Appendix N A contradiction when computing the dissipation

The dissipation for an newtonian incompressible fluid is

$ℰ=-{\displaystyle \frac{1}{V}}{\displaystyle {\int }_V}{\displaystyle \frac{\eta }{2}}{\left({\displaystyle \frac{\partial v_i}{\partial r_j}}+{\displaystyle \frac{\partial v_j}{\partial r_i}}\right)}^2𝑑V$

(N.1)

If we assume a flowfield with $\frac{\partial v_x}{\partial r_x}\ne 0$ (which causes $\frac{\partial v_y}{\partial r_y}+\frac{\partial v_z}{\partial r_z}=-\frac{\partial v_x}{\partial r_x}$, because the divergence of the velocity field must be zero) and vanishing diagonal components of the jacobian $\frac{\partial v_y}{\partial r_x}=\frac{\partial v_x}{\partial r_y}=\frac{\partial v_z}{\partial r_y}=\frac{\partial v_y}{\partial r_z}=\frac{\partial v_x}{\partial r_z}=\frac{\partial v_z}{\partial r_x}=0$ , we compute via the formula above a dissipation of

$ℰ=-{\displaystyle \frac{2\eta }{V}}{\displaystyle {\int }_V}{\left({\displaystyle \frac{\partial v_x}{\partial r_x}}\right)}^2+{\left({\displaystyle \frac{\partial v_y}{\partial r_y}}\right)}^2+{\left({\displaystyle \frac{\partial v_z}{\partial r_z}}\right)}^{2}dV.$

(N.2)

The absolut value of the dissipation must be greater than zero, because all the terms in the integral are quadratic and our assumption was $\frac{\partial v_x}{\partial r_x},\frac{\partial v_y}{\partial r_y},\frac{\partial v_z}{\partial r_z}\ne 0$.

But if we compute the dissipation according to equation (5.19) like

$ℰ=-{\displaystyle \frac{\eta }{V}}{\displaystyle {\int }_V}{\omega }^2𝑑V$

(N.3)

we get for our flow field $ℰ=0$, because the vorticity of our velocity field is zero! This seems to be an obvious contradiction, and either raises some doubts about the validity of the derivation of equation (5.19) for example in Frisch (1995) or implies that our assumed flow field is unphysical.