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# Viscosity and Viscous Force and Its Derivation for JEE     ## What is Viscosity?

Last updated date: 25th Mar 2023
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Consider that you have a cup of oil and a cup of water. You can tell a big difference in how each liquid flows when you pour it into another cup. While the oil pours slowly, the water comes out swiftly. Their varying viscosities are the cause of this variation.

Viscosity is a material property. It measures how easily a fluid can deform at a certain rate. Viscosity denotes opposition to flow. Fluids move with the maximum velocity at the top of their layers. Thus, different layers of fluid have varying velocity gradients. The faster-moving layer wants to accelerate the slower layer it is in touch with, whereas the slower-moving layer wants to slow down the faster-moving layer.

## Shear Stress and Shear Rate

In the mathematical formulation of viscosity, shear stress, and shear rate are important factors. Think of the approximate way of fluid flow as fluid layers pass one another. If we imagine a fluid flowing in this way, the shear stress is the force pushing one layer over another divided by the area of the layers. More formally, this can be stated as the ratio of the force ​$F$ applied with the cross-sectional area ​$A$ of the material that is parallel to the applied force.

Shear stress is often denoted by $\tau$​, and hence the corresponding mathematical expression is:

$\tau =\dfrac{F}{A}$

The rate at which the fluid layers are passing one another is referred to as the shear rate. The formal definition is as follows:

$\dot{\gamma }=\dfrac{\Delta v}{x}$

Where $\Delta v$ is the difference in velocity between the two layers, and ​$x$ is the layer separation.

## Newton’s Law of Viscosity and Coefficient of Viscosity

According to Newton's viscosity law, shear stress between adjacent fluid layers is proportional to velocity gradients. Shear stress to shear rate is constant for a given temperature and pressure, and is known as viscosity or coefficient of viscosity.

## SI Unit of Viscosity

The SI unit of viscosity is Poiseuille $\left( Pl \right)$. Other units of viscosity are newton-second per square meter $\left[ \text{N s }{{\text{m}}^{-2}} \right]$ and pascal-second $\left( Pa\cdot s \right)$. The dimensional formula of viscosity can be given as $\left[ M{{L}^{-1}}{{T}^{-1}} \right]$.

## Types of Viscosity

• Dynamic or absolute viscosity is the ratio of shear stress to shear rate.

$\eta =\dfrac{\tau }{{\dot{\gamma }}}$

• Kinematic viscosity is defined as the absolute viscosity divided by mass density.

$\nu =\dfrac{\eta }{\rho }$

Consider two fluids with differing mass densities but the same dynamic viscosity. Under the influence of gravity, these two fluids pour out at different rates since each will have different gravitational forces acting on it (proportional to their masses). In general, the kinematic viscosity of liquids is calculated by dividing by the mass density, and, therefore, can be calculated as a measure of resistance to flow under gravity alone.

## What is the Viscous Force?

An opposing force between layers is called viscous force. The viscosity of a fluid serves as the proportionality constant, and viscous forces in a fluid are proportional to the rate of change in fluid velocity in space. Viscous force can be considered analogous to frictional force, but those have differences.

## Derivation of Viscous Force Formula

Let's take into account two fluid layers, each with surface area A. The viscous force (F) between them is

$F\propto A$

$F\propto \dfrac{dv}{dx}$

On combining,

$F=-\eta A\dfrac{dv}{dx}$

Where $\eta$ is the coefficient of viscosity of the fluids, this is also known as viscosity formula.

In other words, the viscosity of a liquid is determined by the constant act of viscous force acting tangentially per unit area of the liquid layer with a unit velocity gradient perpendicular to the direction in which the liquid flows. The poise unit measures the coefficient of viscosity.

## Application of Viscosity

• The oil used to lubricate heavy machinery parts should have a high viscosity. It is important to know the viscosity of the lubricant to select the right one.

• Some instruments use this highly viscous liquid to dampen their motions, and hydraulic brake oil uses it as brake oil.

• A fluid's viscosity determines how well blood circulates through arteries and veins.

• To ascertain the charge of an electron, Millikan performed the oil drop experiment. His understanding of viscosity helped him calculate the charge.

## Conclusion

When fluid layers move relative to one another, viscosity develops. More precisely, Viscosity is the property of a fluid by virtue of its offering resistance to the movement of one layer of fluid over an adjacent layer. The viscous force is the force acting across layers and counteracting relative motion. The viscous force per unit area of a liquid layer having a unit velocity gradient operating perpendicularly in a direction perpendicular to the direction of the liquid's flow is the viscosity of the liquid.

## FAQs on Viscosity and Viscous Force and Its Derivation for JEE

1. State the difference between viscosity and friction.

Differences between Viscosity and Friction are explained below.

• Viscosity: It refers to resistive forces that occur between layers of a fluid when fluids attempt to flow. Viscous force depends on velocity gradient and area of contact. The viscosity of liquid decreases with an increase in temperature. Viscosity occurs in liquids and gasses.

• Friction: It refers to forces that resist relative motion, in general. The frictional force depends on the area of contact and relative velocity. Friction is independent of temperature. Friction occurs in solids.

2. Explain Stoke’s law.

Stokes law states that the force that slows down a sphere moving through a viscous fluid is directly proportional to the velocity, radius of the sphere, and viscosity of the fluid.

It can be represented as:

$F\propto {{\eta }^{a}}{{r}^{b}}{{v}^{c}}$

$\Rightarrow F=k{{\eta }^{a}}{{r}^{b}}{{v}^{c}}$                                         -----(1)

Where $k$ is the proportionality constant.

The dimensions of parameters of equation (1) can be written as,

$[ML{{T}^{-2}}]={{[M{{L}^{-1}}{{T}^{-1}}]}^{a}}{{[L]}^{b}}{{[L{{T}^{-1}}]}^{c}}$

Simplifying the above equation, we get

$[ML{{T}^{-2}}]={{M}^{a}}\cdot {{L}^{-a+b+c}}\cdot {{T}^{-a-c}}$           ----(2)

Here, mass, length and time are independent entities.

Equating the superscripts of dimensions from equation (2), we get

$a=1$                                                                     -----(3)

$-a+b+c=1$                                                      -----(4)

$-a-c=-2\Rightarrow a+c=2$                                   ----(5)

Substituting (3) in (5), we get

$1+c=2$

$\Rightarrow c=1$                                                              ------(6)

Substituting the value of (3) & (6) in (4), we get

$-1+b+1=1$

$\Rightarrow b=1$                                                                ------(7)

Substituting the value of (3), (6) and (7) in (1), we get

$F=k\eta rv$

The value of $k$ for a spherical body was experimentally obtained as $6\pi$.

Hence, the viscous force on a spherical body is given by

$F=6\pi \eta rv$