Question Fluid Mechanics(Robert Granger) fluid Velocity comp.

[knockout_artist]

Question "Fluid Mechanics(Robert Granger)" fluid Velocity comp.

Chapter 4. eq. 4.28- 4.30

u = u0 + (x - x0)(du/dx) + (y - y0)(du/dy) + (z - z0)(du/dz)

v= ...

w = ...


Then without any significance explanation these equation take a form of
eq 4.31 - 4.33


u = u0 + 1/2 [(x - x0)(du/dx - du/dx ) + (y - y0)(du/dydx - dv/dx) + (z - z0)(du/dz - dw/dx)] ...

v= ...

w = ...


what does this 1/2 in second set of equation means ?

Thank you.

 

[Chestermiller]

The first three equations you wrote expresses the difference in velocity between two neighboring points in a fluid flow in terms of the components of the velocity gradient (tensor) in their immediate vicinity. In the second set of equations you wrote, the velocity gradient (tensor) is resolved into a rotation rate (vorticity tensor) and a rate of deformation. The terms you have given are the components of the vorticity, and tell how rapidly the fluid elements are rotating, while the other terms you have not shown (your 4th equation has ... in it) describe the rate of deformation of the fluid elements, with the rotation removed.

 

[knockout_artist]

Thank you!