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

or,
or,
or,
or,

• 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)