P M V Subbarao

1
Interaction of Fluid flow with Solid walls

• P M V Subbarao
• Professor
• Mechanical Engineering Department
• I I T Delhi

How many Fluid Particles are getting Affted by
Wall to What extent!?!?!
2
The No-Slip Boundary Condition in Viscous Flows

• The Riddle of Fluid Sticking to the Wall in Flow.
• Consider an isolated fluid particle.

When this particle hits a wall of a solid body,
its velocity abruptly changes.
This abrupt change in momentum of the patricle is
achieved by an equal and opposite change in that
of the wall or the body.
3
Video of the Event

• At the point of collision we can identify normal
  and tangential directions n and t to the wall.
• The time of impact t0 is very brief.
• It is a good assumption to conclude that the
  normal velocity vn will be reversed with a
  reduction in magnitude because of loss of
  mechanical energy.
• If we assume the time of impact to be zero, the
  normal velocity component vn is seen to be
  discontinuous and also with a change in sign.

4
Time variation of normal and tangential
velocitycomponents of the impinging particle.
Whether it is discontinuous or not, the fact that
the particle has to change the sign of its
normal velocity, is obvious, since the particle
cannot continue penetrating into the solid wall.
5
The Tangential Component of Velocity

• The case of the tangential component vt is far
  more complex and more interesting.
• First of all, the particle will continue to move
  in the same direction and hence there is no
  change in sign.
• If the wall and the ball are perfectly smooth
  (i.e. frictionless).
• vt will not change at all.
• In case of rough surfaces vt will decrease a
  little.
• It is important to note that vt is nowhere zero.
• Even though the ball sticks to the wall for a
  brief period t0, at no time its tangential
  velocity is zero!
• The ball can also roll on the wall.

6
The Ideal Interaction Max Well (1879)

• 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

7
An Irreversible Interaction Max Well (1879)

• 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.

8
The condition of fluid flow at the Wall

• Fluid flow at wall is fundamentally different
  from the case of an isolated particle since a
  fluid flow is a field.
• The difference is that a fluid parcel in contact
  with a wall also interacts with the neighboring
  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?

9
Thermodynamic equilibrium

• Thermodynamic equilibrium implies that the
  macroscopic quantities need 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 flow 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.

10
A Peculiar condition for fluid flow at Solid wall

• 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.
• Fails to justify the occurrence of quasi
  thermodynamic-equilibrium.
• Two 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.

11
Knudsen Layer
Surface at a distance of one mean free
path/lattice spacing.
us
uw
Wall
12
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
Lattice Dimension
d diameter of the molecule
V is the molar volume NA Avogadros number.
n molar density of the fluid, number
molecules/m3
13
Velocity Extrapolation Theory
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
14
Boundary Conditions

• Maxwell was the first to propose the boundary
  model that has been widely used in various
  modified forms.
• Maxwells model is the most convenient and
  correct formulation.
• Maxwells model assumes that the boundary surface
  is impenetrable.
• The boundary model is constructed on the
  assumption that some fraction (1-ar) of the
  incident fluid molecules are reflected form the
  surface specularly.
• The remaining fraction ar are reflected diffusely
  with a Maxwell distribution.
• ar gives 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.

15

Slip Boundary conditions
Maxwell proposed the first order slip boundary
condition for a dilute monoatomic gas given by

Where
Tangential momentum accommodation coefficient
( TMAC )
--
--
Velocity of gas adjacent to the wall
--
Velocity of wall
--
Mean free path
Velocity gradient normal to the surface
--
16
Modified Boundary Conditions
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
Maxwell second order slip condition
-----
17
Regimes of Engineering Fluid Flows

• Conventional engineering flows Kn lt 0.001

• Micro Fluidic Devices Kn lt 0.1

• Ultra Micro Fluidic Devices Kn lt1.0

18

Flow Regimes

• Based on the Knudsen number magnitude, flow
  regimes can be classified as follows
    Continuum Regime Kn lt 0.001  Slip Flow
  Regime 0.001 lt Kn lt 0.1  Transition
  Regime 0.1 lt Kn lt 10  Free Molecular
  Regime Kn gt 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.

19
Popular Creeping Flows

• Fully developed duct Flow.
• Flow about immersed bodies
• Flow in narrow but variable passages. First
  formulated by Reynolds (1886) and known as
  lubrication theory,
• Flow through porous media. This topic began with
  a famous treatise by Darcy (1856)
• Civil engineers have long applied porous-media
  theory to groundwater movement.
• http//www.ae.metu.edu.tr/ae244/docs/FluidMechani
  cs-by-JamesFay/2003/Textbook/Nodes/chap06/node17.h
  tml