Flow of mechanically incompressible, but thermally expnsible viscous fluids
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Transcript of Flow of mechanically incompressible, but thermally expnsible viscous fluids
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