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For the velocity field of Prob. 4?63, what relationship

Chapter 4, Problem 60P

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QUESTION:

Problem 60P

For the velocity field of  Prob. 4‒63, what relationship must exist between the coefficients to ensure that the flow field is incompressible?

PROBLEM: A general equation for a steady, two-dimensional velocity field that is linear in both spatial directions (x and y) is

Where U and V and the coefficients are constants. Their dimensions are assumed to be appropriately defined. Calculate the x- and y-components of the acceleration field.

Questions & Answers

QUESTION:

Problem 60P

For the velocity field of  Prob. 4‒63, what relationship must exist between the coefficients to ensure that the flow field is incompressible?

PROBLEM: A general equation for a steady, two-dimensional velocity field that is linear in both spatial directions (x and y) is

Where U and V and the coefficients are constants. Their dimensions are assumed to be appropriately defined. Calculate the x- and y-components of the acceleration field.

ANSWER:

Step 1:

        Consider a steady flow in the two dimensional field. If the flow is incompressible the volumetric strain rate of the flow must be zero.

        The given velocity field is

        = = +----(1)

        The components of velocity field are

 = -----(2)

         = -----(3)

        Where  and  are constants.

        The coefficients are and

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