Thermofluids for Medical Engineering

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Thermofluids for Medical Engineering

• (1) Basic Fluid Mechanics flow descriptions and
  kinematics
• (2) Dynamic similarity
• (3) Momentum theorems
• (4) Introductory dynamics of the cardiovascular
  system
• (5) Pipe Flows

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Difference between a fluid and a solid

• Solid Constant strain results from a constant
  stress (SI unit Pa Pascal N m2)

The familiar 1-dimensional case
Strain (extension) / (original length) No
dimension
Stress (force) / (area) Unit Pa or N/m2
Youngs modulus (material property) Unit Pa or
N/m2
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Kinematics of fluid motion

• Fluid
• Constant rate of strain results from a constant
  stress.
• (i.e. there must be a time element, or time must
  enter the consideration).
• In other words, the material deforms continuously
  under constant stress.

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Shearing stress and strain of a fluid
Munson, Fundamentals of Fluid Mechanics 4th
edition.
Shearing strain angular displacement
(for small )
U velocity of upper plate
Rate of shearing strain
velocity gradient
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Viscosity (friction) of a fluid

• We have thus established that, at least for this
  very simple case of a linear shear, the rate of
  shear strain is equal to the velocity gradient.
• In solid mechanics, it is usual to assume that
  stress is proportional to strain, at least for
  small displacement.
• In fluid, this implies stress or constant
  times the velocity gradient (?).

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Shear stress force per unit area For a
Newtonian fluid Shear stress is proportional to
rate of shear strain
Rate of shear strain Unit 1/s
Shear stress Unit Pa or N/m2
Dynamic viscosity Unit Pas Constant for
Newtonian fluid. Variable for non-Newtonian fluid
(stress depends nonlinearly on strain rate or
other factors)
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Viscosity (side note)

• A red blood cell (RBC) is about 7µm in diameter.
• In large blood vessels, blood is effectively
  Newtonian.
• Since individual RBCs do not interact too much
  with each other.
• In small vessels (capillaries), blood is
  non-Newtonian.
• Size of RBC becomes comparable to vessel
  diameter. RBCs interact with each other and the
  vessel wall. The continuum approximation of fluid
  breaks down and blood can no longer be considered
  Newtonian.

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Eulerian versus Lagrangian descriptionsof fluid
motion

• Eulerian Employ fixed coordinates Do not
  follow individual particles, velocity expressed
  as functions of spatial coordinates. (More
  common)
• Lagrangian Follow individual particles
  Positions of specified particles are the
  objectives. Can employ Newtons laws of motion
  but less convenient for applications.

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Eulerian versus Lagrangian descriptionsof fluid
motion

• What does it really mean by following individual
  particles (Lagrangian) versus using fixed
  coordinates (Eulerian)?
• Lagrangian (probably the one more familiar to
  you)
• Position of particle (x,y,z) to be found as a
  function of time
• Example
• at time 1, the particle is at (3,2)
• at time 3, the particle is at (9,18)
• But this takes care of ONLY ONE PARTICLE.

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Eulerian versus Lagrangian descriptionsof fluid
motion

• Eulerian (more common in fluid dynamics)
• We seek a function describing a certain variable
  in the whole region.
• Example velocity field (very artificial i.e. not
  real, illustration purpose only)
• Thus, at every position (i.e. x,y) in each
  instant (i.e. t), there is an associated
  velocity vector.
• In other words, we fix our region of interest and
  look at what happens in that region as time goes
  by.
• The function can also be scalar the associated
  variable at every position in each instant is
  thus a scalar
• e.g. temperature field, pressure field

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Eulerian versus Lagrangian descriptionsof fluid
motion

• Concept
• Lagrangian
• You need a single equation of motion for each
  particle of interest, (which is impractical for
  fluid motion since a fluid has too many
  particles).
• Particle position x,y,z is a function of time
• Eulerian
• A single field equation contains the information
  for every point in the region.
• Position x,y,z is no longer attached to
  individual particles. It only refers to one point
  in space different particles will flow past
  that point in space as time goes by.
• Position x,y,z and time are independent
  variables

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Flow Kinematics StreamlinesPathlinesStreakl
ines
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• Streamline line (or more precisely, curve)
  which is everywhere tangent to the velocity
  fields.
• Pathline line traced out by a given
  particle as it flows
• from one point to another.
• Streakline curve joining all particles in a
  flow that have previously passed through a given
  point.

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Streamline
Streamlines -everywhere tangent to velocity
vectors This velocity profile is a little bit
artificial. It is given here as an illustration
only.
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Velocity field
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Corresponding streamline
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Pathline
- Line traced out by a given particle
The particle we are looking at
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Pathline
- Line traced out by a given particle
Pathline of the particle
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Streakline

• Curve joining all particles that have previously
  passed through a given point.
• Produced experimentally by continuously injecting
  dye at a point in the flow field.
• Concept
• For steady flow (i.e. time invariant)
• streakline pathline streamline
• For unsteady flow, streamline, pathline,
  streakline are all distinct from each other.
• Usually, the pictures we see in textbooks are
  either pathlines (labelling a single particle by
  dye) or streaklines (continuous injection of dye
  at a point in the flow field)
• The term streamline is often used incorrectly,
  although all three types of lines (or curves) are
  the same in STEADY flow.

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Additional information

• http//www.atmos.washington.edu/durrand/animation
  s/vort505/vortanim2.psp
• Unsteady flow streamlines (in red) vary with
  time
• Pathlines of 3 particles shown
• The (circular) vortex is beyond the scope of this
  course.

