JEE Main & Advanced Physics Fluid Mechanics, Surface Tension & Viscosity / द्रव यांत्रिकी, भूतल तनाव और चिपचिपापन  Equation of Continuity

The equation of continuity is derived from the principle of conservation of mass.

A non-viscous liquid in streamline flow passes through a tube AB of varying cross section. Let the cross sectional area of the pipe at points A and B be \[{{a}_{1}}\] and \[{{a}_{2}}\] respectively. Let the liquid enter with normal velocity \[{{v}_{1}}\] at A and leave with velocity \[{{v}_{2}}\] at B. Let \[{{\rho }_{1}}\] and \[{{\rho }_{2}}\] be the densities of the liquid at point A and B respectively.        

Mass of the liquid entering per second at A = Mass of the liquid leaving per second at B

\[{{a}_{1}}{{v}_{1}}{{\rho }_{1}}={{a}_{2}}{{v}_{2}}{{\rho }_{2}}\] and \[{{a}_{1}}{{v}_{1}}={{a}_{2}}{{v}_{2}}\] [If the liquid is incompressible \[{{\rho }_{2}}={{\rho }_{1}}\]] or     \[av=\]constant

or  \[a\propto \frac{1}{v}\]

This expression is called the equation of continuity for the steady flow of an incompressible and non-viscous liquid.

(1) The velocity of flow is independent of the liquid (assuming the liquid to be non-viscous)

(2) The velocity of flow will increase if cross-section decreases and vice-versa. That is why :          

(a) In hilly region, where the river is narrow and shallow (i.e., small cross-section) the water current will be faster, while in plains where the river is wide and deep (i.e., large cross-section) the current will be slower, and so deep water will appear to be still.

(b) When water falls from a tap, the velocity of falling water under the action of gravity will increase with distance from the tap\[(i.e.,\,{{v}_{2}}>{{v}_{1}})\]. So in accordance with continuity equation the cross section of the water stream will decrease \[(i.e.,\,{{A}_{2}}<{{A}_{1}}),\]i.e., the falling stream of water becomes narrower.