# 10.1 Introduction

The mathematical complication of the Naiver–Stokes equations suggests that a simplified approached can be employed. N–S equations are a second non–linear partial equations. Hence, the simplest step will be to neglect the second order terms (second derivative). From a physical point of view, the second order term represents the viscosity effects. The neglection of the second order is justified when the coefficient in front of the this term, after non–dimensionalzing, is approaching zero. This coefficient in front of this term is \(1/Re\) where \(Re\) is Reynold's number. A large Reynolds number means that the coefficient is approaching zero. Reynold's number represents the ratio of inertia forces to viscous forces. There are regions where the inertia forces are significantly larger than the viscous flow. Experimental observations show that when the flow field region is away from a solid body, the inviscid flow is an appropriate model to approximate the flow. In this way, the viscosity effects can be viewed as a mechanism in which the information is transferred from the solid body into depth of the flow field. Thus, in a very close proximity to the solid body, the region must be considered as viscous flow. Additionally, the flow far away from the body is an inviscid flow. The connection between these regions was proposed by Prandtl and it is referred as the boundary layer. The motivations or benefits for such analysis are more than the reduction of mathematical complexity. As it was indicated earlier, this analysis provides an adequate solution for some regions. Furthermore the Potential Flow analysis provides several concepts that obscured by other effects. These flow patterns or pressure gradients reveal several ``laws'' such as Bernoulli's theorem, vortex/lift etc which will be expanded. There are several unique concepts which appear in potential flow such as Add Mass, Add Force, and Add Moment of Inertia otherwise they are obscured with inviscid flow. These aspects are very important in certain regions which can be evaluated using dimensional analysis. The determination of what regions or their boundaries is a question of experience or results of a sophisticated dimensional analysis which will be discussed later. The inviscid flow is applied to incompressible flow as well to compressible flow. However, the main emphasis here is on incompressible flow because the simplicity. The expansion will be suggested when possible.

### Contributors

Dr. Genick Bar-Meir. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or later or Potto license.