Momentum Heat Mass Transfer

1
Momentum Heat Mass Transfer
MHMT2
Balance equations. Mass and momentum balances.
Fundamental balance equations. General transport
equation, material derivative. Equation of
continuity. Momentum balance - Cauchys equation
of dynamical equilibrium in continua. Euler
equations and potential flows. Conformal mapping.
Rudolf Žitný, Ústav procesní a zpracovatelské
techniky CVUT FS 2010
2
Mass-Momentum-Energy
MHMT2
Mechanics and thermodynamics are based upon the

• Conservation laws
• -conservation of mass
• conservation of momentum M.du/dtF (second
  Newtons law)
• conservation of energy dqdupdv (first law of
  thermodynamics)
• Transfer phenomena summarize these conservation
  laws and applies them to a continuous system
  described by macroscopic variables distributed in
  space (x,y,z) and time (t)

Description of kinematics and dynamics of
discrete mass points is recasted to consistent
tensor form of integral or partial differential
equations for velocity, temperature, pressure and
concentration fields.
3
Transported property ?
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Transfer phenomena looks for analogies between
transport of mass, momentum and energy.
Transported properties ? are scalars (density,
energy) or vectors (momentum). Fluxes are amount
of ? passing through a unit surface at unit time
(fluxes are tensors of one order higher than the
corresponding property ?, therefore vectors or
tensors).
Convective fluxes (?? transported by velocity of
fluid) Diffusive fluxes (?? transported by
molecular diffusion) Driving forces gradients
of transported properties
4
Transported property ?
MHMT2
This table presents nomenclature of transported
properties for specific cases of mass, momentum,
energy and component transport. Similarity of
constitutive equations (Newton,Fourier,Fick) is
basis for unified formulation of transport
equations.
? Property related to unit mass P?? related to unit volume (?? is balanced in the fluid element) P diffusive molecular flux of property ? through unit surface Constitutive laws and transport coefficients having the same unit m2/s P-c?P
Mass 1 ? 0
Momentum Viscous stresses Newtons law (kinematic viscosity)
Total energy Enthalpy E h ?E ?h?cpT Heat flux Fouriers law (temperature diffusivity)
Mass fraction of a component in mixture ?A ?A??A diffusion flux of component A Ficks law (diffusion coefficient)
5
Momentum flux stress
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Isnt it strange that the momentum flux is a
viscous stress? This is an explanation
Consider a viscous fluid flowing in a gap between
parallel plates. The upper plate is moving at a
constant velocity. Follow two adjacent control
volumes with different momentum ?u in the
x-direction exchanging their position due to
random molecular motion in the y-direction
Through the unit surface (at plane y) flows from
above the value of momentum in the x-direction
and the x-momentum flows from below
Momentum in the boxes is preserved during the
exchange, because lm is the mean free path of
molecules and the mutual collisions are not
expected
Resulting x-momentum transported through the unit
surface y (in the direction y, from below) is
and the time necessary for the transport is lm/v
where v is a random velocity of molecules in the
y-direction. Resulting momentum flux (through the
unit surface and unit time) is therefore
x-momentum transport through the plane y
dynamic viscosity ?
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Mass conservation (fixed fluid element)
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Mass conservation principle can be expressed by
balancing of a control volume (rate of mass
accumulation inside the control volume is the sum
of convective fluxes through the control volume
surface). Analysis is simplified by the fact that
the molecular fluxes are zero when considering
homogeneous fluid.
Control volumes can be fixed in space or moving.
The simplest case, directly leading to the
differential transport equations, is based upon
identification of fluxes through sides of an
infinitely small FLUID ELEMENT fixed in space.
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Mass conservation (fixed fluid element)
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Using the control volume in form of a brick is
straightforward but clumsy. However, tensor
calculus is not necessary.
