Question

The rate of flow of liquid in a tube of radius r, length whose ends are maintained at a pressure difference p is \[V = \dfrac{{\pi Qp{r^4}}}{{\eta l}}\] ​, where \[\eta \] is coefficient of the viscosity and Q.
A. 8
B. \[\dfrac{1}{8} \\ \]
C. \[\dfrac{1}{6} \\ \]
D. \[\dfrac{1}{{16}}\]

Answer 1

Hint: When the viscosity of the liquid is not considered then the total energy of the flow is constant at every point on the line of the flow. But, when the viscosity is considered, then the energy is dissipated in the form of frictional work, the same as the energy dissipated in a resistor carrying electric current through itself when electric potential is applied across the resistor.

Complete step by step solution:
The radius of the tube in which the liquid is flowing is given as r. The length of the tube is given as $l$. The pressure difference between the ends of the tube is constant and it is equal to $p$. The rate of flow is given as,
\[V = \dfrac{{\pi Qp{r^2}}}{{\eta l}}\]

We need to determine Q. As in the expression for the flow rate there is the term coefficient of viscosity, so the fluid’s drag is considered while finding the expression for the flow rate of the liquid in the tube.

Hagen-Poiseuille equation gives the relation between the fluid resistance to the flow through the tube, the flow rate, the length of the flow and the pressure between the ends of the tube which is analogous to the flow of electric current in the resistance when electric potential is applied across the ends of the resistor as,
\[V = \dfrac{{\pi p{r^4}}}{{8\eta l}}\]
When we compare both the equations, we find the value of Q as \[\dfrac{1}{8}\].

Therefore, the correct option is B.

Note: When the tube is frictionless then we simply use Bernoulli’s equation to determine the flow rate as total energy of the flow is constant when the surface of the tube is frictionless. The speed of the flow at both the ends is related to the continuity equation. But when it is given that we need to consider the friction of the tube or the viscosity of the fluid then we use the Hagen-Poiseuille equation.