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for the solid constituent
ôø ôø
ôøwsôø
ôø ôø
= 0, (101)
ôø ôø
ôøÁs (vs - ½)ôø·n =0.
ôø ôø
ôø ôø (102)
However, we may not have the freedom to obtain similar jump conditions asso-
ciated with the balances of linear momenta and energy for the solid constituent
due to the problem associated with identifying the momentum and energy of the
electromagnetic field related to the solid constituent and the ambiguity of whether
to decompose some quantities into surface flux terms or source terms. The relation
(102) is met automatically when the interface is a part of the boundary "D of the
aqueous dielectric body since ½ = vs|"D on one side and Ás = 0 on the other
side. Also, the requirement can be imposed that the temperature is continuous
across the interfaces between the region of {s, w} and the region of {s, w, g} and
c
between {s, w, g} and {s, g} and this temperature equals T . The other set of
conditions follows from the entropy equation for the mixture as a whole [12],
ôø ôø
ôø ôø
ôø
ôø ôø
Á±s± (v± - ½) - J±ôø·n =0, (103)
ôø ôø
which is consistent with the assumption that the interface does not possess intrinsic
surface properties of its own, like surface mass density, surface entropy and so on.
4. Summary
We have presented, within the context of Maxwell-Lorentz field equations and mix-
ture theory, the electromagnetic field equations and the equations of motion for the
946 K. R. Rajagopal and L. Tao ZAMP
porous solid, water and gas in an aqueous dielectric corresponding to the process
of microwave drying. Three mixtures are considered: mixture of a porous solid
and water; mixture of a porous solid, water and gas; and mixture of a porous solid
and gas. On the basis of continuum thermodynamics, constitutive relations are
proposed for the macroscopic electric polarization vectors, Cauchy stresses, heat
fluxes, internal momentum supplies and so on to characterize the thermomechan-
ical behaviors of these mixtures. The interfacial jump conditions between these
different mixture regions were derived. This framework is tentative at best, some
revisions might be needed upon further studies, and much more needs to be done
with regard to fixing the specific forms of the constitutive relations based on ex-
perimental data and by solving specific drying problems. The appendix discusses
briefly an issue related to the requirement of material frame indifference.
Appendix
Though a subject of controversy (Edelen and McLennan [1]; Wang [15]), the re-
quirement of material frame indifference (MFI) plays a major role in constitutive
theory (Noll [10]; Oldroyd [11]; Truesdell and Toupin [14]). Part of the disagree-
ment lies in whether MFI should be adopted or a less restrictive requirement, the
principle of Galilean invariance, should be used to formulate constitutive relations.
Confusion has also been caused in averaged turbulence modeling and mixture the-
ory, wherein the situation is worse in that there seems no sharp criterion to classify
some quantities as material properties, due to the averaged fields involving large
spatial and temporal scales. In this appendix, we examine a feature of MFI briefly
from a physical perspective, a perspective related to observers measuring of length
and time interval necessary in obtaining constitutive relations, not merely talking
about these material properties in a purely abstract and conceptual sense. And
the intended purpose is to provide some justification for our adoption of Galilean
invariance in this work.
Let {x, t} be an inertial frame, and let a body in motion be represented in
this frame. Consider another frame x ,
x = Q(t)x + a(t), QQT = QT Q = 1
and assume that an observer is located at x0 in the frame of x . We have to
restrict x0 , Q and a such that
Ù
Ù
QQT (x0 - a) + a
(104)
where c is the speed of light in vacuum and »
that we may neglect relativistic effects. Otherwise, the observer s measuring scales
of length and time would differ nontrivially from that of the observer s located in
the frame of x , and we have to go beyond the framework of Newtonian mechanics
which is needed to establish the corresponding relations between the quantities
Vol. 53 (2002) Modeling of the microwave drying process of aqueous dielectrics 947
Ù
Ù
associated with these two frames. Hence, there is the constraint that Q , a , a
and x0 have to meet.
