A New Continuum Formulation for Materials--Part II. Some Applications in Fluid Mechanics

0.58 MB·

by  Melissa Morris

7 A New Continuum Formulation for
1
0 Materials–Part II. Some Applications in Fluid
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Mechanics
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a
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4 Melissa Morris
2 3420 Campus Blvd. NE

Albuquerque, NM 87106
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[email protected]
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d
January 18, 2017
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s Abstract
c
i InpartIofthispaper,Iproposedanewsetofequations,whichIrefer
s
y to as theM(D,η)-formulation and which differsfrom theNavier-Stokes-
h Fourierdescription offluidmotion. Here,Iusetheseequationstomodel
p severalclassicexamplesinfluidmechanics,withtheintentionofproviding
[ a generalsense of comparison betweenthetwoapproaches. Afew broad

facts emerge: (1) it is as simple–or in most cases, much simpler–to find
1
solutions with the M(D,η)-formulation, (2) for some examples, there is
v
3 notmuchofadifferenceinpredictions–infact,forsoundpropagationand
9 for examples in which there is only a rotational part of the velocity, my
0 transport coefficientsD andη arechosen tomatchNavier-Stokes-Fourier
7 solutions in the appropriate regimes, (3) there are, however, examples in
0 whichpronounceddifferencesinpredictionsappear,suchaslightscatter-
. ing, and (4) there arise, moreover, important conceptual differences, as
1
0 seen in examples like sound at a non-infinite impedance boundary, ther-
7 mophoresis, and gravity’s effect on theatmosphere.
1
:
v Contents
i
X
r 1 Introduction 3
a
2 Equations of Motion from Part I 5
3 Stability Analysis 7
4 Pure Diffusion 10
5 Sound Propagation 12
1
6 Sound at a Boundary 15
7 Hydrodynamical Fluctuations 20
8 Light Scattering 23
9 The Effect of Gravity on the Atmosphere 26
10 Poiseuille Flow 29
11 Thermophoresis 33
12 Shock Waves 37
13 Future Work 43
A Tensors 45
B Equilibrium Thermodynamic Relationships 45
C Values for D 47
List of Figures
1 sound propagationin the noble gases . . . . . . . . . . . . . . . . 16
2 theoreticallightscatteringspectraforaclassicalmonatomicideal
gas (y =20) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3 Poiseuille flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4 thermophoresis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5 shock wave in a fixed coordinate system . . . . . . . . . . . . . . 38
6 shock wave in a coordinate system moving with the front . . . . 39
7 numerically computed shock wave profiles: M(D,η) - formulation 42
8 numerically computed shock wave profiles: NSF formulation . . . 43
List of Tables
1 diffusionparametersforvariousfluidsatnormaltemperatureand
pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2 diffusion coefficient for air at atmospheric pressure and various
temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3 diffusioncoefficientfornitrogengasat0.01atmandvarioustem-
peratures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4 diffusioncoefficientformethanegasat0.01atmandvarioustem-
peratures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5 diffusion parameters for water at atmospheric pressure and vari-
ous temperatures between the freezing and boiling point . . . . . 52
2
6 diffusion coefficient for water at 273 K and 303 K and various
pressures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
7 diffusion coefficient for liquid mercury at atmospheric pressure
and various temperatures . . . . . . . . . . . . . . . . . . . . . . 53
1 Introduction
Inchoosingtheexamplesthatappearinthispaper,Idesiredthesimplestmath-
ematicsandclosed-formsolutionswhereverpossible. Tothis end, (1)variation
is limited to one spatial dimension in all sections but §7 on hydrodynamical
fluctuations and §10 on Poiseuille flow, (2) equations are linearized about a
constant state in all sections but §12 on shock waves, and (3) parameters are
chosen to yield phenomena strictly in the hydrodynamic (small Kn) regime in
all sections but §12. Furthermore, so as not to distract from the main results,
for examples that require any mathematical complexity, such as the stochasti-
cal subjects of §7 and §8, I provide guiding references and outline steps, but
for the most part I merely present and discuss the solutions. I will treat these
problems–andothersthatareexaminedonlysuperficiallyhere,likestabilityand
sound propagation–withrigor in future papers.
