Page 408 - Introduction to Continuum Mechanics

392 Irrotational Flow of an Inviscid Incompressible Fluid of Homogeneous Density

O *J I *\
where v = vf+V2+V3 is the square of the speed. Therefore Eq. (6.21.6) becomes

Thus

where f(t) is an arbitrary function oft.
If the flow is also steady then we have

Equation (6.21.8) and the special case (6.21.9) are known as the Bernoulli's equations. In
addition to being a very useful formula in problems where the effect of viscosity can be
neglected, the above derivation of the formula shows that irrotational flows are always
dynamically possible under the conditions stated earlier. For whatever function #?, so long as
d<f) 2
v, = -~ and V <p = 0, the dynamic equations of motion can always be integrated to give
OXf
Bernoulli's equation from which the pressure distribution is obtained, corresponding to which
the equations of motion are satisfied.

Example 6.21.1

3
2
Given <p=x -3xy .
(a) Show that <p satisfies the Laplace equation.
(b) Find the irrotational velocity field.
(c) Find the pressure distribution for an incompressible homogeneous fluid, if at (0,0.0)
p =p 0 and Q=gz.

(d) If the plane y — 0 is a solid boundary, find the tangential component of velocity on the
plane.
Solution, (a) We have