Couette Flow - 2 | Fluid Mechanics for Mechanical Engineering PDF Download

Applications-

• Equation (26.8) is commonly used in the measurement of viscosity with the help of capillary tube viscometers . Such a viscometer consists of a constant head tank to supply liquid to a capillary tube (Fig. 26.4).

• Pressure drop readings across a specified length in the developed region of the flow are taken with the help of a manometer. The developed flow region is ensured by providing the necessary and sufficient entry length.
• From Eq. (26.8), the expression for viscosity can be written as

• The volumetric flow rates (Q) are measured by collecting the liquid in a measuring cylinder. The diameter (D) of the capillary tube is known beforehand. Now the viscosity of a flowing fluid can easily be evaluated.
• Shear stress profile across the cross-section can also be determined from this information. Shear stress at any point of the pipe flow is given by

                                          (26.9a)

                             (26.9b)

This also indicates that τ varies linearly with the radial distance from the axis.

• At the wall, τ  assumes the maximum value.

as it should be. Negative sign indicates that the force is acting in opposite to the flow direction. 

• However, from Eq. (26.6b), we can write

                                                                                           (26.9c)

                                                                                             (26.10)

Losses and Friction Factors

• Over a finite length l , the head loss                              (26.11) 
  Combining Eqs (26.10) and (26.11), we get

                                                                           (26.12) 

• On the other hand, the head loss in a pipe flow is given by Darcy-Weisbach formula as

                                                                                                     (26.13)

where "f" is Darcy friction factor . Equations (26.12) and (26.13) yield

 

which finally gives , where  is the Reynolds number.

• So, for a fully developed laminar flow, the Darcy (or Moody) friction factor is given by

                                                                                                                        (26.14a)                                                                                                              

Alternatively, the skin friction coefficient for Hagen-Poiseuille flow can be expressed by

With the help of Eqs (26.9b) and (26.9c), it can be written



The skin friction coefficient C_f is called as Fanning's friction factor . From comparison of Eqs (26.14a) and (26.14b), it appears

                                                                                                           (26.14b)

• For fully developed turbulent flow, the analysis is much more complicated, and we generally depend on experimental results. Friction factor for a wide range of Reynolds number (10⁴ to 10⁸) can be obtained from a look-up chart . Friction factor, for high Reynolds number flows, is also a function of tube surface condition. However, in circular tube, flow is laminar for Re ≤ 2300 and turbulent regime starts with  .
• The surface condition of the tube is another responsible parameter in determination of friction factor.
• Friction factor in the turbulent regime is determined for different degree of surface-roughness  of the pipe, where ∈ is the dimensional roughness and D_h is usually the hydraulic diameter of the pipe .
• Friction factors for different Reynolds number and surface-roughness have been determined experimentally by various investigators and the comprehensive
• results are expressed through a graphical presentation which is known as Moody Chart after L.F. Moody who compiled it.
• The hydraulic diameter which is used as the characteristic length in determination of friction factor, instead of ordinary geometrical diameter, is defined as

where A_w is the flow area and P_w is the wetted perimeter .

• Kinetic energy correction factor , α The kinetic energy associated with the fluid flowing with its profile through elemental area    and the total kinetic energy passing through per unit time

This can be related to the kinetic energy due to average velocity   through a correction factor, α

• Here, for Hagen-Poiseuille flow,

                                                          (26.16)