Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics

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Negative Refractive Index Media

Interleaving the structures for a negative "

eff

and 

eff

can create a composite with "

eff

< 0and

eff

< 0ata

common frequency (!) (Smith et al., Shelby et al.

2001), which as predicted by Veselago (1968) should

give rise to a material with negative refractive index.

Although this appears intuitively correct, it is actually

nontrivial that the electromagnetic fields of the two

composites do not interfere with each other’s function

(Pokrovsky and Efros 2002) and this could depend

crucially on the relative placement of the two

structures (Marques and Smith 2004). However,

there is now overwhelming experimental and numer-

ical evidence that such composite structures possess

negative refractive index (see Ramakrishna (2005,

section 6)). Now consider a medium with predomi-

nantly real " and .For">0and>0, we have

our usual optical materials. Only one of " or  lesser

than zero with the other positive would imply a

medium which cannot support any propagating

modes. This is a consequence of Maxwell’s equations:

k  k ¼ "ð!Þð !Þ

½19

which implies that only evanescently decaying waves

with an imagina ry component of k are possible.

Common examples are ordinary metals with "<0

and >0. Now consider a medium with both "<0

and <0, or a negative refractive index medium.

The M axwell’s equations for a plane time-harmonic

wave exp[i (k  r  !t)] are:

k  E ¼

ð!ÞH ½20

k  H ¼

"ð!ÞE ½21

The ‘‘left-handedness’’ of the triad (E, H, k)isclear

from these equations for "(!), (!) < 0. A real

refractive index means that waves propagate with the

direction of energy flow given by the Poynting vector,

S ¼ E  H ½22

opposite to the direction of the wave vector. Since

the group velocity is in the direction of the energy

flow, we conclude that in these left-handed materials

(LHMs) the group velocity and the phase velocity

are oppositely directed. The phase accumulated in

propagating a distance x is  = 

ﬃﬃﬃﬃﬃﬃ

"

!=c

x. Thus,

the refractive index can be taken to be n = 

ﬃﬃﬃﬃﬃﬃ

"

that is, a negative quantity. Mathematically, it is

more reasonable to ask for the sign of the square-

root to determine the wave vector given by eqn [19].

It can be shown by arguments of analytic continuity

in the complex plane that the negative sign has to be

chosen for propagating waves when Re(") < 0 and

Re() < 0(Ramakrishna 2005).

The negative refractive index has real effects on

the behavior of radiation even in basic processes

such as refraction. Consider an interface between

vacuum and a negative refractive index medium

with n < 0 shown in Figure 4. Continuity conditions

on the electromagnetic fields at the interface require

for a plane wave incident from the vacuum side at

an oblique angle that the parallel wave vector k

conserved for the transmitted and reflected wave.

This is the origin of Snell’s law:

sinð

Þ¼sinð

Þ¼n



sinð

Þ½23

where 

, 

and 

are the angles of incidence,

reflection, and transmission, respectively. The flow

of energy across the interface determines the direc-

tion of the group velocity in the material medium as

being away from the interface. Therefore, the

component of the phase velocity vector normal to

the interface must change sign as we pass from

vacuum into the material medium. We are then

forced to conclude that the ray is bent toward the

same side of the su rface normal as the incident

wave. This picture is consistent with Snell’s law with

the interpretation that n< 0 )

<0. Figure 4 illus-

trates this point which has been experimentally

verified by several groups (Shelby et al. 2001,

Parazzoli et al. 2003, Eleftheriades et al. 2002).

As a direct consequence of this, it is seen that a

flat slab of negative refractive medium can act as a

lens as shown in Figure 5. Provided that the slab is

of sufficient thickness, the refracted rays from a

point source come to a focus inside the slab and

upon exiting the slab the rays are redirected again

such that they come to a focus on the opposite side

of the slab (Veselago 1968). Veselago also predicted

a negative Doppler shift in such media and an

obtuse angle cone for Cerenkov radiation.

(a) (b)

VAC LHM

VAC RHM

Figure 4 Illustration of Snell’s law at an interface between

two media with (a) positive refractive index (VAC/RHM) and

(b) negative refractive index (VAC/LHM). The arrows indicate the

wave vectors and the energy flow is opposite to the wave vector

in the negative index medium.

Negative Refraction and Subdiffraction Imaging 487

Perfect Lens: Subwavelength Imaging

A wave analysis of the Veselago lens revealed an

extremely novel aspect: it did not suffer from the

diffraction limit and the image resol ution could be

infinite (Pendry 2000),ifthenegativeindex

material were perfectly nondispersive and nonab-

sorbing. Before we analyze this, let us first brief ly

review the p roblem of imaging and the diffraction

limit.

