Concept explainers
Consider steady, incompressible, two-dimensional shear flow for which the velocity field is
(a) In similar fashion, calculate the location of each of the other three corners of the fluid particle at time t+dt.
(b) From the fundamental definition of linear strain rate (the rate of increase in length per unit length), calculate linear strain rates
(c) Compare your results with those obtained from the equations for
(a)
The location of each of the other three corners of the fluid particle at time
Answer to Problem 66P
The location of the lower left corner after time
The location of the lower right corner after time
The location of the upper left corner after time
The location of the upper right corner after time
Explanation of Solution
Given information:
Two-dimensional shear flow, flow is incompressible, the velocity field is
Write the expression for the two-dimensional velocity field in the vector form.
Here, the constants are
The following figure shows the position of the corners at time
Figure-(1)
Here, the length of the lower edge at time
Write the expression for location of the lower left corner after time
Write the expression for location of the lower right corner after time
Write the expression for location of the upper left corner after time
Write the expression for location of the upper right corner after time
Write the expression for velocity along x direction.
Calculation:
Substitute
Substitute
Substitute
Substitute
Conclusion:
The location of the lower left corner after time
The location of the lower right corner after time
The location of the upper left corner after time
The location of the upper right corner after time
(b)
The linear strain rates.
Answer to Problem 66P
The linear strain rate along x axis is
The linear strain rate along y axis is
Explanation of Solution
Write the expression for the strain rate along x direction.
Write the expression for the strain rate along y direction.
Write the expression for the length of the lower edge at time
Write the expression for the length of the lower edge at time
Calculation:
Substitute
Substitute
Substitute
Conclusion:
The linear strain rate along x axis is
The linear strain rate along y axis is
(c)
The linear strain rates in Cartesian coordinates.
Comparison of the linear strain rate by fundamental principal to the linear strain rates in Cartesian coordinates.
Answer to Problem 66P
The linear strain rate in Cartesian coordinates along x axis is
The linear strain rate in Cartesian coordinates along y axis is
The linear strain rate by fundamental principal and the linear strain rates in Cartesian coordinates are same
Explanation of Solution
Given information:
Linear strain along x axis is
Write the expression for the velocity along y direction.
Write the expression for the linear strain rate along x direction in Cartesian coordination.
Write the expression for the linear strain rate along y direction in Cartesian coordination.
Calculation:
Substitute
Substitute
Conclusion:
The linear strain rate in Cartesian coordinates along x axis is
The linear strain rate in Cartesian coordinates along y axis is
The linear strain rate by fundamental principal and the linear strain rates in Cartesian coordinates are the same.
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Chapter 4 Solutions
Fluid Mechanics: Fundamentals and Applications
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