Document [original]

On the Shock Front Thickness in Water and other Molecular Liquids

Siegfried Hess

Institut für Theoretische Physik, Technische Universität Berlin, PN 7-1,

Hardenbergstr. 36, D-10623 Berlin

Z. Naturforsch. 52 a, 213-219 (1997); received October 29, 1996

Theoretical explanations are presented for the deviation of the shock front thickness from

linear hydrodynamics, as observed by W. Eisenmenger (1964) in water and several molecular

liquids for large driving pressure differences. Two mechanisms are proposed, which are based on

generalizations of the Maxwell relaxation equation for the friction pressure tensor. One is due

to the spatial inhomogeneity and linked with piezo-electric or piezo-tetradic effects. The other is

caused by nonlinearities which account for shear thickening.

1. Introduction

Within the range of applicability of (linear) hydro-

dynamics, the thickness of a shock front (shock width)

in a fluid is closely related to the damping of sound

waves, which, in turn, is determined by the transport

coefficients of thermo-hydrodynamics, viz.: the vis-

cosity, the bulk viscosity and the thermal conductiv-

ity [1]. The first measurements of the shock width in

liquids which were reliable enough to test the hydro-

dynamic theory, showed satisfactory agreement for

small driving pressure differences but significant de-

vations for larger pressure differences [2]. The shock

width is larger than the corresponding hydrodynamic

value. The transport coefficients seem to increase with

increasing deviations from equilibrium. In this article,

two mechanisms are discussed which can account for

the observed deviations from linear hydrodynamis.

These are effects associated with the spatial inho-

mogeneity of the friction pressure tensor and with

a nonlinear generalization of Maxwell's relaxation

model, recently invented to describe shear thickening

[4]. Though these ideas have been presented previ-

ously at conferences [3], it has been over thirty years

since the publication of the article by W. Eisenmenger

[2], which still contains the best experimental data

for water and some other molecular liquids. Non-

equilibrium molecular dynamics (NEMD) computer

simulations, performed for a simple Lennard-Jones

model liquid, yielded similar deviations from hydro-

Reprint requests to Prof. S. Hess,

e-mail: [email protected].

0932-0784 / 97 / 0100-0213 $ 06.00 © - Verlag der

dynamics [5], yet at high temperatures and for driving

pressure differences orders of magnitude larger than

those used in the experiments.

This article proceeds as follows. In Sect. 2, the

pressure profile in the shock front and the meaning

of the shock width L are stated. The expression for L

derived from linear hydrodynamics is reviewed and

a function H is introduced which characterizes the

relative deviation of L from it. Experimental results

for water and methanol as measured by Eisenmenger

[2] are presented graphically. Section 3 is devoted

to the discussion of two conjectures for the explana-

tion of the observed deviations from hydrodynamics.

Both employ generalizations of the Maxwell relax-

ation equation for the friction pressure tensor. The

first one (Sect. 3.1) is the addition of a term propor-

tional to a second spatial derivative of the pressure

tensor. This leads to a relative deviation H of the

shock width from its hydrodynamic value which is

proportional to the square of the driving pressure dif-

ference. The plot of the data for water and methanol

confirm such a behavior within the experimental un-

certainties. A new characteristic length parameter P,

which turns out to be a few hundred to one thausand

times larger than the size of a molecule, can be infered

from this analysis. Some speculations on the origin of

the additional term in the Maxwell relaxation equa-

tion and the meaning of the length £ are presented.

A link with piezo-electric or piezo-tetradic effects is

indicated. The second mechanism (Sect. 3.2) is caused

by terms nonlinear in the friction pressure tensor. In

lowest order in the nonlinearity, again it is found that

H is proportional to the square of the driving pressure

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214 S. Hess • On the Shock Front Thickness in Water and other Molecular Liquids

p/(pl + p2)

0.8

0.6

0.4

0.2

—^

^—

.2 -1 0 1

Fig. 1. Pressure profile in the vicinity of the shock front.

difference. Now, however, the proportionality coeffi-

cient has a different meaning. Results from the full

nonlinear theory [4] are also compared with the ex-

perimental data for water. Good agreement is found.

Some concluding remarks are added.

