An Unified Analysis of Macro & Micro Flow Systems… P M V Subbarao Professor Mechanical Engineering...
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Transcript of An Unified Analysis of Macro & Micro Flow Systems… P M V Subbarao Professor Mechanical Engineering...
An Unified Analysis of Macro & Micro Flow Systems…
P M V SubbaraoProfessor
Mechanical Engineering Department
I I T Delhi
Slip to No-slip in Viscous Fluid Flows
Incident of A Single Fluid Particle on a Smooth Wall• When the particle hits a smooth wall of a solid body, its velocity
abruptly changes.
• This sudden change due to perfectly smooth surface (ideal surface) was defined by Max Well as Specular Reflection in 1879.
• The Slip is unbounded
Φi
ui
iv
vi
Φrrv
ur
vr•
Incident of A Single Fluid Particle on a Rough Wall
Φi
ui
iv
vi
Φ
r
rv
ur
vr
• When the particle hits a rough wall of a solid body, its velocity also changes.
• This sudden change due to a rough surface (real surface) was defined by Max Well as Diffusive Reflection in 1879.
• The Slip is finite.
The condition of fluid flow at the Wall
• This is fundamentally different from the case of an isolated particle since a flow field has to be considered now.
• The difference is that a fluid element in contact with a wall also interacts with the neighbouring fluid.
• The problem of velocity boundary condition demands the recognition of this difference.
• During the whole of 19th century extensive work was required to resolve the issue.
• The idea is that the normal component of velocity at the solid wall should be zero to satisfy the no penetration condition.
• What fraction of fluid particles will partially/totally lose their tangential velocity in a fluid flow?
Thermodynamic equilibrium
• Thermodynamic equilibrium implies that the macroscopic quantities need have sufficient time to adjust to their changing surroundings.
• In motion, exact thermodynamic equilibrium is impossible as each fluid particle is continuously having volume, momentum or energy added or removed.
• Fluid flows heat transfer can at the most reach quasi-equilibrium.
• The second law of thermodynamics imposes a tendency to revert to equilibrium state.
• This also defines whether or not the flow quantities are adjusting fast enough.
Knudsen Layer
• In the region of a fluid flow very close to a solid surface, the occurrence of quasi thermodynamic equlibrium is also doubtful.
• This is because there are insufficient molecular-molecular and molecular-surface collisions over this very small scale.
• This fails to justify the occurrence of quasi thermodynamic-equilibrium.
• Two defining characteristics of this near-surface region of a gas flow are the following:
• First, there is a finite velocity of the gas at the surface ( velocity slip).
• Second, there exists a non-Newtonian stress/strain-rate relationship that extends a few molecular dimensions into the gas.
• This region is known as the Knudsen layer or kinetic boundary layer.
λ
us
uw
Wall
Surface at a distance of one mean free path/lattice spacing.
Knudsen Layer
LayerKnudsen :
Dimension ticCharacteri :L
Molecular Flow Dimensions
• Mean Free Path is identified as the smallest dimensions of gaseous Flow.
• MFP is the distance travelled by gaseous molecules between collisions.
• Lattice Spacing is identified as the smallest dimensions of liquid Flow.
Mean free path : 2
1
dn
d : diameter of the molecule
n : molar density of the fluid, number molecules/m3
Lattice Dimension: 3
1
AN
V
V is the molar volume NA : Avogadro’s number.
Velocity Extrapolation Theory
LayerKnudsen :
Dimension ticCharacteri :L
Knudsen defined a non-dimensional distance as the ratio of mean free path of the gas to the characteristic dimension of the system.This is called “Knudsen number”
LKn
........2 bKnaKnuu ws
Boundary Conditions
• Maxwell was the first to propose the boundary model that has been widely used in various modified forms.
• Maxwell’s model is the most convenient and correct formulation.• Maxwell’s model assumes that the boundary surface is
impenetrable.• The boundary model is constructed on the assumption that some
fraction (1-r) of the incident fluid molecules are reflected form the surface specularly.
• The remaining fraction r are reflected diffusely with a Maxwell distribution.
rgives the fraction of the tangential momentum of the incident molecules transmitted to the surface by all molecules.
• This parameter is called the tangential momentum accommodation coefficient.
Slip Boundary conditions Maxwell proposed the first order slip boundary condition for a dilute monoatomic gas given by:
w
ws y
uuu
2
Where :
--Tangential momentum accommodation coefficient
( TMAC )su Velocity of gas adjacent to the wallwu Velocity of wall--
--
Velocity gradient normal to the surface--y
u
-- Mean free path
w
ws y
uKnuu
2 Non-dimensional form ( I order slip
boundary condition)-----
Using the Taylor series expansion of u about the wall, Maxwell proposed second order terms slip boundary condition given by:
Second order slip boundary condition
ww
ws y
uKn
y
uKnuu
2
22
2
2
Maxwell second order slip
condition-----
Modified Boundary Conditions
Regimes of Engineering Fluid Flows
• Conventional engineering flows: Kn < 0.001
02
22
22
ww
ws y
uKn
y
uKnuu
• Micro Fluidic Devices : Kn < 0.1
0~
~2
w
ws y
uKnuu
• Ultra Micro Fluidic Devices : Kn <1.0
02
22
22
ww
ws y
uKn
y
uKnuu
Based on the Knudsen number magnitude, flow regimes can be classified as follows : Continuum Regime : Kn < 0.001 Slip Flow Regime : 0.001 < Kn < 0.1 Transition Regime : 0.1 < Kn < 10 Free Molecular Regime : Kn > 10
In continuum regime no-slip conditions are valid. In slip flow regime first order slip boundary conditions are
applicable. In transition regime (according to the literature present) higher
order slip boundary conditions may be valid. Transition regime with high Knudsen number and free
molecular regime need molecular dynamics.
Flow Regimes