Use the provided equations of motion and continuity in spherical coordinates. Most of the equation content will simplify out. Show derivations and integrations, solve for all integration constants. Use the provided equations of motion and continuity. There is a hemisphere that is 5 cm in diameter. There is a hole at the apex of the sphere through which the chocolate will be pumped up. This hole takes up the first 2 of the hemisphere. The pump provides enough head to get the chocolate to the top of the hemisphere where it will begin flowing over the sphere with a maximum velocity of v0. We want the chocolate to have a thickness of to ensure an appealing appearance. Using the equations of continuity and motion, find the differential equation that defines the velocity profile of the chocolate film. State all your assumptions and boundary conditions needed to solve this equation. To simplify this equation, assume the mass of the chocolate flowing within this system isnt that great so the gravity term wont carry much weight in the overall equation. If this is the case, then that should allow us to eliminate one of the directions from the equation making it possible to solve for the velocity profile. Additionally, we can assume that the thickness of the chocolate will not vary much in all directions of the hemisphere which should allow use to ignore the impact of the decreasing velocity with increasing angular direction. What is the impact of these two assumptions on your equations? What is the velocity profile for this system under the assumptions made?

Equation of Motion for incompressible, Newtonian fluid (Navier-Stokes equation), 3 components in spherical coordinates (tvr+vrrvr+rvvr+rsinvvrrv2+v2)=rP+(r(r21r(r2vr))+r2sin1(sinvr)+r2sin2122vrr2sin2(vsin)r2sin2v)+gr(tv+vrrv+rvv+rsinvv+rvrvrv2cot)=r1P+(r21r(r2rv)+r21(sin1(vsin))+r2sin2122v+r22vrr2sin2cotv)+g(tv+vrrv+rvv+rsinvv+rvrv+rvvcot)=rsin1P+(r21r(r2rv)+r21(sin1(vsin))+r2sin2122v+r2sin2vr+r2sin2cotv)+g Continuity Equation, spherical coordinates t+r21r(r2vr)+rsin1(vsin)+rsin1(v)=0