Any object is visibl e because it emits or scatters

light. The problem of imaging is then concerned

with reproducing the electromagnetic field distribu-

tion on a 2D object plane in the 2D image plane. If

E(x, y, 0) be the electric field on the object (z = 0)

plane, the fields in free space can be decomposed

into the Fourier components k

and k

,and

polarization defined by :

Eðx; y; z; tÞ¼

;k



; k



 exp i k

x þ k

y þ k

z  !t



½24

where



ðk

; k

Þ¼

x;y



ðx; y; 0Þe

iðk

xþk

yÞ

dx dy ½25

In the above expression, the source is assumed to be

monochromatic of frequency !, k

þ k

, c

is the speed of light in free space, and

z is the optical axis. A conventional lens acts by

applying a phase correction to each of the propaga-

ting components so that they reassemble to a focus

at a point beyond the lens. For these compone nts k

is real, thus a phase change is all that is required to

form an image containing these components. The

higher spatial details in an object, however, are

described by the nonpr opagating near-field compo-

nents with an imaginary k

where k

þ k

A conventional lens cannot restore these

components in the image plane as they decay

exponentially in amplitude as one moves away

from the source. Hence the resolution, , provided

by a conventional lens is limited to those compo-

nents with

þ k

)  

2c

¼  ½26

Now consider the slab of medium with " = 1

and  = 1 and of thickness d

. It can be shown

(Pendry 2000) that the transmission and reflection

coefficients are

lim

"!1

!1

t ¼ exp ik

½ ½27

lim

"!1

!1

r ¼ 0 ½28

respectively, where k

is the component of the wave

vector normal to the interface. Thus, the slab

reverses the phase advance for the propagating

waves as revealed by the ray picture. Analytic

continuation to imaginary wave vectors k

= i

implies that the transmittance

t ! exp(þ

d), that

is, the slab also increases the amplitude of the

evanescent waves in transmission at exactly the

same rate as the rate of the decay in free space

outside. Thus, each wave, propagating or evanes-

cent, arrives at the image plane with its phase or

amplitude restored exactly to the values at the object

plane so as to perfectly reconstruct the image. The

lens is also perfectly impedance matched and has

zero reflection. These incredible properties have led

the phenomenon to be called ‘‘perfect lensing.’’

Note that there is no energy flux associated with

purely evanescent waves, and hence the amplifica-

tion obtained in the steady state corresponds to local

field enhancements which would imply the presence

of localized resonances. In fact, the entire mechan-

ism of the focusing of the near-field components is

due to surface modes that reside on the surfaces of

these negative index materials (Ramakrishna 2005).

" = 1and = 1 are precisely the conditions for

these surface modes of electric and magnetic nature,

respectively. These surface plasmon resonanc es

which are excited resonantly by the evanescent

modes and the secret to the perfect lens is that all

the surface modes are completely degenerate.

Although the conditions for realizing a perfect lens

are easy to specify, in practice these are very difficult to

meet. The requirement of negative values for " and 

implies that these quantities must disperse necessarily

with frequency and be dissipative. Thus, the perfect-

lens condition can only be met approximately at a

single frequency. Any deviation from the ideal

d/ 2

Object

plane

Image

plane

= –1

Figure 5 Steady-state passage of rays (representing the energy

flow) of light from vacuum through a slab made of a LHM with

n = 1. The slab acts as a lens mapping a point on the image plane

to a point on the object plane.

488 Negative Refraction and Subdiffraction Imaging

conditions can then result in the excitation of slab

polariton resonances which can swamp the image. The

effects of absorption, which are always present, can

also seriously degrade the lens performance by damp-

ing out the surface plasmon resonances (Ramakrishna

2005). Consider the transmission for the P-polarized

radiation through a negative index slab:

tðk

Þ¼

4ðk

Þðk



Þe

½29

where

D¼ðk

þ k



ðk

 k



2ik

Under the perfect-lens conditions, the first term in

the denominator goes to zero for evanescent waves

and the exponential in the second term decays faster

than the exponential in the numerator. However, if

there was a mismatch in the conditions, ("

= 1 and



= 1 þ , say) then the first term in the denomi-

nator no longer vanishes. In the large wave vector

limit (k

 !=c

), the two terms in the den ominator

become approximately equal when

¼







½30

thus yielding a criterion for the largest wave vector

for which there is effective amplification. The

dependence through the logarithm on the deviations

(whether real or imaginary) from the resonant

conditions underlines the fact that the perfect lens

effect is indeed very sensitive. In practice, the

periodicity, d, of the strucuture of the meta-

materials comprising the negative index slab itself

imposes an upper wave vector cutoff k

= 2=d. The

material will become spatially dispersive for wave

vectors k !k

, and for k >k

the very description as

a homogeneous material will break down.

