A group invariant solution for a pre-existing fluid-driven fracture in permeable rock

The propagation of a two-dimensional pre-existing fracture in permeable rock by the injection of a viscous, incompressible Newtonian fluid is considered. The fluid flow in the fracture is laminar. By the application of lubrication theory, a partial differential equation relating the half-width of the fracture to the fluid pressure and leak-off velocity is derived. The model is closed by the adoption of the PKN formulation in which the fluid pressure is proportional to the fracture half-width. The partial differential equation admits four Lie point symmetries provided the leak-off velocity satisfies a first order linear partial differential equation. The solution of this equation yields the leak-off velocity as a function of the distance along the fracture and time. The group invariant solution is derived by considering a linear combination of the Lie point symmetries. The boundary value problem is reformulated as a pair of initial value problems. The model in which the leak-off velocity is proportional to the fracture half-width is considered. The working condition of constant pressure at the fracture entry is analysed in detail.     

Group invariant solution for a pre-existing fluid-driven fracture in impermeable rock

The propagation of a two-dimensional fluid-driven fracture in impermeable rock is considered. The fluid flow in the fracture is laminar. By applying lubrication theory a partial differential equation relating the half-width of the fracture to the fluid pressure is derived. To close the model the PKN formulation is adopted in which the fluid pressure is proportional to the half-width of the fracture. By considering a linear combination of the Lie point symmetries of the resulting non-linear diffusion equation the boundary value problem is expressed in a form appropriate for a similarity solution. The boundary value problem is reformulated as two initial value problems which are readily solved numerically. The similarity solution describes a preexisting fracture since both the total volume and length of the fracture are initially finite and non-zero. Applications in which the rate of fluid injection into the fracture and the pressure at the fracture entry are independent of time are considered.     

Group invariant solution for a pre-existing fluid-driven fracture in impermeable rock

The propagation of a two-dimensional fluid-driven fracture in impermeable rock is considered. The fluid flow in the fracture is laminar. By applying lubrication theory a partial differential equation relating the half-width of the fracture to the fluid pressure is derived. To close the model the PKN formulation is adopted in which the fluid pressure is proportional to the half-width of the fracture. By considering a linear combination of the Lie point symmetries of the resulting non-linear diffusion equation the boundary value problem is expressed in a form appropriate for a similarity solution. The boundary value problem is reformulated as two initial value problems which are readily solved numerically. The similarity solution describes a preexisting fracture since both the total volume and length of the fracture are initially finite and non-zero. Applications in which the rate of fluid injection into the fracture and the pressure at the fracture entry are independent of time are considered.     

Perturbation analysis of a modified second grade fluid over a porous plate

A modified second grade non-Newtonian fluid model is considered. The model is a combination of power-law and second grade fluids in which the fluid may exhibit normal stresses, shear thinning or shear thickening behaviors. The flow of this fluid is considered over a porous plate. Equations of motion in dimensionless form are derived. When the power-law effects are small compared to second grade effects, a regular perturbation problem arises which is solved. The validity criterion for the solution is derived. When second grade effects are small compared to power-law effects, or when both effects are small, the problem becomes a boundary layer problem for which the solutions are also obtained. Perturbation solutions are contrasted with the numerical solutions. For the regular perturbation problem of small power-law effects, an excellent match is observed between the solutions if the validity criterion is met. For the boundary layer solution of vanishing second grade effects however, the agreement with the numerical data is not good. When both effects are considered small, the boundary layer solution leads to the same solution given in the case of a regular perturbation problem.     

On the similarity solutions of magnetohydrodynamic flows of power-law fluids over a stretching sheet

A rigorous mathematical analysis is given for a magnetohydrodynamics boundary layer problem, which arises in the two-dimensional steady laminar boundary layer flow for an incompressible electrically conducting power-law fluid along a stretching flat sheet in the presence of an exterior magnetic field orthogonal to the flow. In the self-similar case, the problem is transformed into a third-order nonlinear ordinary differential equation with certain boundary conditions, which is proved to be equivalent to a singular initial value problem for an integro-differential equation of first order. With the aid of the singular initial value problem, the uniqueness and existence results for (generalized) normal solutions are established and some properties of these solutions are explored.     

MHD power-law fluid flow and heat transfer over a non-isothermal stretching sheet

This article presents a numerical solution for the magnetohydrodynamic (MHD) non-Newtonian power-law fluid flow over a semi-infinite non-isothermal stretching sheet with internal heat generation/absorption. The flow is caused by linear stretching of a sheet from an impermeable wall. Thermal conductivity is assumed to vary linearly with temperature. The governing partial differential equations of momentum and energy are converted into ordinary differential equations by using a classical similarity transformation along with appropriate boundary conditions. The intricate coupled non-linear boundary value problem has been solved by Keller box method. It is important to note that the momentum and thermal boundary layer thickness decrease with increase in the power-law index in presence/absence of variable thermal conductivity.     

