Lie-group method for unsteady flows in a semi-infinite expanding or contracting pipe with injection or suction through a porous wall

The unsteady incompressible laminar flow in a semi-infinite porous circular pipe with injection or suction through the pipe wall whose radius varies with time is considered. The present analysis simulates the flow field by the burning of inner surface of cylindrical grain in a solid rocket motor, in which the burning surface regresses with time. We apply Lie-group method for determining symmetry reductions of partial differential equations. Lie-group method starts out with a general infinitesimal group of transformations under which given partial differential equations are invariant, then, the determining equations are derived [Ibragimov, Elementary Lie Group Analysis and Ordinary Differential Equations, Wiley, New York, 1999; Hydon, Symmetry Methods for Differential Equations, Cambridge University Press, Cambridge, 2000; Olver, Applications of Lie Groups to Differential Equations, Springer, New York, 1986; Seshadri, Na, Group invariance in engineering boundary value problems, Springer, New York, 1985; Yi, Fengxiang, Lie symmetries of mechanical systems with unilateral holonomic constraints, Chinese Sci. Bull. 45 (2000) 1354–1358; Moritz, Schwalm, Uherka, Finding Lie groups that reduce the order of discrete dynamical systems, J. Phys. A: Math. 31 (1998) 7379–7402; Nucci, Clarkson, The nonclassical method is more general than the direct method for symmetry reductions. An example of the Fitzhugh–Nagumo equation, Phys. Lett. A 164 (1992) 49–56; Basarab, Lahno, Group classification of nonlinear partial differential equations: a new approach to resolving the problem, Proceedings of Institute of Mathematics of NAS of Ukraine, vol. 43, 2002, pp. 86–92; Burde, Expanded Lie group transformations and similarity reductions of differential equations, Proceedings of Institute of Mathematics of NAS of Ukraine, vol. 43, 2002, pp. 93–101; Gandarias, Bruzon, Classical and nonclassical symmetries of a generalized Boussinesq equation, J. Nonlinear Math. Phys. 5 (1998) 8–12; Hill, Solution of Differential Equations by Means of One-Parameter Groups, Pitman Publishing Co., 1982]. The determining equations are a set of linear differential equations, the solution of which gives the transformation function or the infinitesimals of the dependent and independent variables. After the group has been determined, a solution to the given partial differential equation may be found from the invariant surface condition such that its solution leads to similarity variables that reduce the number of independent variables in the system. Effect of the cross-flow Reynolds number Re and the dimensionless wall expansion ratio $\alpha$ on velocity, flow streamlines, axial and radial pressure drop, and wall shear stress has been studied both analytically and numerically and the results are plotted.     

Homotopy analysis method for the asymmetric laminar flow and heat transfer of viscous fluid between contracting rotating disks

The present work investigates the effects of the disks contracting, rotation, heat transfer and different permeability on the viscous fluids and temperature distribution between two heated contracting rotating disks. Two cases are considered. For the first case, we neglect the viscous dissipation effects in the energy equation and reduce the Navier-Stokes equations and energy equation into nonlinear coupled ODEs by introducing the Von Kármán type similarity transformations. The effects of various physical parameters like expansion ratio, Prandtl number, Reynolds number and rotation ratio on the velocity and temperature are discussed in detail. The second and more general case is that we consider the viscous dissipation in the energy equation. Under this assumption, the energy equation is reduced to a ordinary differential equation including the Eckert number, whose solution also is solved by HAM.     

Exact solutions using symmetry methods and conservation laws for the viscous flow through expanding–contracting channels

We present a range of definitions and methods dealing with the reduction of partial differential equations on the basis of the underlying symmetry structure, conservation laws and a combination of these. The method is used to reduce a complex system to an easy-to-handle second-order ordinary differential equation system independent of restrictions on any physical parameters. In particular, we construct exact solutions of a system modelling viscous flow between slowly expanding and contracting walls.     

Exact solution for flow in a porous pipe with unsteady wall suction and/or injection

This paper presents an extension of the exact solution of the steady laminar axisymmetric flow in a straight pipe of circular cross section with porous wall, given by R.M. Terrill, to the case of unsteady wall injection and/or suction. The cases of the pulsating parabolic profile and of the developed pulsating flow are investigated as examples. The pulsating flow in porous ducts has many applications in biomedical engineering and in other engineering areas.     

Multiple positive solutions for a semilinear equation with prescribed singularity

Variational methods are used to prove the existence of multiple positive solutions for a semilinear equation with prescribed finitely many singular points. Some exact local behavior for positive solutions are also given.     

Application of He's methods for laminar flow in a porous‐saturated pipe

In this paper the problem of fully developed laminar steady forced convection inside a porous‐saturated pipe with uniform wall temperature is presented and the homotopy perturbation method (HPM) and the variational iteration method (VIM) are employed to solve the differential equations governing the problem. The obtained results are valid for the whole solution domain with high accuracy. These methods can be easily extended to other linear and nonlinear equations and so can be found widely applicable in engineering and science. Copyright © 2009 John Wiley & Sons, Ltd.     

Slow laminar flow in a converging or diverging channel with suction at one wall and blowing at the other wall

Résumé On établit et résoud l'équation de l'écoulement laminaire dans un canal convergent ou divergent avec aspiration de la couche limite sur l'une des parois et soufflement sur l'autre.     