An important simplification of the perfect-lens

conditions results when we consider a situation in

which all length scales in the problem are much less

than the wavelength of the light (the quasistatic

approximation). Under these conditions, the electric

and magnetic fields effectively decouple. If we

consider the case of P-polarized fields, it can be

shown (Pendry 2000) that in the quasistatic limit

only the value of the permittivity is important, and

there are essentially no conditions on the value of

the permeability. This brings metals such as silver

into the picture as the permittivity of silver becomes

equal to 1 in the optical region of the spectrum

and with relati vely small losses (Pendry 2000). To

overcome the loss es, a series of refinements of the

simple thin-slab picture have been proposed includ-

ing dividing the lens into a series of layers and using

optical amplification to act against the deleterious

effects of absorption (Ramakrishna 2005).

The Generalized Perfect-Lens Theorem

The negative refractive slab can be considered as

‘ ‘o ptical antimatter’ ’ in the sense that it cancels out the

effects on radiation of the traversal through an equal

amount of positive refractive index medium. This

cancelation is applicable to the phase changes for the

propagating modes and the amplitude changes to the

evanescent modes. In fact, the focussing action can

happen for more general situations where the require-

ment of homogeneity of the slab material can be

relaxed. Now consider the more general situation

where the dielectric permittivity and the magnetic

permeability are arbitrary functions of the spatial

coordinates:

¼ "ðx; yÞ;

¼ ðx; yÞ½31



¼"ðx; yÞ;



¼ðx; yÞ½32

corresponding to the Figure 6. We will consider the

imaging axis to be the z-axis. Thus, we see that the

system is antisymmetric with respect to the z = d

plane. It turns out (Pendry and Ramakrishna 2003 )

that such a system also transfers the image of a

source placed at the z = 0 to the z = 2d plane in the

same exact sense that it includes both the propagat-

ing and evanescent components. In general, the rays

in spatially varying media will not be straight lines

as shown in Figure 6, but the effect of propagating

through the positive medium is nullified by the

negative medium. Thus, to an observer on the right-

hand side, it would appear as if the region between

z = 0 and z = 2d did not exist. We will call such

media with the same sense of transverse spatial

variation but with opposite signs as optical com-

plementary media, and the effect of any such pairs

of complementary media on radiation is null.

d ddd

Figure 6 A pair of complementary optical media nullify the

effect of each other for the passage of light. Spatially varying

positive and negative refractive indices are schematically depicted

by the white or shaded regions.

Negative Refraction and Subdiffraction Imaging 489

The most general conditions on the permittivity

and permeability tensors for such complementary

behavior are:



½33

and



"

þ"

"

þ"

"





þ



þ



½34

and a perfect focus results whenever the two slabs of

positive and negative media hav e such a behavior (see

Pendry and Ramakrishna (2003) and Ramakrishna

(2005) for the proof). This theorem clearly shows that

the dependence along the x- and y-directions trans-

verse to the imaging axis z is completely irrelevant as

long as the two slabs are optically complementary. As

an extension, it can be shown that any system of

optically complementary media will also have a

perfect focus as long as the system has a plane of

antisymmetry normal to the optical axis. The above

effects have also been numerically verified for several

such spatially varying complementary media (Pendry

and Ramakrishna 2003).

Perfect Lens in Other Geometries

The above generalized perfect-lens theorem along with

a method of coordinate transformations can enable us

to now generate a variety of superlenses in different

geometries. In general, if we can find a geometric

transformation that maps a given configuration into

the geometry for the generalized slab lens, then we

would have generated one more arrangement that will

exhibit the property of transferring images of sources

in a perfect sense. If we define the new coordinates

(x, y, z), q

(x, y, z), and q

(x, y, z) (assumed ortho-

gonal), then in the new frame, the material parameters

and fields are given by (Ward and Pendry 1996)

¼ "

;



¼ 

½35

¼ Q

;

¼ Q

½36

where



½37

Note that a distortion of space results in the change

of " and  tensor s in general. Thus, in many cases,

the transformed geometry would involve spatially

varying (inhomogeneous) and anisotropic medium

parameters.