A modification of the invariant imbedding method for a singular boundary value problem

We consider a dynamically-consistent analytical model of a 3D topographic vortex. The model is governed by equations derived from the classical problem of the axisymmetric Taylor–Couette flow. Using linear expansions, these equations can be reduced to a differential sixth-order equation with variable coefficients. For this differential equation, we formulate a boundary value problem, which has a number of issues for numerical solving. To avoid these issues and find the eigenvalues and eigenfunctions of the boundary value problem, we suggest a modification of the invariant imbedding method (the Riccati equation method). In this paper, we show that such a modification is necessary since the boundary conditions possess singular matrices, which sufficiently complicate the derivation of the Riccati equation. We suggest algebraic manipulations, which permit the initial problem to be reduced to a problem with regular boundary conditions. Also, we propose a method for obtaining a numerical solution of the matrix Riccati equation by means of recurrence relations, which allow us to obtain a matrizer converging to the required eigenfunction. The suggested method is tested by calculating the corresponding eigenvalues and eigenfunctions, and then, by constructing fluid particle trajectories on the basis of the eigenfunctions.     

Hopf-Cole transformation to some systems of partial differential equations

In this paper we study a system of nonlinear partial differential equations which we write as a Burgers equation for matrix and use the Hopf-Cole transformation to linearize it. Using this method we solve initial value problem and initial boundary value problems for some systems of parabolic partial differential equations. Also we study an initial value problem for a system of nonlinear partial differential equations of first order which does not have solution in the standard distribution sense and construct an explicit solution in the algebra of generalized functions of Colombeau. Received November 1999     

The impact of finite precision arithmetic and sensitivity on the numerical solution of partial differential equations

In this paper, we address some fundamental issues concerning “time marching” numerical schemes for computing steady state solutions of boundary value problems for nonlinear partial differential equations. Simple examples are used to illustrate that even theoretically convergent schemes can produce numerical steady state solutions that do not correspond to steady state solutions of the boundary value problem. This phenomenon must be considered in any computational study of nonunique solutions to partial differential equations that govern physical systems such as fluid flows. In particular, numerical calculations have been used to “suggest” that certain Euler equations do not have a unique solution. For Burgers' equation on a finite spatial interval with Neumann boundary conditions the only steady state solutions are constant (in space) functions. Moreover, according to recent theoretical results, for any initial condition the corresponding solution to Burgers' equation must converge to a constant as t → ∞. However, we present a convergent finite difference scheme that produces false nonconstant numerical steady state “solutions.” These erroneous solutions arise out of the necessary finite floating point arithmetic inherent in every digital computer. We suggest the resulting numerical steady state solution may be viewed as a solution to a “nearby” boundary value problem with high sensitivity to changes in the boundary conditions. Finally, we close with some comments on the relevance of this paper to some recent “numerical based proofs” of the existence of nonunique solutions to Euler equations and to aerodynamic design.     

Analytical solutions of the boundary layer flow of power-law fluid over a power-law stretching surface

This article discusses analytical solutions for a nonlinear problem arising in the boundary layer flow of power-law fluid over a power-law stretching surface. Using perturbation method analytical solution is presented for linear stretching surface. This solution covers large range of shear thinning and shear thickening fluids and matches excellently with the numerical solution. Furthermore, some new exact solutions are found for particular combination of m (power-law stretching index) and n (power-law fluid index). This leads to generalize the case of linear stretching to nonlinear stretching surface. The effects of fluid index n on the boundary layer thickness and the skin friction for nonlinear stretching surface is analyzed and discussed. It is observed that the boundary layer thickness and the skin friction coefficient increase as non-linear parameter n decreases. This study gives a new dimension to obtain analytical solutions asymptotically for highly nonlinear problems which to the best of our knowledge has not been examined so far.     

A boundary layer problem arising in gravity-driven laminar film flow of power-law fluids along vertical walls

A rigorous mathematical analysis is given for a boundary layer problem for a third-order nonlinear ordinary differential equation, which arises in gravity-driven laminar film flow of power-law fluids along vertical walls. Firstly, the problem is transformed into a singular nonlinear two-point boundary value problem of second order. Next, the latter is proved to have a unique positive solution, for which some estimates are established. Finally, the result above-mentioned is turned over to the original problem. The conclusion of this paper is that the boundary layer problem has a unique normal solution if the power-law index is less than or equal to one and a generalized normal solution if the power-law index is greater than one. Also the asymptotic behavior of the normal solution at the infinity is displayed.The work was supported by NNSF of China.     