2. Basics and Motivation

2.1. Pressure Profile

The pressure p = p(x) in the vicinity of a plane

shock wave, propagating in the —x-direction, is, in

a co-moving coordinate system where x = 0 corre-

sponds to the center of the shock front, given by [1]:

P = +P\)+ ]jiP2 -pi)tanh(2x/L). (1)

Here bp = p2 — p\ is the pressure difference driv-

ing the shock wave, p\ and p2 are the asymptotic

values of the pressure before and behind the shock

front. The shock width is denoted by L. The pressure

p(x), in units of the average pressure (p\ + pi)/2, is

depicted schematically in Fig. 1 as function of x in

units of a convenient reference length. The thickness

L decreases with increasing driving pressure bp. More

specifically, the shock width can be written as

L = H Lref pref (bp)

-l (2)

where the product of the reference values for the

length and pressure, Lref and pref, is determined by

linear hydrodynamics. The quantity H = H(bp/pref),

with H = 1 in the hydrodynamic limit, characterizes

the deviation from linear hydrodynamics.

In the experiments, the shock width is inferred from

the pressure rise time se = L/cs in the shock front,

and cs is the speed of the shock wave, which is prac-

tically equal to the adiabatic sound velocity [2],

2.2. Hydrodynamics

Linear thermo-hydrodynamics [

] yields (2) with

if =1, and

Lref Pref = 16 aV' aV

dp2

with the abbreviation

a = V

4 \ ^fl 1

-Tl + 77v + A

3 /

Cv Cn

(3)

• (4)

Here V = p~l is the specific volume and cs the sound

velocity, The shear viscosity, bulk viscosity and heat

conductivity are denoted by 77, r/v, and A, respectively,

cv and cp are the specific heat at constant volume and

constant pressure.

The decrease of the amplitide Ampler) = Ampl(O)

exp(-a x) of a sound wave with frequency uj = lit u,

propagating in x-direction, is characterized by the

inverse length parameter a, which is related to the

quantity a occuring in (3) and (4) by a =

UJ2

a. By

order of magnitude, one has a « 10-15s2m-1 for

water and the other liquids studied in [2].

2.3. The Experiment

In 1964 W. Eisenmenger published data on the

shock width for liquid water, acetone, methanol,

ethanol, a few other organic substances, as well as

for some mixtures, in the article [2] with the ti-

tle Experimentelle Bestimmung der Stossfrontdicke

aus dem akustischen Frequenzspektrum elektromag-

netisch erzeugter Stosswellen in Flüssigkeiten bei

einem Stossdruckbereich von lOatm bis lOOatm (Ex-

perimental determination of the shock front thick-

ness from the acoustic frequency spectrum ofelectro-

magnetically generated shock waves in liquids in the

shock pressure range of lOatm to lOOatm; 1 atm

~ 105 Pa). In Figs. 2 and 3, his data are used to plot

the shock front thickness L, in units of the conve-

niently chosen reference length Lref = 1 |im, versus

the pressure difference bp for water and methanol.

The quantity bp is expressed in units of a reference

pressure pTef, which is determined by the hydrody-

namic value as it follows from (3) and (4) with the

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215 S. Hess • On the Shock Front Thickness in Water and other Molecular Liquids

. 1

. • • • .

H2ä r1:

\ ;

\ •

\ \ ;

N L. . •

\ •

:>i

1 11 1 1 .1 1.. .

. X -

-1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4

log (5P/DJ

Fig. 2. The shock front thickness L in water as function of

the presssure difference 5p, Lref = 1 )im, pref « 70atm.

Data from Eisenmenger [2],

~0.2

-0.2

-0.4

-0.6

-0.2 0 0.2 0.4 0.6

log (5p / pref)

Fig. 3. The shock front thickness L in methanol as function

of the presssure difference 5p, LKt = 1 Urn, Pref ~ 17 atm.

Data from Eisenmenger [2].

specific choice for Lref. In particular, one has pref ~

70 atm and pref « 17 atm for water and methanol,

respectively. The straight (dashed) lines correspond

to the hydrodynamic result. No adjustable parame-

ters occur in the comparison with the experimental

data. The hydrodynamic limit is approached for small

values of bp, Significant deviations, however, are ob-

served for dp > pTe{. The quantity H of (2) is a mea-

sure of this deviation. Notice that the shock width is

two to four orders of magnitude larger than a molecu-

lar length. Thus a continuum description of the shock

problem is appropriate.

3. Deviations from Hydrodynamics

The density and temperature dependence of the

transport coefficients and of the other thermo-physical

properties within the shock front account for the devi-

ations of the shock width from linear hydrodynamics,

as observed in the strong shock waves of the NEMD

simulations [5] for simple fluids. This explanation

does not work for the relatively weaker shocks em-

ployed in the experiments with the molecular liquids

[2] to be analyzed here.