The change in geometry can also make it possible

forustorealizelenseswithcurvedsurfaces.The

original slab lens maps every point on the object plane

to another point on the image plane. But the size of

the image is identical to that of the source. This is due

to the invariance in the transverse direction and the

transverse wave vector (k

, k

) is preserved. In

general, to change the size of the images, the

translational symmetry would have to be broken and

curved surfaces will necessarily be needed. The

focussing action for the evanescent waves is crucially

dependent on the near degeneracy of the surface

plasmons in the case of the slab, and curved surfaces,

in general, have a completely different dispersion for

the surface plasmons. Thus, one should expect that

inhomogeneous materials will be required for such

curved lenses of negative refractive index. It can be

shown (Ramakrishna 2005) that mapping the slab

lens into cylindrical coordinates

x ¼ r

‘=‘

cos ; y ¼ r

‘=‘

sin ; z ¼ Z

½38

where ‘

is some scale factor(= 1) generates a

cylindrical annulus of inner and outer radii a

and a

, respectively, with the material parameters

given by

¼ 

¼1



¼ 



¼1

¼ 

¼1=r

½39

for the annular region. The positive material outside

the annular region should vary as

¼ 

¼þ1



¼ 



¼þ1

¼ 

¼þ1=r

½40

where r = r

exp(‘=‘

). This system transfers images

in and out of the cylindrical annulus and the image

of a source inside at r = a

will be formed on the

surface a

= a

)

. Thus, there will be a

magnification of the image by the factor

M¼



½41

490 Negative Refraction and Subdiffraction Imaging

Note that these cylindrical lenses are also short-

sighted in the same manner as the slab lens. They

can only focus sources from inside to the outside

only when a

< r < a

, and the other way

around from outside to the inner world when the

source is located in a

< r < a

Similarly the transformation into spherical coor-

dinates (r = r

‘=‘

, , ) can be used to generate a

spherical perfect lens wherein a spherical shell of

negative refractive material with "(r) 1=r and

(r) 1=r with arbitrary dependence along  and 

(which could be constant too!) have the property of

perfectly transferring images of sources in and out of

the shell (Pendry and Ramakrishna 2003). This

spherical lens also has exactly the same magnifica-

tion factor given by eqn [41]. In fact, the solutions in

these two cases of a cylinder and sphere can also be

obtained by a more conventional electromagnetic

calculation in terms of the scattering modes

(Ramakrishna 2005). One can obtain even more

esoteric configurations such as one or two intersect-

ing corners of negative refracting materials that

behave as perfect lenses (Pendry and Ramakrishna

2003).

Other Approaches to Negative Refraction

There is also an approach to negative refractive

materials based on loaded transmission lines

(Eleftheriades et al. 2002), which has been imple-

mented at radio- and microwave frequencies using

lumped circuit elements. These show all the hall-

marks of a negative refractive material within an

effective medium approach.

Effects which can be interpreted as negative

refraction have been observed in certain periodic

photonic crystals (PCs) (Luo et al. 2003). An

incident propagating plane wave from vacuum

appears to undergo negative refraction inside the

PC, and a slab of the PC can even work as a

Veselago lens. The negative refraction in this case is

a result of the curvature of the equifrequency surface

and is present in spite of the right-handed nature of

the propagation. In these instances, an effective

permittivity and permeability cannot be easily

ascribed to the crystal as the long wavelength

condition is not met. It is difficult to homogenize

the PC in the sense of meta-materials, and the

energy transport in these PCs is very sensitive to the

periodicity and the structural arrangements. Thus, it

would be an over-simplification to characterize these

effects in PC as merely due to an effective refractive

index.

Further Reading

Eleftheriades GV, Iyer AK, and Kremer PC (2002) Planar negative

refractive index media using periodically L–C loaded transmis-

sion lines. IEEE Transactions on Microwave Theory and

Techniques 50: 2702–2712.

Garland JC and Tanner DB (eds.) (1978) Electrical Transport and

Optical Properties of Inhomogeneous Media. New York:

American Institute of Physics.

Landau LD, Lifschitz EM, and Pitaevskii LP (1984) Electro-

dynamics of Continuous Media, 2nd edn. Oxford: Pergamon.

Luo C, Johnson SG, Joannopoulos JD, and Pendry JB (2003)

Subwavelength imaging in photonic crystals. Physical Review

B 68: 045115.

Marques R and Smith DR (2004) Comment on ‘‘Electrodynamics

of Metallic Photonic Crystals and the Problem of Left-Handed

Materials’’. Physical Review Letters 92: 059401.

O’Brien S and Pendry JB (2002a) Photonic band-gap effects and

magnetic activity in dielectric composites. Journal of Physics:

Condensed Matter 14: 4035–4044.

O’Brien S and Pendry JB (2002b) Magnetic activity at infrared

frequencies in structured metallic photonic crystals. Journal of

Physics: Condensed Matter 14: 6383–6394.

Parazzoli CG, Greegor RB, Li K, Koltenbah BEC, and Tanelian M

(2003) Experimental verification and simulation of negative

index of refraction using Snell’s law. Physical Review Letters

90: 107401.

