A thermomechanical model of continuous media based on second gradient theory has been used to study the motions in liquid-vapor interfaces. In.
UTILIZATION OF THE SECOND GRADIENT THEORY IN CONTINUUM MECHANICS TO STUDY THE MOTION AND THERMODYNAMICS OF LIQUID-VAPOR INTERFACES Henri Gouin Department of Mathematics and Mechanics University of Aix-Marseille Rue Henri Poincare 13397 Marseille, Cedex 13, France INTRODUCTION A thermomechanical model of continuous media based on second gradient theory has been used to study the motions in liquid-vapor interfaces. In the equilibrium state this model is shown to be fundamentally equivalent to molecular theories. Conservative motions in such fluids verify the first integrals that provide Kelvin's circulation theorems and potential equations. The dynamic surface tension of a liquid-vapor interface has been deduced from equations written with a viscosity factor. The result provides and explains the Marangoni effect. BACKGROUND Liquid-vapor interfaces are generally represented schematically by a material surface endowed with energy related to Laplace's surface tension. Several studies conducted in the fields of fluid mechanics and thermodynamics represent the interface as a surface separating two media. This surface has its own characteristic behavior and energy properties 1- 8 . Detailed theoretical or experimental study shows that when working far from the critical temperature, the capillary layer has a thickness equivalent to a few molecular beams 3 ,8,9. For a fluid defined by two parameters such as temperature and density, molecular models such as those used in gas kinetic theory lead to laws of state associated with nonconvex internal energies, e.g., the Van der Waals modeI 7 ,9-14. These models appear advantageous as they provide an even more precise verification of Maxwell's rule applied to isothermal liquid-vapor transition 1S • Nonetheless, they present two disadvantages. First, for densities that lie between the liquid and vapor bulk densities, the pressure may become negative. Simple physical experiments can be used, however, to cause traction that leads to these pressure values 9 ,12,16. Second, in the field between vapor and liquid, internal energy cannot be represented by a convex surface associated with the variables density and entropy. This fact seems to contradict the existence of the steady equilibrium state of the matter in this type of region. To overcome these disadvantages, the thermodynamic investigation replaces the nonconvex portion corresponding to internal energy with a plane domain. The fluid can no longer be considered as a continuous medium. The interface is represented by a material surface with a null thickness. In this case the M. G. Velarde (ed.), Physicochemical Hydrodynamics © Plenum Press, New York 1988
667
only possible representation of the dynamic behavior of the interface is one of a discontinuous surface, and its essential structure remains unknown. In the equilibrium state it is possible to eliminate the above diba~vantages by appropriately modifying the stress tensor of the capillary l~yer. which is then expressed in an anisotropic form. As a consequence, the energy of the continuous medium must change 10 ,17,18. If the interface fluid is to be represented as a continuous medium, how can the stress tensor of the capillary layer be written in a dynamic expression 4 ? In recent articles, we have proposed a dynamic theory based on the second gradient: internal capillarity 19-Z1. Conceived in the sixties for the static case, this theory shows the advantage of using a threedimensional approach to capillarity in a continuous media mechanical model 17 ,ZZ. The resulting equations of equilibrium provide a satisfactory representation of isothermal liquid-vapor equilibrium states. This approach is not new, and in fact dates back to Van der Waals and Korteweg Z3 ,24. It was taken up again in 1959 by Cahn and Hilliard in reference to free energy10, and corresponds to what is known as the Landau-Ginzburg theory 14. The representation proposed in the present study is based on the notion of internal energy which is more convenient to use when the temperature is not uniform. One of the problems that complicates the study of phase transformation dynamics is the apparent contradiction between Korteweg's classical stress theory24 and the Clausius-Duhen inequality Z5. Proposals made by Dunn