Additional sources for deviations from hydrody-

namics are modifications of the constitutive relations

between the thermodynamic forces and fluxes, gov-

ering the transport processes. Here emphasis is put

on the friction pressure tensor and the shear viscosity.

Similar modifications for the bulk viscosity and the

heat conductivity, which may well be of importance,

are not treated explicitly.

Consequences of two generalizations of the Max-

well relaxation equation for the friction pressure ten-

sor, associated with spatial inhomogeneities and with

terms nonlinear in the friction pressure, are presented.

3.1. Spatial Inhomogeneity

The friction pressure tensor n = p, i.e. the sym-

metric traceless part of the pressure tensor p (nega-

tive stress tensor) is assumed to obey the generalized

Maxwell relaxation equation

TM—TV + T'A"7T + 7T = —27/ Vv . (5)

The symbol tr? indicates the symmetric traceless part

of a tensor. The Maxwell relaxation time TM is related

to the (newtonian) shear viscosity 77 by 7/ = G TM,

where G is the high frequency shear modulus. In (5), A

is the Laplacian and £ stands for a characteristic length

which is considered as a model parameter. Arguments

for the derivation of this additional term are given

later. When spatial inhomogeneities of the friction

pressure tensor over the length i are small, the term

£2

A-7T

can be disregarded. Then (5) reduces to the time

honoured standard Maxwell relaxation equation.

\ 1 . 1 • 1 1 1 1 1 1 1 1

Oh i j

• \

i j

-Vv*

.V %.t. »...».

• >

•

1 1 1 , , . . . . . . . N

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216 S. Hess • On the Shock Front Thickness in Water and other Molecular Liquids

When the rise time tTlse = L/c$ of the pressure

in the shock front is large compared to the Maxwell

relaxation time TM, it suffices to analyze the stationary

solution

7T = —27] V V ~[2

A7T

. (6)

Considering the term involving the length d as a small

perturbation, one obtains, in lowest order in 02,

-2r] (\-£2A)Vv . (7)

The continuity equation, for a stationary state,

d(pvx)/dx = 0, implies

dvx/dx = -p vx(dp/dp)s dp/dx

= dp/dx.

(8)

Notice that cl = (dp/dp)s. Again in lowest order in

C2, one has

?2 Advjdx « ~(£/L)2dvx/dx. (9)

Thus the viscosity rj occuring in the expression (4)

which, in turn, determines the shock front thickness

L, can be replaced by the effective viscosity

Vetf = n (i +(t/L)2 (10)

Assuming, for simplicity, that all transport coeffi-

cients are modified in the same manner, the quantity

H, specifying the deviation of L from its hydrody-

namic value, is given by

H :=

r^ — fir

Lref Pref \Lref

—) -(H)

In Figs. 4 and 5, H = L dp, in units of Z/refPref» is

plotted versus (bp/pref)2 for water and methanol. The

points represent the same data as shown in Figs. 2

and 3. Within the experimental uncertainty, H in-

creases in direct proportion to (bp)2 in accord with

(11). From the slope of the dashed straight lines one

inferes « 0.3, « 0.08 for H20 and CH3OH,

respectively. Thus the characteristic length parame-

ter I, for water and methanol, is approximately equal

to 0.5 pm and 0.3 jam. Of course, with all the ap-

proximations made here, these numbers should just

be regarded as an estimate of the order of magnitude

2.4

_ 2.2

"35

CL 2

(3)

_T 1.8 1.8

1.6

1.4

1.2

0.8

:H2O . . . nr-j • y.

/ •

; • / ;

•

* /

•

;

" . , , , 1 1 1 1 . , , . iiii'

(5p/p,

ref

•

Fig. 4. Shock front thickness times pressure difference L bp

in water as function of the square of the pressure differ-

ence (bp)2, Lref - 1 Jim, pref ~ 70atm. Data from Eisen-

menger [2].

«8-

2,4

2,2

1,8

1,6

1,4

1,2

0,8

i i i j i i .

: CH Of

•,. • 111

.. >

... • i • T ft"

/ •

• •

• >

•

- •

i •. ,

1 I "

0 2 4 6 8 10 12 14 16

(SP'PJ2

Fig. 5. Shock front thickness times pressure difference L bp

in methanol as function of the square of the pressure differ-

ence (bp)2, Lref = 1 pref ~ 17atm. Data from Eisen-

menger [2],

Nevertheless, one is now faced with the question

"where does a characteristic length come from which

is several hundred to one thousand times larger than

the size of a molecule?".