Pendry JB (2000) Negative refraction makes a perfect lens.

Physical Review Letters 85: 3966–3969.

Pendry JB (2004) Contemporary Physics 45: 191–202.

Pendry JB, Holden AJ, Robbins DJ, and Stewart WJ (1998) Low

frequency plasmons in thin-wire structures. Journal of Physics:

Condensed Matter 10: 4785–4809.

Pendry JB, Holden AJ, Robbins DJ, and Stewart WJ (1999)

Magnetism from conductors and enhanced nonlinear phenom-

ena. IEEE Transactions on Microwave Theory and Techni-

ques 47: 2075–2084.

Pendry JB and Ramakrishna SA (2003) Focusing light using

negative refraction. Journal of Physics: Condensed Matter 15:

6345–6364.

Pokrovsky AL and Efros AL (2002) Electrodynamics of metallic

photonic crystals and the problem of left-handed materials.

Physical Review Letters 89: 093901.

Ramakrishna SA (2005) Physics of negative refractive index

materials. Reports on Progress in Physics 68: 449–521.

Shelby RA, Smith DR, and Schultz S (2001) Experimental

verification of a negative index of refraction. Science 292:

77–79.

Smith DR, Padilla WJ, Vier DC, Nemat-Nasser SC, and Schultz S

(2000) Composite medium with simultaneously negative

permeability and permittivity. Physical Review Letters 84:

4184–4187.

Veselago VG (1968) The electrodynamics of substances with

simultaneously negative values of var " and . Soviet Physics–

Uspekhi 10: 509–514.

Ward AJ and Pendry JB (1996) Refraction and geometry in

Maxwells equations. Journal of Modern Optics 43: 773–793.

Negative Refraction and Subdiffraction Imaging 491

Newtonian Fluids and Thermohydraulics

GLabrosseandGKasperski, Universite

Paris-Sud XI,

Orsay, France

Introduction

Thermohydraulics is based on the hypothesis of

continuous medium. This hypothesis is easily satis-

fied since, for instance, a one-thousandth of 1 mm

of a perfect gas at normal temperature and pressure

conditions (300 K, 1 atm) contains about 2.5 10

molecules. Instantaneous balances are made inside a

control volume fixed in the system of axes and

crossed by the flows. The limit where this volume

vanishes leads to the local formulation of the laws

governing the flows. The flow is described by

velocity

v (

r, t), pressure p(

r, t), temperature T(

r, t),

and other fields,

r being the position vector of a

point M,andt the time. The material derivative of

r, t)is



þð

v :

Þ



Let Q (

Q) be one of the scalar (vectorial) extensive

quantities whose balance participates in the flow

dynamics. It can be a quantity of matter, heat,

impulse, or something else. Let Q be the amount of

Q contained in the volume V localized around M,

and q(

r, t) its local representative defined by

ð

r; tÞqð

r; tÞ¼ lim

V!0

Q

V



½1

where  is the density, similarly defined considering

the case where [Q] is taken as the mass m:

ð

r; tÞ¼

½2

Table 1 gives examples of q quantities.

The instantaneous local balance of Q reads

ðqÞþ

 

þ q



¼ S

½3

where S

stands for any possible local source of Q,

and

is the Q conduction flux density. Figure 1

illustrates how these quantities allow us to evaluate the

flux d

S of Q that instantaneously crosses

a surface d

S. Table 2 gathers the physical dimension

of these notions for various Q’s.

For

Q, the flux densities are second-order tensors,

since d

= j

)

d

S is vectorial (Figure 1). Its

balance reads

ð

qÞþ

  j

)

þ 

v 



½4

where

indicates the transposition and  a dyadic

product.

and j

)

are given later.

The governing equations of thermohydraulics are

like [3] and [4]. They are completed by compatible

initial and boundary conditions. The most general

linear expression of the latter ones is of mixed type,

for a scalar field,

q þ 

 



q ¼  on the boundary ½5

, , and  being prescribed data, and

n the outward

normal to the boundary. For a vectorial field,

q and

, respectively, replace q and . The simplest cases

are Dirichlet and Neumann boundary conditions

with, respectively,  = 0or = 0.

Governing Equations

We consider nonisothermal flows of fluids in thermo-

dynamic conditions far from the critical point where

acoustic effects are involved. The fluid is possibly a

binary mixture, the simplest non-pure-fluid case where

modeling does not raise conceptual difficulties. The

Table 1 Some quantities q. T is the absolute temperature, C

the specific heat at constant pressure, and C the solute mass

fraction

Mass Impulse Kinetic energy Heat Mass fraction

T 0 < C < 1

dΦ

Figure 1 Q flux density and

Q flux.

