# How to develop a Parabolic Velocity Profile in 3D tube/channel flow

1. Sep 18, 2014

### ksbiefr

i am trying to develop a Parabolic Velocity Profile in 3D tube/channel flow. for the 2D case i use

u = 1.5*Um *(1-(2y/H)^2)
where

Um= fluid velocity
y = position of solid on "y" axis (x,y)
H = width of channel
The above equation is not possible to used for 3D case (x,y,z). i try to search for 3D case not i am not successful.

Any body suggest me a equation/formula for developing Parabolic Velocity Profile in 3D tube/channel flow.

2. Sep 18, 2014

### Staff: Mentor

Your post is a little confusing. By 2D are you referring to flow between parallel plates, and by 3D are you referring to axial flow in a duct of circular cross section?

Chet

3. Sep 18, 2014

Are you familiar with how the equation you used was derived? If so, you can easily derive the equation for the velocity profile in a circular tube for a steady, fully-developed, laminar flow with constant pressure gradient. It is still parabolic. The flow is called Poiseuille flow. I won't go through the derivation, but here is the velocity profile for a circular tube:
$$u_z = \dfrac{1}{4\mu}\dfrac{dp}{dz}(r^2 -R^2).$$

Here, $u_z$ is the flow velocity, $\mu$ is the dynamic viscosity, $r$ is the distance from the centerline, $R$ is the inner radius of the tube, and $dp/dz$ is the pressure gradient through the tube.

4. Sep 18, 2014

### Staff: Mentor

To expand on what boneh3ad has said, that axial velocity can also be expressed in the same form as your "2D" equation (in terms of the mean velocity) by writing:

$u_z=2u_m\left(1-(\frac{r}{R})^2\right)$

Chet