what is stokes law ? What is terminal velocity?

 Hi,

When a solid body moves through a fluid,the fluid in contact with the solid is dragged with it. Relative velocities are established between the layers of the fluids near the solid so that the viscous forces start operating. The fluid exerts viscous force on the solid to oppose the motion of the solid. The magnitude of the viscous force depends on the shape and size of the solid body, Its speed and the coefficient of viscosity of the fluid.

Suppose a spherical body of radius r moves at speed v through a fluid of viscosity ƞ .the viscous force F acting on the body depends on r,v and ƞ . assuming that the force is proportional to various powers of these quantities we can obtain the dependence  through dimensional analysis.

Let F = kravb ƞc

MLT-2 = kLa(LT-1)b(ML-1 T-1)c

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

a,b,c each = 1 

hence F = krvƞ, value of k =6π,

hence stokes law becomes,

F = 6πƞrv…… which is the required stokes law

Terminal velocity:

When a solid is dropped in a fluid, the forces acting on it are.

  • Weight  W acting downwards
  • Viscous force F acting upwards
  • Buoyant force B acting upwards

 

We know from Stokes law that F is proportional to velocity v.

Initially , F=0 and the solid is accelerated due to force W-B

Because of this acceleration, v increases. At a certain instant

F becomes equal to W-B. net force is 0 and solid falls with constant velocity. This is known as terminal velocity. which is given by the following formula:

V0 = (2r2(p-σ)g)/9ƞ

Wherev0 is terminal velocity,p is the density of the body, σ is density of the liquid and g is acceleration of the body.

  • 13

 Stokes law - the viscous force acting on a spherical ball is directly proportional to the coefficient of viscosity of the fluid, the radius of the ball and the terminal velocity.

 

Terminal Velocity - It is the uniform velocity with which a body falls down through a viscous medium

  • 40

In 1851, George Gabriel Stokes derived an expression, now known as Stokes' law, for the frictional force – also called drag force – exerted on spherical objects with very small Reynolds numbers (e.g., very small particles) in a continuous viscous fluid. Stokes' law is derived by solving the Stokes flow limit for small Reynolds numbers of the generally unsolvable Navier–Stokes equations:

F_d = 6 pi,mu,R,v_s,

where:

  • Fd is the frictional force acting on the interface between the fluid and the particle (in N),
  • μ is the dynamic viscosity (N s/m2),
  • R is the radius of the spherical object (in m), and
  • vs is the particle's settling velocity (in m/s).

  • -4

Terminal velocity is the term for the state an object reaches when the force of drag acting on it is equal to the force of gravity acting on it. When an object reaches its terminal velocity, it no longer accelerates, remaining at whatever velocity it was already traveling or else slowing down.

  • 4

thanx.

  • -15
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