Name: Viscous Force and Coefficient of Viscosity
Uploaded: 2019-07-19
Duration: 6 min 33 s
Description: Quantitatively, comparing and calculating is an important part of the analysis in fields of science. Let's discuss the mathematics involved in viscosity for...

Question

Define terminal velocity. Derive an expression for it. Explain how can you use this concept to determine

η

.

Accepted Answer

The maximum constant velocity acquired by a body while falling through a viscous medium is called its terminal velocity. It is attained when force of resistance of a medium is equal and opposite to the force of gravity. Stoke showed that the retarding force F due to viscous drag for a spherical body of radius r that moves with a velocity v in a fluid, with coefficient of viscosity η, is given as :

$F = 6 π η r v$

This expression is known as Stoke's Law.

Derivation with help dimensions :

$F \propto v$ , velocity

$F \propto r$ , radius of the body;

$F \propto η$ , coefficient of viscosity of the fluid.

⇒F=Aη^ar^bv^c

where A is a constant with no dimensions. Putting the dimensions,

[MLT^-2]=[ML^-1T^-2]^a [L]^b [LT^-1]^c

or [MLT^-2]=[M^a L^-a+b+c T^-a-c ]

Comparing the two sides, we get

a = 1, b = 1 and c = 1

$∴ F = A η r v$

A was determined to be $6 π$ .

$∴ F = 6 π η r v$

Derivation for terminal velocity :

Let p be the density of the body and ρ be the density of the medium. Then,

Weight of body = $\frac{4}{3} π r 3 ρ g$ ........(11.4)

Upthrust of body = $\frac{4}{3} π r 3 σ g$ ........(11.5)

Resultant force (downwards) = $\frac{4}{3} π r 3 (ρ - σ) g$

In equilibrium,

$6 π η r v = \frac{4}{3} π r 3 (ρ - σ) g$ ......(11.6)

⇒ $v = \frac{2}{9} \frac{r^{2} (p - σ)}{η} g$ .....(11.7)

This is the formula for terminal velocity.

It is apparent from the above expression that the terminal velocity is (a) directly proportional to the square of the radius of body, (b) directly proportional to the densities of the body and the medium, (conversely proportional to the coefficient of viscosity of the medium.

Define terminal velocity. Derive an expression for it. Explain how can you use this concept to determine $η$ .

Important Questions on Fluid Motion

Learn and Prepare for any exam you want !

Define terminal velocity. Derive an expression for it. Explain how can you use this concept to determine η.

Important Questions on Fluid Motion

Learn and Prepare for any exam you want !

Define terminal velocity. Derive an expression for it. Explain how can you use this concept to determine $η$ .