Question

a)' Derive Stock's law dimensionally and hence fiod an expression for terminal velocity​

Answer 1

Answer:

Explanation:

According to Stokes’ law, the backward viscous force acting on a small spherical body of radius r moving with uniform velocity v through fluid of viscosity η is given by

                         F=6 πη rv

 Where, r = Radius of the spherical body

v = velocity of the spherical body

It gives the relationship between retarding force and velocity. When viscous force plus buoyant force becomes equal to force due to gravity, the net force becomes zero. The sphere then descends with a constant terminal velocity (vt).

Now,

                          6 π η rv =4/3 πr^3( p-σ )

Where, ρ = Density of the liquid

σ = Density of the spherical body

Terminal velocity of a spherical body of density ρ and radius r moving through a liquid of density ρ’ is

where,

ρ = density of body,

sigma = density of liquid,

η = coefficient of viscosity of liquid and,

g = acceleration due to gravity

1)  If ρ > ρ0, the body falls downwards.

2) If ρ < ρ0, the body moves upwards with the constant velocity.

3) If po << ρ, v = (2r2ρg/9η)

Answer 2

Answer:

Stokes Law Formula

Stokes came up with this formula in 1851 to calculate this drag force or frictional force of spherical objects immersed in viscous fluids. Here, look at the formula mentioned below.  

F = 6 * πηrv  

Where,  

F is the drag force or frictional force at the interface.  

η is viscosity of a liquid.  

r is radius of the spherical body.  

V is velocity of flow.  

Stokes Law Derivation  

Stokes’ proposition regarding this immersion of the spherical body in a viscous fluid can be mathematically represented as,  

F ∝ ηa rb vc  

By solving this proportional expression, we can get Stokes law equation. To change this proportionality sign into equality, we must add a constant to the equation. Let us consider that constant as ‘K’, so the transformed equation becomes  

F = K * ηa rb vc                …………….. (1)

Now, Let us write down the dimensions of this equation on both sides. Please note, K is a constant which has no dimension.  

[MLT-2] = [[ML-1T-1]a . Lb . [LT-1]c ]  

We need to simplify this expression by separating each variable as follows

[MLT-2] = [Ma ] [L-a+b+c]  [T-a-c]

Comparing the value of length, mass and time, following equations can be found.  

a = +1  …………….. (2)

-a+b+c = +1 …………….. (3)

-a-c= -2  …………….. (4)

On solving equation 2,3, and 4, we get a =1, b=1, c=1.  On substituting these values in equation 1, we have  

F = K * η1 r1 v1 = K ηrv

Further, the value of K was found to be 6π for spheres through experimental observation. The above calculations helped in Stokes equation derivation along with its fundamental formula.  

Terminal Velocity Formula  

As explained earlier, terminal velocity is attained at an equilibrium position when the net force acting upon the spherical body and acceleration becomes zero. Here is the formula for terminal velocity derived from Stokes law definition.  

Vt = 2a2 (ρ−σ) g / 9η

Where,  

• ρ is mass density of a spherical object.  

• σ is mass density of a fluid.

TERMINAL VELOCITY:

Terminal velocity is defined as the highest velocity attained by an object that is falling through a fluid. It is observed when the sum of drag force and buoyancy is equal to the downward gravity force that is acting on the object. The acceleration of the object is zero as the net force acting on the object is zero.

The terminal velocity acquired by the ball of radius r when dropped through a liquid of viscosity η and density ρ is, v=2r2(ρo−ρ)g9η.

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