(Solved): Mass conversion and volume flux (a) Let \( \rho(\mathbf{x}, t) \) represent the density of a fluid, ...

(a) Let \( \rho(\mathbf{x}, t) \) represent the density of a fluid, and let \( \mathbf{u}(\mathbf{x}, t) \) represent its velocity field. (i) Give an expression for the total mass of fluid within a fixed arbitrary volume, \( \Omega \). (ii) By considering the rate of change of this mass, use Gauss's Divergence Theorem and the principle of conservation of mass to obtain \[ \frac{\partial \rho}{\partial t}+\nabla \cdot(\rho \mathbf{u})=0 \] (b) In spherical polar coordinates, a velocity field is given by \[ \mathbf{u}(r, \phi, \theta)=\alpha r^{2} \cos (\phi) \mathbf{e}_{r}-r^{2} \sin (\phi) \mathbf{e}_{\phi}-r \sin ^{2}(\phi) \mathbf{e}_{\theta} . \] (i) Find the value of \( \alpha \) that makes \( \mathbf{u} \) incompressible. (ii) Calculate the net volume flux through a cone that has its apex at the origin and expands upwards at an angle of \( \frac{\pi}{6} \) from the \( x-y \) plane to a height of 2 above the \( x-y \) plane. (iii) On average, is the fluid travelling upwards or downwards through the cone? Explain.