Viscosity and Mechanisms of Momentum Transport

8/12/2019 Viscosity and Mechanisms of Momentum Transport

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ChapterViscosity and Mechanisms of MomentumTransport


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CONCEPT OF VISCOSITY

Friction is felt only when you move either slower or faster than

the other passengers.

The extent of friction depends on the type of clothes they are

wearing.

It is this type of clothes that gives rise to the concept of

viscosity.


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Example of two parallel plates

Shear force acting on the second

molecular layer of fluid is due to

the difference in the velocities of

the two adjacent layers

Top layer stationary,

Bottom layer moves with constant velocity V A fluid is filled between the plates

No slip condition between fluid and plates at both the plate

surfaces

Flow is laminar


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x

y

x

y

Y t< 0

t= 0

x

y

x

y

small t

large t

V

V

vx(y)

V

vx(y, t)

Fluid initially

at rest

Lower plate set

in motion

Velocity buildupin unsteady flow

Final velocity

distribution insteady flow


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Mathematical Interpretation Of Common Sense

F V F V

A Y A Y

The force applied, F is the shear force

xdvV

Y dy

V/Yis the gradient or slope


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The shear stress exerted in the x-direction on a fluid

surface of constant y by the fluid in the region of

lesser y is designated as

yx

fluid surface of constant y, Shear

force on unit area perpendicular to

the y-direction

x-direction

Shear Stress

The shear stress is moving in the

direction of y because the bottom layerof fluid exerts a shear stress

on the next layer which then exerts

a shear stress on subsequent layer

Shear stress is induced by the

motion of the plate. Shear stress

can be induced by a pressuregradient or a gravity force.

Pressure force is a force acting on a

surface while the gravity force is the

force acting on a fluid volume


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The shear stress is a function of

1. Velocity gradient

2. Properties of the fluid

xyx

dv

dy

Where, vx= fluid velocity in the x-direction

= fluid viscosity, a property of the fluid, not the physical system

If this functional dependence is linear fluids are called

Newtonian Fluids

If this functional dependence is non-linear fluids are

called Non-Newtonian Fluids


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The momentum goes downhill from a region of high velocityto the region of low velocity, same as heat flows from higher

temperature towards lower.

xyx

dv

dy

This velocity gradient (dvx/dy) is the driving force for the

momentum transport.


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Viscosity divided by density= /

yx = N/m2, x = m/s, y = m

= Pa.s

Units of Quantities


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Viscosity data for water and air, for other gasesand liquids is provided in the tables in the textbook.

See Example 1.1-1


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For gases at low density the viscosity increases with the

increase in temperature.

In gases momentum is transported by the molecules in freeflight between collisions.

For liquids the viscosity usually decreases with increase intemperature.

In liquids the momentum transport takes place by the virtue

of intermolecular forces that pairs of molecules experience.


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In this equation the velocity is considered onlyin x-direction while vyand vzare zero.

xyx

dvdy

But usually three velocity components dependon all the three co-ordinates and time.

Therefore this relation needs to be generalized.


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What do you mean by generalization?

What are the vectors?

What are the tensors?


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Consider a general flow pattern

Velocity may be in various directions at

different places and also depends on time. The velocity components will be:

vx= vx(x,y,z,t)

vy= vy(x,y,z,t)

vz= vz(x,y,z,t)


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A small cube-shaped volume element within theflow field, each face having unit area.

The center is at position x, y, z


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Slice the volume element in such a way as to

remove half the fluid within it. Cut the volume perpendicular to each of the

three coordinates i.e. x , y and z.

Two types of forces will contribute:

Pressure forces

Viscous forces


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Pressure force will always be perpendicular to theexposed surface.

These will be exerted either the fluid is stationary or inmotion.

Pressure Forces


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Force per unit area on this surface is:

Pressure( scalar) Unit vector in x direction


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Pressure( scalar) Unit vector in y direction

Similarly for this face


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Pressure( scalar) Unit vector in z direction

Similarly for this face


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Viscous Forces

These come into play when there exist velocity

gradient in the fluid. Neither perpendicular to the surface element, nor

parallel to it.

Exist at some angle to the surface. In the last figures the viscous forces are: x, y, z

These forces have components, for example:

x has components xx, xy, xzyhas components yx, yy, yz

zhas components zx, zy, zz


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These include both types of stresses i.e.thermodynamic pressure and viscous stresses.

Where i and j may be x, y, or z


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Normal Stresses

Shear Stresses


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The quantities having one subscript associatedwith the coordinate directions are calledvectors.

The quantities having two subscriptsassociated with the coordinate directions arecalled tensors.

So, is viscous stress tensor and is molecularstress tensor.

Appendix A

Viscosity and Mechanisms of Momentum Transport

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