Question 10.14.5: For a nonsteady flow of an incompressible viscous fluid unde...
For a nonsteady flow of an incompressible viscous fluid under conservative body force with curl\ \textbf{w}=\triangledownξ for some scalar function ξ show that the Navier-Stokes equation becomes
\frac{\partial\textbf{v}}{\partial t}+\textbf{w}\times\textbf{v}=-\triangledown H^* (10.14.39)
where
\textbf{H}^*=\frac{p}{\rho}+\frac{1}{2}v^2+\chi+vξ (10.14.40)
Deduce the following.
(i) If the flow is of potential kind, then \textbf{H}^*+(\partial\phi/\partial t)=f(t),\ where\ f(t) is an arbitrary function of t.
(ii) If the flow is steady and \textbf{v}\times\textbf{w}\neq 0,then\ H^* is constant along stream lines and vortex lines.
(iii) If the flow is steady and \textbf{v}\times \textbf{w}=\textbf{0},\ then\ H^* is constant everywhere in the fluid.
