Couette Flow - 1 | Fluid Mechanics for Mechanical Engineering PDF Download

Couette Flow

Couette flow is the flow between two parallel plates (Fig. 26.1). Here, one plate is at rest and the other is moving with a velocity U . Let us assume the plates are infinitely large in z direction, so the z dependence is not there.

The governing equation is

flow is independent of any variation in z-direction. 
The boundary conditions are ---(i)At y = 0, u = 0 (ii)At y = h, u = U.

• We get,

Invoking the condition (at y = 0, u = 0), C₂ becomes equal to zero.

Invoking the other condition (at y = h, u = U),

Equation (26.1) can also be expressed in the form

Where 

Equation (26.2a) describes the velocity distribution in non-dimensional form across the channel with P as a parameter known as the non-dimensional pressure gradient .

• When P = 0, the velocity distribution across the channel is reduced to

This particular case is known as simple Couette flow.

• When P > 0 , i.e. for a negative or favourable pressure gradient    in the direction of motion, the velocity is positive over the whole gap between the channel walls. For negative value of P ( P < 0 ), there is a positive or adverse pressure gradient in the direction of motion and the velocity over a portion of channel width can become negative and back flow may occur near the wall which is at rest. Figure 26.2ashows the effect of dragging action of the upper plate exerted on the fluid particles in the channel for different values of pressure gradient.

Maximum and minimum velocities

The quantitative description of non-dimensional velocity distribution across the channel, depicted by Eq. (26.2a), is shown

in Fig. 26.2b.

• The location of maximum or minimum velocity in the channel is found out by setting du/dy =0. From Eq. (26.2a), we can write

Setting    gives

                                                                                            ( 26.2b)

• It is interesting to note that maximum velocity for P = 1 occurs at y/h = 1 and equals to U . For P > 1, the maximum velocity occurs at a location  .
• This means that with P > 1, the fluid particles attain a velocity higher than that of the moving plate at a location somewhere below the moving plate.
• For P = -1, the minimum velocity occurs,  at . For P < -1,  the minimum velocity occurs at a location .
• This means that there occurs a back flow near the fixed plate. The values of maximum and minimum velocities can be determined by substituting the value of y from Eq. (26.2b) into Eq. (26.2a) as 
• This means that there occurs a back flow near the fixed plate. The values of maximum and minimum velocities can be determined by substituting the value of y from Eq. (26.2b) into Eq. (26.2a) as

                                                                 (26.2c)

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

• Now, from continuity equation, we obtain

• Invoking    in the

Navier-Stokes equations, we obtain

                                                               (26.3)

• For steady flow, the governing equation becomes

                                                                                 (26.4)

The boundary conditions are- (i) At r =0  v_z is finite and (ii) r = R, v_z  = 0 yields

• Equation (26.4) can be written as

• At r =0  v_z is finite which means A should be equal to zero and at r = R, v_z  = 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,

                                                           (26.6b)

                                                                           ( 26.6c)

• Now, the discharge (Q) through a pipe is given by

                                      (26.7)                        

                                                                        (26.8)