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In-Depth Information
Viscous Thin-Film Flow
This diffusion problem is a special case of the incompressible Navier-Stokes equa-
tions, where the fluid flows between two plates, described by the equations x
D
a
and x
D
b. The simplified Navier-Stokes equations take the form
@t
D
k
@
2
u
@
u
C
f:
(7.48)
@x
2
The parameters are as follows:
(a)
u
.x; t /, the velocity directed along the boundary plates
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(b) k, the fluid viscosity
(c) f.t/, the effects of pressure gradient and gravity
The relevant boundary condition is
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(a) Controlled plate velocities:
u
D
U
0
.t /;
U
0
is prescribed.
(7.49)
The flow can be driven by the plates (boundary conditions), by a pressure gradient
(source term) and/or gravity (source term).
Spherical Symmetry
For three-dimensional diffusion problems with spherical symmetry and a constant
diffusion coefficient k, the governing PDE is
@t
D
k
@
2
u
@
u
C
xf .x/;
(7.50)
@x
2
where x denotes the radial coordinate, and the function of physical significance is
v
.x; t /
D
r
1
u
.x; t /.Thex domain is now Œa; b, with a possibility of a
D
0.At
the boundary we normally have conditions on
v
, either
v
known or
k@
v
=@x (radial
flux) known. In the former case,
u
is also known by dividing the boundary values by
x. The latter case is somewhat more involved, since we can have
k
@
v
@x
D
q.x/;
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The velocity direction is perpendicular to the x coordinate.
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We could easily model
free surface
thin-film flow by the boundary condition @
u
=@x
0,but
the applications are somewhat limited, since instabilities in the form of waves (and hence a three-
dimensional problem) often develops at free thin-film surfaces. The wavy thin-film motion of rain
on the front window of a car is an example.
D