and Serrin 11 and a capillary fluid model 19 rectify this apparent anomaly. In the more general case of internal capillarity, by representing energy in terms of the second gradient 26 and by applying a simple algebraic identity, it is possible to draw a relationship between the energy equation, the motion equation, the mass conservation equation, and entropy. Through deduction, and for a conservative fluid, an additional term that has the dimension of a heat flux may be added to the energy equation. In the case of a viscous fluid, results provide a set of equations that do not modify the Clausius-Duhem inequality, thereby making them compatible with the second thermodynamic principle. For the nonviscous case, classical fluid motions and motions of fluids endowed with internal capillarity reveal a common structure that results in identical thermodynamic forms and potential equations 27 - 29 . This leads to the same classification of motion 30 ,31, a generalization of Kelvin's theorem and the Crocco-Vaszonyi equation, and first integrals that keep the same values across the interfaces. By representing internal energy as a function of entropy, density, and the density gradient, using a single constant C for internal capillarity, the resulting equations are thereby identical to those obtained with molecular models in the isothermal case. For a surface area that is relatively large with respect to the thickness of the capillary layer, surface tension is calculated using integration throughout the interface. Lengthwise it is not constant and depends on the dynamic distributions of density and temperature. These dynamic distributions, based on equations of motion, call upon a Navier-Stokes type viscosity. For interfaces in an isothermal equilibrium, the results are classical 9 ,lO,lZ,13,18. The study of motion in the interface without mass transfer introduces the surface tension gradient and the velocity gradients associated with dynamic viscosity. The ensuing Marangoni effect has been interpreted using limit analysis wherein the approximate quantities correspond to the physical dimensions of the interface. (For the case where mass flow across the interface is not null, a general dynamic form of Laplace's equation is given in ref. 32). The method presented herein is completely different from classical calculations based on balance equations established for both sides of a discontinuous surface, which take into account density variations only as a difference across the interface 33 • In the particular case of isothermal liquid-vapor equilibria, an invariant inte-
668
gral of motions compatible with the interface coincides with a generalization of the rule advanced by Maxwell, associated with phase transitions 15 • The model of a viscous fluid endowed with internal capillarity is therefore substantiated by consequences verified in both the equilibrium state and in non-isothermal motion. This model provides a better understanding of the behavior of liquid-vapor interfaces in motion. It gives at least a partial answer to the question: "is the fluid at the interface rigid or moving?,,34 , and proposes a theory that takes into account the stress tensor and dynamics in the essential structure of the interface 4 . The resulting laws of behavior are not the thermodynamic laws governing classical fluids since they include an anisotropic stress tensor in the momentum equation and an additional heat flux term in the energy equation. 1.
EQUATIONS OF MOTION FOR A FLUID ENDOWED WITH INTERNAL CAPILLARITY
1.1
Case of a Conservative Fluid
Second gradient theory 26, conceptually more straightforward than Laplace's theory, can be used to construct a theory of capillarity. In the present text the only addition is an internal mass energy s that is a function of density p and entropy s, as well as grad P. The internal mass energy characterizes both the compressibility and capillarity properties of the fluid, independent of the bodies with which it is in contact. For an isotropic fluid, it is assumed that 17 :
s where
13
f(p,s,13) =
(grad p)
(1)
2
The equation of motion for a conservative fluid is written: p
r
=
diva - p grad
where r denotes the acceleration vector, and a is the general stress tensor:
a or
a
= -
= -
1.J
where and or
(2)
extraneous force potential,
p I - C (grad p ) (grad p ) t
p I + C {(gradP)2 I
a .. = - p 0 .• or
n the
n
1.J
(3a)
(gradP)(gradp)t}
C P . P .
,1.
