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.
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
Where and are constants.
The coefficients are and
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.
Assume that the flow is in two dimensional field
= = 0
Therefore the equation (4) can be written as
= = +-----(5)