, plug flow radius and flow rate are examined. The minimum pressure gradient required to start the flow is found to be independent of the permeability of the porous medium and is equal to the yield stress of the fluid.     

Effect of heat transfer on a third grade fluid in a flat channel

The heat transfer analysis on the laminar flow of an incompressible third grade fluid through a porous flat channel is examined. The lower plate is assumed to be at a higher temperature than the upper plate. Analytical solution for temperature distribution is obtained for various values of the controlling parameters and discussed. The obtained analytical solution is also compared with the numerical solution. The comparison shows the fact that the accuracy is remarkable. Copyright © 2009 John Wiley & Sons, Ltd.     

Domain perturbation method and shape of a bubble in a uniform flow of an inviscid liquid

In many problems with a free boundary there is defined a small parameter, , for which the solution is sometimes known for a particular value, =0, and the general solution is obtained as a series in the parameter. To find this solution, the equations can be written on a reference configuration and solved in a fixed domain. The purpose of this study is to show that this method of domain perturbation is a good one. The range of validity of this method will be studied on the model example of the irrotational flow of a perfect fluid around a bubble. The radius of convergence of the series solution will be determined, as will the nature of the solution in the neighbourhood of the first real singularity.     

Mixed Convection on a Horizontal Surface Embedded in a Porous Medium: the Structure of a Singularity

The mixed convection boundary-layer flow on a horizontal impermeable surface embedded in a saturated porous medium and driven by a local heat source is considered. Similarity solutions are obtained for specific outer flow variations and these are shown to have a solution only for parameter values greater than some critical value. When this is not the case the solution develops a singularity at a finite distance from the leading edge. The nature of this singularity is also discussed.