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Mass conservation (fixed fluid element)
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Using index or symbolic notation makes equations
more compact
Continuity equation written in the index notation
(Einstein summation is used)
Continuity equation written in the symbolic form
(Gibbs notation)
Example Continuity equation for an
incompressible liquid is very simple
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Mass conservation (fixed fluid element)
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Symbolic notation is independent of coordinate
system. However, the continuity equation written
in the component notation looks different in the
cartesian and in the cylindrical coordinate
system (r,?,z). The component form in the
cylindrical coordinate system of the mass balance
can be derived using the following control
volume
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Mass conservation (fixed fluid element)
MHMT2
Example Tornado
Show that the velocity field satisfies continuity
equation for steady incompressible flow
MATLAB alfa1beta10 tlinspace(1,100,500) r(2
alfat) fibeta/(2alfa)log(t) zt xr.cos(f
i) yr.sin(fi) plot3(x,y,z)
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Mass conservation (fixed fluid element)
MHMT2
In a similar way we can derive the continuity
equation for a spherical coordinate system, by
balancing the following control volume
This approach (based upon drawing and balancing a
control volume with 6 faces corresponding to
constant values of new coordinates) is quite
difficult when applied to tensors and in this
case a more general transformation techniques of
tensor calculus (Lameé coefficients, Christoffel
symbols) should be used. Probably the best way is
to look into textbooks where explicit formulae of
gradients, divergence, and Laplace operator are
given (at least for the cylindrical and spherical
coordinate systems).
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Time rate changes of ?
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Observer (an instrument measuring the property ?)
can be fixed in space and then the recorded rate
od change is
fixed observer measuring velocity of wind
10 km/h
Rate of change of property ?(t,x,y,z) recorded by
the observer moving at velocity
Total derivative Time changes of ? recorded by
observer moving at velocity
running observer
20 km/h
Material derivative is a special case of the
total derivative, corresponding to the observer
moving with the particle (with the same velocity
as the fluid particle)
observer in a balloon
0 km/h
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Fluid PARTICLE / ELEMENT
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Fluid element a control volume fixed in space
(filled by fluid)
Accumulation and convection
Convective fluxes through faces
Fluid particle a convected control volume
(group of molecules at a material point,
characterized by a property ? related to unit
mass).
Accumulation and convection
Modigliani
Only diffusional fluxes flow through moving faces
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Balancing ?? in a fixed fluid element and
material derivative
MHMT2
intensity of inner sources or diffusional fluxes
across the fluid element boundary
Accumulation ? in FE Outflow of ? from FE
by convection
This follows from the mass balance
These terms are cancelled
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Balancing ?? in a fixed fluid element and
material derivative
MHMT2
This is very important result, demonstrating
equivalence between the balances in a fixed
control volume and in the moving control volume
of a fluid particle.
Rate of ? increase of convected fluid particle
Flowrate of ?? out of the fixed fluid element
Remark continuity equation follows immediately
for ?1
Accumulation of ?? inside the fixed fluid element
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Integral balance of ? (fixed CV)
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Integral balance in a fixed control volume has
the advantage that it is possible to apply a time
derivative operator to integrand, because the
integral bound is constant (independent of time)
apply Gauss theorem (conversion of surface to
volume integral)
17
Moving Fluid element (Reynolds theorem)
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You can imagine that the control volume moves
with fluid particles, with the same velocity,
that it expands or contracts according to the
changing density (therefore it represents a
moving cloud of fluid particles), however The
same resulting integral balance is obtained in a
moving element as for the case of the fixed FE in
space
Diffusive flux of ? superposed to the fluid
velocity u
Internal volumetric sources of ? (e.g. gravity,
reaction heat, microwave)
Reynolds transport theorem
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Moving Fluid element (proof)
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Fluid element VdV at time tdt
velocity of particle (flow)
velocity of FE
Integral balance of property ?
Fluid element V at time t
Amount of ? in new FE at tdt
Convection inflow at relative velocity
Diffusional inflow of ?
Terms describing motion of FE are canceled
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Moving Fluid element (proof)
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Fluid element VdV at time tdt
this is the last equation from the previous slide
convective term can be converted to volumetric
integral
Fluid element V at time t
and it is seen that the conservation can be
expressed in terms of the material derivative
And these are exactly the same results as those
obtained with the fixed fluid element balance
20
Differential balance of ?
