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Question
Obtain an expression for conservation of mass starting from the equation of continuity.
Solution
Consider a tube of flow as shown in the figure. All the fluid that passes through a tube of flow must pass through any cross section that cuts the tube of flow. We know that all the fluid is confined to the tube of flow. Fluid can not leave the tube or enter the tube. Consider section A1 and A2 located at points A and B respectively. The mass of the fluid within this region is constant over time. That means, if mass m of the fluid enters the section A1 then equal mas of fluid should leave the section A2 . Let the speed of the fluid which crosses the section EFGH at point A in time interval ∆t be v1.
Thus, the mass of the fluid entering the tube through the cross section at point A is
ρA1v1∆t
Similarly, let the speed of the fluid be v2 at point B.
The fluid crosses the section PQRS of area A2 in time interval ∆t. Thus, the mass of the fluid leaving the tube through the cross section at B is
ρA2v2∆t
As fluid is incompressible, the mass of the fluid entering the tube at point A is the same as the mass leaving the tube at B.
Mass of the fluid in section EFGH = mass of fluid in section PQRS
ρA1v1∆t = ρA2v2∆t
A1v1 = A2v2 or
Av = constant
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