Please solve this transport problem using the equations of continuity:

One reason why we study transport phenomena is to gain a better understanding of how to analyze flows in non-traditional systems. An example of this is the flow of chocolate in a chocolate fountain. In this party center piece, the velocity of liquid chocolate has to be slow enough to appear smooth over the fountains surfaces while providing a conformal coating over them but the velocity needs to be fast enough to prevent the chocolate from freezing onto those same surfaces. To find this perfect point, we need to analyze both the equations of motion as well as the equations of energy. For this exam, we will focus on the equation of motion.

Lets start with a hemisphere that is 10 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. Please make sure to state all your assumptions and boundary conditions needed to solve this equation.

You should end up an equation that is rather complex as your velocity should vary in more than one direction due to the complex gravity term. To simplify this equation, we could realize that the mass of the chocolate flowing within this system isnt that great and, as such, 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?