In addition, these particular examples were chosen to exhibit a wide vari-
ety of phenomena–including ones that connect the two theories and so provide
anchoring points to tie M(D,η)-formulation parameters to those of Navier-
Stokes-Fourier (NSF), and ones that reveal major differences both measurably
in experiment and conceptually.
The following is a brief description of the examples this paper contains.
• A one-dimensional linear stability analysis is carried out in §3 to show
that my formulation is unconditionally stable provided that the diffusion
parameter, D, is positive.
• In §4, a mass equilibration problem with no mechanical forces is studied
in order to show that my formulation reduces to one of pure diffusion
governed by Fick’s law.
• Low amplitude sound propagation, which is the subject of §5, is demon-
stratedto be a wayofrelating my longitudinaldiffusionparameter, D,to
the transport parameters of the NSF formulation by matching attenua-
tions in the hydrodynamicregime. I employ this relationshipto estimate
values ofD forseveraldifferentgasesandliquids, andthese arepresented
in appendix C.
• In§6, acousticimpedance atasoundbarrierisdiscussed. Inparticular,I
pointoutthattheM(D,η)-formulationallowsanimpermeableboundary
with a non-zero normal velocity at that boundary, as that which occurs
in the case of non-infinite impedance.
3
• As a step towardsbridging the gapbetween microscopicand macroscopic
scales in my theory, the subject of hydrodynamical fluctuations is ad-
dressed in §7, where it is shown that the thermodynamic interpretation
of a continuum mechanical point (i.e. a point occupied by matter and
moving with velocity v) stemming from the M(D,η)-formulation is fun-
damentally different from that of Navier-Stokes-Fourier. That is, in the
NSF theory,a continuummechanicalpointis viewedasathermodynamic
subsystem in contact with its surrounding material which acts as a heat
reservoir;whereasinmytheory,thesurroundingmaterialactsnotonlyas
a heat–but also a particle–reservoir.
• The light scattering spectra from the M(D,η)-formulation and the NSF
equationsareprovidedin§8. Comparingthese,onefindstheshiftedBril-
louinpeaks,whichcorrespondtothesound(phonon)partofthespectrum,
are virtually indistinguishable between the two theories, as one would
expect, but the central Rayleigh peak predictions are significantly differ-
ent. Forexample,inaclassicalmonatomicidealgasinthehydrodynamic
regime, the M(D,η)-formulation predicts a Rayleigh peak that is about
29%taller and narrowerthan the NSF equations (but with the same area
so that the well-verified Landau-Placzek ratio remains intact). Previ-
ous experimenters, e.g. Clark [7] and Fookson et al. [13], did not study
gases fully in the hydrodynamic regime, Kn . O 10−2 , choosing in-
steadtofocusonmorerarefiedgaseswithKnudsennumbersintherange,
(cid:0) (cid:1)
O 10−1 . Kn . O(1), which extends into the slip flow and transition
regimes. Therefore, in order to verify the M(D,η)-formulation, it is im-
(cid:0) (cid:1)
portant to conduct a high-resolutionlight scattering experiment for a gas
in the hydrodynamic regime.
• In §9, I study the effect of gravity on the Earth’s lower atmosphere, and
show that enforcing no mass flux and no total energy flux conditions in
the M(D,η)-formulation, leads to the isentropic condition that is typi-
cally assumed a priori. In contrast, enforcing the same conditions in the
NSFformulation,leadstoanisothermalcondition,whichisobviouslynot
physical.
• Poiseuille flow is the subject of §10. There, I show that, when compared
totheNSFmassflowrate,whichisduesolelytoconvection,theM(D,η)-
formulation predicts an additional contribution due to diffusion. In the
hydrodynamic regime, however, this contribution is very small, and so
these predictions do not differ by much. Since viscometers may be based
on Poiseuille flow, this–and other examples of this sort–provide justifica-
tion that the shear viscosity appearing in the M(D,η)-formulation may
be taken to be the same as the Navier-Stokes shear viscosity.
