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.
Fig 26.3 - Hagen-Poiseuille flow through a pipe
- Now, from continuity equation, we obtain
[ For rotational symmetry, ] | |
which means
- Invoking in the
Navier-Stokes equations, we obtain
(in the z-direction) | (26.3) |
- For steady flow, the governing equation becomes
| (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
| (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,
| (26.6a) |
- The average velocity in the channel,
| |
or, | |
| (26.6b) |
or | (26.6c) |
- Now, the discharge (Q) through a pipe is given by
| (26.7) |
or, [From Eq. 26.6b] | |
or | (26.8) |