非线性Blasius方程求解的一种新算法

通过一半无限大平板的不可压缩的两维稳定流是一个典型的工程问题,被称为边界层流问题.它是一个由三阶非线性微分方程描述的边值问题,其微分方程称为Blasius方程.首先将该边值问题转化为一对初值问题,然后用状态方程直接积分法和Taylor级数展开法对这对初值问题进行求解.与其它算法相比,具有算法简单,精度高的优点.     

Universal boundary value problem for equations of mathematical physics

A new statement of a boundary value problem for partial differential equations is discussed. An arbitrary solution to a linear elliptic, hyperbolic, or parabolic second-order differential equation is considered in a given domain of Euclidean space without any constraints imposed on the boundary values of the solution or its derivatives. The following question is studied: What conditions should hold for the boundary values of a function and its normal derivative if this function is a solution to the linear differential equation under consideration? A linear integral equation is defined for the boundary values of a solution and its normal derivative; this equation is called a universal boundary value equation. A universal boundary value problem is a linear differential equation together with a universal boundary value equation. In this paper, the universal boundary value problem is studied for equations of mathematical physics such as the Laplace equation, wave equation, and heat equation. Applications of the analysis of the universal boundary value problem to problems of cosmology and quantum mechanics are pointed out.     

Homogenization in the Theory of Viscoplasticity

We study the homogenization of the quasistatic initial boundary value problem with internal variables which models the deformation behavior of viscoplastic bodies with a periodic microstructure. This problem is represented through a system of linear partial differential equations coupled with a nonlinear system of differential equations or inclusions. Recently it was shown by Alber [2] that the formally derived homogenized initial boundary value problem has a solution. From this solution we construct an asymptotic solution for the original problem and prove that the difference of the exact solution and the asymptotic solution tends to zero if the lengthscale of the microstructure goes to zero. The work is based on monotonicity properties of the differential equations or inclusions. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Normal form for the KdV–Burgers equation

The local dynamics of the KdV–Burgers equation with periodic boundary conditions is studied. A special nonlinear partial differential equation is derived that plays the role of a normal form, i.e., in the first approximation, it determines the behavior of all solutions of the original boundary value problem with initial conditions from a sufficiently small neighborhood of equilibrium.     

ASYMPTOTIC BEHAVIOR OF A STOCHASTIC EVOLUTION PROBLEM IN A VARYING DOMAIN

We study the asymptotic behaviour of the initial boundary value problem for a stochastic partial differential equation in a sequence of perforated domains. We prove that the sequence of solutions of the problem converges in appropriate topologies to the solution of a limit stochastic initial boundary value problem of the same type as the original problem, but containing an additional term expressed in terms of some characteristics of the perforated domain.     

The pressure distribution around a growing crack

Exact solutions of the problem of the pressure distribution around an ideal hydraulic fracture are derived. The crack propagates in a permeable porous medium following a square-root growth law. The case of the penetration of the fracturing fluid into a reservoir is also considered.     

On the numerical solution of integral equations with piecewise continuous displacement kernels

Solution of a Fredholm integral equation with a piecewise continuous displacement kernel is considered. It is shown that this problem is equivalent to the solution of an initial value problem for an unusual partial differential equation for continuous functions of two variables. The difference scheme for the numerical solution of the initial value problem is derived. This scheme allows implementation on parallel processors and is of linear complexity. The approach based on the numerical solution of the initial value problem is compared with a corresponding quadrature method and demonstrates certain advantages.This work was supported by the NSF grant DMS-8801961This work was supported by a research grant from the NSERC of Canada     

Global solution to a singular integro-differential system related to the entropy balance

This paper is devoted to the mathematical analysis of a thermodynamic model describing phase transitions with thermal memory in terms of an entropy equation and a momentum balance for the microforces. The initial and boundary value problem is addressed for the related integro-differential system of partial differential equations (PDEs). Existence and uniqueness, continuous dependence on the data, and regularity results are proved for the global solution, in a finite time interval.     

Self-similar solution of the plane problem of the evolution of a hydraulic fracture crack in an elastic medium

The plane problem of the evolution of a hydraulic fracture crack in an elastic medium is considered. It is established that a self-similar solution is only possible at a constant rate of fluid injection. The solution for the value of the crack opening is presented in the form of a series expansion in Chebyshev polynomials of the second kind, and expansion coefficients are obtained as a solution of the algebraic set of equations which arise when projecting the balance equation for injected fluid mass on Chebyshev polynomials. When there is no part of the region unfilled with fluid (a fluid lag), the gradient of the crack opening at the crack tip turns out to be singular when the finiteness of the medium stress intensity factor is taken into account. According to the estimate made, the rate of convergence of the series expansion for the solution in Chebyshev polynomials is fairly rapid for a small injection intensity.