Flow of mechanically incompressible, but

thermally expansible 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:

the density, velocity, specific internal energy, absolute temperature and specific entropy, satisfying the following system of equations:

s,T,e,v,

[1]. H. Schlichting, K. Gersten, Boundary-Layer Theory, 8th edition, Springer, Heidelberg, 2000.

0

:

3

T

q

Dt

Ds

vqDt

De

,egDt

vD

vDt

D

T

TD

TTT

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.

vtDt

D

Tvv/v

21D

q

3eg

j,i

ijijTvv TDTDTD

Tr:

Introducing the specific (i.e. per unit mass) Helmholtz free energy

Tse and using the energy equation, Clausius-Duhem inequality becomes

0:

T

Tqv

Dt

DTs

Dt

D DT

Constitutive equations

Key point of the model is to select suitable constitutive equations for

sand,q,,e

Tin terms of the dependent variables ,T,v

The constitutive equations selection has to be operatedconsidering 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)

T 1 T 2 > T1

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

Shear stress proportional to shear rate

Vin , Tin

infininfin

in

TTT,TVTVV

T

V

V

1

Vfin , Tfin

thermal expansion coefficient

We assume that may depend on temperature, but not on pressure,i.e. 00 TifV

ensityd,V

M

Twith,dT

d

1

So, if, for instance, is constant,

XTt,XTexpXt,X oo

in general we have Tfunction of temperature^

Vin , Tin Vfin , Tfin

refernce configurationactual configuration

TTV

V,

T

VV

V in

fininfin

in

11

FdetJV

V

in

fin i.e. J is a function of T, and canbe expressed in terms of the thermal expansion coeff.

Since J=J(T), continity equation reads as a constraint linkingtemperature variations with the divergence of the velocity field.

vDt

D

1

Dt

DTT

ID :v-

00: vDt

DTv

Dt

DT ID

Remark

If T is uniform and constant the flow is isochoric,v 0

First consequence of our modelling:We have introduced a CONSTRAINT. The admissible flows arethose that fulfill the condition

Dt

DTv

force that maintains the constraint

The constraint T models the fluid as being incompressibleunder isothermal conditions, but whose density changes in responseto changes in temperature.In other words, given a temperature T the fluid is capable to exertany force for reaching the corresponding density.

Physical remark

rigid box

fluid

C

According to the model

According to the reality The box does notbreake

C

C The box breakes

Let us now turn to a discussion of the constraint.We start considering the usual incompressibility constraint

0:0 )v(v

DIWe proceed applying classical procedure. We modify the forces byadding, as in classical mechanics, a term (the so called constraintresponse) due to the constraint itself. We thus consider

rc TTT 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.

0:0: vwhenvc

DIDT

Hece, the above conditions demand that:

1. IT pc

2. t,xpp

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

Fc

x

Smooth constraints in classical mechanics (D'Alembert )

Fc Constarint response

x Displacement compatible with the constraint

0toequivalentis0: xFv cc

DT

In the general case (density-temperature constraint, for instance) it is assumed that the dependent quantities are determined by constitutivefunctions only up to an additive constraint response (see [2], [3])

Adkins, 1958 [3] Green, Naghdi, Trapp, 1970

rcrcrcrc sss,,qqq,

TTT

r constitutive, c constraint

The contraint responses are required to do not dissipate energyRecalling

0:

T

Tqv

Dt

DTs

Dt

D DT

we have

0:

T

Tqv

Dt

DTs

Dt

DT ccc

c DT

no entropy production,i.e. no dissipation

Considering various special subsets of the set of all allowablethermomechanical processes, the above equation leads to

00 cc q,

(1)when T and are such that

0: vDt

DT DI

v

Equation (1) reduces to

0:when0: vDt

DTv

Dt

DTsT cc

DIDT

Ip

Tps,psT

c

cc

T

Hence

pss

p

r

r

ITT

Where p is a function of position and time. Recall: There is noThere is noconstitutive quation for constitutive quation for pp. The function p defined above is usually referred to as mechanical

pressure and is not the thermodynamic pressure that is definedthrough an equation of state.

Remark 1There is also another formulation of the theory. We briefly summarize itWe start considering the theory of a compressible (i.e. unconstrained)fluid, introducing the Helmholtz free energy

T,ˆ and we develop the standard theory considering whichgives . We deduce that:

• There is no equation of state defining the pressure p.

•

T T,TˆˆT

pTss

The distinctions of the two constraint theory approaches are asfollows: The first approach assumes assumes the existence of an additiveconstraint response, postulated to produce no entropy, whereas theabove approach deducesdeduces the additive constraint response.

Remark 2

Constraints that are nonlinear cannot, in general, be considered by theProcedure prevuously illustrated.

For example, consider this constraint(1)

Now the conditions to be imposed are:

1. whenever

2. does not depend on the state variables

We can no longer demand that condition 2 is fulfilled !!

0i.e.0 32 DDD tr,:

0DT :c 02 DD :

cT

(1) Such a constraint is for illustrative purposes only and does not necessarily correspond to a physicallymeaningful situation.

Deduction of the constitutive equations

0:

T

Tqv

Dt

DTs

Dt

Drrr

r DT

Assumption 1.

Assumption 2. Linear viscous fluid

Dt

DT

dT

d

Dt

TDT rr

rr

IIDDT :2 r

Setting so that we have IIDDD :ˆ3

1 0: IDD ˆˆtr

IIDDIIDIIDDT :3

22::

3

12

ˆˆ

r

0::3

22

T

Tqˆ

Dt

DTs

dt

drr

r DIIDD

rT IIDD :

3

1ˆ

0:3

23:2 2

T

Tqˆˆ

Dt

DTs

dt

drr

r IDDD

0

,

dT

Tds r

r

0

3

2

So, we are left with

0:2

T

Tqˆˆ

r

DD

and assume lawFourierTqr K

0:22 Tˆˆ KDD

OK !!

Summarizing

Tqqq

dT

Tdpsss

p

T

rc

rc

rc

rrc

K

IIDDITTT :3

22

ASSUMTION

Remark

In formulating constitutive models, such as, for instance, we must have in mind some inferential method forquantifying them.

Concerning ) we shall return to this point later.