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 - Cauchy´s equation of dynamical equilibrium in continua. Euler equations and potential flows. Conformal mapping.
Rudolf Žitný, Ústav procesní a zpracovatelské techniky ČVUT FS 2010

2. Mass-Momentum-Energy
MHMT2
Mechanics and thermodynamics are based upon the
Conservation laws
-conservation of mass
conservation of momentum M.du/dt=F (second Newton’s law)
conservation of energy dq=du+pdv (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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Transported property 
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 
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Transported property 
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 
Momentum
Viscous stresses
Newton’s law (kinematic viscosity)
Total energy Enthalpy E h
E h=cpT
Heat flux
Fourier’s law (temperature diffusivity)
Mass fraction of a component in mixture A
A=A
diffusion flux of component A
Fick’s law (diffusion coefficient)

5. Momentum flux = stress MHMT2
Isn’t 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
this is situation at time t
this is situation at time t+lm/v lm
red line represents unit surface x y
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 

6. 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.

7. 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.
Accumulation of mass
Mass flowrate through sides W and E x x y z z
South
West
Top
East
North
Bottom y

8. 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

9. 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:  r z
r r z u v w
Summing the fluxes through the 6 faces of fluid element (volume ) gives:

10. Mass conservation (fixed fluid element)
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Example: Tornado
Show that the velocity field satisfies continuity equation for steady incompressible flow
MATLAB
alfa=1;beta=10;
t=linspace(1,100,500);
r=(2*alfa*t);
fi=beta/(2*alfa)*log(t);
z=t;
x=r.*cos(fi);
y=r.*sin(fi);
plot3(x,y,z)

11. Mass conservation (fixed fluid element)
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In a similar way we can derive the continuity equation for a spherical coordinate system, by balancing the following control volume r
r
r   x y z
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).

12. Time rate changes of  MHMT2
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

13. Fluid PARTICLE / ELEMENT
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Fluid PARTICLE / ELEMENT
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

14. Balancing  in a fixed fluid element and material derivative
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Balancing  in a fixed fluid element and material derivative
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

15. Balancing  in a fixed fluid element and material derivative
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Balancing  in a fixed fluid element and material derivative
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

16. Integral balance of  (fixed CV)
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Integral balance of  (fixed CV)
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) V ds

17. Moving Fluid element (Reynolds theorem)
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Moving Fluid element (Reynolds theorem)
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
Reynolds transport theorem
Internal volumetric sources of  (e.g. gravity, reaction heat, microwave…)

18. Moving Fluid element (proof)
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Moving Fluid element (proof)
Fluid element V+dV at time t+dt
velocity of particle (flow)
velocity of FE
Integral balance of property 
Fluid element V at time t
Amount of  in new FE at t+dt
Convection inflow at relative velocity
Diffusional inflow of 
Terms describing motion of FE are canceled

19. Moving Fluid element (proof)
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Moving Fluid element (proof)
Fluid element V+dV at time t+dt
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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Differential balance of 
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

21. Momentum conservation
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Momentum balance = balance of forces is nothing else than the Newton’s law m.du/dt=F applied to continuous distribution of matter, forces and momentum.
Newton’s law expressed in terms of differential equations is called
CAUCHY’S equation
valid for fluids and solids (exactly the same Cauchy’s equations hold in solid and fluid mechanics).
Modigiani

22. external forces, like gravity
Momentum integral balance
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MOMENTUM integral balances follow from the general integral balances
for
total stress
flux
external forces, like gravity
source

23. Momentum integral balance
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Alternative formulations (using Gauss theorem, Material derivative)
Decomposition of stress flux
viscous stress
total stress
isotropic pressure

24. 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, Cauchy’s 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.

25. Sum of forces on fluid particle
Balance of forces
MHMT2
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)
Newton’s law (mass times acceleration=force)
mass
acceleration
Sum of forces on fluid particle

26. Pressure forces on fluid element surface
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Resulting pressure force acting on sides W and E in the x-direction z y
Top
North
West x
East
South y z x
Bottom x

27. 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 x z x B S

28. Cauchy’s Equations MHMT2
Taking all forces together we arrive to the Cauchy’s equation, written in the component form for Cartesian coordinate system

29. Cauchy’s Equations MHMT2
The same technique can be applied for the cylindrical coordinate system (r,,z, velocities u,v,w). For example the Cauchy’s equation for the balance of forces in the axial direction z follows from summing of fluxes through the 6 faces of the fluid element  r z
r r z u v w
This equation will be later on necessary when calculating axial velocity profiles in pipes
Inertial term is the product of density times acceleration
Result

30. Cauchy’s Equations MHMT2
Cauchy’s equation holds for solid and fluids (compressible and incompressible)
formulation with primitive variables,u,v,w,p. Suitable for numerical solution of incompressible flows (Ma<0.3)
Ma-Mach number (velocity related to speed of sound)
Making use the previously derived relationship the Cauchy’s 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.

