Flow of mechanically
incompressible, but
thermally expnsible viscous
fluids
A. Mikelic, A. Fasano, A. Farina
Montecatini, Sept. 9 - 17
LECTURE 1.
Basic mathemathical modelling
LECTURE 2.
Mathematical problem
LECTURE 3.
Stability
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:

 , v , e, T , s
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.
D
 
Dt
where:

  v


Dv

Dt



De
    q  D v  : T
Dt


Ds
 q 
  
  0
Dt
T 
D
 
  v 
•
Dt t


ge3    T ,
T  TT
denotes the material derivative.



T
• Dv   1 / 2 v  v 
is the rate of strain tensor.

 T
•T
is
the
Cauchy
stress
tensor.
Further,



T    Dij Tij

D
v
:
T

Tr
D
v

i, j
• q is the heat flux vector.
•  ge3 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.
Introducing the specific (i.e. per unit mass) Helmholtz free energy
  e  Ts
and using the energy equation, Clausius-Duhem inequality becomes
DT
 D

s
Dt
 Dt
  T

0
  T : Dv   q 
T

Constitutive equations
Key point of the model is to select suitable constitutive equations for

e , T , q , and s
in terms of the dependent variables

v, T , 
The constitutive equations selection has to be operated
considering the physical properties of the fluids we dial with.
Main physical properties
•
The fluid is mechanically incompressible but thermally
dilatable (i.e. the fluid can sustain only isochoric motion in
isothermal conditions)
T1
•
T 2 > T1
The fluid behaves as a linear viscous fluid (Newtonian fluid)
Shear stress proportional to shear rate
Vin , Tin
1 V
 
Vin T
Vfin , Tfin
thermal expansion coefficient
V  V T fin   V Tin  , T  T fin  Tin
We assume that  may depend on temperature, but not on pressure,
i.e.
V  0 if T  0
M
 
, density
V
1 d
 
,
 dT
with
   T 
So, if, for instance,  is constant,
 
       




 X ,t  o X exp   T X ,t  To X
in general we have
  ^  T   function of temperature
Vin , Tin
Vfin , Tfin
refernce configuration
actual configuration
V fin
1 V fin  Vin
 
, 
 1   T T
Vin
T
Vin
V fin
Vin
 J  det F i.e. J is a function of T, and can
be expressed in terms of the
thermal expansion coeff. 
Since J=J(T), continity equation reads as a constraint linking
temperature variations with the divergence of the velocity field.

1 D
   v
 Dt
DT
  T 
Dt

-Dv  : I


DT
DT

 Dv  : I  0   
 v  0
Dt
Dt
Remark

If T is uniform and constant   v  0 , the flow is isochoric
First consequence of our modelling:
We have introduced a CONSTRAINT. The admissible flows are
those that fulfill the condition

DT
v  
Dt
force that maintains the constraint
Physical remark
The constraint     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.
rigid box
T  20 C
The box brakes
fluid
T  0 C
T  20 C
The box does not
brake
Let us now turn to a discussion of the constraint.
We start considering the usual incompressibility constraint

v  0 

I : D( v )  0
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
T  Tc  Tr
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:
(i) to have no dependendance on the state variables
(ii) to produce no entropy in any motion that satisfies the constarint, i.e.


Tc : Dv   0 when I :Dv   0
Hece, the above conditions demand that:
1.
2.
Tc   p I

p  p  x ,t 
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.
Remark
Smooth constraints in classical mechanics (D'Alembert )
Fc
dx
Fc Constarint response
dx Displacement compatible with the constraint



Tc : Dv   0 is equivalent to Fc  d x  0
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])
  
T  Tc  Tr , q  qc  qr ,    c  r , s  sc  sr
r constitutive, c constraint
The contraint responses are required to do not dissipate energy
Recalling
DT
 D

s
Dt
 Dt
2 Adkins, 1958
  T

0
  T : Dv   q 
T

[3] Green, Naghdi, Trapp, 1970
no entropy production,
i.e. no dissipation
we have
  T
DT 
 D c
 T 
 sc
0
  Tc : Dv   qc 
Dt 
T
 Dt

when T and v are such that

DT

 I : Dv   0
Dt
Considering various special subsets of the set of all allowable
thermomechanical processes, the above equation leads to
 c  0,

qc  0
(1)
Equation (1) reduces to

DT
 T sc
 Tc : Dv   0 when
Dt
 T sc   p , 
Tc   pI

DT

 I : Dv   0
Dt
sc   p

 T 
Hence
T  Tr  pI

s  sr  p

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.
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
  ˆ  ,T 
and we develop the standard theory considering
ˆ T ,T  . We deduce that:
gives    T   ˆ 
  ̂ T  which
• There is no equation of state defining the pressure p.

• s  ŝ T   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.
Remark 2
Constraints that are nonlinear cannot, in general, be considered by the
Procedure prevuously illustrated.
For example, consider this constraint(1)
D2 : D  0,
i.e.
 
tr D3  0
Now the conditions to be imposed are:
1. Tc : D  0
whenever D 2 : D  0
2. Tc 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.
Deduction of the constitutive equations
  T
DT 
 D r

 sr
0
  Tr : Dv   qr 
Dt 
T
 Dt
Assumption 1.
D r T  d r DT
 r   r T  

Dt
dT Dt
Assumption 2. Linear viscous fluid
ˆ  D  1 D : II
Setting D
3
Tr  2D   D : II

ˆ D
ˆ :I  0
so that tr D
we have
ˆ 1

2

ˆ
Tr  2  D  D : II   D : II  2 D      D : II
 3

3


 T
 d r
 DT  ˆ  2

 sr 
 2 D      D : II : D  qr 
0
T
 dt
 Dt 
3



Tr
ˆD  1 D : II
3
 T
 d r
 DT
2

2
ˆ
ˆ

 sr 
 2 D : D  3    D : I  qr 
0
T
 dt
 Dt
3

0
d r T 
sr  
,
dT
0
2
 
3
So, we are left with
 T
ˆ
ˆ
 2  D : D  qr 
0
T

and assume qr  K T Fourier law
2
ˆ
ˆ
 2 D : D  K T  0
OK !!
Summarizing
   c  r   r T 
ASSUMTION
2
T  Tc  Tr   pI  2 D   D : I I
3
 d T 
s  sc  s r   p 

dT
 

q  qc  qr  K T
Remark
In formulating constitutive models, such as, for instance,
T, we must have in mind some inferential method for
quantifying them.
Concerning T ) we shall return to this point later.