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Integral balance should be satisfied for
arbitrary volume V
Therefore integrand must be identically zero
Remark special case is the mass conservation for
?1 and zero source term
and using this the differential balance can be
expressed in the alternative form
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Momentum conservation
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Momentum balance balance of forces is nothing
else than the Newtons law m.du/dtF applied to
continuous distribution of matter, forces and
momentum. Newtons law expressed in terms of
differential equations is called CAUCHYS
equation valid for fluids and solids (exactly the
same Cauchys equations hold in solid and fluid
mechanics).
Modigiani
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Momentum integral balance
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MOMENTUM integral balances follow from
the general integral balances
for
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Momentum integral balance
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Alternative formulations (using Gauss theorem,
Material derivative)
Decomposition of stress flux
viscous stress
total stress
isotropic pressure
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Momentum conservation
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Differential equations of momentum conservation
can be derived directly from the previous
integral balance
which must be satisfied for any control volume V,
therefore also for any infinitely small volume
surrounding the point (x,y,z) and
N/m3
This is the fundamental result, Cauchys equation
(partial parabolic differential equations of the
second order). You can skip the following shaded
pages, showing that the same result can be
obtained by the balance of forces.
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Balance of forces
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Differential equations of momentum conservation
can be derived also from the force balances this
approach is not so elegant as the previous one,
but avoids the tedious conversion of divergence
of tensors at different coordinate systems (e.g.
cylindrical or spherical)
Newtons law (mass times accelerationforce)
Sum of forces on fluid particle
acceleration
mass
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Pressure forces on fluid element surface
MHMT2
Resulting pressure force acting on sides W and E
in the x-direction
z
y
Top
North
x
West
East
South
?z
?y
?x
Bottom
?x
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Viscous forces on fluid element surface
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Resulting viscous force acting on all sides
(W,E,N,S,T,B) in the x-direction
z
N
y
T
E
W
x
?y
?z
?x
?x
B
S
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Cauchys Equations
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Taking all forces together we arrive to the
Cauchys equation, written in the component form
for Cartesian coordinate system
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Cauchys Equations
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The same technique can be applied for the
cylindrical coordinate system (r,?,z, velocities
u,v,w). For example the Cauchys equation for the
balance of forces in the axial direction z
follows from summing of fluxes through the 6
faces of the fluid element
This equation will be later on necessary when
calculating axial velocity profiles in pipes
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Cauchys Equations
MHMT2
Cauchys equation holds for solid and fluids
(compressible and incompressible)
formulation with primitive variables,u,v,w,p.
Suitable for numerical solution of incompressible
flows (Malt0.3)
Ma-Mach number (velocity related to speed of
sound)
Making use the previously derived relationship
the Cauchys equation can be expressed in form
conservative formulation using momentum as the
unknown variable is suitable for compressible
flows, shocks. Passage through a shock wave is
accompanied by jump of p,?,u but (?u) is
continuous.
These formulations are quite equivalent
(mathematically) but not from the point of view
of numerical solution CFD.
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Eulers Equations inviscid flows
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Inviscid flow theory of ideal fluids is very
highly mathematically developed and predicts
successfully flows around bodies, airfoils, wave
motion, Karman vortex street, jets. It fails in
the prediction of drag forces.
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Eulers Equations and velocity potential
MHMT2
Eulerss equations are special case of Cauchys
equations for inviscible fluids (therefore for
zero viscous stresses)
Vorticity vector describes rotation of velocity
field and is defined as
for example the z-coordinate of vorticity is
Using vorticity the Euler equation can be written
in the alternative form
this formulation shows, that for zero vorticity
the Eulers equation reduces to Bernoullis
equation accelerationkinetic energypressure
dropexternal forces
Proof is based upon identity
see lecture 1.
