Question

For the flow of an incompressible viscous fluid the Navier-Stokes equations lead to $$-\nabla \times(\mathbf{v} \times(\nabla \times \mathbf{v}))=\frac{\eta}{\rho_{0}} \nabla^{2}(\nabla \times \mathbf{v})$$ Here $\eta$ is the viscosity and $\rho_{0}$ the density of the fluid. For axial flow in a cylindrical pipe we take the velocity $\mathbf{v}$ to be $$\mathbf{v}=\mathbf{z} v(\rho)$$ From Example $2.4 .1$ $$\nabla \times(\mathbf{v} \times(\nabla \times \mathbf{v}))=0$$ for this choice of $\mathbf{v}$. Show that leads to the differential equation $$\frac{1}{\rho} \frac{d}{d \rho}\left(\rho \frac{d^{2} v}{d \rho^{2}}\right)-\frac{1}{\rho^{2}} \frac{d v}{d \rho}=0$$ and that this is satisfied by $$v=v_{0}+a_{2} \rho^{2}.$$

          For the flow of an incompressible viscous fluid the Navier-Stokes equations lead to
$$-\nabla \times(\mathbf{v} \times(\nabla \times \mathbf{v}))=\frac{\eta}{\rho_{0}} \nabla^{2}(\nabla \times \mathbf{v})$$
Here $\eta$ is the viscosity and $\rho_{0}$ the density of the fluid. For axial flow in a cylindrical pipe we take the velocity $\mathbf{v}$ to be
$$\mathbf{v}=\mathbf{z} v(\rho)$$
From Example $2.4 .1$
$$\nabla \times(\mathbf{v} \times(\nabla \times \mathbf{v}))=0$$
for this choice of $\mathbf{v}$. Show that leads to the differential equation
$$\frac{1}{\rho} \frac{d}{d \rho}\left(\rho \frac{d^{2} v}{d \rho^{2}}\right)-\frac{1}{\rho^{2}} \frac{d v}{d \rho}=0$$
and that this is satisfied by
$$v=v_{0}+a_{2} \rho^{2}.$$

Added by Jamie W.

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