Stokes’ Law, Terminal Velocity

The branch of physics that deals with switching the properties of fluid motion is called Hydrodynamics. Fluids are defined as those substances which can flow from one place to another. Liquids and gases are generally known as fluids.

The property of moving fluid to oppose the relative motion between the layers during its flow is known as viscosity. This is similar to the concepts of friction which we studied in mechanics. It indicates that there exists some form of friction at the atomic level. The greater the magnitude of the viscosity larger will be the resistance between the layers. For example between honey and water, Honey has a larger magnitude of friction and hence it is more difficult to make the Honey flow compared to water.

This factor not only determines the ability of resistance between the fluids but also gives an idea about the resistance any solid will experience when it moves through the fluid. For example, it is much more difficult to move our hands through the air compared to the water where both are fluids.

Causes of viscosity

Viscosity is the internal friction of the fluid that officers the motion between the layers. So, the viscosity of any fluid depends on the following factors:

• The nature of the fluid.

• Area of the layers in contact. The greater the area between the layers the greater will be the dragging Force.

• Velocity gradient between the two layers. The velocity gradient is a mathematical term that is defined as dv/dx, where dv is the change of velocity and dx is the change in the distance along the same direction of velocity.

The viscosity always tends to retard the motion of the liquid. To maintain the original liquid flow a  force of magnitude exactly equal to the drag force must be applied.

Coefficient of viscosity

Let A be the area of contact between the two layers of the fluids. dv/dx be the gradient of the force. 

Then Force ,

F∝ dv/dx

F∝ A* dv/dx

F = η A *dv/dx

Here, η is a constant of proportionality and is called the coefficient of viscosity which depends on the nature of the fluids.

Hence the coefficient of viscosity is defined as the tangential force required to maintain a unit velocity gradient between two parallel layers each of unit area.

Stokes’ law

Stokes’ law asserts that the force that opposes a sphere’s descent in a viscous fluid is exactly proportional to the sphere’s velocity, radius and viscosity.  If you’ve ever studied physics, you’re probably aware of the well-known rule that states that every response has opposite and equal responses or reactions. This rule is used in physics to describe forces, or the forces that drive things to move, slow down and finally halt. This physics law is applied to spherical solid structures falling in fluids using Stokes’ law.

According to Stokes’ law, F = 6πrηv.  The variables in Stokes’ law are as follows, from left to right:  The drag force is denoted by the letter F. This is the force that acts opposite to the sphere’s motion and tries to bring it to a halt. The diameter of the sphere is r. It is a measure of distance between the sphere’s centre and any place on its surface.  The viscosity of a fluid is represented by η (the Greek letter eta). Simply put, viscosity is indeed a measurement of a fluid’s “thickness.” Honey, for example, has a viscosity substantially greater than water.  The sphere’s velocity is given by v. This is the rate at which the sphere is travelling. Gravity pulls the sphere down, thus velocity considers it as well.  So, if you know the sphere’s radius, velocity, and viscosity, you can apply Stokes’ equation to calculate the pressure resisting the sphere.

Stokes’ law, on the other hand, has a wide range of applications. It may be used to investigate sediments in water, determine the viscosity of fluids and analyse rain clouds and fog as some gases, such as water vapour, are also viscous fluids.

The drag force opposing the fall of a sphere object through a liquid medium is expressed by Stoke’s law formula,

F=6πηrv

• Here f is the drag force of an object
• π is the value of pi
• n is the viscosity of the fluid
• r is the radius of the object
• v is the velocity of a sphere

Conclusion

The branch of physics that deals with fluid motion is known as hydro-dynamics. The substances which can move from one place to another by some external influence are known as fluid. Usually, the gases and solids are considered fluids. The property of the fluid to oppose the relative motion between the two layers when there is the state of motion is known as viscosity. This is similar to the concept of friction in mechanics. This is also known as the internal friction of the fluid and depends on the area between the two layers, nature of the fluids, velocity gradient. 

The coefficient of viscosity of a fluid is defined as the tangential force required to maintain a unit velocity gradient between two parallel layers of each unit area. Stoke’s law states that the magnitude of the viscous force depends on the shape and size of the solid body, speed of the body, coefficient of viscosity of the fluid. The maximum constant velocity acquired by a solid during its free-fall motion in any fluid is known as terminal velocity.