MEDE 3005 Transport Phenomena for Biological Systems

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MEDE 3005Transport Phenomena for Biological
Systems

• Dr. K. W. Chow (3 weeks) Basic principles of
  fluid dynamics Conservation laws of mass and
  momentum Continuity of equations Eulers and
  Navier Stokes equations of motion.
• Dr. C. O. Ng (3 weeks)
• Dr. L. Q. Wang (the rest of the course)

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• Basic Concepts in Fluid Flows
• Dependence of time and space
• (a) Steady uniform flows properties independent
  of time and space
• (b) Steady non-uniform flows properties
  independent of time but depend on space (e.g.
  converging channels)
• (c) Unsteady uniform flows properties depending
  on time but not on space (e.g. turning on a
  faucet slowly)
• (d) Unsteady non-uniform flows properties
  depending on space and time.

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Real and ideal fluids(1) Ideal fluid no
friction, fluid can slide tangentially along
the solid boundary.(2) Real fluid will possess
friction (or viscosity), fluid cannot slide
along boundary no slip boundary
condition.Tangential velocity zero if the wall
is at rest.(3) Velocity component perpendicular
to the wall must be the same as that of the wall
no penetration condition ( zero if the wall is
at rest).
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(1) Incompressible flows density of the fluid
remains constant. Otherwise compressible
fluid.(Strictly speaking incompressible flow
refers to the material derivative of the density
being zero)In practice, compressible if the
Mach number about 0.5 or so.Sound speed 340 m
per s. (2) 1D, 2D, 3D (dimensional) flows
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• (1) Differential (versus integral or macroscopic,
  i.e. control surfaces, control volume types)
  Analysis of Fluid Motions
• (2) Conservation of mass, momentum and energy.
• Incompressible fluids no need to use the energy
  equation.
• (3) Material derivative Derivative following
• the particle as it flows

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Eulerian versus Lagrangian descriptionsof fluid
motion

• Eulerian fixed coordinates, do not follow
  particles, velocity expressed as functions of
  spatial coordinates, i.e. different particles
  will flow through the same point at different
  times. Most fluid mechanics textbooks and papers
  use this system, i.e. more common than the
  Lagrangian method.

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Eulerian versus Lagrangian descriptionsof fluid
motion (contd)

• Lagrangian follow individual particles,
  positions of specified particles are the
  objectives. Can employ the more familiar Newtons
  laws of motion but less convenient for
  applications. Main difficulty is that we have
  billions of fluid particles and not just one or
  two.

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Vorticity curl of the velocity field twice
the local angular velocity of the fluid
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Differential analysis of the motion of a fluid
element(1) Translation,(2) Angular
distortion,(3) Rotation (related to the
vorticity),(4) Volume distortion (zero if the
fluid is of constant density).
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Continuity Equation Conservation of Mass

• Mass outflux mass influx
• flow due to source(s)
• flow due to sink(s)
• Volume flux velocity X (area)
• velocity X (length) X depth normal to page
• Mass flux density X (volume flux)

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Incompressible flows (constant density)Derivati
on of the continuity equation in Cartesian
coordinates.
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Incompressible flows (constant density)Continuit
y equation divergence of the velocity field
0.
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Continuity equation (1) in three
dimensions(2) in polar coordinates(3) in
summation convention.
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Reynolds number (reference velocity) X
(reference length)/(kinematic viscosity)Shear
stress (Dynamic viscosity) X (velocity
gradient)Kinematic viscosity (Dynamic
viscosity)/density
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Reynolds number (inertial force)/(viscous
force)Inertial force Mass X AccelerationRe
gtgt 1, viscosity not importantRe ltlt 1, viscous
effects dominant.What is driving the fluid
motion?
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Laminar flow slow, regular motion.Turbulent
flow fast, chaotic motion.Transition from
laminar to turbulent flows.
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Inviscid Equations of Motion Eulers equation
of motion

• Mass X (acceleration)
• Mass X (MATERIAL DERIVATIVE of the velocity
  field)
• Force
• (usually due to force from the pressure
  gradient alone)
• (Exceptions additional body force due
  stratification, rotation, electric charge etc)

