# Preface

This section concerns about first order partial differential equations.

Introduction to Linear Algebra with Mathematica

# Fluid Problems

Liquids and gases are similar in the sense that they have no fixed shape like solids do. A liquid or a gas will shape themselves to fit perfectly to any container we pour them into. This similarity of liquids abd gases makes it possible to present their mathematical description in a unified way---called fluid dynamics. Liquids and gases are certainly different too. The former is for example very hard to compress in opposite to the latter. The part of fluid dynamics that concerns itself with easily compressible substances is callrd gas dynamics. Their modeling is based on the following three conservative laws;

1. conservation of mass;
2. conservation of moments;
3. conservation of energy.

During the past few decades, the flow of non-Newtonian fluids has several tech-nical and industrial applications, especially in many real life applications like: fuel combustion, engineering process, polymer solutions or melts, drilling mud, hydro-carbon oils, paper, and textile industries. Out of many models which have been used to describe the non – Newtonian behavior exhibited by certain fluids, there is not a single constitutive equation available by which all the non-Newtonian fluids can be analyzed. Because of this fact, several constitutive equations for such fluids have been studied and suggested by many researchers, among them were Garg and Rajagopal (1990), Parand and Babolgham (2012). The steady flow is the flow in which the properties at every point in the flow do not depend upon time. There is a slow change with time in the steady flow. When water flow out of a tap which has just been opened, this flow is unsteady to start with, but with time it becomes steady. Some flows, though unsteady, become steady under certain frames of reference which are referred to as pseudosteady flow (Bussuioc and Ratiu, 2003).

he fluids of the differential type have received special attention. Fluid of thesecond and third grade have been studied in various types of flow situations whichform a subclass of the fluids of the differential type. Boundary layer theories forfluid similar to a second grade fluid have been formulated by Garg and Rajagopal(1990) developed a boundary layer approximation for a second grade fluid.The third grade fluid is a subclass of non-Newtonian fluid which solved the sys-tem of non-linear differential equations governing the problem on the semi-infinitedomain without truncating it to a finite domain. The second grade fluid modelis able to predict the normal stress difference but it does not take into accountthe shear thinning or shear thickening phenomena that many fluids show for ex-ample, in water (Garg and Rajagopal, 1990). The third grade fluid models evenfor steady flow exhibits such characteristics such as honey which have higher vis-cosity and has porous medium, which reduces fluid flow in honey comb.

Korteweg–de Vries equation

The Korteweg–de Vries (KdV) equation is a mathematical model of waves on shallow water surfaces and it is written as
$\phi_t + \phi_{xxx} -6\,\phi\,\phi_x =0 .$

Harry Dym (HD) Equation

The nonlinear Dym equation (HD equation, for short) is
$u_t = u^3 u_{xxx} , \qquad$

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