A "shell" is a differential element of the flow. By looking at the momentum and forces on one small portion, it is possible to integrate over the flow to see the larger picture of the flow as a whole. The balance is determining what goes into and out of the shell. Momentum is created within the shell through fluid entering and leaving the shell and by shear stress. In addition, there are pressure and gravitational forces on the shell. From this, it is possible to find a velocity for any point across the flow.
Applications
Shell Balances can be used in many situations. For example, flow in a pipe, the flow of multiple fluids around each other, or flow due to pressure difference. Although terms in the shell balance and boundary conditions will change, the basic set up and process is the same.
A fluid is flowing between and in contact with two horizontal surfaces of contact area A. A differential shell of height Δy is utilized (see diagram below).
Diagram of the shell balance process in fluid mechanics
The top surface is moving at velocity U and the bottom surface is stationary.
(Rate of momentum in) - (rate of momentum out) + (sum of all forces) = 0
To perform a shell balance, follow the following basic steps:
Find momentum from shear stress.(Momentum from Shear Stress Into System) - (Momentum from Shear Stress Out of System). Momentum from Shear Stress goes into the shell at y and leaves the system at y+Δy. Shear stress = τyx, area = A, momentum = τyxA.
Find momentum from the flow. Momentum flows into the system at x= 0 and out at x= L. The flow is steady state. Therefore, the momentum flow at x= 0 is equal to the moment of flow at x= L. Therefore, these cancel out.
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