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In-Depth Information
Flow in Tubes
Viscous fluid flow in a straight tube with a circular cross section can be modeled
analogously to the flow between two flat plates. The key idea is to introduce cylin-
drical coordinates .y;r;/,wherethey axis coincides with the tube axis, i.e., the
flow direction. The r and coordinates represent polar coordinates in the circular
cross section of the tube. The velocity field can then be written as
v
D
v
.y;r;;t/
j
.
Due to symmetry, we do not expect
v
to vary with . Inserting
v
D
v
.y;r;t/
j
in
(
7.30
) yields @
v
=@y
D
0, i.e.,
v
D
v
.r; t /
j
. With this
v
(
7.30
) leads to a slight
modifications of the Cartesian counterpart (
7.31
)-(
7.33
):
0
D
@p
@r
;
(7.35)
r
@
v
@r
;
%
@
v
@t
D
@p
@y
C
1
@
@r
(7.36)
r
0
D
1
@p
@
:
(7.37)
The details of the derivation of (
7.35
)-(
7.37
) requires familiarity with vector calcu-
lus in cylindrical coordinates. We have omitted gravity effects, since these involve
somewhat complicated calculations in cylindrical coordinates without adding any
interesting aspects of the PDE we want to derive.
Again, differentiation of (
7.36
) with respect to y reveals that
@p = @y cannot
depend on y. Equations (
7.35
)and(
7.37
) imply that p is independent of r and
. Therefore p can only depend on time (in a general way) and be linear in y.We
denote
@p = @y as C.t/, assumed to be a prescribed function, as in the flow between
flat plates. The PDE governing
v
.r; t / then follows from (
7.36
):
r
@
v
@r
C
C.t/:
%
@
v
@t
DC
1
@
@r
(7.38)
r
We mention this model since flow in a tube or pipeline is quite common and the
model is so closely related to flow between plates. In the fluid mechanics literature,
(7.34)and(7.38) are known as Couette
13
and Poiseuille
14
flows, respectively.
13
Maurice Marie Alfred Couette, 1858-1943, was a French physicist known for his studies in fluid
mechanics.
14
Jean Louis Marie Poiseuille, 1797-1869, was a French physician and physiologist. He was par-
ticularly interested in the flow of human blood in narrow tubes and performed physical experiments
and mathematical formulations of flow in pipelines.
