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

Tqqq

dT

Tdpsss

p

T

rc

rc

rc

rrc

K

IIDDITTT :3

22

Following standard mechanics arguments we have obteined:

We now write explicitly the equations governing the flow.

1. Energy equation

TD : qDt

De

vtDt

D where

and recall the constraint 0 vDt

DT

From definition e –T s, we have

p

dT

dTTTse r

r

and

Dt

DpT

Dt

DTp

Dt

DT

dT

dpT

dT

dTp

dT

dT

Dt

De

12

2

2

vp

DD

DIDIDT

ˆˆvp

vp

:2

:3

12:

IIDDD :ˆ3

1

t

VP

This term gives the classical

DD ˆˆDT

DpTT

Dt

DT

dT

dpTTp

dT

dT

:2

2

2

2

K

mechanical energy converted intoheat by the internal friction

p,Tcp

Remark 1.

The coefficient in front of represents, from the physical point

of view, the isobaric specific heat.

The fluid we are modelling admits only the isobaric specific heat.Indeed any change of body's temperature implies a change in volume.Hence it is not possible to work with the isochoric specific heat cv

Dt

DT

Remark 2.

Experiments show that the variations of cp with respect to pressureare generally quite small. Hence we impose that cp (p,T) is constantwith respect to the pressure field p. Thus we require

dT

dTp

dT

dTcp

22

2

11

02

dT

d

is of this form RR

R

TT

1

with TR reference temperature and R=TR )

As a consequence, from we have the

following law for the density

,dT

d

1

We will consider the linearized version, namely

We however remark that, from the mathematical point of view such aSimplifcation is not crucial and it is consistent with the data reportedin the experimental literature.

Remark 3.

We remark that in the framework of the mechanical incompressibilityassumption, the term

vp

is necessarily compensated by the mechanical work associated withdilation. Thus it does not appear in the energy balance. Indeed we havedeveloped the theory assuming that the constraint response does notdissipate energy.

Remark 4.

Measuring cp we can reconstruct the Helmoltz free energy Indeed

TcTdT

dp

12

2

We have a method for “quantifying”

2. Momentum equation

ID vvpeg

Dt

vD

3

223

Next, we introduce the hydraulic head 3gxpP R so that

ID vvPeg

Dt

vDR

3

223

thus getting

Concerning the viscosity we assume theVogel-Fulcher-Tamman's(VFT) formula

BT

ACTlog

In particular, is monotonically decreasing with T. For more detailswe refer to [4], chapter 6.

[4]. J.E. Shelby, Introduction to Glass Science and Technology, 2005.

3. Complete system

TBAT

ˆˆDT

DpTT

dt

DTTc

vvPegDt

vD

vDt

DT

p

R

DD :2

3

22

(1)

3

K

ID

Non-DimensionalizationThe scaling of model (1) has to be operated paying particular attentionto the specific problem we are interested in.

x3

0

H

R (x3, lat

in

We are considering a gravity drivenflow of melted glass through anozzle in the early stage of a fibermanufacturing process.The inlet and outlet temperature ofthe fluid are prescribed. In particular,the fluid temperature on in is higherthan the one on out.

out

wRout

win

TTT

TT

Concerning the temperature, we introduce

RW

R

TT

TT

so that 0and1 outin ,

Moreover we introduce also

R

ww T

TT~

In the phenomena we are considering is small but not negligible.Typically is of order 10-1.

1wT~

1wT~

The characteristic of the problem we are analyzing is that there

exists a reference velocity VR . This makes our approach different

from the ones presented in [5] and in [6] where there is no velocityscale defined by exterior conditions.

[5]. Rajagopal, Ruzicka, Srinivasa, M3AS 1996.[6]. Gallavotti, Foundations of Fluid Dynamics, 2002

The flow takes place in a nozzle of radius R and length H, with

R/H=O (1). Hence we take H as length scale.

Concerning the time scale we take tR=H/VR

As the reference pressure PR we take the point of view that flows

of glass or polymer melts are essentially dominated by viscous effects.Accordingly we set

Notice that PR 0 as VR tends to 0 and, as a consequence ptends to the hydrostatic pressure. This is consistent with the fact that P “measures” the deviation of the pressure from the hydrostatic-onedue to the fluid motion.

RR

R VH

P

Summarizing, we have the following dimensionless quantities

Rp

pp

R c

cc~,

~

KK

K

Suppressing tildas to keep notation simple, model (1) rewrites

k

We now list the non-dimensional characteristic numbers appearing inthe previous model

•

•

•

•

•

•

•

We may write

As mentioned, we are interested in studying vertical slow flows of veryviscous heated fluids (molten glasses, polymer, etc.) which are thermallydilatable. So, introducing the so-called expansivity coefficient(or thermal expansion coefficient)

1 wTKWe will consider the mathematical system in the realistic situationin which the parameter is small. Typically (e.g. for molten glass)

23 1010 In particular, can be rewritten

as 1

Next, we define the Archimedes' number

gH

V

TK

R

w

22

1

FrAr

So that the mathematical system rewrites

1

Dt

Dvdiv

1

k

We consider a flow regime such that

111 OOO PeReAr ,,

and

910tipically1 - EcEc

The terms in energy equation containing the Eckert are dropped.So such an equation reads as follow

KdivDt

Dcp Pe

1

KdivDt

Dc

vvdivPegDt

vD

Dt

Dvdiv

p Pe

ReAr

1

3

22

1

1

3 ID

We consider the stationary version of system

k

with the following BC