Flow of mechanically incompressible, but thermally expnsible viscous fluids

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Flow of mechanically incompressible, but
thermally expnsible viscous fluids

• A. Mikelic, A. Fasano, A. Farina
• Montecatini, Sept. 9 - 17

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• LECTURE 1.
• Basic mathemathical modelling
• LECTURE 2.
• Mathematical problem
• LECTURE 3.
• Stability

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Definitions and basic equations
We start by recalling the mass balance, the
momentum and energy equations, as well as the
Clausius-Duhem inequality in the Eulerian
formalism. Following an approach similar to the
one presented in 1, pages 51-85, we denote
by the density, velocity, specific internal
energy, absolute temperature and specific
entropy, satisfying the following system of
equations
1. H. Schlichting, K. Gersten, Boundary-Layer
Theory, 8th edition, Springer, Heidelberg, 2000.
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• where
• denotes the
  material derivative.
• 
  is the rate of strain tensor.
• T is the Cauchy stress tensor. Further,
• is the heat flux vector.
• is the gravity acceleration.
  Indeed is the unit vector relative to the x3
• axis directed upward.
• In the energy equation the internal heat sources
  are disregarded.

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Introducing the specific (i.e. per unit mass)
Helmholtz free energy
and using the energy equation, Clausius-Duhem
inequality becomes
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Constitutive equations
Key point of the model is to select suitable
constitutive equations for
in terms of the dependent variables
The constitutive equations selection has to be
operated considering the physical properties of
the fluids we dial with.
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Main physical properties

• The fluid is mechanically incompressible but
  thermally
• dilatable (i.e. the fluid can sustain only
  isochoric motion in
• isothermal conditions)

T 1
T 2 gt T1

• The fluid behaves as a linear viscous fluid
  (Newtonian fluid)

Shear stress proportional to shear rate
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Vin , Tin
Vfin , Tfin
thermal expansion coefficient
We assume that b may depend on temperature, but
not on pressure, i.e.
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So, if, for instance, b is constant,
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Vin , Tin
Vfin , Tfin
refernce configuration
actual configuration
i.e. J is a function of T, and can be expressed
in terms of the thermal expansion coeff. b
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Since JJ(T), continity equation reads as a
constraint linking temperature variations with
the divergence of the velocity field.
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Remark
First consequence of our modelling We have
introduced a CONSTRAINT. The admissible flows
are those that fulfill the condition
force that maintains the constraint
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Physical remark
The constraint r r ( T ) models the fluid as
being incompressible under isothermal conditions,
but whose density changes in response to changes
in temperature. In other words, given a
temperature T the fluid is capable to exert any
force for reaching the corresponding density.
T 20 C
The box brakes
According to the model
T 20 C
The box does not brake
According to the reality
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Let us now turn to a discussion of the
constraint. We start considering the usual
incompressibility constraint
We proceed applying classical procedure. We
modify the forces by adding, as in classical
mechanics, a term (the so called
constraint response) due to the constraint
itself. We thus consider

• Where the subscript r indicates the portion of
  the stress given by
• constitutive equations and subscript c refers to
  the constraint response.
• The contraint response is required
• to have no dependendance on the state variables
• to produce no entropy in any motion that
  satisfies the constarint, i.e.

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Hece, the above conditions demand that 1.

There is no constitutive equation

for p, i.e. p does not depend on any variable
defining the state of the system. p, usually
referred to as mechanical pressure, is a new
state variable of the system.
2.
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Remark
Smooth constraints in classical mechanics
(D'Alembert )
Constarint response
Displacement compatible with the constraint
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In the general case (density-temperature
constraint, for instance) it is assumed that the
dependent quantities are determined by
constitutive functions only up to an additive
constraint response (see 2, 3)
r constitutive, c constraint
The contraint responses are required to do not
dissipate energy
Recalling
2 Adkins, 1958 3 Green, Naghdi, Trapp,
1970
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we have
(1)
when T and are such that
Considering various special subsets of the set of
all allowable thermomechanical processes, the
above equation leads to
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Equation (1) reduces to
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Hence
Where p is a function of position and time.
Recall There is no constitutive quation for p.
The function p defined above is usually referred
to as mechanical pressure and is not the
thermodynamic pressure that is defined through an
equation of state.
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Remark 1 There is also another formulation of the
theory. We briefly summarize it We start
considering the theory of a compressible (i.e.
unconstrained) fluid, introducing the Helmholtz
free energy

• and we develop the standard theory considering
  which
• gives . We
  deduce that
• There is no equation of state defining the
  pressure p.

The distinctions of the two constraint theory
approaches are as follows The first approach
assumes the existence of an additive constraint
response, postulated to produce no entropy,
whereas the above approach deduces the additive
constraint response.
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• Remark 2
• Constraints that are nonlinear cannot, in
  general, be considered by the
• Procedure prevuously illustrated.
• For example, consider this constraint(1)
• Now the conditions to be imposed are
• whenever
  
• does not depend on the state variables
• We can no longer demand that condition 2 is
  fulfilled !!

(1) Such a constraint is for illustrative
purposes only and does not necessarily correspond
to a physically meaningful situation.
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Deduction of the constitutive equations
Assumption 1. Assumption 2. Linear viscous
fluid
Setting so
that we have
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(No Transcript)
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So, we are left with
and assume
OK !!
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Summarizing
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Remark
In formulating constitutive models, such as, for
instance, yy(T), we must have in mind some
inferential method for quantifying
them. Concerning y(T ) we shall return to this
point later.