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Viscosity- Stoke's Law, Terminal Velocity, Practice Problems, FAQs

Viscosity- Stoke's Law, Terminal Velocity, Practice Problems, FAQs

When we throw a ball in a water tank, we can observe that its speed gets reduced. This reduction in the velocity of the ball in the fluid is due to viscosity. Viscosity is somewhat similar to frictional force as both oppose the relative motion and both are due to molecular attractions. On the other hand there are some differences between these two forces that we will see in this article. We are also going to study Stoke's law and terminal velocity.

Table of Contents

  • Similarity and differences between viscosity and friction
  • Application of viscosity
  • Stoke’s law
  • Terminal velocity
  • Practice problems
  • FAQs

Similarity and Differences between Viscosity and Friction

Similarities: 

  • Viscosity and sliding friction oppose relative motion. The viscosity of a liquid layer opposes the relative motion of two adjacent liquid layers. The relative motion of two solid layers is opposed by sliding friction.
  • Kinetic friction and viscosity come into play whenever there is relative motion between layers of liquid or solid surfaces as the case may be.
  • Both are due to molecular attraction.

Dissimilarities:

Viscosity Friction
Viscosity between layers of liquid is directly proportional to the area of the liquid layers. Friction between two solids is independent of the area of solid surfaces in contact.
Viscous force is proportional to the relative velocity between two layers of liquid. The relative velocity between two surfaces has no bearing on friction.
Viscous force is independent of normal reaction between two layers of liquid. Friction is directly proportional to the normal reaction between two surfaces in contact.

Application of Viscosity

The viscosity of various liquids and gases has been applied in everyday life. Here are a few examples of applications:

  • Because liquid viscosity varies with temperature, proper lubricant selection is dependent on the season. 
  • At railway stations, high viscosity liquids are employed in shock absorbers and buffers. 
  • The phenomena of air and liquid viscosity is employed to dampen the motion of various sensors. 
  • The molecular weight and shape of organic molecules are determined using knowledge about the coefficient of viscosity of organic liquids.
  • It finds an important use in the circulation of blood through the arteries and veins of the human body.

Stoke’s Law

The streamlines for a fluid flowing slowly past a stationary solid sphere are shown in the figure. 


When the sphere moves slowly as compared to the fluid, the pattern is similar but the streamlines then flow in such a way that the apparent motion of the fluid particles is as seen by someone on the moving sphere. In this latter case, it is known that the layer of fluid in contact with the sphere moves with it, thus creating a velocity gradient between this layer and the other layers of fluid. Viscous force thereby brought into play and constitutes the resistance experienced by the moving sphere. If we make the plausible assumption that the viscous retarding force F depends on the size of the body, the velocity with which it moves, the viscosity of the fluid and mass density of the fluid, then an expression can be derived for F by the method of dimensional analysis. So,

Where x, y, z are the indices to be found and k is a dimensionless constant. The dimensional equation is

Equating the indices of M, L, T on both sides, we get

Solving we get, , , . Hence

With the experiments the value of k is found to be , so

This expression is called Stoke’s law. It holds true only for steady motion in a fluid of infinite extent. 

Terminal Velocity

When a body is dropped in a viscous fluid, it is first accelerated and then its acceleration becomes zero and it attains a constant velocity called terminal velocity. Consider a small ball of radius r, which is gently dropped into the liquid of infinite extent. As the steel ball falls, it experiences three forces: the force of gravity W; the buoyant force ; the viscous force .

The free body diagram of the ball is shown in the figure below. If be the density of the body and be the density if the liquid, then


As the body falls under gravity, its net weight is opposed by liquid resistance . Initially the viscous force is small and as the body gains velocity the viscous force also increases and at a particular velocity, called terminal velocity, the net force on the body becomes zero and moves with constant velocity, .

Balancing the forces on the ball,

So when means net force on an object becomes zero then acceleration will also be zero and body will move with constant velocity.

By solving above equation wee get,

If we plot the variation of velocity of the falling sphere with time, we obtain following graph:


Note:

Initially the velocity of the ball increases at a fast rate. Then the rate of increase of the velocity decreases with time. Finally the rate of increase of velocity becomes zero and the sphere acquires a terminal velocity .

Practice Problems

Q. A raindrop with a diameter of descends through the air. If the viscosity of the air is . Find viscous force acting on the drop when its speed is ?

A. Radius of raindrop

Velocity

From Stoke’s law,

Q. Find the terminal velocity of a ball of radius and density in a fluid of density ?

A.

We know,

Q. A powder comprising particles of various sizes is stirred up in a vessel filled to a height of with water. Find the largest particle that will remain in suspension after 1 hour, assuming the particles are spherical? [density of powder, viscosity of water 0.01 poise]

A. Terminal velocity of largest particle which is just about to settle at the bottom of the vessel is,

Let r be the radius of particle, then

On solving, we get

Q. A small sphere of radius r falls from rest in a liquid of viscosity . Due to friction, heat is produced. Find the rate of production of heat in terms of its terminal velocity at steady state?

A. We know,

At steady state sphere will attain terminal velocity

So at steady state,

Rate of production of heat is nothing but power, so

Rate of production of heat is .

FAQs

Q. When do we use Stoke’s Law?
A. We use Stoke’s law to determine the terminal velocity, the size and the density of sphere and liquid, respectively. The viscosity of a fluid can also be calculated using Stoke's law.

Q. How does terminal velocity of a spherical ball falling through viscous liquid is proportional to its radius?
A. The terminal velocity of a spherical ball falling through viscous liquid is proportional to the square of its radius.

Q. Comment on the acceleration and velocity of the raindrop falling towards the earth.
A. Raindrop attains its terminal velocity near earth surface so it has zero acceleration and constant velocity. 

Q. Is Stokes applicable to any shape of body?
A. No, Stoke’s law is only applicable for spherical objects.

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