Multiple solutions of mixed convection boundary-layer approximations in a porous medium

This note deals with a theoretical analysis of the existence, non-uniqueness and non-existence of similarity solutions of the two-dimensional mixed convection boundary-layer flow over a vertical surface with a power law temperature. Here, it is assumed that the surface is embedded in a saturated porous media. The results depend on the power law exponent and the ratio of the Rayleigh to Péclet numbers. It is shown, under certain circumstance, that the problem has an infinite number of solutions.     

Some exact solutions of the fin problem with a power law temperature-dependent thermal conductivity

This study investigates the exact solutions of a nonlinear fin problem with temperature-dependent thermal conductivity and the heat transfer coefficient. Both the conduction and heat transfer terms are given by the same power law in one case and the distinct power law in the other. Classical Lie symmetry techniques are employed to construct the exact solutions which satisfy the realistic boundary conditions. The effects of the physical applicable parameters such as the thermo-geometric fin parameter and the fin efficiency are analyzed.     

Abundant explicit exact solutions to the generalized nonlinear Schrödinger equation with parabolic law and dual‐power law nonlinearities

This paper is concerned with the generalized nonlinear Schrödinger equation with parabolic law and dual‐power law. Abundant explicit and exact solutions of the generalized nonlinear Schrödinger equation with parabolic law and dual‐power law are derived uniformly by using the first integral method. These exact solutions are include that of extended hyperbolic function solutions, periodic wave solutions of triangle functions type, exponential form solution, and complex hyperbolic trigonometric function solutions and so on. The results obtained confirm that the first integral method is an efficient technique for analytic treatment of a wide variety of nonlinear systems of partial DEs. Copyright © 2014 John Wiley & Sons, Ltd.     

Existence and estimate of positive solutions to a nonlinear singular boundary value problem in the theory of dilatant non-Newtonian fluids

This paper presents a rigorous proof of existence and uniqueness of solutions to laminar boundary layer flow in power law non-Newtonian fluid. A theoretical estimate for skin friction coefficient is given, which is characterized by a power law exponent. The reliability and efficiency of the proposed estimate formula are verified by numerical results with a good agreement. The estimate formula can be successfully applied to give the value of the skin friction coefficient.     

Reliable analysis for nonlinear Schrödinger equations with a cubic nonlinearity and a power law nonlinearity

The tanh method and the sine–cosine method are effectively used for reliable analysis for the nonlinear Schrödinger equations with cubic and power law nonlinearities. A variety of exact solutions with distinct structures are formally derived for each equation. The study reveals the power of the two proposed algorithms.     

Modeling of primary and secondary creep for a wide stress range

Many materials exhibit a stress range dependent creep behavior. The power–law creep observed for a certain stress range changes to the viscous type creep if the stress value decreases. Recently published experimental data for advanced heat resistant steels indicate that the high creep exponent (in the range 5–12 for power–law behaviour) may decrease to the low value of approximately 1 within the stress range relevant for engineering structures. The aim of this paper is to confirm the necessity of the assumption of the stress range dependent power–law–viscous creep transition for the solution of stress relaxation problems affected by creep behavior at elevated temperatures. A constitutive model for the minimum creep rate is introduced to describe both the linear and the power law creep depending upon the stress level. The proposed constitutive model includes a strain hardening function to describe the primary creep stage. To demonstrate the existence of the linear creep behaviour in the low stress range of application area and the influence of the primary creep behaviour on relaxation, several solutions of a uniaxial stress relaxation problem are presented for the loading values relevant to engineering applications. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Steady heat transfer through a radial fin with rectangular and hyperbolic profiles

We construct some exact solutions for thermal diffusion in a fin with a rectangular profile and another with a hyperbolic profile. Both the thermal conductivity and the heat transfer coefficient are assumed to be temperature dependent. Moreover, the thermal conductivity and the heat transfer terms are given by the same power law in one case and distinct power laws in the other. A point transformation is introduced to linearize the problem when the power laws are equal. In the other case, classical Lie symmetry techniques are employed to analyze the problem. The exact solutions obtained satisfy the realistic boundary conditions. The effects of applicable physical parameters such as the thermo-geometric fin parameter and the fin efficiency are analyzed.     

Quasi-norm error bounds for the finite element approximation of a non-Newtonian flow

Summary. We consider the finite element approximation of a non-Newtonian flow, where the viscosity obeys a general law including the Carreau or power law. For sufficiently regular solutions we prove energy type error bounds for the velocity and pressure. These bounds improve on existing results in the literature. A key step in the analysis is to prove abstract error bounds initially in a quasi-norm, which naturally arises in degenerate problems of this type. Received May 25, 1993 / Revised version received January 11, 1994     

A finite element method for heat transfer of power‐law flow in channels with a transverse magnetic field

Heat transfer of a power‐law non‐Newtonian incompressible fluid in channels with porous walls has not been carefully studied using a proper numerical method despite a few constructions of approximate analytic solutions through the similarity transformation and perturbation method for Newtonian fluids (i.e. power‐law index being one). In this paper, we propose a finite element method for the thermal incompressible flow equations. The incompressible condition is treated by a penalty formulation. Numerical solutions are validated by comparing them with an approximate analytic solution of the Navier–Stokes equation in the Newtonian fluid case. Then, the method is used to simulate the heat transfer of various power‐law fluids. Additionally, unlike previous studies, we allow the thermal diffusivity to be a function of temperature gradient. The effect of different values of the parameters on the temperature and velocity is also discussed in this paper. Copyright © 2013 John Wiley & Sons, Ltd.     