We now take a = C (t + t0) with C and t0 being constant in order to
simplify the analysis and we have
Ù
QQT (x0 - C (t + t0)) + C
(105)
The independence between Q , x0 and C which is necessary for applying MFI
to formulate constitutive relations results in
|C|
Ù
by picking Q = 0 ;
Ù
QQT x0
(107)
by picking C = 0 ; and
Ù
QQT C (t + t0)
(108)
from (105) to (107) and the triangle inequality. Furthermore, as a proper estima-
tion, we can infer from (106) and (108) that
"
Ù
QQT |t + t0| d"3 2.
(109)
"
Ù
That is, the product of the rotating speed QQT / 2 and time t + t0 is bounded
Ù
above. Specifically, if we choose t0 = 12 (hours) (or both t0 = 0 and QQT
constant) and t " [0, 12] (hours), the rotating speed will be smaller than the spin
of Earth. This implies essentially that Q should be treated as constant or the
frame x should be approximated as not rotating, because we neglect the effect of
Earth s spin in measuring material properties in most cases.
Acknowledgement
We thank the National Science Foundation and the National Institutes for Health
for support of this work.
References
[1] D. G. B. Edelen and J. A. McLennan, Material indifference: a principle or a convenience,
Int. J. Engng. Sci. 11 (1973), 813-817.
[2] A. C. Eringen, Mechanics of Continua. Wiley, New York, 1980.
[3] A. C. Eringen and G. A. Maugin, Electrodynamics of Continua I. Springer-Verlag, New
York, 1989.
[4] S. R. de Groot and P. Mazur, Non-Equilibrium Thermodynamics. Dover Publications, Inc.,
New York, 1984.
948 K. R. Rajagopal and L. Tao ZAMP
[5] S. R. de Groot and L. G. Suttorp, Foundations of Electrodynamics. North-Holland Publish-
ing Co., Amsterdam, 1972.
[6] J. B. Hasted, Aqueous Dielectric. Chapman and Hall, London, 1973.
[7] A. R. von Hippel, Dielectrics and Waves. The MIT Press, Cambridge, Massachusetts, 1954.
[8] I-Shin Liu, Method of Lagrange multipliers for exploitation of the entropy principle, Arch.
Rat. Mech. Anal., 46 (1972), 131-148.
[9] A. C. Metaxas and R. J. Meredith, Industrial Microwave Heating. Peter Peregrinus Ltd,
London, 1983.
[10] W. Noll, On the continuity of the solid and fluid state, J. Ratl. Mech. Anal. 4 (1955), 13-81.
[11] J. G. Oldroyd, On the formulation of rheological equations of state, Proc. Roy. Soc. A200
(1950), 523-541.
[12] L. Tao, L., J. D. Humphrey and K. R. Rajagopal, A mixture theory for heat-induced alter-
ations in hydration and mechanical properties in soft tissues, Int. J. Engng. Sci. 39 (2001),
1535-1556.
[13] C. Truesdell, Rational Thermodynamics. Springer-Verlag, New York, 1984.
[14] C. Truesdell and R. Toupin, The Classical Field Theories. In Handbuch der Physik Vol.
III/1, W. Flugge, ed, Springer-Verlag, New York, 1960.
[15] C. C. Wang, On the concept of frame-indifference in continuum mechanics and in the kinetic
theory of gases, Arch. Ratl. Mech. Anal. 58 (1975), 381-393.
[16] S. Whitaker, Coupled transport in multiphase systems: A theory of drying. In Advances in
Heat Transfer, Vol. 31, 1-104. Academic Press, New York, 1998.
K. R. Rajagopal and L. Tao
Department of Mechanical Engineering
Texas A&M University
College Station, Texas 77843-3123
USA
e-mail: Krajagopal@mengr.tamu.edu
To access this journal online:
http://www.birkhauser.ch
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