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

fountain's 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.

Let's start with a hemisphere that is $10cm$ 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 $v_0.$ 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 isn't that great and, as such, the gravity term won't 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?