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14.2 Accelerated and Decelerated Fluid Flows 397
14.2 Accelerated and Decelerated Fluid Flows
14.2.1 Stokes First Problem
Stokes (1851) was one of the first scientists who provided an analytical solu-
tion for an unsteady, one-dimensional flow problem, namely the solution for
the fluid motion induced by the sudden movement of a plate and the related
momentum diffusion into an infinitely extended fluid lying above the plate.
In order to understand better the induced fluid movement, also observed
in practice, in this and the subsequent sections flow processes are discussed
which occur in fluids due to imposed wall movements. The simplest examples
on the subject discussed in this chapter on wall-induced fluid motions concern
the movement of plane plates. However, the general physical insights gained
from these examples are not limited to the plate-induced fluid motions only,
but can also be transferred to axially symmetrical flows (rotating cylinders).
Figure 14.1 shows schematically the velocity distribution which takes place
in a fluid due to a wall moved at a velocity U
0
. The flow setting in, due to
the movement of the plate, can be expressed mathematically as follows:
For t<0:U
1
(x
2
,t)=0
For t ≥ 0:U
1
(x
2
=0,t)=U
0
U
1
(x
2
→∞,t)=0
(14.29)
As a consequence of the fluid viscosity (molecular momentum transport), the
momentum of the fluid layer, moved in the immediate vincinity of the plate,
is transferred to the layers that are further away. With progress of time, layers
that are further away from the moved wall are also included in the induced
fluid motion. The differential equation describing this entire process reads:
∂U
1
∂t
= ν
∂
2
U
1
∂x
2
2
(14.30)
Increase in time
t
U
0
x
2
x
1
U
0
For
x
2
no
fluid motion
Fig. 14.1 Sketch of the flow induced by a plane wall suddenly set in motion