Hagen Poiseuille Flow
- Consider fully developed laminar flow through a straight tube of circular cross-section as in Fig. 26.3. Rotational symmetry is considered to make the flow two-dimensional axisymmetric.
- Let us take z-axis as the axis of the tube along which all the fluid particles travel, i.e.
![](images/Equation_lec%2026/image040.gif)
![](images/8-7.gif)
Fig 26.3 - Hagen-Poiseuille flow through a pipe
- Now, from continuity equation, we obtain
[ For rotational symmetry, ] |
|
which means ![](images/Equation_lec%2026/image044.gif)
- Invoking
in the
Navier-Stokes equations, we obtain
(in the z-direction) |
(26.3) |
- For steady flow, the governing equation becomes
![](images/Equation_lec%2026/image050.gif) |
(26.4) |
The boundary conditions are- (i) At r = 0, is finite and (ii) r = R, yields
- Equation (26.4) can be written as
- At r =0,
is finite which means A should be equal to zero and at r = R, = 0 yields
![](images/Equation_lec%2026/image066.gif) |
(26.5) |
- This shows that the axial velocity profile in a fully developed laminar pipe flow is having parabolic variation along r.
- At r = 0, as such,
![](images/Equation_lec%2026/image070.gif) |
(26.6a) |
- The average velocity in the channel,
![](images/Equation_lec%2026/image072.gif) |
|
or, ![](images/Equation_lec%2026/image074.gif) |
|
![](images/Equation_lec%2026/image076.gif)
|
(26.6b) |
or ![](images/Equation_lec%2026/image078.gif) |
(26.6c) |
- Now, the discharge (Q) through a pipe is given by
![](images/Equation_lec%2026/image080.gif) |
(26.7) |
or, [From Eq. 26.6b] |
|
or ![](images/Equation_lec%2026/image084.gif) |
(26.8) |
|