Note on exact solutions for a class of nonlinear diffusion equations

An effective characterization is given for a class of generalized nonlinear diffusion equations with power law dependent terms. Further, a new auxiliary equation ansatz is derived. Consequently, new exact traveling wave trigonometric function, solitary-like and Weierstrass elliptic solutions to a subclass are obtained by means of an auxiliary equation method and a generalized Riccati equation expansion method.     

Blowup in a mass-conserving convection-diffusion equation with superquadratic nonlinearity

A nonlinear convection-diffusion equation with boundary conditions that conserve the spatial integral of the solution is considered. Previous results on finite-time blowup of solutions and on decay of solutions to the corresponding Cauchy problem were based on the assumption that the nonlinearity obeyed a power law. In this paper, it is shown that assumptions on the growth rate of the nonlinearity, which take the form of weak superquadraticity and strong superlinearity criteria, are sufficient to imply that a large class of nonnegative solutions blow up in finite time.

     

Finite element error analysis of a quasi-Newtonian flow obeying the Carreau or power law

Summary We consider the finite element approximation of a quasi-Newtonian flow, where the viscosity obeys the Carreau or power law. For sufficiently regular solutions we prove energy type error bounds for the velocity and pressure. These bounds improve on existing results in the literature.Supported by SERC research grant GR/F81255     

Nature of the growth of positive solutions of a parabolic equation

For a second-order uniformly parabolic equation with uniformly oscillating coefficients we obtain a power law estimate of the growth of positive solutions in unbounded domains in the lower half-plane.Translated from Matematicheskie Zametki, Vol. 12, No. 4, pp. 393–402, October, 1972.     

Power series solutions of Tarski’s associativity law and of the cyclic associativity law

In the present paper we study the associativity law of Tarski and the cyclic associativity law. We characterize the power series solutions of these equations in the complex domain and we give the connection to the solutions of the ordinary associativity equation.     

Construction of upper and lower solutions for flow past a non-uniformly heated plate

The Oseen linearization and the modified Oseen linearization are often used in studying fluid mechanical problems, but whether the linearized solution is accurate is usually difficult to assess. For the sample problem of uniform flow past a plate, we use a comparison theorem to show that the Oseen linearization, used in two ways, gives both an upper and a lower solution. Further, we make use of the comparison theorem and the modified Oseen linearization to construct a sharper upper solution valid in the boundary layer. We then go on to consider the case when the plate temperature increases along the plate according to a power law. Upper and lower solutions for the temperature equation are constructed, and bounds on the temperature gradient at the plate are obtained. With a minor modification, similar results for the case when the logarithmic derivative of the plate temperature lies between two power law curves are obtained.     

幂律流体边界层方程的近似解析解和壁摩擦因数的近似值

对幂率流体层流平板边界层的解析解进行了研究．对该问题提供了Adomian分解方法并且推导出了问题的级数形式的近似解析解,该近似解析解具有快速收敛性和易于计算性．对不同的幂率给出了方程的近似解析解和相应的壁摩擦因数近似值,最后对近似解所推出结果和所得壁摩擦因数与文献中的数值解进行了比较验证,证实了该文提出的解析近似方法的准确性和可靠性,说明了该近似解能够应用于提供所研究问题的壁摩擦因数．     

Some solutions of karman''s equation describing the transonic flow past a corner point on a profile with a curvilinear generatrix

Transonic flows in the neighbourhood of a corner point on a profile are investigated in a class of self-similar solutions of Karman's equation. The corner point is formed by the intersection of two smooth curves, the tangents to which make a convex angle. The generatrix, which lies in the subsonic part of the flow is assumed to be curvilinear and to vary according to a power law. Values of the self-similarity index are found for which transonic flows are possible either with a free streamline or with a rarefaction wave.     

Existence of weak solutions for a diffuse interface model of non-Newtonian two-phase flows

We consider a phase field model for the flow of two partly miscible incompressible, viscous fluids of non-Newtonian (power law) type. In the model it is assumed that the densities of the fluids are equal. We prove the existence of weak solutions for general initial data and arbitrarily large times with the aid of a parabolic Lipschitz truncation method, which preserves solenoidal velocity fields and was recently developed by Breit, Diening, and Schwarzacher.     

Continuous symmetric perturbations of planar power law forces

We show the existence of periodic solutions for continuous symmetric perturbations of certain planar power law problems.     

New exact solutions for some power law nonlinear diffusion equation

An extended auxiliary equation method for exact traveling wave solutions of constant coefficient nonlinear partial differential equations of evolution is proposed. This, together with a convenient characterization, affords new exact traveling wave solutions of some classes of nonlinear power law diffusion equations to be obtained.