Introduction: The conduit shown in Figure length is L and its area is A. Express the mass flow in and/or out, and the mass in the conduit as function of time.Here it is very convenient to choose a non-deformable control volume that is inside the conduit dV is chosen as ¼ R2 dx. Using equation, the flow out (or in) is
The density is not a function of radius, r and angle, µ and they can be taken out the integral as
This result in
The flow out is a function of length, L, and time, t, and is the change of the mass in the control volume.
Non Deformable Control Volume: For this case the volume is constant therefore the mass is constant, and hence the mass
Change of the control volume is zero. Hence, the net flow (in and out) is zero. This condition can be written mathematically as
Or in a more explicit form as
Notice that the density does not play a role in this equation since it is canceled out. Physically, the meaning is that volume flow rate in and the volume flow rate out have to equal.
Deformable Control Volume: The left hand side of question can be examined further to develop a simpler equation by using the extend Leibniz integral rule for a constant density and result in
Where Ub is the boundary velocity and Ubn is the normal component of the boundary velocity.