Information Technology Reference
In-Depth Information
0
D
@p
@x
C
%g
x
;
(7.31)
@y
C
@
2
v
%
@
v
@t
D
@p
C
%g
y
;
(7.32)
@x
2
0
D
@p
@
z
C
%g
z
:
(7.33)
The gravity vector g is decomposed as
g D
g
x
i C
g
y
j C
g
z
k
. Component (
7.32
)
is of main interest here. Differentiation of (
7.32
) with respect to y shows that
@
2
p=@y
2
D
0,sop can only depend on y in a linear way. The two other equa-
tions, (
7.31
)and(
7.33
), imply that p is linear in x and
z
too. Thus, @p = @y is either
a constant or a function of time only. The latter case allows a pulsating pressure and
thereby a time-dependent velocity field too. Let us denote
@p = @y as ˇ.t/.Since
%g
y
is a constant, we can add this constant to ˇ.t/ and name the new function C.t/.
Physically, (
7.32
) expresses that the flow is driven by a pressure gradient and gravity
(boundary conditions can in addition express that the flow can be driven by mov-
ing plates, i.e., moving boundaries). The C.t/ function is hence the combined effect
of the pressure of gravity that drives the flow. Equation (
7.32
) is now a standard
diffusion equation:
@t
D
@
2
v
@
v
C
f.t/;
(7.34)
@x
2
where we have divided by % and introduced a new function f.t/
D
C.t/=%.The
parameter equals =% and is actually a common viscosity coefficient easily found
in the fluid dynamics literature.
What can the diffusion model (7.34) be used for in a fluid flow context? Normally,
(7.34) is used to study friction between lubricated surfaces. The fluid of interest is
then some kind of greasy oil. Engineers designing machinery may want to compute
the relation between the friction force (from the fluid) on the surfaces, since this is
relevant for the required applied force in the machinery. The friction force is cal-
culated by evaluating @
v
=@x at the boundaries, i.e., the plates x
D
0 and x
D
H .
If the fluid is also driven by pressure differences (some kind of pumping, for
instance) and perhaps also gravity, these effects are trivially incorporated in f .One
should notice that model (7.34), derived by assuming infinitely long flat plates, may
well describe the fluid flow and friction between rotating surfaces in machinery,
e.g. in a car engine, as long as the gap H is very small compared with the curvature
of the surfaces.
Boundary Conditions
The boundary conditions are simple, because viscous fluids stick to walls: The
velocities
v
.0; t / and
v
.H; t / must equal the velocity of the planes x
D
0 and
x
D
H , respectively.
