Omid Ejtehadi

Ehsan Roohi, Javad Abolfazli Esfahani

Department of Mechanical Engineering

Ferdowsi university of Mashhad, Iran

Overview

Micro/Nano Couette Flow

DSMC Algorithm

Code Validation

Entropy and Entropy Generation

Results

Compressibility Effects

Rarefaction Effects

Conclusion

Micro-Fluidic Systems

Micro-Channel

Micro-Pump Micro-Motor Micro-Nozzle Micro-Valve

Micro-Turbine

Micro-Fluidic Systems

Thermal ink-jet operation

Micro-Beam

Micro-Propulsion

System

Flow Regimes

Boltzmann Equation (BE) / DSMC Collisionless BE /

DSMC Molecular

Models

Continuum

Models

Euler Equations

Navier-Stokes Equations

Burnett Equations

Kn 0.001 0.01 0.1 1 10 0

MEMS NANO

Slip-Flow Regime

Transition

Regime Free Molecular Regime

Continuum

Regime

Knudsen

Number = Mean Free Path

System Length

Micro-Fluidics: Governing Equation

• Boltzmann Equation: temporal-spatial changes of number of molecules in a velocity class c

Local rate of change of

number of molecules

11

*

1

*

4

0

2 )()(.)(.)( dcdcffffnnfc

Fnfr

cnft

r

Binary Collision term Influx of molecules due to

external force

Influx of molecules due to

convection

Assumptions i) Dilute Gas: Binary Collision

ii) Molecular Chaos: colliding particles are uncorrelated

DSMC Algorithm

-Developed by Bird (1960’s)

Initialize system with particles

Loop over time steps

Create particles at open boundaries

Move all the particles

Cell size < λ, time step < 1/ν

Process any interactions of particle &

boundaries

Sort particles into cells

Sample statistical values

Select and execute random collisions

Slide taken from Alejandro L. Garcia, Department of Physics, San Jose State University

DSMC Applications

Hypersonic Flight at High Altitudes

Vacuum Technology

Micro/nano Scale Devices

Micro/Nano Couette Flow

Schematic of the Planar Couette flow

Micro-Bearing A Hard Disk Drive and its Schematic

Platter

Convergence Check

Problem is essentially 1-D, periodic BC’s on the sides

At least 300 particles per each cell

Heat transfer coefficient Temperature

Grid Independency Study

Code Validation

Current Gu-Emerson DSMC NS Analytical

Kn= 0.01

Mw= 0.16

Kn= 0.1

Mw= 0.16

Entropy

Tendency of a process to proceed in a particular

direction

Proceeding from order into disorder (randomness )

Boltzmann Relation

the number of possible microstates in the system

In DSMC:

Sorting particles to achieve velocity distribution function f

and integrating the resulting distribution

Entropy Generation

Irreversibilities

Quantifying Non-equilibrium

Optimum Design

Positive Definite

Naterer and Camberos

Equilibrium Based Definition

Myong Formula

Entropy: Compressibility Effects

Similar trends in entropy and temperature profiles

T depends on variance of velocity

In probability theory, the variance is a measure of how far a set of

numbers is spread out (Wikipedia)

Increase of Mach number results in increase of entropy (disorder) in

the domain

Mach Mach

Entropy Generation: Compressibility

effects

Similar trends in density and entropy generation shows this parameter can

be used for quantifying non-equilibrium

Increase of Mach number results in increase of entropy generation

Also as Mach increases entropy profiles become non-uniform

Entropy Generation: Different

Approaches

Role of Entropy Flux

Entropy Generation: Compressibility

Effects

Generation of entropy is mainly due to viscous dissipation

Viscous Thermal

Entropy: Rarefaction Effects

Similarities between entropy and temperature profiles

Entropy Generation: Rarefaction

effects

In a more rarefied flow less entropy is generated

Entropy Generation: Rarefaction

Effects

Generation of entropy is mainly due to viscous dissipation

Viscous Thermal

Concluding Remarks

Increase of wall Mach number results in non-uniform entropy

profiles

Entropy and temperature profiles are following an identical

trend

Increase of Knudsen number results in more uniform entropy

profiles

Entropy generation can be properly applied in quantifying

non-equilibrium phenomena

In the micro-Couette flow the generation of entropy is mainly

due to viscous dissipation term

Thanks