CONTINUUM MECHANICS for ENGINEERS

and thus
which describes the pressure in the fluid.
Next, by integrating the middle equation above (for i = 2) twice with
respect to x 3, we obtain
with a and b constants of integration. But from the boundary conditions,
1. v 2 = 0 when x 3 = 0, therefore b = 0
2. σ 23 = 0 when x 3 = h, therefore
Finally, therefore, from the equation for v 2 we have by the substitution of
gh
a = ρ
µ
* sin
β ,
= 0 +( ) ( 3 − )
p p ρgcos β x h
2
v2 = 3 a 3 b
2
−ρgsin
β
x + x +
*
µ
gh
a = ρ
µ
having the profile shown in Figure E7.4-1B.
* sin
ρgsin β
v = ( 2h−x
) x
*
2µ
If the velocity field of a fluid is one for which the tensor W vanishes
identically, we say the flow is irrotational. In this case the vorticity vector w,
which is related to W by Eq 4.10-22, is also zero everywhere, so that for
irrotational flow
(7.4-7)
Finally, from the identity curl(grad φ ) = 0, we conclude that, for a flow
satisfying Eq 7.4-7, the velocity field may be given in terms of a velocity
potential, which we write as
(7.4-8)
Indeed, it may be shown that the condition curl v = 0 is a necessary and
sufficient condition for irrotationality and the consequence expressed in
β
2 3 3
1
1 1
wi= ε v = 0 or w = × v = curl v = 0
ijk k, j
2
2 2
vi , i or v
= φ = φ