Fluid Flow: Principal Rates of Deformation/Principal Axes

[jhuleea]

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!

 

[Rope]

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