Page 299 - Intro to Tensor Calculus

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We now consider various special cases of the Navier-Stokes-Duhem equations.
~
~
Special Case 1: Assume that b is a conservative force such that b = −∇ φ. Also assume that the viscous
force terms are zero. Consider steady flow ( ∂~v = 0) and show that equation (2.5.28) reduces to the equation
∂t
−1
(~v ·∇)~v = ∇ p −∇ φ% is constant. (2.5.32)
%
Employing the vector identity
1
(~v ·∇)~v =(∇× ~v) × ~v + ∇(~v · ~v), (2.5.33)
2
~
we take the dot product of equation (2.5.32) with the vector ~v. Noting that ~v · [(∇× ~v) × ~v]= 0weobtain

p 1 2
~v ·∇ + φ + v =0. (2.5.34)
% 2
This equation shows that for steady flow we will have

p 1 2
+ φ + v = constant (2.5.35)
% 2
along a streamline. This result is known as Bernoulli’s theorem. In the special case where φ = gh is a
p v 2
force due to gravity, the equation (2.5.35) reduces to + + gh = constant. This equation is known as
% 2
Bernoulli’s equation. It is a conservation of energy statement which has many applications in fluids.

~
Special Case 2: Assume that b = −∇ φ is conservative and define the quantity Ω by
1
~ ω = Ω (2.5.36)
Ω= ∇× ~v =curl~v
2
as the vorticity vector associated with the fluid flow and observe that its magnitude is equivalent to twice
the angular velocity of a fluid particle. Then using the identity from equation (2.5.33) we can write the
Navier-Stokes-Duhem equations in terms of the vorticity vector. We obtain the hydrodynamic equations
∂~v 1 2 1 1
~
~
+ Ω × ~v + ∇ v = − ∇ p −∇ φ + f viscous , (2.5.37)
∂t 2 % %
~
where f viscous is defined by equation (2.5.30). In the special case of nonviscous flow this further reduces to
the Euler equation
∂~v 1 2 1
~
+ Ω × ~v + ∇ v = − ∇ p −∇ φ.
∂t 2 %
If the density % is a function of the pressure only it is customary to introduce the function
Z
p dp dP 1
P = so that ∇P = ∇p = ∇p
c % dp %
then the Euler equation becomes
∂~v 1 2
~
+ Ω × ~v = −∇(P + φ + v ).
∂t 2
Some examples of vorticies are smoke rings, hurricanes, tornadoes, and some sun spots. You can create
a vortex by letting water stand in a sink and then remove the plug. Watch the water and you will see that
a rotation or vortex begins to occur. Vortices are associated with circulating motion.