Basic Equations of Fluid Mechanics

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Basic Equations of Fluid Mechanics and Their
Derivations Momentum Equation and Energy Equation
by Prof. Dr. Dr.h.c. F. Durst
Institute of Fluid Mechanics University of
Erlangen-Nürnberg Cauerstr. 4, D-91058 Erlangen,
FRG
Title of Lecture
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The present lecture carrys our considerations
to derive the molecular transport terms for
Newtonian media, shows that tij are molecular
caused transport terms, explains the individual
terms in the tij-equations for Newtonian media,
derives the mechanical energy equations and
explains the physical meaning of the
individual terms, considers the thermal energy
equation and presents this in various forms,
underlines that the mechanical energy equation is
not an independent equations.
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Contents of Lecture
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The equations we have derived so far
are Continuity equation
Momentum equation ( j 1,2,3)
For Newtonian media we obtain
Continuity- and Momentum Equation
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Addition of continuity equations, after
multiplication with Uj, to the left hand side of
the momentum equation yields
or rewritten
If one introduces for the total velocity, the
fluid velocity and the molecular velocity, we
obtain
Molecular Caused Terms, 1
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Molecular Caused Terms, 2
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Molecular Caused Terms, 3
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Molecular Caused Terms, 4
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Molecular Caused Terms, 5
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Molecular Caused Terms, 6
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There is an additional term that expresses the
increase of the volume of a fluid element with
time. For this volume increase with time at a
point xi at time t we can write
For the surface enlargement of the fluid element
we can write
Because of the increase of the surface of the
fluid element, we get an additional input of
momentum
This term has to be added to the two other terms
derived in order to get the complete formula for
?ij.
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Molecular Caused Terms, 7
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Hence, we can write for Newtonian fluids
Derivations were carried out for an ideal gas.
They are valid in a similar manner for liquids
the molecules oscillate around a mean position
and transfer momentum in this way.
In the momentum equation we have the following
term
This is the difference between momen-tum input
and momentum output into a fluid element.
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for Newtonian Fluids
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For the considered fluid element we can write
Input through area A Output through area C
In the momentum equation the gradient of ?ij in
the xi-direction occurs. The double index i dens
for the summation in all three directions of
molecular transport, i.e. to obtain the total
amount of j-momentum, transported by the
molecules into the fluid element, all transport
directions have to be consi-dered and the
appropriate summation has to be taken.
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in Momentum Equation
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Continuity and momentum equation

• Continuity equation
• Momentum equation
• For Newtonian media.

These equations contain five unknowns P, r, Uj
with j 1,2,3
Further equations are needed to permit general
solutions for fluid flow problems. The energy
equation is necessary and in addition transports
equations for chemical species. Furthermore, the
equations of states are necessary.
Basic Equations r const.
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If one multiplies the jet momentum equation with
Uj, one obtains the mechanical energy equation
This equation tells all the mechanical energy of
a fluid element is changing. This equation was
deduced from the momentum equation.
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Mechanical Equation, 1
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Further considerations to the mechanical energy
equations are necessary. For the potential energy
we can write
Hence, we can introduce
For most cases considered in fluid mechanics
This equation expresses how the kinetic energy
and the potential energy of a fluid element
change with time. The individual terms that cause
these changes can be explained physically.
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Mechanical Equation, 2
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The terms I to IV have the following physical
meaning Term I
Difference of input and output of pressure energy
Input of pressure energy
Output of pressure energy
Taylor series expansion and forming the
difference of input and output yields
(from molecular considerations we obtain
hence an energy.)
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Meaning of Individual Terms, 1
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The other terms have the following physical
meaning
Term II
expansion work pro unit volume
Term III
difference of molecularly caused input and output
of kinetic energy
Term IV
dissipation of mechanical energy into heat
Again, the mechanical energy equation can be
deduced from the momentum equation and is
therefore not an independent relationship
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Meaning of Individual Terms, 2
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The derivation of the total energy equation reads
as follows
The difference of input and output of heat yields
the following term
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Total Energy Equation, 1
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The total energy equation and the mechanical
energy equation represent the basis for the
thermal energy equation. By difference one
deduces
tij for Newtonian media reads
Furthermore, we can write
(Fourier law)
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Total Energy Equation, 2
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Using the equations of state as additional
information, a complete set of equation results
for flow computations of non-isothermal fluids.
Hence, the system of equations is closed and
solutions are possible, if sufficient and correct
boundary conditions are imposed.
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Equations of State
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The derived energy equation can be rewritten.
Starting with the equation
and utilising the following relationships
one obtains
This form for obtaining solutions to flow
problems with heat transfer, the best form of the
energy equation for a particular flow problem is
usually chosen.
Different Forms of Energy Equations, 1
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Utilising the continuity equation one obtains
Introduced into the energy equation we obtain
The temperature form of the energy equation for
fluids reads in general.
Different Forms of Energy Equations, 2
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For an ideal liquid r const is valid and,
hence, we obtain
For a solid material, we can write
There are many forms of the energy equation and
they are documented in many books.
Special Forms of the Energy Equation
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The derivation of the Bernoulli equation can be
obtained from the mechanical energy equation
given in previous slides.
From slide 15 we obtain
For stationary processes we can write
or rewritten
Hence, one obtains
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Bernoulli Equation, 1
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The derivation of the Bernoulli equation can also
be based on considerations of the total energy of
a fluid element
const, i.e.
the total energy of a fluid element remains
constant if there is no additional energy
introduced. (The tij terms are neglected in this
relationship and hence, also the molecularly
caused energy losses.)
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Bernoulli Equation, 2
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The basic equations of fluid mechanics read
These equations can be written in form of a
general transport equation which reads for
Newtonian fluids as follows
Sf source term for quantity f
Gf molecular transport coefficient
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General Transport Equations, 1
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For continuity equation we obtain
For momentum equation we can write
For energy equation the following holds
To write down a general transport equation is
very advantageous for deriving the equations for
numerical solutions of fluid flows. All the
mathematics behind numerical solutions can be
explained in terms of the general transport
equation and are then applicable to all equations.
In an analogue manner, the equations for the
transport of chemical species can be derived.
These equations can also be written in form of a
general transport equation.
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General Transport Equations, 2
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• The presentation showed
• the tij -term in the Navier-Stokes equation are
  molecularly caused momentum transport terms,
• for Newtonian media, the following relationship
  holds

• the mechanical energy equation can be deduced
  from the momentum equation and is therefore no
  independent equation,
• the thermal energy equation is required if fluid
  flow problems of compressible media are
  needed. The equation of states need to be
  considered also,
• the Bernoulli equation can be deduced from the
  general energy equation of fluid mechanics,
• a general transport equation can be derived for
  all quantities.

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Final Conclusions