$2\mathrm{.}3$

The shaft in Problem $2\mathrm{.}2$ is rotated in the hole and it is found that a torque of $1$ N$$m is required sustain a speed

of rotation of $60$ rpm$.$ Calculate the viscosity of the lubricant in the annulus between the shaft and the block.

Assume a linear velocity gradient through the lubricant and ignore any end effects.

Momentum Transport and Laminar Flow of Newtonian Fluids

$2\mathrm{.}4$

A $5-$mm$-$thick film of oil of density $888$ kg$/$m$3$ and viscosity $0\mathrm{.}8$ Pa$$s flows down a vertical wall. $($a$)$ Calculate the

mass flow rate. If the wall were inclined at $45\backslash$deg to the vertical and the mass flow rate were the same, what would

$($b$)$ the film thickness and $($c$)$ the average flow velocity be$?$

CFD & Materials Processing Lab.

Problems

$2\mathrm{.}5$

A layer of molten slag of density $2700$ kg$/$m$3$ and viscosity $0\mathrm{.}3$ Pa$$s is being transferred from one reverberatory

furnace to another by flow down a plane between the two furnaces which is inclined at an angle of $80\backslash$deg to the

vertical. The inclined plane is $5$ m in length and the mass flow rate of the slag is $7\mathrm{.}5$ kg$/$s$.$

Calculate:

$($a$)$ The thickness of the layer

$($b$)$ The average linear flow velocity of the slag

$($c$)$ The mean residence time of the slag on the plane

$($d$)$ The fraction of the slag that remains on the plane for times equal to or greater than the mean residence time

Neglect end effects.

Momentum Transport and Laminar Flow of Newtonian Fluids

$2\mathrm{.}6$

A capillary flowmeter is being designed to measure the volumetric flow rate of air. The diameter of the capillary

is $0\mathrm{.}5$ mm and the maximum value of h that can be read on the manometer is $1$m$.$ Calculate the minimum length

of capillary at which, with a reading of h $=1$ m$,$ the flow through the capillary is still laminar. The manometric

fluid is carbon tetrachloride of density $1594$ kg$/$m$3$ and the density and viscosity of air at $298$ are, respectively,

$1\mathrm{.}19$ kg$/$m$3$ and $1\mathrm{.}8610-5$ Pa$$s$.$

CFD & Materials Processing Lab.

Problems

$2\mathrm{.}7$

When air at $300$ K is flowing through a horizontal tube of inner diameter $1$ cm$,$ the local flow velocity measured

at a radius of $3$ mm for the centerline of the tube is found to be $1$ m$/$s$.$ Calculate:

$($a$)$ The rate of pressure drop in the flowing air

$($b$)$ The average linear velocity of the flow

$($c$)$ The Reynolds number

$($d$)$ The shear stress exerted on the tube by the air

At $300$ K the density and viscosity of air are, respectively, $1\mathrm{.}177$ kg$/$m$3$ and $1\mathrm{.}8510-5$ Pa$$s

Momentum Transport and Laminar Flow of Newtonian Fluids

$2\mathrm{.}8$

Show that in laminar fluid flow in a tube of circular cross section, half of the fluid remains in the tube for

times equal to or greater than the mean residence time, and half remains in the tube for times equal to or

less than the mean residence time.

$2\mathrm{.}9$

For laminar flow in a tube of circular cross section of radius R$,$ by what factor would R have to be increased to:

$($a$)$ Double the average linear velocity

$($b$)$ Double the volume flow rate?

$2\mathrm{.}10$

A horizontal pipeline $20$ miles long delivers petroleum at a certain flow rate when the pressure drop over the

line is $3000$ kPa. If the pressure drop over the $20$ miles remains at $3000$ kPa, which of the three following

procedures provides the largest increase in the volume flow rate of the oil? Assume that the flow is laminar.

$(1)$ Doubling of the last $10$ miles of the pipe with an identical pipe.

$(2)$ Replacement of the last $10$ miles of the pipe with a pipe that has twice the diameter of the original.

$(3)$ Replacement of the last $10$ miles of pipe with a pipe that has twice the flow area of the original.