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217 S. Hess • On the Shock Front Thickness in Water and other Molecular Liquids

3. 1.1 Origin of the length parameter £

Various causes can lead to an additional term in

the Maxwell relaxation equation which involve the

Laplacian. These are

1) diffusive processes,

2) energetic effects in analogy to Frank elasticity

of liquid crystals,

3) piezo-electric and piezo-tetradic couplings.

Cause 1), referred to as the "Burnett" contribution in

connection with gases [6], can be ruled out because

the characteristic length involved there is of the or-

der of a mean free path in gases, or of the order of

a molecular size in liquids [7] and, even more im-

portant, because of an opposite sign in front of t1

in (5). Burnett-like terms would imply a decrease of

the shock width with increasing driving pressure dif-

ference rather than the observed increase. The same

remark with regard to the sign applies to cause 2)

unless one makes the rather unlikely assumption that

a spatially inhomogeneous state of the friction pres-

sure tensor is energetically more favourable than a

homogeneous one. Cause 3), favoured here, deserves

some additional explanation. The relaxation equation

for the friction pressure tensor 7r is written in the form

a —

tm — 7T+

<P=

—2r] V v , (12)

where is the derivative of a Landau type poten-

tial function <P with respect to 7r. The simple choice

= (1 /2)7t : 7T, e. g. yields <P=

corresponding to

the standard Maxwell model. The above mentioned

piezo-electric and piezo-tetradic couplings mean the

use of the Landau potential

1 , , 1

# = x7r :

+ -a\d • d + -a3t:t

2 2 (13)

+ Ci tt : ('Vd) + C3Tr : (V • t).

Here the vector d and the third rank tensor t are the

dipolar and the tetradic order parameters associated

with the dipole moment and the tetrahedral coordina-

tion of the molecules. The quantities a

> 0, 0:3 > 0,

and are phenomenological coefficients. The lat-

ter ones, which may have either sign, characterize

the coupling between the friction pressure tensor, the

dipolar and the tetradic order parameters. The poten-

tial (13) implies

i=7T + Ci V d +£3 V • t. (14)

In analogy to (12) with (14), the dipolar order param-

eter obeys the relaxation equation

T\ ^-d + a\d - Ci V

• TV

= 0, (15)

where the relaxation time T\ is an additional phe-

nomenological coefficient. In a stationary situation,

one obtains

(16)

which underlies the piezo-electric effect. A similar

relation, now involving the ratio Q3/0:3, is found for

the tetradic order parameter. Insertion of these expres-

sions for d and t into (14) leads to terms involving

second spatial derivatives of the friction pressure ten-

sor which, for the geometry under consideration, are

equivalent to the Laplacian used in (5). Then one has,

apart from factors of the order of 1,

C1 C1

+ (17)

CK3

Hence the piezo-electric or piezo-tetradic coupling

yields the correct sign irrespective of the sign of £1

and £3. No estimate of the order of magnitude of

the coefficients occuring on the r.h.s. of (17) can be

given. The coefficients a 13 in the denominator of

(17) could become quite small, and consequently i1

rather large, due to collective effects as encountered in

pretransformational phenomena, e.g. in liquid crystals

[8], [9]. Consequences of the piezo couplings for the

sound absorption should be analyzed.

A test of the importance of the piezo effects ex-

pected in the polar liquids could be provided by a

comparison with data for liquids where such effects

do not exist due to the symmetry of the molecules.

Indeed, the only substance of that kind measured by

Eisenmenger [2], viz. liquid CCI4, does not show any

deviation from the preditions of linear hydrodynam-

ics. Since the range of driving pressure differences

was rather limited in that case, no definitive conclu-

sions can be drawn, however.

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218 S. Hess • On the Shock Front Thickness in Water and other Molecular Liquids

log|E|

Fig. 6. The relative deviation from hydrodynamics H ~

L 5p as function of the logarithm of the dimensionless ex-

tension rate E ~ (8p)2. Comparison with the experimental

data for water (Eisenmenger [2]) with curves of the full

nonlinear theory (full curves) and the simple approxima-

tion (dashed).