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Eulers Equations and velocity potential
MHMT2
Inviscid flows are frequently solved by assuming
that velocity fields and volumetric forces f can
be expressed as gradients of scalar functions
(velocity potential)
Vorticity vector of any potential velocity field
is zero (potential flow is curl-free) because
to understand why, remember that for the Levi
Civita tensor holds ?imn - ?inm
Velocities defined as gradients of potential
automatically satisfy Kelvins theorem stating
that if the fluid is irrotational at any instant,
it remains irrotational thereafter (holds only
for inviscible fluids!).
Because vorticity is zero the Euler equation is
simplified
integrating along a streamline gives
Bernoullis equation
34
Eulers Equations and stream function
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In 2D flows it is convenient to introduce another
scalar function, stream function ?
Velocity derived from the scalar stream function
automatically satisfies the continuity equation
(divergence free or solenoidal flow) because
Curves ?const are streamlines, trajectories of
flowing particles. For example solid boundaries
are also streamlines. Difference ?? is the fluid
flowrate between two streamlines.
Advantages of the stream function ? appear in the
cases that the flow is rotational due to viscous
effects (for example solid walls are generators
of vorticity). In this case the dynamics of flow
can be described by a pair of equations for
vorticity ? and stream function ? In this way
the unknown pressure is eliminated and instead of
3 equations for 3 unknowns ux uy p it is
sufficient to solve 2 equations for ? and ?.
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Eulers Equations vorticity and stream function
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Let us summarize For incompressible
(divergence-free) flows the velocity potential
distribution is described by the Laplace equation
(ensures continuity equation)
For irrotational (curl-free) flow the stream
function should also satisfy the Laplace equation
36
Eulers Equations flow around sphere
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Example Velocity field of inviscid
incompressible flow around a sphere of radius R
is a good approximation of flows around gas
bubbles, when velocity slips at the sphere
surface. Velocity potential can be obtained as a
solution of the Laplace equation written in the
spherical coordinate system (r,?,?)
Velocity potencial satisfying boundary condition
at r?? and zero radial velocity at surface is
The solution is found by factorisation ? to
functions rn (n1,-2) and cos ? (sin doesnt work)
and velocities (gradient of ?)
Velocity profile at surface (rR) determines
pressure profile (Bernoullis equation)
37
Eulers Equations flow around cylinder
MHMT2
Example Potencial flow around cylinder can be
solved by using velocity potencial function or by
stream function. Both these functions have to
satisfy Laplace equation written in the
cylindrical coordinate system (the only
difference is in boundary conditions).
Stream function satisfying boundary condition at
r?? (uniform velocity U) and constant ? at
surface is
see the result obtained by using complex functions
giving radial and tangential velocities
Compare with the previous result for sphere the
velocity decays with the second power of radius
for cylinder, while with the third power at
sphere (which could have been expected).
38
Eulers Equations and complex functions
MHMT2
Many interesting solutions of Eulers equations
can be obtained from the fact that the real and
imaginary parts of ANY analytical
function satisfy the Laplace equation (see next
page). zxiy is a complex variable
(i-imaginary unit) and w(z)?(x,y)i?(x,y) is
also a complex variable (complex function), for
example
This is important statement Quite arbitrary
analytical function describes some flow-field.
Real part of the complex variable w is velocity
potential and the imaginary part Im(w) is stream
function!
Simple analytical functions describe for example
sinks, sources, dipoles. In this way it is
possible to solve problems with more complicated
geometries, for example free surface flows, flow
around airfoils, see applications of conformal
mapping.
