Concept explainers
A steady, two-dimensional velocity field in the ay-plane is given by
(a) What are the primary dimensions (m. L,t. T,.. .) of coefficients a, b. c, and ci?
(b) What relationship between the coefficients is necessary in order for this flow to be incompressible?
(c) What relationship between the coefficients is necessary in order for this flow to be irrotational?
(d) Write the strain rate tensor for this (e) For the simplified case of d = -b, derive an equation for the streamlines of this tiow, namely, = function(x, a, b, c).
(a)
The primary dimensions (
Answer to Problem 120P
The primary dimension for
Explanation of Solution
Given information:
The flow is steady and two-dimensional.
The velocity field in the
Here,
Write the expression of the comparison for
Here, the velocity vector is
Write the expression for the velocity
Here, the distance is
Substitute
Substitute
Here, the length is
Write the expression of the comparison for
Here, the coefficient is
Substitute
Substitute
Write the expression of the comparison for
Here, the coefficient is
Substitute
V).
Write the expression of the comparison for
Substitute
Substitute
Here, the coefficient is
Conclusion:
The primary dimension for
(b)
The relationship between the coefficients in order for the given flow to be incompressible.
Answer to Problem 120P
The relation between the coefficients is
Explanation of Solution
Write the expression for the flow to be incompressible.
Here, the del operator is
Write the expression for del operator.
Here, the vector along
Substitute
Conclusion:
The relation between the coefficients is
(c)
The relationship between the coefficients in order for the given flow to be irrotational.
Answer to Problem 120P
The relation between the coefficients for the flow being irrotational is 0.
Explanation of Solution
Write the expression for the flow to be irrotational.
Here, the del operator is
Substitute
Here, the
Substitute
Conclusion:
The relation between the coefficients for the flow being irrotational is 0.
(d)
The strain rate tensor for the given flow.
Answer to Problem 120P
The strain rate tensor for two dimensional flow is
Explanation of Solution
Write the expression for the strain tensor for two-dimensional flow.
Here, the derivative in
Substitute
Conclusion:
The strain rate tensor for two dimensional flow is
(e)
An equation for the streamlines of the given flow, namely,
Answer to Problem 120P
The equation for the streamlines for
Explanation of Solution
Write the expression for the streamline equation.
Here, the differential with respect to
Substitute
Substitute
Integrate
Here, the integration constant is
Conclusion:
The equation for the streamlines for
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Chapter 4 Solutions
Fluid Mechanics: Fundamentals and Applications
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