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• What will the students of the class of 2010 be
  doing in 2011?
• Streamline Many students, ONE moment in time
  (i.e. many particles, one moment in time, in a
  time evolving situation).

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• What will you be doing in 2011, 2012, 2013?
• Pathline ONE student, DIFFERENT moments in
  time, or a sequence of time steps (i.e. one
  particle, different moments in time).

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• What will the top ranked students of the classes
  of 2008, 2009, 2010 be doing in 2011?
• Streakline Many students who share definite
  characteristics (being top ranked in MedEng of
  HKU), ONE moment in time (i.e. many particles,
  one moment in time).

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Dynamic Similarity of Flows

• We are looking for Dynamic Similarity and NOT
  Geometric Similarity.
• Testing airplanes in a wind tunnel in the
  development phase
• Suppose the length scales of the prototype
  (real thing) to the model is 100 1, should
  the velocities be also reduced by a factor of 100
  in the model study ?
• NO. We seek dynamic similarity by keeping
  certain non-dimensional parameters the same.

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Nondimensionalizing the equations ofmotion (a)
Choosing characteristic time, length, velocity
(and so on) scales. (b) Express equations in
non-dimensional forms.(c) Keep certain key
parameters the same.
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Dynamic similarity - concept

• Directly from the governing differential
  equations (if known)
• Original equation in the y direction (u, v
  velocities in x, y directions)
• Then, rescale variables with suitable reference
  quantities and make them dimensionless (for
  details please refer to notes)
• e.g
• x x/L (x L, L L)
• u u/U (u L/T, U L/T)
• p p/(?U2) (p Pa M/LT2, ? M/L3, U
  L/T)
• (These constants should be some meaningful
  parameters in the problem. For example, in pipe
  flow, L can be the diameter for the pipe. Since x
  is the distance from the tube inlet, the physical
  meaning of x is a normalized x, or, how many
  diameter downstream from inlet.)

Note M,L and T inside brackets refer to
dimensions of mass, length and time respectively
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Dynamic similarity - concept

• After some algebra
• where
• The whole equation is also dimensionless.
• Thus, if you have different scales for the same
  physical problem (e.g. micrometres vs kilometres,
  seconds vs years ), as long as your R and Fr are
  the same in the two scales, you end up with
  identical governing equations and boundary
  conditions in dimensionless form.
• Non-dimensional parameters in this case R
  (Reynolds number) and Fr (Froude number)
• Note the physical meaning of these numbers. e.g.
  Reynolds number is the ratio of inertial force to
  viscous drag force.

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Important nondimensionalizing the equations of
motion does NOT help us in solving the
differential equations (it should not, the
mathematical difficulty is always there).(2)
However, if you have one set of solutions (by
numerical simulations, laboratory experiments
etc), you can apply that solution (or data set)
to other configurations.
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Two coefficients of viscosity (a) Dynamic
viscosity and (b) Kinematic viscosity
(Dynamic Viscosity)/Density(2) Inertial force
mass (acceleration)
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Common non-dimensional parameters(1) Reynolds
number Ratio of inertial force to viscous
force (2) Froude number Ratio of inertial
force to gravity force(3) Mach number Ratio
of inertial force to elastic force(4) Weber
number Ratio of inertial force to surface
tension force.
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Inertial force (mass)(acceleration)(2)
Strictly speaking there is no physical mechanism
called inertial force. In practice, what is
driving the fluid to move?
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Dynamic similarity application

• To test the aerodynamics of an aircraft wing we
  can set up experiments in a smaller scale to save
  .
• In practical situations, it is almost always
  impossible to set up experiments such that ALL
  dynamic similarities are satisfied.
• We want to satisfy the most important aspect(s)
  of the problem
• e.g. in pipe flow, we want to satisfy the
  Reynolds number since the problem is dominated by
  viscous drag force and inertia force of the fluid.

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An alternative approach (without using
differential equations)

• (1) We first list or identify the variables
  (velocity, pressure, or length scales) relevant
  to the problem (without differential equations,
  but require more insight).
• (2) Form certain non-dimensional parameters.
• (3) Buckingham Pi theorem prescribe or limit
  the number of such nondimensional parameters.

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Inertial force (mass)(acceleration)
(density)(volume)(velocity/time)(2) Viscous
force (viscous stress)(area) (coefficient of
viscosity)(velocity gradient)(area)
(coefficient of viscosity)(velocity/length)(area)
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by assuming volume (length)3 area (length)2.
Hence(Inertial force)/(Viscous force)
(density)(volume)(velocity/time)/(viscosity
coefficient)(velocity)(length)
(density)(area/time)/viscosity coefficient
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(Inertial force)/(Viscous force)
(density)(length)(length/time)/coefficient of
viscosity (velocity)(length)(density)/coeffic
ient of viscosity Reynolds numberkinematic
viscosity dynamic viscosity/density
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An example on the importance of non-dimensional
parametersForce on a cylinder in a fluid will
depend on (a) the density of fluid, (b) the
diameter of the cylinder, (c) the free stream
velocity, and(d) the viscosity of the fluid.
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For each variable, consider ten data points.(2)
Each experiment say takes 1/2 hour. Thus we need
10,000 (0.5) 5,000 working hours.(3) We work 8
hours per day, 365 days a year. (4) The whole
process will take 2.5 years.(5) Time consuming,
expensive and NOT illustrative (4dimensional
space).
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Thus, dynamic similarity is not only an efficient
method to compare data/ numerical
simulations/experimental results around the
world,it is a very compact means to present
results / insight in a a complicated flow
configuration.