CONTINUUM MECHANICS for ENGINEERS

where dx i is a differential tangent vector along the streamline. This is the
well-known Bernoulli equation. If the motion is steady, the time function G(t)
resulting from the integration reduces to a constant G, which may vary from
one streamline to another. Furthermore, if the flow is also irrotational, a
unique constant G 0 is valid throughout the flow.
When gravity is the only force acting on the body, we write Ω = gh where
g = 9.81 m/s 2 is the gravitational constant and h is a measure of the height
above a reference level in the fluid. If h p = P/g is defined as the pressure head
and h v = v 2 /2g as the velocity head, Bernoulli’s equation for incompressible
fluids becomes
(7.5-3)
Recall that by Eq 2.8-5 in Chapter Two we introduced Stoke’s theorem,
which relates the line integral around a closed curve to the surface integral
over its cap. By this theorem we define the velocity circulation Γ c around a
closed path in the fluid as
(7.5-4)
where n i is the unit normal to the surface S bounded by C and dx i is the
differential tangent element to the curve C. Note that, when the flow is
irrotational, curl v = 0 and the circulation vanishes. If we take the material
derivative of the circulation by applying Eq 5.2-7 to Eq 7.5-4 we obtain
(7.5-5)
For a barotropic, inviscid fluid with conservative body forces, this integral
may be shown to vanish, leading to what is known as Kelvin’s theorem of
constant circulation.
Problems
p v
h h h h
g g G
2
+ p + v = + + =
ρ 2
= vdx = v ndS ∫ ∫
Γc i i εijk
k,j i
S
∫
( )
˙ = vdx ˙ + vdv
Γ c i i i i
7.1 Introduce the stress deviator S ij and the viscous stress deviator
1
Tij = τij − δ τ
into Eq 7.1-5 to prove that =
.
3 ij kk Sij Tij
0