Conformal mapping
?const streamlines
w(z)
z(w)
?const Equipotential lines
39
Eulers Equations and complex functions
MHMT2
Derivative dw/dz of a complex function
w(zxiy)?i? with respect to z can be a complex
analytical function as soon as both Re(w), Im(w)
satisfy the Laplace equation
Result should be independent of the dx, dy
selection, therefore
dy0?
dx0?
and this requirement is fulfilled only if both
functions ?,? satisfy Cauchy-Riemann conditions
and therefore
40
Eulers Equations and complex functions
MHMT2
The real and the imaginary part of derivative
dw/dz determine components of velocity field
w(z)?i? ux??/?x uy ??/?y streamlines






y
x
y
x
y
x
dipole
x
y
source
x
y
circulation
x
41
Eulers Equations and complex functions
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Example Lets consider the transformation
w(z)az2 in more details
Equipotential lines
Stream lines
The same graph can be obtained from inverse
transformation z(w)
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Eulers Equations and complex functions
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Example Lets consider the transformation
w(z)az3
Equipotential lines
function psy ylinspace(0.05,10) ps0.1 hold
off for i110 x(0.333(ps./yy.2)).0.5 plot(x
,y) psps2 hold on end
Stream lines
43
Eulers Equations and complex functions
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• The following examples demonstrate the most
  important techniques used for construction of
  conformal mappings
• Potential flow around circular cylinder with
  circulation (using directory of basic
  transformations, see previous slide application
  of superposition principle sum of analytical
  functions is also an analytical function)
• Potential flow around an elliptical cylinder
  (making use conformal mapping of ellipse to
  circle, based upon Laurent series expansion
  this is application of the substitution
  principle analytical function of an analytical
  function is also an analytical function)
• Cross flow around a plate (or how to transform an
  arbitrary polygonal region into upper half plane
  of complex potential Schwarz Christoffel
  theorem)
• Flows with free surface (contraction flow from an
  infinitely large reservoire through a slit)

44
Eulers Equations cylinder with circulation
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Example Potential flow around cylinder with
circulation can be assumed as superposition of
linear parallel flow w1(z)Uz, dipole w2(z)UR2/z
and potential swirl w3(z)?/(2?i) ln z (see the
previous table).
Substituting coordinates x,y by radius r and
angle ? results into (xiyr ei?)
Comparing real and imaginary part potential and
stream functions are identified
velocity potential is the real part of the
analytical function w(z)
stream function is the imaginary part of the
analytical function w
without circulation, I have a problem in Matlab
45
Eulers Equations elliptic cylinder
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Example Potential flow around elliptic cylinder.
Previous example solved the problem of potential
flow around a cylinder with radius R, described
by the conformal mapping
The analytical function transforming outside of
an elliptical cylinder to the plane of complex
potential w ?i? can be obtained in two steps
First step is a conformal mapping ?(z)
transforming ellipse with principal axis a,b to a
cylinder with radius ab. The second step is
substitution of the mapping ?(z) to the velocity
potential
There exist many techniques how to identify the
conformal mapping ?(z) transforming a general
closed region in the zxiy plane into a unit
circle, for example numerically or in terms of
Laurent series
this is the way how to solve the problems of
flow around profiles, for example airfoils. It is
just only necessary to find out a conformal
mapping transforming the profile to a circle.
46
Eulers Equations elliptic cylinder
MHMT2
Im?
y
Z-plane
?-plane
x
Re?
b
ab
a
For the conformal mapping of ellipse only three
terms of Laurents series are sufficient
with
Inversion mapping ?(z) is the solution of
quadratic equation
Complex potential (potential and stream function)
is therefore
47
Eulers Equations conformal mapping
MHMT2
Generally speaking it does not matter if we
select analytical function w(z) mapping the
spatial region (zxiy) to complex potential
region w?i?, or vice versa. This is because
inverse mapping is also conformal mapping.
48
Eulers Equations Schwarz Christoffel theorem
MHMT2
Schwarz Christoffel theorem explicitly defines a
conformal mapping between the half plane (Im zgt0)
to a polygonal region in the space w(z). The
region enclosed by linear segments is typical for
flow-field regions delineated by straight walls,
wedge flows, obstacles, branched channels and so
on.
Im z
Im w
w-plane
z-plane
w6
Re w
a1 a2 a3 a4
a5 a6
?1?
Re z
w1
?2?
w2
The transformation is described by integral
where exponents are inner angles in vertices
(divided by ?) and ai , C1,C2 values must be
determined from specified coordinates of vertices
wi.