• Thermophoresis, or the down-temperature-gradient motion of a macro-
scopic particle in a resting fluid, is discussed in §11. For an idealized
4
problem, it is shown that, through the steady-state balancing of the con-
vective and diffusive terms of the mass flux, the M(D,η)-formulation
provides a mechanism for thermophoresis that is not present in the NSF
formulation. In the latter type, a thermal slip boundary condition at
the particle’s surface is needed in order to model thermophoretic motion,
whereas the former type may be used to describe this motion with a no-
slip boundary condition, i.e. one in which the tangential velocity of the
particle at its surface equals that of the fluid. The concept of thermal
slip is based on kinetic gas theory arguments in the slip flow regime, ap-
propriate for Kn & O 10−2 , yet thermophoresis is observed in gases for
much smaller Knudsen numbers in the hydrodynamic regime and also in
(cid:0) (cid:1)
liquids. Therefore, there are obvious advantages to being able to model
this problem without the thermal slip condition.
• A non-linear steady-state shock wave problem is considered in §12. It
is well-known that for this problem, the NSF formulation–and all other
accepted theories, for that matter–produces normalized density, veloc-
ity, and temperature shock wave profiles that are appreciably displaced
from one another with the temperature profile in the leading edge of the
shock,velocitybehind it,anddensitytrailinginthe back. TheM(D,η)-
formulation, however, predicts virtually no displacement between these
three profiles in the leading edge of the shock and much less pronounced
displacements in the middle and trailing end of the shock.
2 Equations of Motion from Part I
AllformulasreferencedfrompartIofthispaperarelabeledwith”I.”preceding
their number. As derived in part I, the M(D,η)-formulation consists of the
balance laws (I.43):
∂m
=−∇·(mv)
∂t
∂m
=−∇· q +mv (1)
∂t M
∂(mv) (cid:16) (cid:17)
=−∇· P +mvv +mf
∂t M
∂u (cid:0) (cid:1)
=−∇· q +uv −P: ∇v
∂t U
(cid:16) (cid:17)
with constitutive equations (I.74):
q =−D∇m
M
− mD− 4η (∇·v)1−
P =P1+P with P = 3 (2)
visc visc 2η(∇v)sy,dev
(cid:20) (cid:0) (cid:1) (cid:21)
q =−D∇u.
U
5
The Navier-Stokes-Fourierformulation is given by the balance laws (I.45):
∂m
=−∇·(mv)
∂t
∂(mv)
=−∇· P +mvv +mf (3)
∂t M
∂u (cid:0) (cid:1)
=−∇· q +uv −P: ∇v
∂t U
(cid:16) (cid:17)
with constitutive equations (I.46):
P =PI +P with P =−ζ (∇·v)I −2η(∇v)sy,dev (4)
NS
visc visc
q =−k ∇T.
U F
Other quantities employed in this paper that hold for both formulations
when the proper constitutive equations are used are the total energy balance
law (I.8)/(I.20),
∂e
=−∇· q +P ·v+ev , (5)
∂t U
(cid:16) (cid:17)
the total energy flux,
j =q +P ·v+ev, (6)
E U
and the total mass flux,
j =q +mv. (7)
M M
To solve the steady-state examples of §9-11, one may derive the following
convenient forms for the two formulations considered here. First, taking the
balance laws (1 a-c) and (5) with constitutive equations (2), setting the time-
derivatives equal to zero for a steady state, using thermodynamic relationships
(233) and (234) to express the equations in terms of the variables m, P, v, and
T, and linearizing about a constant state,1
(m,P,v,T)=(m ,P ,0,T ), (8)
∗ ∗ ∗
one obtains
0=−m ∇·v
∗
γ
0=D −(mα ) ∇2T + ∇2P −m ∇·v (9)
∗ P ∗ c2 ∗ ∗
h (cid:16) (cid:17) i
4η
0=−∇P + mD− ∇(∇·v)+2η ∇· (∇v)sy,dev
∗
3
(cid:18) (cid:19)∗ h i
[m(c −h α )] ∇2T+
P M P ∗
0=D −(mh ) ∇·v.
