Molecular transport equation

7/25/2019 Molecular transport equation

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Problem

From the given illustration below, calculate

the steady-state shear stress yx and the

shear rate dvx/dy when the distancebetween the plates is 0.500 cm and the

viscosity of the fluid in-between is 1.77 cp.

V= 1.98 in/s


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Assumption

1. No-slip condition between the plates and the fluid

2. Fluid properties are constant all throughout

3. Plate and fluid motion is unidirectional only

4. No other external forces acting on the fluid

5. Both plates have the same uniform surface area

x

yx

dv

dy yx xdy dv


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Solution

2 2

1 1

2 1

2 1

y v

yx x yx

y v

v vdy dv

y y

-2 g in 2.540 cm

10 1.98 0cm s s 1 in1.77 cp1 cp 0 0.500 cm

yx

2 2

g dyne0.178 or 0.178

cm s cmyx

2 1

2 1

in 2.540 cm1.98 0

s 1 in

0 0.500 cm

xdv v v

dy y y

-110.1 s

xdv

dy


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Viscosity of Fluids


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As a fluid moves, shear stress develops.

No-Slip condition

The layer of fluid adjacent to the boundary

surface has zero velocity relative to that

surface.

Movement of a Fluid

What if the surface is stationary?

What if it is moving?


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Movement of a Fluid


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Velocity at wall should

be zero

But we know that there

is flow, i.e. nonzero

velocity

Movement of a Fluid


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LAMINAR FLOW

Movement of a Fluid


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Viscosity of a Fluid

x

yx

dv

dy


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x

yx

dv

dy

Viscosity (dynamic, )

- Constant of proportionality

- Resistance to flow- Gives rise to viscous forces that resist the

relative movement of adjacent layers in the fluid


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Newtonian Fluids

x

yx

dv

dy

A linear

relationship exists

between the shear

stress and thevelocity gradient

dv/dy.


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GASES

Increases with increasing temperature

Independent of pressure (up to 1000 kPa)

At > 1000 kPa, viscosity increases with increasing

pressure

LIQUIDS

Decreases with increasing temperature

Independent of pressure

Viscosity of Fluids


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Outline

1.Molecular Transport Equations

2.Viscosity of Fluids

3.Fluid Flow


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Fluid Flow


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Fluid Flow

LAMINAR

Low velocity

No lateral mixing

TURBULENT

High velocity

Formation of eddies


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Fluid Flow


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Viscous forces are dominant

Occur at low velocities

Layers of fluid seem to slide by one another

without eddies or swirls

No lateral mixing in the fluid

Fluid travels smoothly and in regular paths

Laminar Flow


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Inertial forces are dominant

Fluid travels in random, chaotic paths

Heavy and lateral mixing occurs

Turbulent Flow


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Ratio of inertial forces (v2) on a fluid to the viscous forces

(v/D) acting on it.

Used to characterize different flow regimes

2

Re

v DvN

v D

NRe < 2100 : laminar

2100 < NRe < 4000 : transition

NRe > 4000 : turbulent

Reynolds Number

Molecular transport equation

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