3.2. Nonlinearity

In general, the relaxation equation for the friction

pressure tensor 7r will also contain terms nonlinear

in 7T. Disregarding effects associated with spatial in-

homogeneities, the generalized Maxwell relaxation

equation

Tm "r-7T + . . . +

7T —

\/6 B 7T

•

°t (18)

+ CTT (3TT : Tr) = -2 R/ VV

has been used to study the nonliear flow behavior

for simple shear, planar biaxial and uniaxial flows

[4], The ellipses ... stand for terms involving prod-

ucts of components of the velocity gradient tensor

with 7v which, however, are disregarded here. De-

pending on the values of the model parameters B

and C, both shear thinning and shear thickening

were found. It should be mentioned that a slightly

different scaling of the variables, e.g. 7r expressed

in units of the high frequency shear modulus, was

used in [4]. Here, the nonlinear relaxation equation is

applied for the appropriate uniaxial extensional or

compressional flow. As before, the extension rate is

e = dvx/dx « -p-1 Cg 1 (8p/L) ~

—

(bp)2. In lowest

order in the nonlinearity (C = 0) one finds, for a

stationary situation,

H = I + ( . (19)

Lref Pref P CS Lref VPref/

Again, the deviation of H from its value 1 based on

linear hydrodynamics is directly proportional to (8p)2.

Thus the slope of the straight lines in Figs. 4 and 5

can be used to determine the coefficient B. For water

one finds B « 6 (atm)-1.

In Fig. 6 the experimental data of [2] for water,

in particular H ~ Lbp plotted versus the logarithm

of the dimensionless extension rate E ~ (bp)2, are

compared with curves of the full nonlinear theory

(full curves) where C/B2 = 0.52, 0.53, 0.54. The

agreement seems to be rather good. The data point

of Fig. 4 at the largest value of bp, however, is not

included in Figure 6. The dashed curve follows from

the simple approximation C = 0, which yields (19).

4. Concluding Remarks

Two mechanisms have been presented which can

account for the experimentally observed deviation of

the shock width from its (linear) hydrodynamic value.

For specific cases, the relevant new phenomenolog-

ical coefficients have been inferrred from the data

upon the assumption that one process dominates. To

predict the relative importance of these two contri-

butions, however, independent experimental data or

estimates from model calculations are needed for the

coefficients determining the magnitude of the effects

discussed here. Non-equilibrium molecular dynam-

ics (NEMD) computer simulations of shock waves in

molecular liquids could also be helpful to clarify this

issue.

A ckn owledgement

I thank W. Eisenmenger for having, a long time ago,

drawn my attention to his shock wave experiments

and for reminding me, over many years, at the Spring

meetings of the Deutsche Physikalische Gesellschaft

(DPG), that the observed deviations of the shock front

thickness from hydrodynamics requires a theoretical

explanation.

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219 S. Hess • On the Shock Front Thickness in Water and other Molecular Liquids

[1] L. D. Landau und E. M. Lifschitz, Lehrbuch der The-

oretischen Physik, Bd. VI, Hydrodynamik, Akade-

mie-Verlag, Berlin 1966, p. 394 ff.; Fluid Mechanics,

Pergamon Press, London 1959.

[2] W. Eisenmenger, Acustica 14, 187 (1964).

[3] 150 Jahre DPG, 59. Physikertagung Berlin 1995, Ver-

handl. DPG(IV) 30, 1115 (1995); EPS, 15th Gen-

eral Conference of the Condensed Mater Division,

Baveno-Stresa 1996, Europhysics Conference Ab-

stracts 20 A, 265 (1996).

[4] O. Hess and S. Hess, Physica A 207, 517 (1994); in:

StatPhys 19, ed. Hao Bailin, World Scientific, Singa-

pore 1996, p. 369.

[5] W. G. Hoover, Phys. Rev. Lett. 42, 1531 (1979); B. L.

Holian, W.G. Hoover, B. Moran, and G.K. Straub,

Phys. Rev. A 22, 2798 (1980); B.L. Holian, in: Mi-

croscopic Simulations of Complex Hydrodynamic

Phenomena, eds. M. Mareschal and B.L. Holian,

Plenum, New York 1992, p. 75.

[6] S. Chapman and T. G. Cowling, Mathematical Theory

of Nonuniform Gases, University Press, Cambridge

1964; L. Waldmann, in: Handbuch der Physik, Vol.

XII, ed. S. Flügge, Springer-Verlag, Berlin 1959.

[7] G. Schmidt, W. E. Köhler, and S. Hess, Z. Naturforsch.

36 a, 545 (1981).

[8] P. G. de Gennes, The Physics of Liquid Crystals,

Clarendon Press, Oxford 1974; H. Kelker and R. Hätz,

Handbook of Liquid Crystals, Verlag Chemie, Wein-

heim 1980; G. Vertogen and W. deJeu, Thermotropic

Liquid Crystals, Fundamentals, No. 45 in Chemical

Physics, Springer-Verlag, Berlin 1988.

[9] S. Hess, Z. Naturforsch. 30a, 728, 1224 (1975); 31a,

1034, 1507 (1976).

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