∗ hMγ −Tα ∇2P M ∗
( c2 P )
∗
(cid:16) (cid:17)
1Thesubscript”∗”isusedtoindicatethatthequantity,writtenasafunctionofP andT,
isevaluatedatthethermodynamicstate(P∗,T∗).
6
If we then substitute (9 a) into (9 c-d) and employ the tensor identity (215),
the above equations imply that for the M(D,η)-formulation,
0=∇·v
0=∇2P (10)
0=−∇P +η ∇2v+m f
∗ ∗ M
0=∇2T.
In addition, by substituting constitutive relations (2) into equations (6) and
(7),employingthermodynamicrelations(233),(234),and(227),andlinearizing
about constant state (8), one finds the following total energy and mass fluxes
for the M(D,η)-formulation:
[m(c −h α )] ∇T+
P M P ∗
j =−D +(mh ) v (11)
E ∗( hcM2γ −TαP ∇P ) M ∗
∗
(cid:16) (cid:17)
and
γ
j =−D −(mα ) ∇T + ∇P +m v (12)
M ∗ P ∗ c2 ∗ ∗
h (cid:16) (cid:17) i
where,inadditiontoallofthe quantitiesdefinedinpart I,wehaveintroduced
the isobaric specific heat per mass c , isobaric to isochoric specific heat ratio
P
γ =c /c , and adiabatic soundspeed c. Carryingout similar steps, yields for
P V
the NSF formulation, the steady-state equations,
0=∇·v
0=−∇P +η ∇2v+m f (13)
∗ ∗ M
0=∇2T,
and the fluxes,
j =−(k ) ∇T +(mh ) v (14)
E F ∗ M ∗
and
j =m v. (15)
∗
M
Notice that the set of equations (13) is identical to the set (10), except for the
absenceofLaplace’sequationforthe pressure(10 b). Also, the fluxes (14)and
(15)havethesameconvectivepartsastheM(D,η)-formulationfluxes(11)and
(12), but differing diffusive parts (with the NSF mass flux having no diffusion
at all).
3 Stability Analysis
Let us consider the Cartesian one-dimensional problem in which variation is
assumed to be in the x ≡ x direction only with v ≡ v as the only non-zero
1 1
component of the velocity and there are assumed to be no body forces. If we
7
use the thermodynamic relationships (231) and (232) to recast the M(D,η)-
formulation(1)/(2)intermsofthevariables,m,m,v,andT,andthenlinearize
about the constant equilibrium state,
(m,m,v,T)=(m ,m ,0,T ), (16)
eq eq eq
via
m=m +δm (17)
eq
m=m +δm (18)
eq
v =δv (19)
T =T +δT, (20)
eq
assuming
|δm|,|δm|≪m
eq
|δT|≪T
eq
|δv|≪c ,
eq
then we arrive at the following system of linear equations:
∂δm ∂δv
=−m (21)
eq
∂t ∂x
∂δm ∂2δm ∂δv
=D −m (22)
∂t eq ∂x2 eq∂x
∂δv 1 ∂δm α ∂δT ∂2δv
P
=− − +D (23)
∂t (m2κ ) ∂x mκ ∂x eq ∂x2
T eq (cid:18) T(cid:19)eq
∂δT ∂2δT Tα ∂δv
P
=D − , (24)
∂t eq ∂x2 mκ c ∂x
(cid:18) T V (cid:19)eq
where the subscript ”eq” indicates that the parameter is evaluated at the con-
stantequilibriumstate(16). Notethatforthislinearizedproblem,themechan-
icalmassequation(21)maybeuncoupledfromtherestofthesystem,(22)-(24).
Postulating a solution,
δm
δv ,
 
δT
 
to (22)-(24) that is proportional to
exp(iκx+ωt),
for κ real and ω complex, one obtains the dispersion relation,
ω3+3D κ2ω2+ c2 +3D2 κ2 κ2ω+
eq eq eq =0. (25)
D c2 +D2 κ2 κ2
(cid:20) eq eq (cid:0) eq (cid:1) (cid:21)
(cid:0) (cid:1)
8
In the above, I have employed the equilibrium thermodynamic relationships
(228) and (229). Equation (25) may be solved for ω to obtain the three exact
roots,2
ω (κ)=−D κ2 (26)
1 eq
ω (κ)=−D κ2+ic κ (27)
2 eq eq
ω (κ)=−D κ2−ic κ. (28)
3 eq eq
Clearly, if
D >0 (29)
eq
is satisfied, then the real parts of ω (κ), ω (κ), and ω (κ), are negative for all
1 2 3
κ, resulting in the unconditional stability of linearized system (22)-(24).