31. Euler’s Equations inviscid flows
MHMT2
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.

32. Euler’s Equations and velocity potential
MHMT2
Eulers’s equations are special case of Cauchy’s 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 Euler’s equation reduces to Bernoulli’s equation: acceleration+kinetic energy=pressure drop+external forces
Proof is based upon identity: see lecture 1.

33. Euler’s 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 Bernoulli’s equation

34. Euler’s Equations and stream function
MHMT2
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 .

35. Euler’s Equations vorticity and stream function
MHMT2
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
Problem of inviscid incompressible flows can be reduced to the solution of two Laplace equations for stream and potential functions, satisfying boundary conditions of impermeable walls ( ) and zero vorticity at inlet/outlet ().

36. Euler’s Equations flow around sphere
MHMT2
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,,)  r U
Velocity potencial satisfying boundary condition at r and zero radial velocity at surface is
The solution is found by factorisation  to functions rn (n=1,-2) and cos  (sin doesn’t work)
and velocities (gradient of )
Velocity profile at surface (r=R) determines pressure profile (Bernoulli’s equation)

37. see the result obtained by using complex functions
Euler’s 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).  r U
Stream function satisfying boundary condition at r (uniform velocity U) and constant  at surface is
giving radial and tangential velocities
see the result obtained by using complex functions
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. Euler’s Equations and complex functions
MHMT2
Many interesting solutions of Euler’s equations can be obtained from the fact that the real and imaginary parts of ANY analytical function
satisfy the Laplace equation (see next page).
z=x+iy 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. x y  
Conformal mapping
=const streamlines
w(z)
z(w)
=const Equipotential lines

39. Euler’s Equations and complex functions
MHMT2
Derivative dw/dz of a complex function w(z=x+iy)=+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
dy=0
dx=0
and this requirement is fulfilled only if both functions , satisfy Cauchy-Riemann conditions
and therefore

40. Euler’s 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. Euler’s Equations and complex functions
MHMT2
Example: Let’s consider the transformation w(z)=az2 in more details
Equipotential lines
=-3
=3
=1
=0.5
=0.1
=0
Stream lines
The same graph can be obtained from inverse transformation z(w)

42. Euler’s Equations and complex functions
MHMT2
Example: Let’s consider the transformation w(z)=az3
Equipotential lines
function psy
y=linspace(0.05,10);
ps=0.1;
hold off
for i=1:10
x=(0.333*(ps./y+y.^2)).^0.5;
plot(x,y)
ps=ps*2;
hold on
end
Stream lines

43. Euler’s Equations and complex functions
MHMT2
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. Euler’s Equations cylinder with circulation
MHMT2
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 (x+iy=r 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. Euler’s Equations elliptic cylinder
MHMT2
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 a+b. 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 z=x+iy 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. Euler’s Equations elliptic cylinder
MHMT2
Im y
-plane
Z-plane
Re b x
a+b a
For the conformal mapping of ellipse only three terms of Laurent’s series are sufficient
with
Inversion mapping (z) is the solution of quadratic equation
Complex potential (potential and stream function) is therefore

47. Euler’s Equations conformal mapping
MHMT2
Generally speaking it does not matter if we select analytical function w(z) mapping the spatial region (z=x+iy) to complex potential region w=+i, or vice versa.
This is because inverse mapping is also conformal mapping.

48. Euler’s Equations Schwarz Christoffel theorem
MHMT2
Schwarz Christoffel theorem explicitly defines a conformal mapping between the half plane (Im z>0) 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
z-plane
w-plane w6
Re w
a a a 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. Euler’s 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).
Re z
Im z
Im w
z-plane
a a a a4 a5 a6
1
2 w1 w2 w6
w-plane
Re w w3
w4* w5 
-4
4’
4’’
a4’ a4’’
w4’
w4’’
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. Euler’s 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 w2=w4=0) and inner angles divided by 
Im z
z-plane
w-plane
Im w
3=2
a1= a2=-1 a3= a4=1
w3=ih
2=1/2
Re z
w1=
4=1/2
w2=0
w4=0
Re w
It is possible to select 3 points ai arbitrarily (you must believe that it is generally true), for example a1= a2=-1 a3= and the point a4=1 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 C1=h and C2=0 follow from coordinates w3=ih and w4=0.
See M.Sulista: Analyza v komplexnim oboru, MVST, XXIII, 1985, pp

51. Euler’s 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 h=1 by MATLAB
=1
fi=linspace(-10,10,1000);
for psi=0.1:0.1:1
z=complex(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

52. Euler’s Equations 3D stream function
MHMT2
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. Euler’s Equations Simple questions MHMT2
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.
Is it possible that a velocity field satisfying the Euler’s equations and the continuity equation is rotational (therefore cannot be expressed as a gradient of potential)? Answer is positive again.

54. Moment of momentum
MHMT2

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 Cauchy’s 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)
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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 Cauchy’s equations
Euler’s equation
Bernoulli’s 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