Determine the volumetric dilation rate and the horticulture

Determine the volumetric dilatation rate and the vorticity for the following flow field, and interpret the results. 5. v=xy + yz+22 w=-3x-x\'/2 + 4

Solution

Given :

u = x² + y² + z²

v = xy +yz + z²

w = -3xz - z²/2 +4

Formula for volumetric dilatation rate = du/dx + dv/dy + dw/dz

d/dx(x² + y² + z²) = 2x

d/dy(xy +yz + z²) = x + z

d/dz(-3xz - z²/2 +4) = -3x-z

put the values of du/dx, dv/dy and dw/dz in above equation, we get ;

(2x) + (x+z) + (-3x-z) = 0

this result indicates that there is no change in volume of fluid element as it moves from one location to another.