For comparison,carryingout a similar procedure with the NSF formulation
(3)/(4) yields the linearization,
∂δm ∂δv
=−m (30)
eq
∂t ∂x
− 1 ∂δm − αP ∂δT+
∂δv = (m2κT)eq ∂x mκT eq ∂x (31)
∂t  ζNS + 4η(cid:16) ∂2(cid:17)δv 
m 3m ∂x2
eq
 (cid:16) (cid:17) 
∂δT k ∂2δT Tα ∂δv
F P x,1
= − , (32)
∂t mc ∂x2 mκ c ∂x
(cid:18) V (cid:19)eq (cid:18) T V (cid:19)eq
with the dispersion relation,
ω3+ ζNS + 4 +Eu η κ2ω2+
m 3 m
eq
 c2 + hEu ζNS(cid:0)+ 4η (cid:1)η i κ2 κ2ω+ =0, (33)
eq m 3m m
eq
 (cid:26) h (cid:16) Eu ηc2 (cid:17) κ4i (cid:27) 
 γ m 
 eq 
 (cid:16) (cid:17) 
where γ is the ratio of specific heats and Eu is the Euken ratio defined to be
k
F
Eu= . (34)
c η
V
Let us also define the quantities,
Eu η
Σ= (35)
γ m
and
1 ζ 4 1 η
NS
Γ= + + 1− Eu . (36)
2 m 3 γ m
(cid:26) (cid:20) (cid:18) (cid:19) (cid:21) (cid:27)
2Notethattheserootsbeingexact enablesustoconstructexact Green’sfunctionsonthe
infinitedomain.
9
Using these, equation (33) yields the following three approximate roots,
ω (κ)≈−Σ κ2 (37)
1 eq
ω (κ)≈−Γ κ2+ic κ (38)
2 eq eq
ω (κ)≈−Γ κ2−ic κ, (39)
3 eq eq
for
η
|κ|≪1, (40)
mc eq
(cid:16) (cid:17)
which corresponds to the low Knudsen number, or hydrodynamic,regime. For
an equilibrium thermodynamically stable fluid,
γ ≥1 and c >0 (41)
V
aresatisfied,andtherefore,ifthe standardassumptionthatη ,(k ) >0and
eq F eq
(ζ ) ≥ 0 is made, then the one-dimensional linearized NSF formulation is
NS eq
stable in the hydrodynamic regime, as well.
In two future papers, [24] and [25], I will examine linear stability in more
detail by constructing Green’s functions on one and three-dimensional infinite
domains for the general M-formulation, which includes both the M(D,η) and
NSFformulations. In[25],itwillbeshownthatthe three-dimensionalstability
requirements on the transport parameters are
D,η >0 (42)
for the M(D,η)-formulation and
4
η, η+ζ ,k >0 (43)
NS F
3
(cid:18) (cid:19)
for NSF. In another future paper [27], I will demonstrate the above criteria
to be identical to those obtained by a particular version of the second law of
thermodynamics,whichIargueshouldbeusedinplaceoftheversionemployed
by de Groot and Mazur [10, ch. IV].
4 Pure Diffusion
If we take the limit as the transport parameters go to zero (D,η → 0 in the
M(D,η)-formulation or η,ζ ,k → 0 in the NSF formulation), then we are
NS F
left with the Euler equations of pure wave motion for which, on an infinite
domain, perturbations travel at the adiabatic sound speed and never decay.
However, in the present section, we study a problem at the opposite extreme,
one of pure diffusion.
For this, it is instructive to bear the following example of thermodynamic
equilibrationinmind. LetusconsideraCartesianone-dimensionalproblemon
10