## Fluid Flow: Principal Rates of Deformation/Principal Axes

Hi all,

I've been stumped on this problem for over a month. Any guidance would alleviate my overwhelming frustration. Here is the original problem statement:

Find the principal rates of deformation and principal axes for the flow given by: u = (x,y) and v = 0, satisfying the continuity equation (density = rho = constant)
$$\frac{\partial u_i}{\partial x_i} = 0$$

Attached to this post is my attempt to work out the solution. I'm not sure how to proceed on, so your help would be greatly appreciated. Thanks!
Attached Thumbnails

 PhysOrg.com science news on PhysOrg.com >> Hong Kong launches first electric taxis>> Morocco to harness the wind in energy hunt>> Galaxy's Ring of Fire

 Quote by jhuleea Hi all, Find the principal rates of deformation and principal axes for the flow given by: u = (x,y) and v = 0, satisfying the continuity equation (density = rho = constant)
I guess that continuity should be du_i/dx_i=0 instead of div(u_3)=0.

Anyhew, your solution seems correct up till and including the principal directions (with the note that u=u(y) only, i.e. 2-D shear flow). When I calculate the principal values and directions in the old fashion way (as eigenvalues/eigenvectors of the strain rate tensor), I get the same directions, but principal values are +/- du/dy.

Cheers //Rope