Detailed Insight into the Derivation of NS Equations

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• Detailed Insight into the Derivation of N-S
  Equations
• Barbaros CETIN
• Microfluidics and Lab-on-a-chip Lab.
• Mechanical Eng., Vanderbilt University
• September 11, 2007

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Detailed Insight into the Derivation of N-S
Equations _______________________________________
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• Why infinitesimal element?

Local Thermodynamic Equilibrium Density,
temperature, pressure are assumed to be
uniform i.e. we can define density, temperature
and pressure by averaging the molecular
information
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__________________________________________________
__________________ Microfluidics and
Lab-on-a-chip Lab., Mechanical Eng., Vanderbilt
University
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Detailed Insight into the Derivation of N-S
Equations _______________________________________
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• When we can ensure LTE?

• Sufficiently large number atoms inside the
  infinitesimal element
• If the molecular relaxation (or adjustment) time
  is very small compared to the characteristic time
  of the flow.
• Relaxation time is directly proportional with the
  collision frequency of the molecules

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__________________________________________________
__________________ Microfluidics and
Lab-on-a-chip Lab., Mechanical Eng., Vanderbilt
University
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Detailed Insight into the Derivation of N-S
Equations _______________________________________
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• When we can ensure LTE?

For liquids, it is extremely small due to the
compact packaging of the molecules. For gases,
different for different energy modes e.g. for
air, at moderate temperature f 1010
collisions/sec Is relatively small at standard
applications.
__________________________________________________
__________________________________________________
__________________ Microfluidics and
Lab-on-a-chip Lab., Mechanical Eng., Vanderbilt
University
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Detailed Insight into the Derivation of N-S
Equations _______________________________________
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• Keeping mind the LTE

Continuity Eqn.
Momentum Eqn.
of Eqns 4 of unknowns 10 NOT a closed SET!
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__________________________________________________
__________________ Microfluidics and
Lab-on-a-chip Lab., Mechanical Eng., Vanderbilt
University
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Detailed Insight into the Derivation of N-S
Equations _______________________________________
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• Newtonian Fluid Stress tensor is assumed to be
  linearly related to the rate of strain

Stress tensor 2nd Rank Rate of strain 2nd
Rank Constant of proportionality 4th Rank
__________________________________________________
__________________________________________________
__________________ Microfluidics and
Lab-on-a-chip Lab., Mechanical Eng., Vanderbilt
University
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Detailed Insight into the Derivation of N-S
Equations _______________________________________
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• For an isotropic fluid and symmetric stress
  tensor, 81 coefficients reduce to two
  coefficients
• Dynamic viscosity (µ)
• Second coefficient of viscosity (?)

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__________________________________________________
__________________ Microfluidics and
Lab-on-a-chip Lab., Mechanical Eng., Vanderbilt
University
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Detailed Insight into the Derivation of N-S
Equations _______________________________________
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Define a mechanical pressure
bulk viscosity
What Stokes did? He simply assumed
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__________________________________________________
__________________ Microfluidics and
Lab-on-a-chip Lab., Mechanical Eng., Vanderbilt
University
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Detailed Insight into the Derivation of N-S
Equations _______________________________________
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Is Stokes hypothesis valid?
For liquids, incompressible fluid,
NO PROBLEM!
For gases, we have to understand the mechanical
and thermodynamic pressures concept from
microscopic point of view!!!
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__________________________________________________
__________________ Microfluidics and
Lab-on-a-chip Lab., Mechanical Eng., Vanderbilt
University
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Detailed Insight into the Derivation of N-S
Equations _______________________________________
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the measure of the translational energy
the measure of the total energy (translational
vibrational rotational)
In a flow, energy can be transferred from one
mode to another, and bulk viscosity is the
measure of the transfer of the energy from
translational mode to other mode. For dilute
monatomic gases, bulk viscosity is zero!
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__________________________________________________
__________________ Microfluidics and
Lab-on-a-chip Lab., Mechanical Eng., Vanderbilt
University
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Detailed Insight into the Derivation of N-S
Equations _______________________________________
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E.g. N2,
at moderate temperatures
The bulk viscosity is expected to be important if
the relaxation time of the molecules is not small
compared to the characteristic time of the
flow E.g. polyatomic gas through a shock wave
E.g. by inclusion of the bulk viscosity,
hypersonic laminar boundary layer flow deviates
significantly from N-S Eqns
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__________________________________________________
__________________ Microfluidics and
Lab-on-a-chip Lab., Mechanical Eng., Vanderbilt
University
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Detailed Insight into the Derivation of N-S
Equations _______________________________________
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Mohamed Gad-El-Hak, Questions in Fluid Mechanics
Stokes Hypothesis for a Newtonian, Isotropic
Fluid, J Fluids Engineering, 117, 1995. the
author would like to thank the three anonymous
reviewers who gently showed me how little I knew
about this rich subject
__________________________________________________
__________________________________________________
__________________ Microfluidics and
Lab-on-a-chip Lab., Mechanical Eng., Vanderbilt
University