# Question 10.14.2: For steady flow of an incompressible viscous fluid under a c...

For steady flow of an incompressible viscous fluid under a conservative body force, prove the following:

\begin{matrix}(i)&\textbf{v}\times \textbf{w}\left(\frac{p}{\rho}+\frac{1}{2}v^2+\chi\right)+v\ curl\ \textbf{w}&(10.14.17)\\ (ii)&(\textbf{v}.\triangledown)\textbf{w}-(\textbf{w}.\triangledown)\textbf{v}=v\triangledown^2\textbf{w}&(10.14.18)\\ (iii)If\\&\textbf{v}=\psi _{,2}e_{1}-\psi_{,1}e_{2}&(10.14.19)\end{matrix}

where ψ = ψ(x_{1},x_{2}), then

\left|\begin{matrix}(\triangledown^2\psi)_{,1}&(\triangledown^2\psi)_{,2}\\ \psi_{,1}&\psi_{,2}\end{matrix}\right|=v\triangledown^4\psi (10.14.20)

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