Page 136 - Bird R.B. Transport phenomena

§4.2 Solving Flow Problems Using a Stream Function 121

Here v° is chosen to be a complex function of y, so that v (y, t) will differ from v (0, t) both in
x
x
amplitude and phase. We substitute this trial solution into Eq. 4.1-44 and obtain
^Г е'А (4.1-49)
dy 2 J
Next we make use of the fact that, if Щг^ги} = 9^{zw}, where z and z are two complex quan-
2
}
2
tities and w is an arbitrary complex quantity, then z x = z . Then Eq. 4.1-49 becomes
2
_ (^j o = о (4.1-50)
v
with the following boundary conditions:
B.C1: aty = 0, v° = v o (4.1-51)
B.C. 2: aty = °°, ° = 0 (4.1-52)
v
Equation 4.1-50 is of the form of Eq. C.l-4 and has the solution
o ^ y -Vu^>y (4.1-53)
v = c + Cie
Since V/ = ±(1/V2)(1 + /), this equation can be rewritten as
° = q^vW^u+oy ^-VWWHOJI (4.1-54)
v +
The second boundary condition requires that Q = 0, and the first boundary condition gives
C 2 = v . Therefore the solution to Eq. 4.1-50 is
0
o ^-\Яы\ ,у (4.1-55)
v = у +
From this result and Eq. 4.1-48, we get

— 71 p-vw/2*/y«j(--/(Vw/2i^-wOl (Л Л Ц,&\

or finally
v (y, t) = vtf'^^v cos (cot - Va>/2vy) (4.1-57)
x
In this expression, the exponential describes the attenuation of the oscillatory motion—that is,
the decrease in the amplitude of the fluid oscillations with increasing distance from the plate.
In the argument of the cosine, the quantity -\/o)/2vy is called the phase shift; that is, it de-
scribes how much the fluid oscillations at a distance у from the wall are "out-of-step" with the
oscillations of the wall itself.
Keep in mind that Eq. 4.1-57 is not the complete solution to the problem as stated in Eqs.
4.1-44 to 47, but only the "periodic-steady-state" solution. The complete solution is given in
Problem 4D.1.

.2 SOLVING FLOW PROBLEMS USING A STREAM FUNCTION

Up to this point the examples and problems have been chosen so that there was only one
nonvanishing component of the fluid velocity. Solutions of the complete Navier-Stokes
equation for flow in two or three dimensions are more difficult to obtain. The basic pro-
cedure is, of course, similar: one solves simultaneously the equations of continuity and
motion, along with the appropriate initial and boundary conditions, to obtain the pres-
sure and velocity profiles.
However, having both velocity and pressure as dependent variables in the equation
of motion presents more difficulty in multidimensional flow problems than in the sim-
pler ones discussed previously. It is therefore frequently convenient to eliminate the