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Acceleration in terms of the Eulerian
descriptionacceleration (velocity at t dt
velocity at t)/dt as dt tends to zero,but
velocity a function of (t, x, y, z) in the
Eulerian description. For 2 D flows, by using a
Taylors expansion (say x direction)?u/?t
(?u/?x)(dx/dt) (?u/?y)(dy/dt) ?u/?t
u(?u/?x) v(?u/?y)
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Stream functionDefinition of streamlinesA
line (or more precisely a curve) such that the
tangent to the curve is PARALLEL to the velocity
vector.As such the flow or the particles will
move along the streamlines.
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Mathematicallyu ??/?y v ??/?xas
follows from a consideration of ? constant and
take the differential d? 0. Analytically, the
stream function is a mathematical device to
satisfy the continuity equation identically (note
that ux vy 0 automatically)
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The Navier Stokes equations equations of motion
for a viscous fluid (Note the principle of
conservation of mass, or continuity equation,
holds whetherthe fluid is viscous or not.

• Eulerian description
• Force mass (acceleration)
• mass (MATERIAL DERIVATIVE of the velocity)
• net forces

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Viscous versus Ideal Fluids

• (1) For a viscous fluid (a fluid with friction),
  there will be tangential as well as normal
  stresses.
• (2) Net Forces Small differences due to
  differential changes in stresses (similar to the
  treatment in solid mechanics).

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Relation(s) between stress and strain
Constitutive equations.(Analytical details in
notes)In terms of (usual) symbols(i) Solid
mechanics u, v, w displacements(ii) Fluid
mechanics u, v, w velocities
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Non-dimensionalizing the equations of
motionConvective acceleration termsPressure
gradient termsBody force termsNew
ingredients VISCOUS terms
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Boundary Conditions(1) Ideal Fluid no
penetration, or the normal velocitiesmust match.
However, the fluid can still slide along the
wall, i.e. the tangential velocities of the fluid
and the wall need not match.

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• (2) Viscous Fluid no penetration boundary
  condition, PLUS
• NO SLIP condition fluid canNOT slide along
  the wall.

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Flow of a viscous fluid along an inclined plane
gravity acts as the body force, and NO pressure
gradient

• x, y axes along and normal to the inclined plane
• u U(y), v 0
• No shear stress at free surface
• No slip at the wall

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• Flow along a horizontal, RECTANGULAR channel
• No body force, and therefore must apply a
  pressure gradient.
• Otherwise the analysis is the same u U(y),
  v 0, p p(x)
• NO free surface, and hence no conditions
  involving the shear stress, instead, just no slip
  conditions at both walls.

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• Flow along a CIRCULAR pipe under constant
  pressure gradient
• No body force, and therefore we must apply a
  pressure gradient.
• The analysis and reasoning are the same as those
  in the rectangular channel case, but we must use
  polar coordinates.

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• Circular Pipe (contd)
• No radial nor tangential velocities.
• Only the axial velocity is nonzero
• The continuity equation implies that the axial
  velocity does not depend on the axial coordinate.
• Equations of motion in the r and T directions
  imply that the pressure depends on the axial
  coordinate only.

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• Usual separation of variables approach to find
  the velocity profile as a function of radius and
  pressure gradient (which must be constant).
• Boundary conditions no slip conditions at the
  wall, and the velocity must be finite at the
  center. (Two no slip conditions for fluids in an
  annular region).

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• Flow of a viscous fluid between corotating or
  counterrotating cylinders
• NO radial velocity
• NO axial velocity
• Only tangential velocity.

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• Continuity equation implies that this tangential
  velocity can only be a function of the radius.
  Can also be deduced from the axisymmetric
  requirement.
• Boundary conditions No slip will imply that
  the fluid will have an angular velocity at the
  walls of the moving cylinder(s).

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• Coutte flow of a layered fluid
• Consider a 2-layer fluid between two rigid walls
  and NO pressure gradient is present.
• Motion is then driven by forcing one (or two)
  rigid wall(s) into motion.
• Equations of motion will continue to hold for
  both fluids, but we must also insist on some
  matching conditions at the interface.

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• Layered fluid (contd)
• Equations of motion will imply the second
  derivative of the velocity must be zero, and
  hence the velocity profile must be piecewise
  linear.
• No slip boundary conditions at both walls.
• At the interface, the velocity and the shear
  stress must be continuous.
• (Note if the viscosities are different, gt
  velocity gradients NOT continuous.)