Page 154 - Bird R.B. Transport phenomena

138 Chapter 4 Velocity Distributions with More Than One Independent Variable

The boundary conditions for this equation for ф(х, у) are
дф
B.C.I: aty = 0 -^ = v = 0 forx>0 (4.4-21)
/ y
дф
B.C. 2: at у = 0, ^ - = -v x = 0 for x > 0 (4.4-22)

B.C. 3: asy->oo, -£-= - Vx^>-v x forx>0 (4.4-23)

B.C. 4: at x = 0, j - = -v x = -v x for у > 0 (4.4-24)

Inasmuch as there is no characteristic length appearing in the above relations, the
method of combination of independent variables seems appropriate. By dimensional argu-
ments similar to those used in Example 4.1-1, we write

g = Щту), where V = yJ\^ (4.4-25)

The factor of 2 is included to avoid having any numerical factors occur in the differential
equation in Eq. 4.4-27. The stream function that gives the velocity distribution in Eq. 4.4-25 is
fy
ф(х,у) = -^/2v vxf{r)), where /(77) = U'(rj)drj (4.4-26)
x
Jo
This expression for the stream function is consistent with Eq. 4.4-25 as may be seen by using
the relation v x = -дф/ду (given in Table 4.2-1). Substitution of Eq. 4.4-26 into Eq. 4.4-20 gives
-//"=/'" (4.4-27)
Substitution into the boundary conditions gives

B.C. 1 and 2: at 77 = 0, / = 0 and /' = 0 (4.4-28)
B.C. 3 and 4: asrj-^cx), /'-> 1 (4.4-29)

Thus the determination of the flow field is reduced to the solution of one third-order ordinary
differential equation.
This equation, along with the boundary conditions given, can be solved by numerical in-
tegration, and accurate tables of the solution are available. 34 The problem was originally
solved by Blasius 7 using analytic approximations that proved to be quite accurate. A plot of
his solution is shown in Fig. 4.4-3 along with experimental data taken subsequently. The
agreement between theory and experiment is remarkably good.
The drag force on a plate of width W and length L may be calculated from the dimen-
.
sionless velocity gradient at the wall,/"(0) = 0.4696 . . as follows:

^- )\ dxdz
i
T

= 1.328 VpfxLW VI (4.4-30)
This result has also been confirmed experimentally. ' 3 4
Because of the approximations made in Eq. 4.4-10, the solution is most accurate at
large local Reynolds numbers; that is, Re = xv^/v » 1. The excluded region of lower
v
Reynolds numbers is small enough to ignore in most drag calculations. More complete