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Vxdr = 0 
or, for example, for flow only in two dimensions 
(v,i+v,j)x(dxi+dyj) = (v,dy-v,dx 1 k = 0 
which implies 
(v,dy-v,dx) = 0 
(6.2-3) 
(6.2-4) 
(6.2-5) 
There is a very nice demonstration of the differences among these lines using flow 
past an oscillating flat plate in the NCFMF film Flow Visualizufion, Steven J. Kline, 
principal. 
290 Chapter 6: Momentum Transfer in Fluids 
or, in terms of the stream function 
( - g d y - g d x ) = 0 
(6.24) 
However, for functions with continuous first partial derivatives near an 
arbitrary point in question (which, to reiterate, is true for functions of interest 
here), the function (here, the stream function) has a total differential. We 
consider a particular instant in time, say to, so the stream function is not a 
function of time and therefore we can write, again for two dimensions, the total 
differential as 
(6.2-7) 
We can see, then, that our collinearity condition requires that the total 
differential of the stream function be zero. 
dY = 0 (6.2-8) 
This is another way of saying that the value of the stream function does 
not change along a given streamline. 
Since the stream function has an exact differential, we can integrate between 
two arbitrary points in the flow field to obtain 
Iv:dY = Y z - Y , = constant (6.2-9) 
that is, the integral is independent of path. A function whose integral 
between any two arbitrary points is independent of path is called a point 
function. Common examples of point functions are the functions referred to in 
thermodynamics as state functions, e.g., the enthalpy, internal energy, entropy, 
etc. - as opposed to heat and work, which are path-dependent. 
To interpret the meaning of difference in the value of the stream function 
between streamlines, consider the two-dimensional (area) rate of flow across an 
Chapter 6: Momentum Transfer in Fluids 291 
arbitrary line, 0 connecting two streamlines. By definition, the area rate of flow 
across such a line is 
but this can be written as 
(6.2-10) 
(6.2- 1 1) 
where Q is measured along the line connecting the two streamlines. 
X 
Figure 6.2-1 Paths between streamlines 
Now consider this integral for a particular path between two streamlines, 
as illustrated in Figure 6.2-1, indicated by P1 (at constant y), where the unit 
normal is directed in the positive y-direction 
(6.2- 12) 
(6.2- 13) 
(6.2- 14) 
In other words, the difference in the values of the stream functions on the 
two streamIines corresponds to the volumetric flow rate between these 
streamlines. Because of independence of path, we know that this is true for any 
292 Chapter 6: Momentum Transfer in Fluids 
other path we might choose between the streamlines, whether it be at constant 
x, as is true for P2 (which we could have chosen for the illustration rather than 
P1 with equal ease), or some arbitrary convoluted path such as P3. 
6.2.1 Irrotational flow 
The relative velocity between two fluid particles P and P' whose coordinates 
are Xk and x k + dxk, respectively, in a velocity field instantaneously described by 
the vector field Vj(Xk), is expressed by the total derivative 
where the partial derivatives are evaluated at xk. We can write the vector gradient 
as the sum of its antisymmetric and symmetric parts 
and define 
to obtain 
(6.2.1-3) 
(6.2.1-4) 
We now proceed to show that dvk* corresponds to the relative velocities of a 
instantaneous rigid body rotation of the neighborhood of the particle P about an 
axis through P.5 
We first define the dual vector of the antisymmetric part of the vector 
gradient as 
(6.2.1-5) 
We can also show that d q * * corresponds to the instantaneous pure deformation of 
this neighborhood, for example, see Prager, W. (1961). Introduction to Mechanics of 
Continua. Boston, MA, Ginn and Company, p. 62 ff.; however, this is not essential 
to our argument at the moment. 
Chapter 6: Momentum Transfer in Fluids 293 
which implies 
1 
a i = 2 &ijk (6.2.1-6) 
The vector Wi is called the vorticity of the velocity field, and lines of constant 
vorticity are called vortex lines, terminology similar to that used for streamlines. 
