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It is important to note that it is only the spatial derivatives of ! that matter, an arbitrary

function of time can always be added to the velocity potential without changing its

physical content. The momentum equation (7.7.3) is, again ignoring friction,

!
!u ! ! %p 1 !
2
+ " # u = $ $ %' $ % | u | (10.2.26)
a
!t & 2
If

1) The fluid is irrotational so that ω =0 ( and so (8.2.25) applies,
a
!p p d # p
2) The fluid is barotropic so that = ! $
" " p ( #)

then,

( "# 1 p d $ p +
2
! ) + | !# | + ' +& , = 0 (10.2.27)
* "t 2 %( $ p ) -

Since the gradient of the quantity in the curly brackets is zero, that quantity must be a
function, at most, only of time, i.e.,

!" 1 p d $ p
2
+ | #" | + ' +& = C(t) (10.2.28)
!t 2 %( $ p )

It is not hard to show that the function C(t) can be taken to be zero. Simply adding a

function of time only to ! leaves the velocity unchanged so that

t
"
! = ˆ ! + C(t ')dt ' (10.2.29)

leaves the velocity unaltered but the equation (8.2.28) no longer contains the “constant”

on the right hand side. Therefore, the Bernoulli equation for irrotational motion is,

p
!" 1 d $ p (10.2.30)
+ | #" | + ' +& = 0
2
!t 2 %( $ p )

Chapter 10 12