Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev

Mechanical Engineering : Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev

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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 . Let us assume the plates are infinitely large in direction, so the dependence is not there.

The governing equation is

Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev

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.

Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev

  • We get,

Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev

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

Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev

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

Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev

Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev(26.1)

Equation (26.1) can also be expressed in the form

Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev(26.2a)

Where

Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev

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

Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev

This particular case is known as simple Couette flow.

  • When P > 0 , i.e. for a negative or favourable pressure gradient (-dp/ dx) 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.

Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev

FIG 26.2a - Velocity profile for the Couette flow for various 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

Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev

Setting Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev gives

Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev (26.2b)

  • It is interesting to note that maximum velocity for P = 1 occurs at y/h = 1 and equals to . For P > 1, the maximum velocity occurs at a location Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev .
  • 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 Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev = 0. For P < -1,  the minimum velocity occurs at a location Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev > 1.
  • 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 from Eq. (26.2b) into Eq. (26.2a) as

Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev (26.2b)

Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev

FIG 26.2b - Velocity distribution of the Couette flow
 

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

 

Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev

Fig 26.3 - Hagen-Poiseuille flow through a pipe

  • Now, from continuity equation, we obtain

Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev

Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev

Navier-Stokes equations, we obtain

Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev  (in the z-direction)             (26.3)

  • For steady flow, the governing equation becomes

  Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev            (26.4)

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

  • Equation (26.4) can be written as

Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev

Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev

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

Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev           (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, v= vmax

Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev           (26.6a)

  • The average velocity in the channel,

Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev

Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev           (26.6b)

Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev           (26.6c)

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

Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev           (26.7)

Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev            (26.8)


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).

Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev

FIG 26.4 Schematic diagram of the experimental facility for determination of viscosity

  • 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

Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev

  • 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

Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev              (26.9a)

Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev               (26.9b)

This also indicates that Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev varies linearly with the radial distance from the axis.

  • At the wall, Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev assumes the maximum value.

Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev

Again, over a pipe length of , the total shear force is

Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev

or

Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev

or

Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev[Pressure drop between the specified length]

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

Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev              (26.9c)

Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev                (26.10)

 

Losses and Friction Factors

Over a finite length l , the head loss  Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev              (26.11)

Combining Eqs (26.10) and (26.11), we get

Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev              (26.12)

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

Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev              (26.13)

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

Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev

which finally gives Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev is the Reynolds number.

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

Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev              (26.14a)

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

Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev

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

Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev              (26.14b)

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

Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev

  • 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 Re ≥ 4000.
  • 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 Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev of the pipe, where ∈ is the dimensional roughness and Dh 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

 

Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev              (26.15)

where Aw is the flow area and Pw is the wetted perimeter

Kinetic energy correction factor α The kinetic energy associated with the fluid flowing with its profile through elemental area Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev and the total kinetic energy passing through per unit time

Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev

This can be related to the kinetic energy due to average velocity(v zav), through a correction factor, α as

Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev     

 Here, for Hagen-Poiseuille flow

Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev              (26.16)


 

Flow between Two Concentric Rotating Cylinders

  •  Another example which leads to an exact solution of Navier-Stokes equation is the flow between two concentric rotating cylinders.
  • Consider flow in the annulus of two cylinders (Fig. 26.5), where r1 and r2 are the radii of inner and outer cylinders, respectively, and the cylinders move with different rotational speeds ω1 and ω2 respectively

 

Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev

FIG 26.5 - Flow between two concentric rotating cylinders

  • From the physics of the problem we know, .vz = 0, vr = 0
  • From the continuity Eq. and these two conditions, we obtain

Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev

which means vθ is not a function of θ. Assume dimension to be large enough so that end effects can be neglected and  Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev  (any property) = 0.

  • This implies  vθ  = vθ (r). With these simplifications and assuming that " θ symmetry" holds good, Navier-Stokes equation reduces to

Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev                        (26.17)

Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev                      (26.18)

  • Equation (26.17) signifies that the centrifugal force is supplied by the radial pressure, exerted by the wall of the enclosure on the fluid. In other words, it describes the radial pressure distribution. 
    From Eq. (26.18), we get

Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev                      (26.19)

  • For the azimuthal component of velocity, vθ, the boundary conditions are: Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev  Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev
  • Application of these boundary conditions on Eq. (26.19) will produce

Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev

and

Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev

  • Finally, the velocity distribution is given by

Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev                 (26.20)
 

Calculation of Stress and Torque Transmitted

Now, Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev is the general stress-strain relation.

 

Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev

  • In our case,

Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev                          (26.21)

  • Equations (26.20) and (26.21) yields

Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev                          (26.22)

  • Now,

Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev

and

Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev

  • For the case, when the inner cylinder is at rest and the outer cylinder rotates, the torque transmitted by the outer cylinder to the fluid is

Couette Flow - Viscous Incompressible Flows Mechanical Engineering Notes | EduRev                         (26.23)

where is the length of the cylinder.

  • The moment T1, with which the fluid acts on the inner cylinder has the same magnitude. If the angular velocity of the external cylinder and the moment acting on the inner cylinder are measured, the coefficient of viscosity can be evaluated by making use of the Eq. (26.23)
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