Table 2 Physical dimension of fluxes, flux densities, and

=(flux density) for some q quantities

Q q Flux Flux density

=(flux density)

Volume undefined m

1

[velocity] s

1

Mass 1 kgs

1

kgs

1

2

kgs

1

3

Energy,

heat

[velocity]

WWm

2

3

Electrical

charge

Coulomb kg

1

AAm

2

3

Impulse [velocity] [force] [pressure] [pressure] m

1

492 Newtonian Fluids and Thermohydraulics

local composition is described by the solute (say) mass

fraction,

CðM; tÞ¼ lim

V!0

m

solute

m



solute



with 0  C  1. Only thermodiffusion is treated,

and the influence the solutal gradient has on the heat

flux is not considered, being negligible in liquid

mixtures. The coupling between the heat and species

molecular transports then comes only in the solutal

flux density relation

solute

¼

ðT; CÞ

C þ Cð1  CÞS

T

½6

with 

> 0, and S

(T, C), the solute Soret

coefficient, which is positive or negative. The

order of magnitude of the Soret coefficient in the

molecular solutions does not exceed few 10

2

1

while for colloidal solutions (ferrofluids) jS

j can

be in the range 0.03–0.5 K

1

. Even if small, the

induced mass fraction separation, C ’ S

T,

generates a solutal buoyancy of significant dyna-

mical influence.

Equation of State for the Density

One must first describe the sensitivity of the density,

 (p, T, C), upon pressure, temperature, and mass

fraction in static conditions. The pressure and

temperature effective ranges, p and T, are

assumed small enough compared to their respective

mean values, p

and T

, for the local (at



 (p

, T

, C

)) tangent to (p, T, C)tobea

good approximation in most cases,

  



¼ ðp  p

Þ

ðT  T

Þþ

ðC  C

Þ½7

where

 ¼



@





and



¼



@





;



@





are the compressibility, thermal, and solutal

expansion positive c oefficients, and C

is the solute

mean mass fraction. Thermodynamic properties of

some fluids are given in Table 3. Equation [7] is

valid if p, 

T,and

jCj are 1. More-

over, in laboratory experiments and industrial

processes, one generally has p=p

 T=T

.The

pressure term i n [7] can thus be neglected in

thermohydraulics.

Notice that water density exhibits a maximum

around 4



C. A quadratic term in T must then be

added to [7].

The Boussinesq Approximations

The parameter 

T  1 is the primary source of

thermohydraulics. Therefore, the

v, p, T and C fields

can be expanded in series of terms of increasing

power in 

T. The leading term of each series

contains an important part of the interesting

dynamics. The forthcoming equations are given in

the corresponding approximation framework. They

contain many simplifications, due to Boussinesq. For

instance, the conductivities and diffusivities are

taken as constant, as well as C(1  C)S

in eqn [6].

The next approximation step, the low-Ma ch model,

keeps the leading compressibility and expansion

effects, while discarding the associated acoustic

waves. This gives access to thermo-soluto-acoustic

phenomena. Expansion oscillations are indeed able

to trigger, and sustain, acoustic waves provided

phase agreements are fulfilled. This second-order

model is not presented here.

The compl iance with the criteria 

T 1 and



j Cj1 must be checke d case by case. The

section ‘‘Steady paral lel-flow model’’ brie fly illus-

trates this point with an example of thermally driven

flow. Furthermore, the T- and C-sensitivity of S

an experimental fact that requires a generic

approach of the problem. The C-sensitivity of the

physical propert ies is generally more pronounced,

nonmonotonic, for instance, over C 2 [0, 1], than

their T-sensitivity.

Boussinesq Local Balances

Mass It reads @=@t þ

 (

v) = 0, or equivalently

(1=)(D=Dt) = 

 

v. The fluid particle density

varies along its trajectory by compressibility and

thermo-solutal expansion. At the leading order in



T and 

jCj, the latter is negligible, whereas

the former is associated with acoustics effects, also

negligible when the fluid velocity is much smaller

Table 3 Some values of density, thermal expansion and

compressibility coefficients, specific heat at constant pressure,

and sound speed at p = 1 atm and T = 293 K; in SI units

Fluid T pC

Air 1.205 1 1 1005 344

Helium 0.167 1 1 5227 1010

1.841 1 1 832 269

Water 1000 0.0607 4:91  10

5

4182 1461

Glycerol 1250 0.148 2:2  10

6

2333 2044

Mercury 13579 0.0533 3:76  10

6

1391 1409

Newtonian Fluids and Thermohydraulics 493

than the sound speed. The mass balance equati on

then reduces to

 

v ¼ 0 ½8

Only transverse velocity waves (or shear waves) are

allowed by this equation,

v ’ e

k

rþ!t)

with

k 

v = 0,

since acoustics contributions are discarded.

