Page 300 - Intro to Tensor Calculus

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Pick an arbitrary simple closed curve C and place it in the fluid flow and define the line integral
I
K = ~v · ˆ t ds, where ds is an element of arc length along the curve C, ~v is the vector field defining the
e
C
velocity, and ˆ t is a unit tangent vector to the curve C. The integral K is called the circulation of the fluid
e
around the closed curve C. The circulation is the summation of the tangential components of the velocity
field along the curve C. The local vorticity at a point is defined as the limit
Circulation around C
lim = circulation per unit area.
Area→0 Area inside C
~
By Stokes theorem, if curl~v = 0, then the fluid is called irrotational and the circulation is zero. Otherwise
the fluid is rotational and possesses vorticity.
If we are only interested in the velocity field we can eliminate the pressure by taking the curl of both
sides of the equation (2.5.37). If we further assume that the fluid is incompressible we obtain the special
equations
∇· ~v = 0 Incompressible fluid, % is constant.
~
Ω=curl~v Definition of vorticity vector.
(2.5.38)
~
∂Ω µ ∗ 2 ~
~
+ ∇× (Ω × ~v)= ∇ Ω Results because curl of gradient is zero.
∂t %
Note that when Ω is identically zero, we have irrotational motion and the above equations reduce to the
~
Cauchy-Riemann equations. Note also that if the term ∇× (Ω × ~v) is neglected, then the last equation in
equation (2.5.38) reduces to a diffusion equation. This suggests that the vorticity diffuses through the fluid
once it is created.
Vorticity can be caused by a rigid rotation or by shear flow. For example, in cylindrical coordinates let
~
~
~
V = rω ˆ θ , with r, ω constants, denote a rotational motion, then curl V = ∇× V =2ω ˆe z , which shows the
e
vorticity is twice the rotation vector. Shear can also produce vorticity. For example, consider the velocity
~
~
field V = y ˆ 1 with y ≥ 0. Observe that this type of flow produces shear because |V | increases as y increases.
e
~
~
For this flow field we have curl V = ∇× V = − ˆ 3. The right-hand rule tells us that if an imaginary paddle
e
wheel is placed in the flow it would rotate clockwise because of the shear effects.
Scaled Variables
In the Navier-Stokes-Duhem equations for fluid flow we make the assumption that the external body
~
forces are derivable from a potential function φ and write b = −∇ φ [dyne/gm] We alsowant towrite the
Navier-Stokes equations in terms of scaled variables
~v % φ y
~ v = % = φ = , y =
v 0 % 0 gL L
p t x z
p = t = x = z =
p 0 τ L L
which can be referred to as the barred system of dimensionless variables. Dimensionless variables are intro-
duced by scaling each variable associated with a set of equations by an appropriate constant term called a
characteristic constant associated with that variable. Usually the characteristic constants are chosen from
various parameters used in the formulation of the set of equations. The characteristic constants assigned to
each variable are not unique and so problems can be scaled in a variety of ways. The characteristic constants