7/30/2019 Lect 2 - 4 - Momentum Transport

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Mechanism of Momentum

Transport

By

Amol Deshpande

04/08/2011 Transport Phenomena

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Outline

Viscosity and Newtons Law of Viscosity

Generalization of Newtons Law of viscosity

Pressure and Temp dependence of viscosity

Determination of viscosity

Gases

Liquids

Convective momentum transport

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Newtons Law of Viscosity

Example - Flow between two large parallel plates

Expression

Flux of x momentum in y - direction

Momentum flux / Momentum transport

Newtonian/Non-newtonian fluids

Kinematic viscosity

P & T effects on viscosity

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Molecular Stress Tensor

Molecular Stress

Pressure

Normal stresses

Viscous (Shear) stresses

Representations

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Generalization of Newtons Law

Viscous stress (generalized)

Time derivatives / Time integrals should not

appear in the expression No viscous forces are present, if fluid is in a state

of pure rotation

Fluid is isotropic

Generalized form

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Dilatational Viscosity (k)

Fluid dynamics problem it is discarded

Ideal mono-atomic gas : k =0

Incompressible liquid : Divergence is zero

Important in describing sound absorption in

polyatomic gases and fluid dynamics of liquids

containing gas bubbles

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Viscosity T and P Dependence

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Viscosity

Critical viscosity

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Determination of Viscosity - Gases

Maxwells expression of viscosity

Rigid sphere model

Viscosity is independent of P and proportional to

sqrt(T) Gives good results for pressures upto 10 atms (T > Tc)

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Determination of Viscosity - Gases

Chapman Enskog Equation

Intermolecular potential energy Empirical expression is

Lennard-Jones potential (for non-polar gases)

Lennard-Jones Parameters ( and )

Collision diameter, Characteristic energy (Eqs. 1.4-11, 12,13)

Collision integral for viscosity (Accounts for the details of the paths

that the molecules take during binary collision)

Valid for Monatomic /Polyatomic gases

Viscosity of gas mixtures

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Determination of Viscosity - Liquids

Based on kinetic theory of transport properties ofmonatomic liquids developed by Kirkwood and

co-workers

h : Plancks constant

N : Avogadros No

V : Volume of a mole of a liquid

Tb: Normal boiling point

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Convective Momentum Transport

Momentum transport by bulk flow of the fluid

Convective momentum flux vectors

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Convective Momentum Transport

Convective momentum flux tensorvv

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Combined Momentum Flux

Used for setting up general momentum balances

Interpretation

xy = combined flux of y-momentum across a surface

perpendicular to x direction by molecular and

convective mechanisms

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Examples

Determination of viscosity of gases at low density

CO2 at 200C, 300C, 800C and at 1 atm.

Using Chapman-Enskog equation

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Problems

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