Fluid Flow in Circular Tube
Fluid Flow in Circular Tube
Fluid Flow in Circular Tube
Solution.
a)
Flow in pipes occurs in a large variety of situations in the real world and
is studied in various engineering disciplines as well as in physics,
chemistry, and biology.
For a circular tube, the natural choice is cylindrical coordinates. Since the
fluid flow is in the z-direction, vr = 0, v = 0, and only vz exists. Further, vz
is independent of z and it is meaningful to postulate that velocity vz = vz(r)
and pressure p = p(z). The only non-vanishing components of the stress
tensor are rz = zr, which depend only on r.
p 0 p L + g L cos
L
(2)
(3)
Equation (3) on integration leads to the following expression for the shear
stress distribution:
rz =
P
C
r+ 1
2L
r
(4)
dvz
P
C
=
r+ 1
dr
2L
r
(5)
BC 2: at r = R,
vz = 0
(8)
From BC 1 (which states that the momentum flux and velocity at the tube
axis cannot be infinite), C1 = 0. From BC 2 (which is the no-slip
condition at the fixed tube wall), C2 = P R2 / (4 L). On substituting C1
= 0 in equation (4), the final expression for the shear stress (or
momentum flux) distribution is found to be linear as given by
rz =
P
r
2L
(9)
P R 2
r
vz =
1
4L
R
(10)
P R2
=
4L
(11)
(ii) The average velocity is obtained by dividing the volumetric flow rate
by the cross-sectional area as shown below.
R
R
vz 2 r dr
0
2
P R2
1
vz,avg =
= 2
=
= vz,max
(12)
R
vz r dr 8 L
2
0 R 2 r dr
0
Thus, the ratio of the average velocity to the maximum velocity for
Newtonian fluid flow in a circular tube is .
(iii) The mass rate of flow is obtained by integrating the velocity profile
over the cross section of the circular tube as follows.
R
w = vz 2 r dr = R2 vz,avg
0
(13)
Thus, the mass flow rate is the product of the density , the crosssectional area ( R2) and the average velocity vz,avg. On substituting vz,avg
from equation (12), the final expression for the mass rate of flow is
P R4
w =
8L
(14)
The flow rate vs. pressure drop (w vs. P) expression above is wellknown as the Hagen-Poiseuille equation. It is a result worth noting
because it provides the starting point for flow in many systems (e.g., flow
in slightly tapered tubes).
c)
The z-component of the force, Fz, exerted by the fluid on the tube wall is
given by the shear stress integrated over the wetted surface area.
Therefore, on using equation (9),
Fz = (2 R L) rz| r = R = R2 P = R2 p + R2 g L cos
(15)
integrating
leading to
it can be
and
Note that every term in the above equation has the dimension of
length. As a fully developed flow is being considered , then 1 = 2
. There are no external features between (1) and (2) which could
add or remove energy. As a result, loss of head is given by,
Note that,
Dividing by
gives,
Dimensional Analysis
Through a dimensional analysis it is possible to derive an
expression for "loss of head" in a pipe flow. Assuming that
pressure drop is proportional to pipe length it can be shown that
with