Question

Osborn Reynolds discovered that the flow regime depends mainly on the ratio of the inertia forces to viscous forces in the fluid. With your own understanding and justification, analyze the effects of frictional factor to the boundary layer.

Answer 1

A boundary layer is defined as the region adjacent to the fluid over which the fluid motion is affected by the presence of a wall.Also the reynolds number is defined as tha ratio of inertia force to viscous force and is given by the expression Re=(rho x VxD/mu)= [V x D/(mu/rho)] or Re=VxD/md where md represents the term momentum diffusivity and is given by the relation (mu/rho) where mu is viscosity of fluid and rho is its density.

Now what happens when a fluid flows over some solid surface like a flat plate ,the part of the fluid which is in close contact with the plate comes to rest because of no slip boundary condition.However the top of fluid still moves with free stream velocity.But as fluid flows further in the downstrem,a larger part of fluid comes into contact with the wall and thus greater part of the fluid experience the effect of the wall which tries to bring down the fluid motion at rest.However,because of inertia,the top layer of the fluid tries to drag away the bottom layer adhered to it in further direction along with it.SO the motion of the fluid is combinely dependent on its inertia and the wall effect which is manifested in the form of friction effect.Therefore whether a part of fluid will move at faster or greater speed or will come to rest not only depends on the frictional effect of wall but also the how the momentum of fluid gets transmitted from fast moving stream to the lower moving stream in downward direction. As a result the denominator term in Reynold number which is md i.e. momentum diffusivity becomes larger and hence the Reynolds number .Therefore the boundary layer formed in this region attains particular thickness which is inversely proportional to square root of Reynolds number and similarly a coeeficient of friction is also defined having the same correlation.

Now as the fluid further passes to the downstream or it covers more length of the plate,the perturbation arising due to the disturbance created by wall stream gets amplified and thus this disturbance propogates to the main stream of fluid causing the eddy formation and resulting loss of momentum of the fluid.However when the flow regime changes from laminar to turbulent because of change in length traversed ,the lesser part of fluid gets affected because of the wall and thus the gradient of velocity term at the wall becomes sufficinetly larger which means we will have higher viscous force at the wall.As a result of this ,the friction factor which is a function of shear stress at the wall becomes sufficiently high in case of Turbulent flow and fluid adjusts to this change because of its momentum.However the relation of boundary layer thickness can be well justified by solving the fluid flow equations,but it is clear that boundary thickness depends on Reynolds number either for laminar or turbulent flow.