We can rewrite this expression as 
(6.2.1-7) 
(6.2.1-8) 
(6.2.1-9) 
We can obtain the dual tensor of the vorticity vector and use it to define 
the vorticity tensor 
(6.2.1 - 10) 
(6.2.1-1 1) 
(6.2.1 - 1 2) 
(6.2.1 - 1 3) 
(6.2.1-14) 
(6.2.1 - 15) 
(6.2.1 - 1 6) 
(6.2.1 - 1 7) 
This permits us to write the antisymmetric part of the relative velocity, 
dVk*, a~ 
dv; = Eijk mi dxj (6.2.1 - 18) 
From kinematics we can recognize that the velocity of points on a solid body 
rotating about an axis through a point x(O) which can be seen from the sketch 
294 Chapter 6: Momentum Transfer in Fluids 
is described by 
where w is the angular velocity, the angular velocity vector is normal to the 
plane of Xi and Xio and points in the direction of a right-hand screw turning in the 
same direction as the rigid body. Comparing with the equation for dvk* we can 
see that dvk* represents a rigid body rotation about an axis through P. 
If the vorticity vanishes identically, the motion of a continuum is called 
irrotational. For such a flow 
However, because of the identity 
&ilk 3, &@ = 0 3 curlgradQ = V x V@ = 0 (6.2.1-21) 
which is valid for any scalar function F, a velocity field with a vanishing curl is 
a gradient field, i.e., 
vi = aio v = VQ, 
Equating components 
(6.2.1-22) 
(6.2.1-23) 
(6.2.1-24) 
Chapter 6: Momentum Transfer in Fluids 295 
6.3 Potential Flow 
In Section 6.1 we considered static fluids. In such fluids, viscosity played no 
part because there were no velocity gradients, the fluid being everywhere 
stationary. There is another important class of fluid behavior in which viscosity 
effects play no part, even though this time the fluid is in motion - behavior in 
which the viscosity of the fluid can be assumed to be zero. Such an assumption 
is useful in two situations: 
. where the viscosity of the fluid is so small, or 
where the velocity gradients are so small 
that viscous forces play a relatively unimportant role in the problem to be 
solved. 
Many real-world flow problems can be successfully modeled as being 
steady state 
irrotational flow of an 
incompressible and 
inviscid (viscosity of zero) fluid. 
Such fluids are called ideal fluids, and the corresponding flow model is called 
potential flow. We now assemble the basic equations constituting the model 
for such flows. 
For constant density the continuity equation reduces to 
( v - p v ) + - & dP = 0 
p ( v . v ) + - & aP = 0 
v - v = 0 
and the irrotationality condition implies that 
(6.3-1) 
v x v = 0 (6.3 -2) 
296 Chapter 6: Momentum Transfer in Fluids 
The equation of motion for a constant density Newtonian fluid - the Navier 
Stokes equation 
(6.3-3) = - v p + p v * v + p g P D , 
when applied to flow of an inviscid fluid reduces to Euler's equation 
ar 
P [ g + ( v * V ) v ] = - v p + p g (6.34) 
which at steady state becomes 
p ( v * v ) v = - v p + p g (6.3-5) 
The left-hand side of this relation can be rewritten by defining6 
( v * v ) v = $V(V.V)-[V x ( v x v ) ] (6.3-6) 
c.f . Bird, R . B., W. E. Stewart, et al. (1960). Transport Phenomena. New York, 
Wiley, pp. 726, 731. This definition is applicable in both rectangular and curvilinear 
coordinates; however, for simplicity illustrating the equivalence using index 
notation (rectangular coordinates) 
( v . v ) v = + v ( v - v ) - [ v x ( v x v ) ] j 
v j a j v i = a i ( 3 v j v j ) - e i j k v j e ~ , a I v , 
'j a j vi = (4 'j Vj)-EkljEklmvjal vm 
v j a j v, = a, (3 v j Vj) - (6, 6jm- 6im6p)