Homotopy analysis method for the heat transfer in a asymmetric porous channel with an expanding or contracting wall

The flow of a viscous incompressible fluid between two parallel plates due to the normal motion of the porous upper plate is investigated and an analysis is made to determine the heat and mass transfer. The unsteady Navier–Stokes equations are reduced to a generalization of the Proudman–Johnson equation retaining the effect of wall motion using a suitable similarity transformation. The analytical solution for stream function and heat transfer characteristics are obtained by employing homotopy analysis method. The effects of various physical parameters like expansion ratio, Prandtl number, Reynolds number on various momentum and heat transfer characteristics are discussed in detail.     

Multiple soliton solutions for the sixth-order Ramani equation and a coupled Ramani equation

In this work, the completely integrable sixth-order nonlinear Ramani equation and a coupled Ramani equation are studied. Multiple soliton solutions and multiple singular soliton solutions are formally derived for these two equations. The Hirota’s bilinear method is used to determine the two distinct structures of solutions. The resonance relations for the three cases are investigated.     

Falkner-Skan equation for flow past a stretching surface with suction or blowing: Analytical solutions

The simultaneous effects of suction and injection on tangential movement of a nonlinear power-law stretching surface governed by laminar boundary layer flow of a viscous and incompressible fluid beneath a non-uniform free with stream pressure gradient is considered. The self-similar flow is governed by Falkner-Skan equation, with transpiration parameter γ, wall slip velocity λ and stretching sheet (or pressure gradient) parameter β. The exact solution for β = −1 and three closed form asymptotic solutions for β large, large suction γ, and λ → 1 have also been presented. Dual solutions are found for β = −1 for each value of the transpiration parameter, including the non-permeable surface, for each prescribed value of the wall slip velocity λ. The large β asymptotic solution also dual with respect to wall slip velocity λ, but do not depend on suction and blowing. The critical values of γ, β and λ are obtained and their significance on the skin friction and velocity profiles is discussed. An approximate solution by integral method for a trial velocity profile is presented and results are compared with the exact solutions.     

Multiple solutions for a class of biharmonic equations with a nonlinearity concave at the origin

In this paper, we investigate the existence of multiple solutions for a class of biharmonic equations where the nonlinearity involves a concave term at the origin. The solutions are obtained from the versions of mountain pass lemma and linking theorem.     

半渗透涨缩管道内微极性流动解析求解

分析了半渗透涨缩管道内的微极性流体的流动．应用合适的相似变换,将控制方程转化为常微分方程组．为了得到该问题的解析解,应用同伦分析方法得到该问题的速度表达式．并且用图形分析了各个不同参数,特别是膨胀系数对速度场和微旋转角速度的影响．     

Existence of two solutions for the Bahri-Coron problem in an annular domain with a thin hole

We will show that the problem

     

Multiple solutions to the pure critical exponent problem in domains with a hole of arbitrary size

We consider the problem \(-\Delta u = \left\vert u\right\vert ^{2^\ast-2} u\,{\rm in}\,\Omega, \quad u = 0\,{\rm on}\,\partial\Omega,\) where Ω is a bounded smooth domain in \(\mathbb{R}^{N}\), N ≥ q3, and \(2^{\ast}=\frac{2N}{N-2}\) is the critical Sobolev exponent. We assume that Ω is annular shaped, i.e. there are constants R ₂ >  R ₁ >  0 such that \(\{x \in \mathbb{R}^{N} : R_{1} < |x| < R_{2}\} \subset \Omega\) and \(0 \not\in \Omega.\) We also assume that Ω is invariant under a group Γ of orthogonal transformations of \(\mathbb{R}^{N}\) without fixed points. We establish the existence of multiple sign changing solutions if, either Γ is arbitrary and R ₁/R ₂ is small enough, or R ₁/R ₂ is arbitrary and the minimal Γ-orbit of Ω is large enough. We believe this is the first existence result for sign changing solutions in domains with holes of arbitrary size. The proof takes advantage of the invariance of this problem under the group of Möbius transformations.     

Nonsimilar solutions for double-diffusion boundary layers on a sphere in micropolar fluids with constant wall heat and mass fluxes

This work presents nonsimilar boundary layer solutions for double-diffusion natural convection near a sphere with constant wall heat and mass fluxes in a micropolar fluid. A coordinate transformation is employed to transform the governing equations into nondimensional nonsimilar boundary layer equations and the obtained boundary layer equations are then solved by the cubic spline collocation method. Results for the local Nusselt number and the local Sherwood number are presented as functions of the vortex viscosity parameter, Schmidt number, buoyancy ratio, and Prandtl number. Higher vortex viscosity tends to retard the flow, and thus decreases the local convection heat and mass transfer coefficients, raising the wall temperature and concentration. Moreover, the local convection heat and mass transfer coefficients near a sphere in Newtonian fluids are higher than those in micropolar fluids.     

The uniqueness and existence of solutions for the 3D Helmholtz equation in a step‐index waveguide with unbounded perturbation

In this paper, we study the 3D Helmholtz equation in a step‐index waveguide with unbounded perturbation, allowing the presence of guided waves. Our assumptions on the perturbed and source terms are too few. On the basis of the Green's function for the 3D homogeneous Helmholtz equation in a step‐index waveguide without perturbation, we introduce a generalized (out‐going) Sommerfeld–Rellich radiation condition, and then we prove the uniqueness and existence of solutions for the studied 3D Helmholtz equation satisfying our radiation condition. Copyright © 2012 John Wiley & Sons, Ltd.