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Here,
v
is the velocity of the fluid, p is the pressure, % is the density, is the vis-
cosity, and
g
denotes the acceleration of gravity (i.e., gravity forces). The velocity
and the pressure are the primary unknowns in these PDEs; all the other quantities
are assumed known. The Navier-Stokes equations are in general difficult to solve,
but in some cases they reduce to simpler equations. One type of simplification ends
up with the diffusion equation.
Consider the flow of a fluid between two straight, parallel, infinite plates. The
flow can be driven by a pressure gradient, by gravity, or by moving the plates.
Infinite plates do not occur in nature or in technological devices, of course. Nev-
ertheless, sometimes one encounters two almost parallel surfaces with a small gap
filled with a fluid. Surfaces in machinery constitute an example. Although these
surfaces are curved, the gap between them can be so small that a model based on
assuming the surfaces to be infinite, parallel, flat plates can be very good. Figure
7.9
depicts the type of problem we are addressing.
It is reasonable to assume that the flow between two straight, parallel plates is par-
allel to the plates. To express our assumptions with mathematics, we introduce the
plates as the planes x
D
0 and x
D
H , and direct the y axis such that it is aligned
with the flow. The velocity field is now of the form
v
.x; y;
z
;t/
D
v
.x; y;
z
;t/
j
.
The fact that the velocity has this uni-directional form leads to a substantial sim-
plification of the Navier-Stokes equations. The mathematical details and physical
arguments of the simplification can be somewhat difficult to grasp for a novice mod-
eler, but that should be of less concern here. Our message is that diffusion occurs
in many different physical contexts. The derivation below is therefore compact such
that you will not lose track and miss the main result, i.e., yet another diffusion
equation.
It is reasonable to assume that there is no variation with
z
in this problem (infinite
planes and no flow directed in the
z
direction); thus
v
D
v
.x;y;t/. Equation (
7.30
),
r
v
D
0, simplifies to @
v
=@y
D
0, leaving
v
as a function of x and t only. Inserting
v
D
v
.x; t /
j
in (
7.30
) requires some algebra. The result is that the term .
v
r
/
v
van-
ishes, the time derivative term becomes
j
%@
v
=@t,and
r
2
v
is reduced to
j
r
2
v
.
Equation (
7.30
)isa
vector
equation, giving us three scalar equations:
x
V
H
(t)
H
β
(t)
gravity
y
V
0
(t)
Fig. 7.9
Sketch of viscous fluid flow between two flat plates