49
Eulers Equations Schwarz Christoffel theorem
MHMT2
Schwarz Christoffel theorem can be applied also
for mapping to generalized polygons having one or
more vertices in infinity. In this way it is
possible to calculate the whole potential flow
field around obstacles (see next example).
This generalization can be derived by introducing
fictive points (for example w4 and w4) instead
of infinitely distant point w4 and by shifting
these points to infinity
Final formula is the same as for the finite
polygon, with the exception of the changed sign
in ?4
50
Eulers Equations cross flow around a plate
MHMT2
Example application of Schwarz Christoffel
theorem to flow around a plate of the height h.
See figure describing location of vertices (in
this case we consider infinitely thin plate,
therefore w2w40) and inner angles divided by ?
Im z
z-plane
w-plane
Im w
?32
a1? a2-1 a30 a41
w3ih
?21/2
?41/2
Re z
w1?
w40
w20
Re w
It is possible to select 3 points ai arbitrarily
(you must believe that it is generally true), for
example a1? a2-1 a30 and the point
a41 follows from symmetry. Also the point z0 can
be chosen arbitrarily (for example 0).
(you must also believe, that the term z-a1
disappears when a1??)
Values C1h and C20 follow from coordinates
w3ih and w40.
See M.Sulista Analyza v komplexnim oboru, MVST,
XXIII, 1985, pp.100-101.
51
Eulers Equations cross flow around a plate
MHMT2
Please notice the fact, that in this case the
role of z and w is exchanged, complex variable w
is spatial coordinates x,y, while z?i? is
complex potential of velocity field.
Solution for h1 by MATLAB
?1
filinspace(-10,10,1000) for psi0.10.11 zcomp
lex(fi,psi) w(z.2-1).0.5 plot(w) hold
on end
?0.1
See M.Sulista Analyza v komplexnim oboru, MVST,
XXIII, 1985, pp.100-101.
52
Eulers Equations 3D stream function
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Disadvantage of the approach using stream
function, complex variables and conformal mapping
is its limitation to 2D flows. While in the 3D
flow the irrotational velocity field can be
described by only one scalar function ?,
description of 3D solenoidal field (satisfying
continuity equation) by stream function is not so
simple. It is necessary to use a generalized
stream function vector and to decompose
velocities into curl free and solenoidal
components (dual potential approach)
Curl free (potential flow)
Divergence free (solenoidal flow)
Vorticity vector is expressed in terms of the
stream function vector
using identity
The dual potential approach increases number of
unknowns (3 stream functions and 2 vorticity
transport equations are to be solved) and is not
so frequently used.
53
Eulers Equations
MHMT2
Simple questions

1. Let the scalar function ?(t,x1,x2) satisfies
   Laplace equation. Does it mean that the gradient
   of this function represents velocity field
   satisfying both Euler equations in the directions
   x1,x2? The answer is positive.
2. Is it possible that a velocity field satisfying
   the Eulers equations and the continuity equation
   is rotational (therefore cannot be expressed as a
   gradient of potential)? Answer is positive again.
   

54
Moment of momentum
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55
Moment of momentum
MHMT2
Moment of a material point moving with velocity u
with respect to an arbitrary point (e.g. origin
of a coordinate system) is a vector product
Conservation of this moment can be expressed in
the integral form derived directly from general
integral balance of P, or from Cauchys equation
vector multiplied by x and integrated
This equation is useful for calculation of
rotational machines, like pumps, turbines
56
EXAM
MHMT2
Transport equations
57
What is important (at least for exam)
MHMT2
You should know what is it material derivative
Balancing of fluid particle Balancing of
fixed fluid element
Reynolds transport theorem
58
What is important (at least for exam)
MHMT2
Continuity equation and Cauchys equations
Eulers equation
Bernoullis equation
59
What is important (at least for exam)
MHMT2
What is it vorticity, stream function and
velocity potential
Special case for 2D flows
Complex potential, analytical functions and
conformal mapping