Impulse The impulse molecular flux density is

)



¼ p 1

)

 

 

v þð

 

vÞ

where 

is the impulse conductivity and 1

)

the

Kronecker tensor. A Newtonian fluid is defined as

having 

constant with respect to the rate-of-

strain tensor

 

v. The impu lse b alance then

reads

ð

vÞþ

 ð

v 

vÞþ

  j

)



¼ 

In the source term 

G =

g for gravity-driven

buoyant flows.

With the aforementioned approximations, the

impulse balance becomes

¼



P þ

  



g þ 



v ½9

with

  



¼

ðT  T

Þþ

ðC  C

 ¼





the impulse diffusivity, and the pressure P = p  p

0, h

satisfying the hydrostatic relation

p

0;h

¼

In the rotating frame of vector

W(t),

  



W ^ð

W ^

rÞþ2

W ^

v þ

must be subtracted from the right-hand side of [9]

and p

0, h

redefined by

p

0; h

¼ 

g 

W ^ð

W ^

rÞ



On a free surface, a particular velocity bounda ry

condition is to be established. Let d

S = dS

n be a

surface element located around M. The tangential

component (

t 

n = 0) of the impulse flux across d

t  d

f ¼

t  j

)



 d

S ¼

t 

 

v þð

 

vÞ

 d

must be continuous. Surface tension (T, C) inho-

mogeneities make the free surface a source of

impulse which diffuses in the fluid core. A flow

occurs even with

G = 0. For the fluid located where

S points to, the velocity boundary condition on the

free surface then reads



t 

 

v þð

 

vÞ



n ¼ð

 

tÞ ½10

with

 

tÞ ¼

@

 

tÞT þ

@

 

tÞC

For most fluids, @=@T < 0. In the Boussinesq

framework @=@T and @=@C are constant. Equa-

tion [10] cou ples the impulse balance with the heat

and composition ones.

Heat Local thermodynamic equilibrium is

assumed. The molecular heat flux density is

heat

= 

T,with

the thermal conductivity.

The approximate heat balance reads

¼ 



T þ S

heat

½11

where 

= 

=(

) is the heat diffusivity and

heat

a possible local (Joule, radioactive, ...) heat

source. Thermohydraulics can simply be driven by

nonuniform thermal conditions imposed along the

fluid boundary, and in this article we henceforth

take S

heat

= 0.

Mass fraction Approximating [6] yields the mass

fraction balance,

¼ 



C þ C

ð1  C

ÞS



T ½12

where 

and S

are evaluated at T

and C

. The

normal flux condition

C 



¼C

ð1  C

ÞS

T 



is imposed on impervious boundaries.

The Hydrostatic State

Knowing whether the fluid can be in static state

with respect to its presupposed rigid container helps

for a first understanding of thermohydraulic

dynamics. This raises two problems: (1) the exis-

tence of this state and (2) its stability, discussed

494 Newtonian Fluids and Thermohydraulics

later. Point (1) requires the fulfilment of three

relations,

p ¼ ðp; T; CÞ

G ½13

¼ 



¼ 



C þ C

ð1  C

ÞS



½14

The curl of [13] yields

ðp; T; CÞ^

G þ ð p; T; CÞ

 ^

G ¼ 0

which has no reason to be generical ly satisfied since

(p, T, C) and

G are totally uncorrelated. The

hydrostatic state cannot exist if

G does not derive

from a scalar potential, as with

G ¼

g 

W ^ð

W ^

rÞ

r if

6¼ 0

The Earth’s rotation axis is known to precess with a

period of about 26 000 years. This generates a

component of 26 000 years timescale in the atmo-

spheric, oceanic, and internal flows.

Considering now that

G ¼



the existence of a hydrostatic state only depends on

the simultaneous verification of [14] and

ðp; T; CÞ^

 ¼ 0 ½15

Iso- surfaces must therefore coincide with iso-

pycnal, isobaric, iso-T, and iso-C surfaces since the

p, T,andC sensitivities of  are uncorrelated. The

compatibility of this condition with [14] is the key

for concluding about the existence of the hydro-

static state. Considering again our planet as an

example (forgetting about precession), the iso-

surfaces are almost ellipsoidal. Such T and C

distributions cannot satisfy [14]. Thus, the atmo-

spheric and oceanic dynamics, and thermohydrau-

lics as well, are due to a nonvanishing thermal

torque,

T ^

 .

