Group Theoretical Analysis and Invariant Solutions for Unsteady Flow of a Fourth-Grade Fluid over an Infinite Plate Undergoing Impulsive Motion in a Darcy Porous Medium

2015 ◽  
Vol 70 (7) ◽  
pp. 483-497 ◽  
Author(s):  
Taha Aziz ◽  
Aeeman Fatima ◽  
Asim Aziz ◽  
Fazal M. Mahomed
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Lie Group ◽  
Group Analysis ◽  
Fourth Grade ◽  
Invariant Solutions ◽  
Conditional Symmetry ◽  
Partial Differential ◽  
Grade Fluid ◽  
Fourth Grade Fluid

AbstractIn this study, an incompressible time-dependent flow of a fourth-grade fluid in a porous half space is investigated. The flow is generated due to the motion of the flat rigid plate in its own plane with an impulsive velocity. The partial differential equation governing the motion is reduced to ordinary differential equations by means of the Lie group theoretic analysis. A complete group analysis is performed for the governing nonlinear partial differential equation to deduce all possible Lie point symmetries. One-dimensional optimal systems of subalgebras are also obtained, which give all possibilities for classifying meaningful solutions in using the Lie group analysis. The conditional symmetry approach is also utilised to solve the governing model. Various new classes of group-invariant solutions are developed for the model problem. Travelling wave solutions, steady-state solution, and conditional symmetry solutions are obtained as closed-form exponential functions. The influence of pertinent parameters on the fluid motion is graphically underlined and discussed.

scholarly journals Shock Wave Solution for a Nonlinear Partial Differential Equation Arising in the Study of a Non-Newtonian Fourth Grade Fluid Model

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Taha Aziz ◽  
A. Fatima ◽  
F. M. Mahomed
Keyword(s):  
Differential Equation ◽  
Shock Wave ◽  
Partial Differential Equation ◽  
Wave Solution ◽  
Fluid Model ◽  
Nonlinear Partial Differential Equation ◽  
Fourth Grade ◽  
Shock Wave Solution ◽  
Grade Fluid ◽  
Fourth Grade Fluid

This study focuses on obtaining a new class of closed-form shock wave solution also known as soliton solution for a nonlinear partial differential equation which governs the unsteady magnetohydrodynamics (MHD) flow of an incompressible fourth grade fluid model. The travelling wave symmetry formulation of the model leads to a shock wave solution of the problem. The restriction on the physical parameters of the flow problem also falls out naturally in the course of derivation of the solution.

scholarly journals Generating Functions for Bessel Functions

1959 ◽  
Vol 11 ◽  
pp. 148-155 ◽  
Author(s):  
Louis Weisner
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Generating Function ◽  
Lie Group ◽  
Generating Functions ◽  
Bessel Functions ◽  
Partial Differential ◽  
Systematic Determination ◽  
Image Position

On replacing the parameter n in Bessel's differential equation1.1by the operator y(∂/∂y), the partial differential equation Lu = 0 is constructed, where1.2This operator annuls u(x, y) = v(x)yn if, and only if, v(x) satisfies (1.1) and hence is a cylindrical function of order n. Thus every generating function of a set of cylindrical functions is a solution of Lu = 0.It is shown in § 2 that the partial differential equation Lu = 0 is invariant under a three-parameter Lie group. This group is then applied to the systematic determination of generating functions for Bessel functions, following the methods employed in two previous papers (4; 5).