,J
(3b)
i,j E {I,2,3}
- po .. + C P k P k 0 .. - C P . P . a .. , , 1.J ,1. ,J 1.J 1.J C = 2 P s' 13 2 p = p s' - p div( C grad p) P 2 2 = s' div( C grad p) - C (grad p) P P P p
(4) (5a) (5b)
(Depending on the calculations, either one of the forms of the stress tensor associated with p and p may be used). It should be noted that:
e = s's
(6)
is the Kelvin temperature expressed as a function of p , sand 13. Demonstration. The equation of motion is obtained in the clearest manner by using the virtual work principle 26 ,35. Virtual displacement, denoted ox, is that which has been defined by J. Serrin in ref. 35 (page 145). For a fluid endowed with internal capillarity, the virtual work
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principle (see ref. 35, IV, Section 14) is stated as follows. The motion of a fluid is such that
af
f
p r ox dv = 0 dV where V is an arbitrary material volume. The variation of 35, page 148) must satisfy the condition Os = 0,
v
p ( E + Q ) dv -
a
while taking into account that
dP dX
=
entropy (ref.
dOP _ ~ dOX ax dX dX
2 ( dO p _ ~ aox ) gr ad p dX dX dX
which implies OE
and
=
E' op + E' oS + E' oS P s 13
as well as equations (14.5) and (14.6) of the cited reference. For virtual displacement where ox is null at the edge of V , (integrations by parts being taken, the integrals calculated on the edge of V having a null contribution), we obtain:
f
V
(p
r - diva
+
p grad Q) ox dv
=
0
where a is expressed by eq. (3). Classical methods of the calculus of variations then lead to eq. (2). It appears that a single term, C, accounts for the effects of the second gradient in the equation of motion. C, like E, depends on p, s, and 13. In a study of surface tension based on gas kinetic theory, Rocard 9 obtained the same expresssion (3) for the stress tensor, but with C constant. If C is constant (see ref. 36), specific energy E is written E(p,s,13)
a(p,s) +
=
C 2P
13
i. e., the second gradient term C13/2p is simply added to the energy a (p, s) of the classical compressible fluid. Pressure for this fluid is written P = p2 a " and temperature 8 = a'.
s
p
This gives For
p = P - C (3: + p lip)
p = P - C (~ + p lip)
8
8
P, Rocard uses either Van der Waal's pressure equation P
R8
= P I-bp -
a p
2
or other laws, of which he provides a comparison. It should be noted that if C is constant, this implies that 8 = 8, and that there exists a relationship between 8, p and s that is independent of the second gradient terms. 1.2
Case of a Viscous Fluid
If the fluid is also endowed with viscosity, the equation of motion includes not only the stress tensor a, but also the classical stress tensor 0 v ' due to viscosity:
a
where D is the field:
(7) = A tr( D) + 2 \l D v deformation tensor, a symmetrical gradient of the velocity D
It would of course be coherent in second gradient theory to add to viscosity those terms accounting for the influence of higher order derivatives
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of this velocity field. This has not been done in the present case. Equation (2) is written by adding the virtual work of the forces of viscosity: P
2.
r
=
di v (0 + 0
) -
v
P grad r2
(8)
ENERGY EQUATION FOR A VISCOUS FLUID ENDOWED WITH INTERNAL CAPILLARITY Let e be total volumic energy: e
1 2
(Z-V
= P
q the heat flux vector, r radiant heating, and h Let
+ E + r2 ), E + ~
=
M
P r - dive 0 + 0 v ) + P grad r2
B
P + P div V
S
pes + div q - r - tr( 0 D) v dr2 de + di v [ ( e - 0 - 0 ) V ] - dive C p grad p) + div q - r Pat dt v
E
Theorem. For internal energy written as in (1) the fluid, the relation E -
M. V -
is an identity.
