In this lesson, we will derive the equation
for the velocity profile of fully-developed
laminar flows in pipes.
Most flows in everyday life are turbulent,
and often they are not fully-developed.
However, we can derive an exact expression
of the velocity profile only for fully-developed
laminar flows, although there are approximate
expressions available for fully-developed
turbulent flows as well.
We begin the derivation by examining the forces
on a small cylinder of fluid in a fully-developed
flow.
In a previous video, we applied Newton’s
second law in the axial direction on a cylinder
of radius lowercase r and length L, and found
an expression relating the pressure drop delta
P and shear stress tau.
We can rewrite mass m as density rho times
the volume V, and volume can be further rewritten
as pi r squared times L.
rho and g can be combined into gamma, and
the equation can be solved for tau.
Since delta P, L, gamma, and phi are not functions
of radius, this means that tau varies linearly
with r.
The shear stress is zero at the center of
the pipe, and reaches a maximum value at the
walls where velocity gradients are largest.
These conclusions are valid for both fully-developed
laminar flows and fully-developed turbulent
flows.
So far, velocity does not appear in this equation
yet, but we can relate the shear stress to
velocity gradients in the flow.
For fully-developed laminar flows, the velocity
vector only has an axial component, Vx, which
only varies in the radial direction.
For Newtonian fluids, we can relate the velocity
to shear stress through the equation tau equals
negative viscosity mu, times the velocity
gradient dVx/dr.
The negative sign appears because the velocity
decreases in the radial direction making the
velocity gradient negative.
However, tau must be a positive quantity.
The negative sign accounts for this.
Substitute this expression for tau into Newton’s
second law and solve for dVx.
Now we will integrate both sides of the equation.
Most of the terms on the right side of the
equation are not functions of r and can be
removed from the integral, leaving r dr.
The integral of dVx is simply Vx, and the
integral of r dr is r-squared over 2.
We also must include a constant of integration,
C.
Let’s move the equation to the top of the
screen and bring the factor of one-half into
the parentheses.
To determine the constant, we must apply a
boundary condition.
At the walls, we have the no-slip boundary
condition.
When the radius is capital R, the velocity
is zero.
This allows us to solve for the constant C,
and plug this expression into the equation
for Vx.
Vx is equal to the quantity delta P over L
minus gamma sin(phi), divided by 4 mu, times
capital R squared, times the quantity 1 minus
lowercase r squared over capital R squared.
This equation describes a parabola.
We can further simplify this equation by rewriting
it in terms of the centerline velocity, Vc.
This is the speed at the center of the pipe,
where the speed is the greatest.
Plugging in zero for the radius lowercase
r, and comparing the expression for Vc to
the expression for Vx, we now have that the
velocity profile is equal to Vc times the
quantity 1 minus lowercase r squared divided
by uppercase R squared.
This means if you know the centerline speed,
you can calculate the entire velocity profile.
With knowledge of the velocity profile, you
can calculate useful quantities such as volumetric
flowrate Q.
Q is equal to Vx dA, integrated over the entire
cross section.
Since we are dealing with a circular pipe,
we will integrate from the center of the pipe
where the radius is zero, to the pipe surface
which is a distance capital R from the center.
The small area dA is the area of a thin annular
strip of thickness dr.
This strip is a distance lowercase r from
the center of the pipe.
This means the area of the strip is the circumference
2 pi r times thickness dr.
Plug in the expressions for Vx and dA in the
integral, and remove 2 pi Vc out of the integral
because they are constant.
After evaluating the remaining integral, we
have Vc over 2 times pi capital R squared.
The volumetric flowrate Q is also equal to
the average speed Vavg times the total cross
sectional area A, which is pi capital R squared.
Comparing the two expressions for the volumetric
flowrate, we see that the average speed is
half of the centerline speed.
If we plug in the expression for the centerline
velocity into the volumetric flowrate equation,
and substitute D over 2 for the pipe radius,
we obtain a relationship between volumetric
flowrate, the pressure drop per length of
pipe, pipe diameter, pipe orientation, and
fluid properties.
This equation is called Poiseuille’s law.
It is important to remember that this equation
is only valid for fully-developed laminar
flows through circular pipes.
