Bernoulli Equation

Ignoring the frictional losses in steady flow energy equations, one can obtain the precise relation of pressure, velocity and elevation. This equation is called as Bernoulli equation developed in the year 1755. This equation is very famous and widely used with lot of restrictions. In general, all fluids are viscous and flows are associated with certain component of friction. In order to use Bernoulli equation correctly, one must confine the regions of flow which are nearly frictionless.

Consider an elemental fixed stream tube control volume of variable area A(s) and length ds as shown in Fig. 2.4.1. The fluid properties p, V and ρ vary along the streamline direction s and t while they are assumed to be uniform over the cross section A. The streamtube is oriented at any arbitrary angle θ with an elevation change .

Fig. 2.4.1: Schematic representation of frictionless flow in a streamtube.

Now, applying the principle of conservation of mass to this elemental control volume, one can write,

(2.4.11)

The linear momentum equation can also be applied in the stream-wise direction i.e.

(2.4.12)