JEE Main & Advanced Physics Fluid Mechanics, Surface Tension & Viscosity / द्रव यांत्रिकी, भूतल तनाव और चिपचिपापन Poiseuille's Formula

Poiseuille studied the stream-line flow of liquid in capillary tubes. He found that if a pressure difference (P) is maintained across the two ends of a capillary tube of length \['l'\]  and radius r, then the volume of liquid coming out of the tube per second is

(i) Directly proportional to the pressure difference (P).

(ii) Directly proportional to the fourth power of radius (r) of the capillary tube

(iii) Inversely proportional to the coefficient of viscosity \[(\eta )\] of the liquid.

(iv) Inversely proportional to the length \[(l)\] of the capillary tube.

i.e.   \[V\propto \frac{P\,{{r}^{4}}}{\eta l}\] or \[V=\frac{KP\,{{r}^{4}}}{\eta l}\]

\[\therefore \]  \[V=\frac{\pi P\,{{r}^{4}}}{8\eta l}\]  

[Where \[K=\frac{\pi }{8}\] is the constant of proportionality]

This is known as Poiseuille's equation.

This equation also can be written as, 

\[V=\frac{P}{R}\] where \[R=\frac{8\eta l}{\pi \,{{r}^{4}}}\]

R is called as liquid resistance.

(1) Series combination of tubes

(i) When two tubes of length \[{{l}_{1}},\,{{l}_{2}}\] and radii \[{{r}_{1}},\,\,{{r}_{2}}\] are connected in series across a pressure difference P,

Then \[P={{P}_{1}}+{{P}_{2}}\]                               ...(i)

Where \[{{P}_{1}}\] and \[{{P}_{2}}\] are the pressure difference across the first and second tube respectively

(ii) The volume of liquid flowing through both the tubes i.e. rate of flow of liquid is same.

Therefore \[V={{V}_{1}}={{V}_{2}}\]

i.e.,  \[V=\frac{\pi {{P}_{1}}r_{1}^{4}}{8\eta {{l}_{1}}}=\frac{\pi {{P}_{2}}r_{2}^{4}}{8\eta {{l}_{2}}}\]              ...(ii)

Substituting the value of P1 and P2 from equation (ii) to equation (i) we get

\[P={{P}_{1}}+{{P}_{2}}\]\[=V\left[ \frac{8\eta {{l}_{1}}}{\pi r_{1}^{4}}+\frac{8\eta {{l}_{2}}}{\pi r_{2}^{4}} \right]\] \[\therefore \] \[V=\frac{P}{\left[ \frac{8\eta {{l}_{1}}}{\pi r_{1}^{4}}+\frac{8\eta {{l}_{2}}}{\pi r_{2}^{4}} \right]}\]\[=\frac{P}{{{R}_{1}}+{{R}_{2}}}=\frac{P}{{{R}_{eff}}}\]        

Where \[{{R}_{1}}\] and \[{{R}_{2}}\] are the liquid resistance in tubes

(iii) Effective liquid resistance in series combination

\[{{R}_{eff}}={{R}_{1}}+{{R}_{2}}\]

(2) Parallel combination of tubes

(i) \[P={{P}_{1}}={{P}_{2}}\]

(ii) \[V={{V}_{1}}+{{V}_{2}}\]\[=\frac{P\pi r_{1}^{4}}{8\eta {{l}_{1}}}+\frac{P\pi r_{2}^{4}}{8\eta {{l}_{2}}}\]

\[=P\left[ \frac{\pi r_{1}^{4}}{8\eta {{l}_{1}}}+\frac{\pi r_{2}^{4}}{8\eta {{l}_{2}}} \right]\]

\[\therefore \]\[V=P\left[ \frac{1}{{{R}_{1}}}+\frac{1}{{{R}_{2}}} \right]\,=\frac{P}{{{R}_{eff}}}\]

(iii) Effective liquid resistance in parallel combination

\[\frac{1}{{{R}_{eff}}}=\frac{1}{{{R}_{1}}}+\frac{1}{{{R}_{2}}}\] or \[{{R}_{eff}}=\frac{{{R}_{1}}{{R}_{2}}}{{{R}_{1}}+{{R}_{2}}}\]