Concept explainers
Consider dimensionless velocity distribution in Couette flow (which is also called generalized Couette flow) with an applied pressure gradient which is obtained in the following from in Example 9-16 as
where
FIGURE P9-102
(a)
The velocity distribution is a superposition of Couette flow with a linear velocity distribution and Poiseuille flow with a parabolic velocity distribution.
Explanation of Solution
Given information:
The following figure shows that two infinite plates.
Figure-(1)
At the point of wall and fluid, the velocity of the fluid is equal to zero.
Write the expression for the pressure gradient.
Here, the arbitrary locations along the
Write the expression for
Here, the viscosity is
Write the expression for
Here, the density of the fluid is
Integrate Equation (II) with respect to
Here, the constant is
Integrate Equation (IV) with respect to
Here, the constant is
Integrate Equation (III) with respect to
Here, the constant is
The following figure represents planar Poiseuile flow.
Figure (2)
Write the expression for dimensionless form of velocity field.
Here, the non- dimensional velocity along the arbitrary location along the
The Equation (VII) indicates that the pressure gradient is positive. If the both walls are stationary and pressure gradient is also there, then the flow should be planar Poiseuile flow.
Write the expression for non- dimensional velocity.
Write the expression for non- dimensional pressure.
Write the expression for non -dimensional distance
Calculation:
Substitute
Substitute
Substitute
The Equations (VII) indicates super position of the linear velocity profile
Substitute
(b)
The plot the
Answer to Problem 102P
The following Figure (4) represents the
Explanation of Solution
At the point of wall and fluid, the velocity of the fluid is equal to zero.
Write the expression for the pressure gradient.
Here, the arbitrary locations along the
The following figure represents planar Poiseuile flow.
Figure (2)
Write the expression for dimensionless form of velocity field.
Here, the non dimensional velocities is
The Equation (VII) indicates that the pressure gradient is positive. If the both walls are stationary and pressure gradient is also there, then the flow should be planar Poiseuile flow.
Write the expression for non -dimensional velocity.
Write the expression for non -dimensional pressure.
The following figure represents the Couette flow between two parallel plates.
Figure-(3)
The pressure gradient is less than two, then pressure is decreasing in
Write the expression for non dimensional distance
Substitute
The following table represents the pressure gradient and x or y non- dimensional distance.
| | |
| | |
| | |
| | |
The following figure represents between
Figure (4)
The Figure (4) indicates that the pressure gradient is positive. If the both walls are stationary and pressure gradient is also there, then the flow should be planar Poiseuile flow.
Conclusion:
The following Figure (4) represents the
(c)
The position and magnitude of maximum dimensionless velocity.
Answer to Problem 102P
The expression for the pressure of fluid 1 is
The expression for the pressure of fluid
Explanation of Solution
Assume, at the point of wall and fluid, the velocity of the fluid is equal to zero.
Write the expression for velocity of the fluid 1,
Here, the velocity of fluid 1 is
Assume, the velocity of the fluid 2 at the free surface of the wall is equal to the velocity of the moving plates.
Write the expressions for the velocity of fluid 2.
Here, the velocity of fluid 2 is
Write the expression for velocity at interface.
Write the expression for rate of shear stress.
Here, the kinematic coefficient of fluid is
Write the expression for the shear stress acting on fluid
Here, the kinematic coefficient of fluid 1 is
Write the expression for the shear stress acting on fluid 2.
Here, the kinematic coefficient of fluid 2 is
Write the expression for the rate of shear stress at interface.
Write the expression for pressure at the bottom of the flow,
Here, the pressure is
Write the expression for the pressure at the interface of fluid 1.
Here, the pressure at the fluid 1 is
Write the expression for the pressure at the interface of fluid 2.
Here, the pressure at the fluid 1 is
At the interface of the fluid the pressure cannot have discontinuity and the surface is ignored.
Write the expression for the pressure at interface of fluid.
Here, the velocity of flow for fluid
Write the expression for
Here, the density of the fluid 1 is
Write the expression for
Here, the density of the fluid 2 is
Calculation:
Integrate Equation (XXVII) with respect to
Here, the constant is
Substitute
Substitute
Integrate Equation (XXVIII) with respect to
Here, the constant is
Substitute
Substitute
Substitute
Conclusion:
The expression for the pressure of fluid 1 is
The expression for the pressure of fluid 2 is
(d)
The position and magnitude of the maximum dimensionless velocity.
Answer to Problem 102P
The position of the maximum dimensionless velocity is
The magnitude of the maximum dimensionless velocity is
Explanation of Solution
Given information:
Write the expression for the position of the velocity.
Here, the viscosity of the fluid is
The following figure represents positions of the velocities.
Figure-(5)
Write the expression for the radial velocity
Here, the maximum velocity is
The following figure represents the magnitude of the velocity.
Figure-(6)
Write the expression for magnitude.
Here the magnitude is
Substitute
Conclusion:
The position of the maximum dimensionless velocity is
The magnitude of the maximum dimensionless velocity is
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Chapter 9 Solutions
Fluid Mechanics: Fundamentals and Applications
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