Can material time derivative be objective?

T. Matolcsi Dep. of Applied Analysis, Institute of Mathematics,

Eötvös Roland University, Budapest, HungaryP. Ván

Theoretical Department, Institute of Particle and Nuclear Physics, Central Research Institute of Physics, Budapest, Hungary

– Introduction

– About objectivity

– Covariant derivatives

– Material time derivative

– Jaumann derivative, etc…

– Conclusions

Classical irreversible

thermodynamics

Local equilibrium (~ there is no microstructure)

Beyond local equilibrium:

Nonlocality in time (memory and inertia)

Nonlocality in space (structures)

constitutive space

Nonlocality in time (memory and inertia)

Nonlocality in space (structures)

constitutive space(weakly nonlocal)

Nonlocality in spacetime

Basic state space: a = (…..)

???

aa),...,( 2 aa n ),...,( )( naa

Rheology

Jaumann (1911)Oldroyd (1949, …)…

Kluitenberg (1962, …),Kluitenberg, Ciancio and Restuccia (1978, …)

Verhás (1977, …, 1998)Thermodynamic theory with co-rotational time derivatives.

Experimental proof and prediction: - viscometric functions of shear hysteresis- instability of the flow !!

Thermodynamic theory:

Material frame indifference

Noll (1958), Truesdell and Noll (1965)Müller (1972, …) (kinetic theory)Edelen and McLennan (1973)Bampi and Morro (1980)Ryskin (1985, …)Lebon and Boukary (1988)Massoudi (2002) (multiphase flow)Speziale (1981, …, 1998), (turbulence)Murdoch (1983, …, 2005) and Liu (2005)Muschik (1977, …, 1998), Muschik and Restuccia (2002)……..

Objectivity

About objectivity:

,ˆˆ jij

i cJc

xQhx )()(ˆ,ˆ tttt

3,2,1,0),(ˆˆ ixxx ii

QxQh 01

J

3,2,1,,ˆ,ˆ 00

xQhxxx

is a four dimensional objective vector, if

j

ii

j x

xJ

ˆˆ

),( 0 ccci

where

Noll (1958)

QcxQhcQxQhc 0

000

)(

01

ˆ

ˆ

c

ccc

Spec. 1: Qcc ˆ00c

Spec. 2: motion vrr )(t

rQrQhvrQrhr ˆˆˆ

))(,1())(,( ttt vr

vQxQhv

101

ˆ

1

is an objective four vector

Covariant derivatives:as the spacetime is flat there is a distinguished one.

kijk

ij

ij

ii

cccDDc

aaDDa

~

~ covector field

mixed tensor field

The coordinates of the covariant derivative of a vector field do not equal the partial derivatives of the vector field if the coordinatization is not linear.

If are inertial coordinates, the Christoffel symbol with respect to the coordinates has the form:

xx

k

l

j

m

lm

i

m

i

kj

mijk x

x

x

x

xx

x

x

x

xx

x

ˆˆ

ˆˆˆ

ˆ 22

xQhx )()(ˆ,ˆ tttt 3,2,1,,ˆ,ˆ 00

xQhxxx

,0,,0 000

jk

where is the angular velocity of the observerQQΩ 1

)))(())(( 1100 xΩΩΩhQxQhQ

Material time derivative:

is the point at time t of the integral curve V passing through x.)(xFt

t0 t

x Ft(x)

V(x)

Flow generated by a vector field V.

))(( xFt is the change of Φ along the integral curve.

t0 t

x Ft(x)

V(x)))(())(( xFDxFdt

dtVt

is the covariant

derivative of according to V.

DVDV

aVaDVaD jj

jj

V

aaaDV )( 0 v

!),1( vV

substantial time derivative

Spec. 1: is a scalar

)()( kijk

ij

jij

jiV ccVcDVcD

cΩvc )( 0 VD

The material time derivative of a vector – even if it is spacelike –is not given by the substantial time derivative.

Spec. 2: is a spacelike vector field),0( c c

Jaumann, upper convected, etc… derivatives:

In our formalism: ad-hoc rules to eliminate the Christoffel symbols. For example:

)(

)(ki

jki

jji

jj

kijk

ij

jij

j

VVcVDc

ccVcDV

ikj

ijk

ij

jij

jij

jij

j VccVVDccDV

for a spacelike vector ),0( c

vccv )( 0i

jji

jj VccV

upper convected (contravariant) time derivative

One can get similarly Jaumann, lower convected, etc…

Conclusions:

– Objectivity has to be extended to a four dimensional setting.

– Four dimensional covariant differentiation is fundamental in non-relativistic spacetime. The essential part of the Christoffel symbol is the angular velocity of the observer.

– Partial derivatives are not objective. A number of problems arise from this fact.

– Material time derivative can be defined uniquely. Its expression is different for fields of different tensorial order.

space + time ≠ spacetime

Thank you for your attention.