Past exam of the mathematics and the environment and mathematical physiology course of the University of Oxford / 2008 / A / 3 / iii

The reduced mass equation gives $hu=1$ after normalizing the reference discharge. The reduced momentum equation integrates to $\eta +F^2{u}^2/2=1+F^2/2$ with the reference free-surface level one. Since $h=1/u$,Also $hu=1$ reduces the concentration equation toWith $E^{\ast }(1)=1$, a steady uniform reference state is $\boxed{u=h=c=\eta =1,s=0}$. The integration constants above specify this reference normalization; a different imposed discharge or head changes them.