Unsteady magnetohydrodynamic flow of a fourth grade fluid caused by an impulsively moving plate in a Darcy porous medium ߞ A group-theoretical analysis

2016 ◽  
Vol 30 (28n29) ◽  
pp. 1640007
Author(s):  
A. H. Carrim ◽  
Taha Aziz ◽  
F. M. Mahomed ◽  
Chaudry Masood Khalique
Keyword(s):  
Porous Medium ◽  
Flow Model ◽  
Invariant Solution ◽  
Nonlinear Partial Differential Equation ◽  
Fourth Grade ◽  
Conditional Symmetry ◽  
Moving Plate ◽  
Symmetry Approach ◽  
Grade Fluid ◽  
Fourth Grade Fluid

The effects of non-Newtonian fluids are investigated by means of an appropriate model studying the flow of a fourth grade fluid. The geometry of this model is described by the unsteady unidirectional flow of an incompressible fluid over an infinite flat plate within a porous medium. The fluid is electrically conducting in the presence of a uniform applied magnetic field. The classical Lie symmetry approach is utilized in order to construct group invariant solutions to the governing higher-order nonlinear partial differential equation (PDE). The conditional symmetry approach has also been utilized to solve the governing model. Some new classes of conditional symmetry solutions have been obtained for the model equation in the form of closed-form exponential functions. The invariant solution corresponding to the nontraveling wave type is considered to be the most significant solution for the fluid flow model under investigation since it directly incorporates the physical behavior of the flow model.

scholarly journals Permeability Models for Magma Flow through the Earth's Mantle: A Lie Group Analysis

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
N. Mindu ◽  
D. P. Mason
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Volume Fraction ◽  
Solitary Wave Solution ◽  
Nonlinear Partial Differential Equation ◽  
Group Analysis ◽  
Partial Differential ◽  
Magma Flow ◽  
Exponential Law ◽  
Flow Through

The migration of melt through the mantle of the Earth is governed by a third-order nonlinear partial differential equation for the voidage or volume fraction of melt. The partial differential equation depends on the permeability of the medium which is assumed to be a function of the voidage. It is shown that the partial differential equation admits, as well as translations in time and space, other Lie point symmetries provided the permeability is either a power law or an exponential law of the voidage or is a constant. A rarefactive solitary wave solution of the partial differential equation is derived in the form of a quadrature for the exponential law for the permeability.

scholarly journals Algebraic aspects of evolution partial differential equation arising in the study of constant elasticity of variance model from financial mathematics

Open Physics ◽  
2018 ◽  
Vol 16 (1) ◽  
pp. 31-36 ◽  
Author(s):  
Tanki Motsepa ◽  
Taha Aziz ◽  
Aeeman Fatima ◽  
Chaudry Masood Khalique
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Conservation Laws ◽  
General Theorem ◽  
Group Analysis ◽  
Partial Differential ◽  
Cev Model ◽  
Constant Elasticity ◽  
Constant Elasticity Of Variance ◽  
Evolution Partial Differential Equation

AbstractThe optimal investment-consumption problem under the constant elasticity of variance (CEV) model is investigated from the perspective of Lie group analysis. The Lie symmetry group of the evolution partial differential equation describing the CEV model is derived. The Lie point symmetries are then used to obtain an exact solution of the governing model satisfying a standard terminal condition. Finally, we construct conservation laws of the underlying equation using the general theorem on conservation laws.

FLOW OF A FOURTH GRADE FLUID

2002 ◽  
Vol 12 (06) ◽  
pp. 797-811 ◽  
Author(s):  
T. HAYAT ◽  
Y. WANG ◽  
K. HUTTER
Keyword(s):  
Magnetic Field ◽  
Differential Equation ◽  
Nonlinear Equation ◽  
Fourth Grade ◽  
The Other ◽  
Parallel Plates ◽  
Suction Velocity ◽  
Material Parameters ◽  
Grade Fluid ◽  
Fourth Grade Fluid

The governing nonlinear equation for the unsteady flow of an incompressible fourth grade fluid is modelled. The fluid is also subjected to a magnetic field. In addition, we investigate steady flow between parallel plates with one plate at rest and the other moving parallel to it at constant speed with a suction velocity normal to the plates. Boundary conditions play a significant role. We construct the numerical solution to the sixth order nonlinear differential equation. It is found that the velocity increases with the increase in the material parameters of the fourth grade terms of the fluid.

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