Question

Derive velocity distribution, maximum velocity, average velocity, and force in z-direction for the following flow conditions, and state your assumptions

Flow of Newtonian fluid in the form of film over flat plate due to gravity, if kept vertical.
Flow of Newtonian fluid in the form of film over flat plate due to pressure force, if kept horizontal
Flow of non-Newtonian fluid in the form of film over flat plate due to gravity, if kept vertical.
Flow of non-Newtonian fluid in the form of film over flat plate due to pressure force, if kept horizontal
Flow of Newtonian fluid between two parallel flat plate due to gravity, if kept vertical; Plate are fixed
Flow of Newtonian fluid between two parallel flat plate due to pressure and gravity, both Plates are fixed.
Flow of Newtonian fluid between two parallel flat plate due to pressure and gravity, if kept vertical; one Plate is fixed, the other is moving downwards at constant velocity, Vo.
Flow of Newtonian fluid between two parallel flat plate due to pressure and gravity, if kept vertical; one Plate is fixed, the other is moving upwards at constant velocity, Vo.

Answer 1

Fig 1 Laminar flow innarrow slit

Assumptions

Density and viscosity are constant. Steady state. Laminar Flow(simple shear flow). Newton's law of viscosity is applicable.

Fluid is flowing in the z direction due to both gravity and pressure difference. Therefore, vz is the only important velocity component. As the slit is very narrow (B<<W<<L ), we may assume that end effects are negligible in y direction and vz is not a function of y.

Thus, intuitively we assume the velocity profile as,

Now, using the equation of continuity in cartesian coordinate system

or

Therefore,

From above velocity profile, we may conclude that is the only important shear stress component. We now select a cuboidal control volume of dimensions L, W, Δx, as shown in Fig. 13.2 (Note: differential thickness is chosen in x direction)

Fig 2 Control volume for laminar flow in a narrow slit.

Momentum balance in z-direction

Convective momentum entering the CV at z=0 is

Convective momentum leaving the CV at z=L is

Momentum entering CV by viscous transport at x=x is

Momentum leaving the CV by viscous transport at x=x+Δx is

Pressure force at z=0 is

Pressure force at z=L is

Gravity force on CV is

Substituting these terms into the momentum balance in z direction, we get

Since, vz is not a function of z, the first two convective momentum terms represented by Equations (13.5) and (13.6) are equal and hence cancel out from the above equation and we get

Dividing Equation (13.13) by the volume of the control volume ΔxLW , we obtain

Combining the pressure force with gravity, and taking the limit as Δx→0, we have

or

where,

Substituting Newton’s law of viscosity, we have

or

and finally after integration, we get

Boundary conditions are

1. At x=0 , the velocity profile must be symmetric. Therefore,

or

2. At x=B , no slip boundary condition is applicable. Thus,

or

Thus, velocity profile may be written as

Equation (13.23) describes the velocity profile in the narrow slit.

Mass flow rate and average velocity

Mass flow rate = Volumetric flow rate × Density

By substituting the value of velocity from Equation (13.23), we have

or

Average velocity = Volumetric flow rate/ Area of cross section

or

Annular flow with inner cylinder moving axially

In a wire coating machine, a wire of radius kR is moving into a cylindrical hollow die. The radius of the die is R, and the wire is moving with a velocity v0 along the axis. The die is filled with a Newtonian fluid, a coating material. The pressure at both ends of the die is same. Find the velocity distribution in the narrow annular region. Obtain the viscous force acting on the wire of length L . Also, find the mass flow rate through the annular region.

Fig3 Annular flow with the inner cylinder moving axially

Assumptions

density and viscosity are constant steady state. laminar (simple shear flow). Newton's law of viscosity is applicable.

Velocity components

The fluid is moving due to the motion of the wire in z direction so vz is the only important velocity component. There is no solid boundary in θ direction, and the flow is steady, therefore vz will not depend on θ and t. Hence,

Now, applying the equation of continuity in cylindrical coordinates

or

Thus,

This result indicates that is the only significant shear stress among the 9 components for momentum balance in z direction. Now, consider a control volume of differential thickness dr and length L at a distance r away from the center. We may write the momentum balance in z direction.

Fig 13.4 Control volume for annular flow with the inner cylinder moving axially

Convective momentum entering at z=0 is

Convective momentum leaving at z=L is

Momentum entering control volume by viscous transport at r = r is

Momentum leaving control volume by viscous transport at at r = r +Δr is

Now, the momentum balance over the control volume is below

Since velocity vz is not dependent on z coordinate therefore the convective terms represented by equations (13.29) and (13.30) are equal and hence cancelled out. Leaving with the following equation,

Dividing equation (13.34) by volume of the control volume,

Taking the limit as dr→0, we have

and after integration

where is an integration constant. Now, using Newton’s law of viscosity, we get

or

where is another integration constant.

Boundary conditions are

at r = kR ,

or

and at r = R,

or

From Equation (13.41)

or

or

By substituting the value of c1 into Equation (13.39), the velocity profile may be obtained as

or

Mass flow rate in the annular region

or

or

Drag force acting on the wire may be calculated as

or

By substituting the value of velocity vz, we obtain

Finally, we obtain the expression for drag force as