Page 455 - Modelling in Transport Phenomena A Conceptual Approach

10.1. MOMENTUM TRANSPORT 435

10.1.1 Solution for Short Times

Once the lower plate is set in motion, only the thin layer adjacent to the lower
plate feels the motion of the plate during the initial stages. This thin layer does
not feel the presence of the stationary plate at x = B at all. For a fluid particle
within this layer, the upper plate is at infinity. Therefore, the governing equation
together with the initial and boundary conditions are expressed as

(10.1-49)

at t=O 21, = 0 (10.1-50)
at x=O v, = v (10.1-51)
at x=ca v, = 0 (10.1-52)
In the literature, this problem is generally :nown as Stc.&s’first prdem’. Note
that there is no length scale in this problem. Since the boundary condition at
x = 00 is the same as the initial condition, the problem can be solved by the
similarity analysis. The solution of this problem is given in Section B.3.6.2 in
Appendix B and the solution is

(10.1-53)

The drag force exerted on the plate is given by

-- APV
-a (10.1-54)

Finally, note that when z/m = 2, Eq. (10.1-53) becomes

v,
- = 1 - erf(2) = 1 - 0.995 = 0.005
V
indicating that v, 21 0. Therefore, the penetration distance for momentum, 6, is
given by
S=4G (10.1-55)

The penetration distance changes with the square root of the momentum dif€mivity
and is independent of the plate velocity. The momentum diffusivities for water and
air at 20°C are 1 x and 15.08 x 10-6m2/s, respectively. The penetration
distances for water and air after one minute are 3.1 cm and 12 cm, respectively.
Some authors refer to this problem as the Rayleigh probfem.