BERNOULLI’S EQUATION FOR THE PRESSURE

The incompressible Euler equation (Eq. (1.31)) can be rewritten with the use of Eq. (2.5) as

^-4X^ + V^=/-V^ (2.29)

For irrotational flow £ = 0 and the time derivative of the velocity can be written as

^ = |v«I> = v(^) (2.30)

at at at)

Let us also assume that the body force is conservative with a potential E,

f = – VE (2.31)

If gravity is the body force acting and the z axis points upward, then E = ~gz. The Euler equation for incompressible irrotational flow with a conservative body force (by substituting Eqs. (2.30) and (2.31) into Eq. (2.29)) then becomes

V(£ + ^+2+f)-° <2’32>

Equation (2.32) is true if the quantity in parentheses is a function of time only:

n q2 ЗФ

Е + Е + + — =т (2.33)

p 2 at

This is the Bernoulli (Dutch/Swiss mathematician, Daniel Bernoulli (1700-1782)) equation for inviscid incompressible irrotational flow. A more useful form of the Bernoulli equation is obtained by comparing the quantities on the left-hand side of Eq. (2.33) at two points in the fluid, an aribtrary point and a reference point at infinity, say. The equation becomes

If the reference condition is chosen such that E„ = 0, Фоо = const., and q,» = 0 then the pressure p at any point in the fluid can be calculated from

€^z£ = — + e + —

p dt 2

If the flow is steady, incompressible but rotational the Bernoulli equation (Eq.

(2.34) ) is still valid with the time-derivative term set equal to zero if the constant on the right-hand side is now allowed to vary from streamline to streamline. (This is because the product q x £ is normal to the streamline d and their dot product vanishes along the streamline. Consequently, Eq. (2.34) can be used in a rotational fluid between two points lying on the same streamline.)