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.

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

Step 2:

The volumetric strain rate is defined as the rate of increase in volume of a fluid element per unit volume. The expression for volumetric strain given by the cartesian coordinate system is

= = ++ ------(4)

Where ,and are the linear strain rates in the cartesian coordinates.

=

=

=

Step 3:

Assume that the flow is in two dimensional field

= = 0

Therefore the equation (4) can be written as

= = +-----(5)