Physical Concepts and Basic Fluid Mechanics
65
Under equilibrium, these forces balance each other, i.e. equal and
opposite,
dv
(p,
-
p2)m2
=
-2m17-
dr
(3.3.6)
The velocity gradient for the particular laminar layer of fluid is therefore,
(3.3.7)
Substituting for the pressure gradient from equation (3.3.3), we have the
relation between the velocity gradient and pressure gradient,
dv
-
1
dp
dr
27
dz
-
-
(3.3.8)
Notice that the velocity gradient is in the radial direction, i.e. across the
vessel, whereas the pressure gradient is in the longitudinal direction, i.e.
along the vessel axis. Velocity at radius r across the vessel can be
readily obtained by integration of (3.3.7)
(3.3.9)
where
k
is the constant
of
integration, obtainable by applying the
boundary condition that the velocity of fluid at the vessel wall (r=ri) is
zero, i.e. v(r=ri)
=
0.
We have,
Thus, velocity is maximum, or vmax, when
r=O,
or along the axis,
(3.3.1
0)
(3.3.1
1)
