General

We now move to the Differential analysis of fluid motion. This is sharply different from the analysis done in the previous chapter using control volumes. Our interest now will be on a description of flow at any given point in the flow than on overall effects of flow on a control volume. Reasons why we require such an analysis are evident. There are numerous situations where one needs a point to point description of flow. One may require the distribution of shear stress on the surface of an aeroplane wing or the distribution of heat transfer coefficient in a piping in an air-conditioning system. This necessitates a more complete knowledge of the flow field than that provided by the Integral Approach of the previous chapter.

Even here we appeal to the general laws of mechanics to give us the equations to solve. These are the conservation laws for mass, momentum and energy as before. The resulting equations now are Differential Equations (and hence the name Differential Approach) while the integral analysis gave us algebraic equations to solve. From this end, differential equations are more complicated.

Remember we need to take into account all the forces acting when we write the momentum equation. We will see that the equations get very involved when viscous forces are considered along with other forces. This leads to what are called the Navier-Stokes Equations. We postpone the discussion of these equations to a later chapter and limit ourselves to simple flows which lend themselves to simple equations which are easily solved. As a result we consider only inviscid flows in this chapter. This leads to a simple analysis. In fact we will be solving only the continuity equation for mass to calculate velocity components. Pressure is obtained from Bernoulli equation. Of course, various assumptions will have to made to make the analysis easier. We discuss these in course of the text.

We first derive the continuity equation which is a statement of the fact that mass is conserved. Then we introduce the stream function which is a powerful concept in fluid dynamics. By anaysing the kinematics of fluid motion we proceed to introduce concepts of Circulation and Irrotationality. Definition of Velocity Potential follows. We then write down the stream functions and velocity potentials for some of the simple flows like a uniform flow, source and sink flow and vortex flow. These flows are then superposed to arrive at solutions for complicated flows. Flow about a circular cylinder is then analysed in some detail.

(c) Aerospace, Mechanical & Mechatronic Engg. 2005
University of Sydney