State and Prove Stoke law
Answers
Explanation:
What is Stoke’s Law?
Stoke’s Law is a mathematical equation that expresses the settling velocities of the small spherical particles in a fluid medium. The law is derived considering the forces acting on a particular particle as it sinks through the loquid column under the influence of gravity. The force that retards a sphere moving through a viscous fluid is directly proportional to the velocity and the radius of the sphere, and the viscosity of the fluid. Sir George G. Stokes, an English scientist expressed clearly the viscous drag force F as:
F=6πηrv
Stokes’s law finds application in several areas such as:
Settling of sediment in freshwater
Measurement of the viscosity of fluids
In the next section, let us understand the derivation of Stoke’s Law.
Stoke’s Law Derivation
The viscous force acting on a sphere is directly proportional to the following parameters:
the radius of the sphere
coefficient of viscosity
the velocity of the object
Mathematically, this is represented as
F∝ηarbvc
Now let us evaluate the values of a, b and c.
Substituting the proportionality sign with an equality sign, we get
F=kηarbvc (1)
Here, k is the constant of proportionality which is a numerical value and has no dimensions.
Writing the dimensions of parameters on either side of equation (1), we get
[MLT–2] = [ML–1T–1]a [L]b [LT-1]c
Simplifying the above equation, we get
[MLT–2] = Ma ⋅ L–a+b+c ⋅ T–a–c (2)
According to classical mechanics, mass, length and time are independent entities.
Equating the superscripts of mass, length and time respectively from equation (2), we get
a = 1 (3)
–a + b + c = 1 (4)
–a –c = 2 or a + c = 2 (5)
Substituting (3) in (5), we get
1 + c = 2
c = 1 (6)
Substituting the value of (3) & (6) in (4), we get
–1 + b + 1 = 1
b = 1 (7)
Substituting the value of (3), (6) and (7) in (1), we get
F=kηrv
The value of k for a spherical body was experimentally obtained as 6π
Therefore, the viscous force on a spherical body falling through a liquid is given by the equation
F=6πηrv
Terminal Velocity Formula
In the case of raindrops, initially, it is due to the gravity that it accelerates. As the velocity increases, the retarding force also increases. Finally, when viscous force and the buoyant force is equal to the force due to gravity, the net force becomes zero and so does the acceleration. The raindrop then falls with a constant velocity. Thus, in equilibrium, the terminal velocity vt is given by the equation
vt=2a2(ρ−σ)g9η
where ρ and σ are mass densities of sphere and fluid respectively.
From the equation above, we can infer that the terminal velocity depends on the square of the radius of the sphere and inversely proportional to the viscosity of the medium
Answer:
Proof of Stokes’ Law
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Introduction:
The viscous force acting on a spherical body falling in a liquid with velocity v is F = 6πη r v. when a body falls in a liquid, it drags the layers of liquid with it downward while the fluid at large distance from the body remains unaffected. Thus a relative motion is produced among the layers of liquid.
Stoke’s Law
On the basis of experimental observations proof George stokes law concluded that the viscous force on a small sphere in a liquid depends upon:
Viscosity of liquid(η)
Radius ( r )
Speed of spherical body
By the method of dimensions
Or
Where K is a dimensionless constant.
Writing dimensions of F, η,r and v we get
Or
proof stokes law equating powers of both sides, we get
Substituting these values in equation (1), we get
F=K η r v
By experiments K = 6 π
Viscous force, F=6 π η r v
This proof is well known stokes’ law. It holds for small spherical bodies in a homogenous viscous fluid.
Terminal velocity
Let us see about proof stokes’ law,
Suppose a small spherical body of radius r, volume V and density ρ is released in a long column of liquid of density σ, viscosity η. The body accelerates downwards due to its weight. Now following three forces act on the body.
Weight of body W=mg=V ρ g(downward)
The weight of liquid displaced or force of buoyancy B=V σ g(upward)
These two forces combine to form the apparent weight of the body in the liquid given by
Viscous force of liquid, F = 6 π η r v (upward) where v is instantaneous velocity of body relative to liquid.
The effective weight of body acts downward, due to which the body accelerates. Let initial acceleration of body be a, then from Newton’s second law (F = ma) we have
Initial acceleration,
This shows that the proof stokes law when the body is placed on liquid, then it accelerates initially in downward direction. Due to which the velocity (v) of body increases and so the viscous force F= 6 π η r v also increases. A stage comes when the viscous force (F0 on body acting in upward direction becomes equal to the effective weight (W ’ ) of the body.
Explanation: