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Properties-of-fluid

A solid cone of maximum radius R and vertex angle `2ta`is to rotate at angular velocity `omega`. An oil of viscosity `mu` and thickness t fills gap between cone and housing. Derive an expression for torque required and rate of dissipation of heat in bearing.

SOLUTION:

Given,

Maximum radius of the cone =R

Vertex angle = `2 theta`

Viscosity of the oil= `mu`

Thickness of the oil layer = t

A solid cone of maximum radius R and vertex angle `2theta`is to rotate at angular velocity `omega`. An oil of viscosity `mu` and thickness t fills the gap between the cone and the housing. Derive an expression for the torque required and the rate of dissipation of heat in the bearing.

Let us consider an elementary area `dA` at a radius r of the cone as shown in figure.Let ds be the inclined length of element and dy be the thickness of element. Now,the elementary area is 

`dA=2pir*ds`

or, `dA=2pi*r*(dr)/(sin theta)`

Now,

Shear stress,  `tau=mu*(du)/(dy)`

Let us assume the gap `t` to be small....Show More

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This Chapter Properties-of-fluid consists of the following topics

A circular disc of diameter D is slowly rotated in a liquid of large viscosity `mu` at a small distance h from a fixed surface. Derive an expression of torque (T) necessary to maintain an angular velocity `omega`.

A solid cone of maximum radius R and vertex angle `2ta`is to rotate at angular velocity `omega`. An oil of viscosity `mu` and thickness t fills gap between cone and housing. Derive an expression for torque required and rate of dissipation of heat in bearing.

Oil of viscosity `mu=0.20`Ns/`m^2` fills the small gap of thickness y=0.2 mm as shown in the figure. Neglecting the fluid stress exerted on the circular underside, find the torque required to rotate the truncated cone at speed of 90 rpm. What is the rate of heat generation, in joules per second, if the value of `alpha=45^0` a=45 mm and b=60 mm.

A solid cone of maximum radius R and vertex angle `2theta`is to rotate at angular velocity `omega`. An oil of viscosity `mu` and thickness t fills the gap between the cone and the housing. Derive an expression for the torque required and the rate of dissipation of heat in the bearing.

Properties of fluid

Fluid-Mechanics