A free surface in hydrostatic state is isothermal

and isocompositional, by eqn [10], whatever

Dimensionless Local Balances

In buoyancy-driven thermohydraulics, we consider

four velocity scales – three of molecular origin, and

the fourth is the free-fall velocity in the buoyancy,



; V



; V



ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ



TgL

L being a fluid container size scale. Thence come the

Rayleigh, Prandtl and Lewis numbers,

Ra ¼

¼ 

T



Pr ¼





; Le ¼



Ra being the experimental control parameter, and

Le  1. Table 4 gives Pr orders of magnitude for

usual fluids. Let V be the fluid velocity amplitude.

The importance of the thermal, solutal, and impulse

convections with respect to the corresponding

diffusions is, respectively, estimated by the thermal,

compositional Pe´clet and Reynolds numbers,



; Pe



Re ¼



with

Pr ¼

; Le ¼

; Ra ¼ðPe

ReÞ

V¼V

Capillary thermohydraulics introduces one velo-

city scale and the Marangoni number,

jj



; Ma ¼

¼ Pe

with  = (d=dT)T in pure fluid. A small

capillary number, Ca = jj=, indicates a weak

influence of the dynamics upon the free-surface

curvature.

Let V

, =

,  = L=V

, T and

C ¼C

ð1  C

ÞS

T

be the velocity, pressure, time, temperature, and

mass fraction scales, with

 ¼

T  T

T

and C¼

C  C

C

the reduced temperature and mass fraction, respec-

tively. The other quantities, coordinates included,

are similarl y reduced and noted identically.

Table 4 Orders of magnitude of the Prandtl number for the

usual fluids. Air and water are in normal conditions

Liquid metals Gases Water Oils

Several 10

3

– 10

2

’1, 0.7 for air 6.7 >10

Newtonian Fluids and Thermohydraulics 495

Equation [8] does not change and [9], [11] and [12]

become, respectively,

¼

P þ Pr Rað þ 

CÞ



½16

D



 ½17

¼ Le



C



 ½18

where



¼



C



T

is the buoyancy separation ratio and

= 

g=j

gj.

A 

< 0(> 0) corresponds to opposite (coopera-

tive) thermal and solutal buoyancies. The reduced

mass fraction boundary condition on impervious

walls is

C



 



½19

In rotating frame, scaling

W(t)by



W(t) =

W(t)=

Ra Fr ð þ 

CÞ



W ^ð



W ^

rÞ





W ^

v þ



must be added inside the square-bracket term of

[16]. The Froude and Ekman number s appear as

Fr ¼



; Ek ¼





The dimensionless capillarity stress condition [10]

reads

t 

 

v þð

 

vÞ



¼Ma ð

 

tÞ þ 

 

tÞC



½20

with



@=@C

@=@T

C

T

the capillarity separation ratio, and

Ma ¼



@

T





These equations show that, in the Boussinesq

framework, the flow physics does not depend on

, T

,andC

, except through the material proper-

ties which enter the numbers.

Linear Stability

Given a base state S= (

v, , C), a solution of [8],

[16]–[18], how does it behave in presence of an

infinitesimal disturbance (

v, , C)? Applying [8],

[16]–[18] to (

v þ 

v,  þ , CþC) and discarding

the quadratic terms in perturbation provide the

disturbance temporal evolution,

 : ð

vÞ¼0 ½21





C

¼Fþ 

v 







þA





C

½22

where F = (

(P), 0, 0)

, and

A¼

Ra Pr

Ra Pr 

0 B

0 



½23

with B

= (

v 

) þ a



. The perturbations (

v, ,

C) have the (

v, , C) bounda ry conditions, but

homogeneous. On a free surface, the perturbation

capillary stress condition is

t 

  

v þð

  

vÞ



¼Ma ð

 

tÞ þ 

 

tÞC



½24

Recasting [21]–[23] provides





C

¼ LðSÞ





C

½25

whose solution is



vðtÞ

ðtÞ

Cðt Þ

¼ e

LðSÞt



vðt ¼ 0Þ

ðt ¼ 0Þ

Cðt ¼ 0Þ

½26

Direct System

L(S) is made of

 acting on the initial pertur bation.

Conclusions about S stability depend on the sign of



max

, the real part of the leading eigenvalue of L

found with all the possible perturbations. There is

stability if 

max

< 0. At 

max

= 0, the marginal

stability, the bifurcation threshold is located at

Ra (Pr, Le, 

, 

, X) = Ra

, Ra

-being the critical

value of the control parameter, X containing all

the other parameters of the problem (container

aspect ratios, etc.). The nonlinear-stability analysis

in the vicinity of Ra

supplies  in 

max

(Ra  Ra

)



, which is character istic of the

bifurcation.

496 Newtonian Fluids and Thermohydraulics