can material time derivative be objective? t. matolcsi dep. of applied analysis, institute of...
Post on 29-Mar-2015
216 Views
Preview:
TRANSCRIPT
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.
top related