Consider a fluid in steady or streamline flow, that is its density is constant. The velocity of the fluid within a flow tube, while everywhere parallel to the tube, may change its magnitude.
Suppose the velocity is v1 at point P and v2 at point. Q. If A1, and A2, are the cross-sectional areas of the tube at these two points,
the volume flux across A1, \(\frac{d}{dt}(V1) = A1v1\)
and that across A2, \(\frac{d}{dt}(V2) =A2v2\)
By the equation of continuity of flow for a fluid,
A1v1 = A2v2
i.e. \( \frac{d}{dt}(V1) = \frac{d}{dt}(V2)\)
If p1 and p2, are the densities of the fluid at P and Q, respectively, the mass flux across A1,
\(\frac{d}{dt}(m1) = \frac{d}{dt}(p1v1) = A1p1v1\)
and that across A2,
\(\frac{d}{dt}(m2) = \frac{d}{dt}(p2v2) = A2p2v2\)
Since no fluid can enter or leave through the boundary of the tube, the conservation of mass requires the mass fluxes to be equal, i.e.
\(\frac{d}{dt}(m1) = \frac{d}{dt}(m2)\)
A1p1v1 = A2p2v2
i.e. Apv = constant
which is the required expression.
