Question

A liquid having coefficient of viscosity of $0.02$ decapoise is filled with a container of cross sectional area of $20{{m}^{2}}$ . If the viscous drag between two adjacent layers in flowing is $1N$. Then the viscosity gradient is:
$A.2{{s}^{-1}}$
$B.2.5{{s}^{-1}}$
$C.3{{s}^{-1}}$
$D.3.5{{s}^{-1}}$

Answer 1

Hint: We will use the relationship of drag force in terms of velocity gradient, cross-section area, and coefficient of viscosity to get the correct answer. The viscosity of a fluid is defined as the measure of the resistance offered to flow. Viscosity can be seen as an internal frictional force between adjacent layers of fluid. The velocity gradient is defined as the difference in velocity between adjacent layers of the fluid.
Formula Used:
We are using the following formula to get the correct answer:-
$F=\eta A\dfrac{dv}{dx}$ .

Complete step-by-step solution
From the problem above we have the following parameters with us:-
The drag force, $F=1N$
Area of cross section, $A=20{{m}^{2}}$
Coefficient of viscosity, $\eta =0.02$ decapoise which is equal to $0.02Ns{{m}^{-2}}$.
We have to calculate the viscosity gradient, $\dfrac{dv}{dx}$.
The velocity gradient is defined as the difference in velocity between adjacent layers of the fluid. It is given as $\dfrac{dv}{dx}$ where $dv$ is the difference in velocity and $dx$ is the distance between the adjacent layers.
We will use the following formula:-
$F=\eta A\dfrac{dv}{dx}$
$\Rightarrow \dfrac{dv}{dx}=\dfrac{F}{\eta A}$………………. $(i)$
Putting the respective values of the parameters in equation $(i)$ we get
$\dfrac{dv}{dx}=\dfrac{1}{0.02\times 20}$
$\Rightarrow \dfrac{dv}{dx}=\dfrac{1}{0.4}$
$\Rightarrow \dfrac{dv}{dx}=2.5{{s}^{-1}}$.
Therefore, velocity gradient is $2.5{{s}^{-1}}$ and option $(B)$ is correct.
Additional Information:
We know that the viscosity of a fluid describes its resistance to flow. In other words, the viscosity of a fluid is a measure of its deformation at a given rate. Thicker liquids are more viscous than thinner liquids. For example, tar and honey are more viscous than water.

Note: We should always take care of the unit of coefficient of viscosity and convert it into $Ns{{m}^{-2}}$ from decapoise. We often get confused between surface tension and viscosity. These two are different terms. Viscosity describes the fluid’s resistance to flow and surface tension is the tendency of the liquid surface to shrink into the minimum possible surface area.