Bernoulli Equation

The momentum equation we have just derived allows us to develop the Bernoulli Equation after Bernoulli (1738). This equation basically connects pressure at any point in flow with velocity. It is one of the widely used equations in fluid dynamics to calculate pressure with the knowledge of velocity. We derive the equation for a stream tube and consider its generalisation , its applicability and limitations later.

Since we are interested in the fluid behaviour at a point consider a differential stream tube within a flow and a small control volume within it as shown in Fig.3.21. Since we are considering a stream tube, any flow takes place only along it and through the ends of it. The flow is therefore one dimensional in nature and takes places in a direction s along the stream tube. Accordingly, we denote the velocity by V_s. There is no flow across the tube. Let the length of the stream tube be ds.

Figure 3.21: Differential Control Volume for an one-dimensional steady flow

 

Since it is a small stream tube any property changes only slightly along it. If the area, velocity, density and pressure at the left hand end i.e., the inlet end, (1) be $A,V_s,\rho ~and ~p$. Let us treat the flow as incompressible (this restriction can be removed later). We assume that at the outlet end (2) the corresponding properties to be $A+dA, V+dV_s, \rho+d\rho~~and~~ p+dp$.

Let us now apply the momentum equation to the differential control volume we have considered. In any derivations or while solving problems involving fluid flows it helps to list out the assumptions made. Accordingly we start with a listing of the assumptions.

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