Question

Force of viscosity acting on a spherical body of radius $r$ moving with velocity $v$ through a fluid of viscosity is given by ?

Answer 1

Hint:A force may cause a mass object to change its velocity (which involves starting to move from a standstill), i.e. accelerate.The rate at which the fluid velocity changes in space is proportional to viscous forces in a fluid; the proportionality constant is the viscosity. The property of a fluid that opposes relative motion between its different layers is known as viscosity, and the force that opposes relative motion between the layers is known as viscous force.

Complete step by step answer:
The force of viscosity acting on a spherical body of radius $r$ moving at velocity $v$ through a viscous fluid can be found as follows:
Let, \[F\, \propto \,{r^a}\]
\[F \propto \,{v^b} \\
\Rightarrow F \propto {\eta ^c} \\
\Rightarrow F \propto {r^a}{v^b}{\eta ^c} \\
\Rightarrow F = k{r^a}{v^b}{\eta ^c} - - - - - - - (i) \\ \]
Here, $k$ is constant of proportionality.

Substitute the dimensional formula of each quantity in eqn (i)
\[\left[ {{M^1}{L^1}{T^{ - 2}}} \right] = \left[ {{M^0}{L^1}{T^0}} \right]\left[ {{M^0}{L^1}{T^{ - 1}}} \right]{\left[ {{M^1}{L^{ - 1}}{T^{ - 1}}} \right]^c} \\
\Rightarrow \left[ {{M^1}{L^1}{T^{ - 2}}} \right] = \left[ {{M^c}{L^{a + b - c}}{T^{ - b - c}}} \right] \]
Equating the power of M,L,T on both sides,
On solving, we get,
\[a = 1,\,b = 1,\,c = 1 \]
Substituting a,b,c in (i), we get:
\[\therefore F = kvr\eta \] is the viscous force, a measurement of a fluid's flow resistance.

Hence, the force of viscosity acting on a spherical body of radius $r$ moving with velocity $v$ through a fluid of viscosity is given by $F = kvr\eta$.

Note:viscous force and frictional force are similar, they are not the same. In a moving fluid, there are multiple layers to remember. As compared to the neighbouring layers, each layer has a noticeable difference in speed. As a result, there is a power in action. This force, like frictional force, decreases the relative motion between the layers. The viscous force is the force that acts between two layers of a moving fluid. The property that causes this force is known as viscosity.