(.!..v 2 + h + r2) 2
B -
S
and for
any motion in
o
(9)
Demonstration. In the first member of (9), the dissipative terms q, r and 0v cancel out. The same is true for the extraneous force potential and the inertia terms. After having replaced 0, p, and e by their respective values (3)-(6), it remains to be proved that the terms from internal energy E also cancel out. These terms include the following: a) in E: Le.,
ddt (pE) + div(p E V) - div(o V) - dive C p grad p)
dV tr (0 dX) - dive C p grad p)
p E + E ( P + p div V) - div (0) V P (E' s
or, finally
s + E'
p
P + E'
B
+ p div V + C (gradp) b) in M. V:
E(p + p divV)- div(o)v
dV dX gradp - dive C p gradp)
- dive 0) V
c) in (.!..v 2 + h + r2) B:
(E+E.)( p + p div V) p
2
d) in S: p E' s
t
B) +
S
This leaves the following in the first member of (9):
P E'p P +
C 2
B-
E. p
p + C ( grad p)
t
dV dX grad p - dive C p grad p)
671
Finally, replacing p by its expression (5), we obtain C (grad p)
t
- div (C
d
dt grad p +
p div( C grad p) + C (grad p)
t
aV
ax grad p
p grad p)
which is identically null. Corollary 1. In motion of a conservative fluid endowed with internal capillarity, conservation of entropy on the trajectories is equivalent to:
~~
+ div [( e - a) V] -
div( C
~
grad p) - p ~~ = 0
(10)
Eq. (10) is naturally designated the energy conservation equation. This relation is a direct result of (9) where a v = 0, div q - r = 0 and M = 0, B = 0, S = O. This brings us to add the additional term div ( C P grad p) to the energy equation. The vector C P grad p has the dimension of a heat flux vector and occurs even in the conservative case. Corollary 2. In motion of a viscous fluid endowed with internal capillarity, the energy equation ae
]
an
at + div [(e - 0- a v ) V - div(C pgradp) + div q -r -Pat
0
(11)
is equivalent to the entropy equation p9s+divq-r-tr(a D) =0 v
(12)
Equation (12) corresponds to the classical version of the entropy variation expressed by the" function of dissipation of stress due to viscosity: IjI
3.
=
tr( aD) v
THE PLANCK INEQUALITY AND THE CLAUSIUS-DUHEM INEQUALITY
For any motion in a viscous fluid endowed with capillarity, been assumed that 1jI?- O. Equation (12) implies that p9s+divq-r~0
it has (13)
This represents the Planck inequality37. Supposing that the Fourier principle is expressed in a very general manner: q • grad 9
~
0
(14)
the Clausius-Duhem inequality can be deduced directly:
er
~ 0
The present analysis reveals that the second thermodynamic principle leads to the existence of a heat flux vector within a fluid in motion if the fluid is endowed with internal capillarity. This heat flux vector adds an additional term to the classical equation of energy. It is present even if the fluid is not viscous and the motion is conservative. Generalization allows these results to be applied to non-fluid continuous media which display an internal energy that contains second gradient terms.
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4.
TRANSFORMATION OF MOTION EQUATIONS FOR A FLUID ENDOWED WITH INTERNAL CAPILLARITY This procedure consists of writing equation (2) in other forms.
4.1
Special Case where C is Constant Let us note
w
n-
=
Eq. (2) may then be written P
C t,p
(15)
r + grad P + p grad w = 0
(16)
This is a formal representation of a perfect fluid where P, for example, is the pressure of Van der Waals. All capillarity terms are grouped together in w. Verification is achieved directly: eqs. (3) and (5) give o .. ~J
thus
-P.+Cpp ...
0 .•. ~J
i.e.,
div
0
oJ
, ~J J
,~
- grad P + C p grad( t,p )
It may be observed that for a viscous fluid, (16) is written: p 4.2
r + grad P + p grad( n - C t,p) - div
0
v
=0
(17)
Thermodynamic Form of the Equation of Motion
In general, and not only when C is constant, the equation of motion (2) may be written in the following form:
r
e
=
grad s - grad ( h + n)
(18)
which in the present case will be designated as the thermodynamic form of the equation of motion. Eq. (18) is widely recognized (see ref. 35, p.171) when capillarity does not exist ( s = a(p,s) , = 0); h is then the enthalpy. It is significant to observe that this equation remains valid even in the present case. Verification consists simply of writing out the second member of eq. (18). Let us note
Ss
A
=
e grad
s - grad h
i. e. ,
Ai
=
es ,i
- h, i
According to (6), this gives A.
~
Thus
A.~
s's.-
