Title: MEDE 3005 Transport Phenomena for Biological Systems
1MEDE 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)
2- 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.
3Real 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).
4(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
5- (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
-
6Eulerian 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.
7Eulerian 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.
8Vorticity curl of the velocity field twice
the local angular velocity of the fluid
9Differential 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).
10Continuity 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)
11Incompressible flows (constant density)Derivati
on of the continuity equation in Cartesian
coordinates.
12Incompressible flows (constant density)Continuit
y equation divergence of the velocity field
0.
13 Continuity equation (1) in three
dimensions(2) in polar coordinates(3) in
summation convention.
14Reynolds number (reference velocity) X
(reference length)/(kinematic viscosity)Shear
stress (Dynamic viscosity) X (velocity
gradient)Kinematic viscosity (Dynamic
viscosity)/density
15Reynolds 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?
16Laminar flow slow, regular motion.Turbulent
flow fast, chaotic motion.Transition from
laminar to turbulent flows.
17Inviscid 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)
18Acceleration 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)
19Stream 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.
20Mathematicallyu ??/?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)
21The 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
-
22Viscous 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).
23Relation(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
24Non-dimensionalizing the equations of
motionConvective acceleration termsPressure
gradient termsBody force termsNew
ingredients VISCOUS terms
25Boundary 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.
26-
- (2) Viscous Fluid no penetration boundary
condition, PLUS - NO SLIP condition fluid canNOT slide along
the wall.
27Flow 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
28- 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.
29- 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.
30- 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.
31- 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).
32-
- Flow of a viscous fluid between corotating or
counterrotating cylinders - NO radial velocity
- NO axial velocity
- Only tangential velocity.
33-
- 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).
34- 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.
35- 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.)