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Appendix
Elements of Tensor Calculus
A. Vector and Tensor Designations
The following tensor designations are used in the book:
a tensor of zero rank (scalar),
a ðakÞ tensor of first rank (vector),
AðAkjÞ tensor of second rank,
UðdkjÞ unit tensor (dkj — Kronecker symbol),
J ðJijkÞ tensor of third rank.
Simmetric and Antisymmetric Tensors
Simmetric and antisymmetric tensors are defined as follows:
simmetric:
A¼Atransp ðA ¼ A Þ (A.1)
kj jk
antisymmetric:
A¼Atransp ðA ¼A Þ (A.2)
kj jk
Trace of tensor is defined as the sum of its diagonal elements:
SpA¼XAkk (A.3)
k
M.Y. Marov and A.V. Kolesnichenko, Turbulence and Self-Organization: 607
Modeling Astrophysical Objects, Astrophysics and Space Science Library 389,
DOI10.1007/978-1-4614-5155-6, # Springer Science+Business Media New York 2013
608 Elements of Tensor Calculus
Scalar and Tensor (Internal) Product
Scalar product:
of two vectors ab¼Pakbk ðÞscalar
k
of vector and tensor Ab¼PAjkbk ðÞvector
k
of tensor and vector bA¼PbkAkj ðÞvector (A.4)
k
of two tensors AB¼PAjkBki ðÞtensor
k
Dual scalar product oftensors:
A:B¼XAjkBkj ðscalarÞ (A.5)
k;j
Internal (dyad) tensor product:
of two vectors ðabÞjk ¼ ajbk ðÞtensor of second rank
of vector and tensor ðaBÞijk ¼ aiBjk ðÞtensor of third rank
(A.6)
of tensor and vector ðBaÞijk ¼ Bijak ðÞtensor of third rank
of two tensors ðABÞijkl ¼ AijBkl ðÞtensor of fourth rank :
Vector product of twovectors and tensor andvector:
ða bÞk ¼ Xeijkaibj ðvectorÞ; ðBaÞik ¼ XejklBijal ðtensorÞ (A.7)
i;j j;l
where symbol of permutation eijk takes the values:
8þ1under even permutation of indexes ði:e: 123; 231; 312Þ
>
eijk ¼ <1 under odd permutationof indexes ði:e: 321; 132; 213Þ (A.8)
>
:0underrecurring indexes:
B. Cylindrical Coordinates
Expressions for the different operators used in the equations of heterogeneous
mechanics are represented here, for the convenience, in the cylindrical coordinate
system r;’;z (in axisymmetric case, @=@’ ¼ 0). Here are the expressions of
operators acting on:
Elements of Tensor Calculus 609
(1) scalars:
dB@BþurB¼@Bþu @Bþu @B; rB¼i @Bþi @B (B.1)
dt @t @t r @r z @z r @r z @z
1 @ @B @2B
r2B¼ r þ ; (B.2)
r @r @r @z2
(2) vectors:
rA¼1@ðrArÞþ@Az; (B.3)
r @r @z
@A @A @A A A @A @A @A
rA¼ii r þi i ’ þi i z þ i i ’ þi i r þ i i r þ i i ’ þi i z ;
r r @r r ’ @r r z @r ’ r r ’ ’ r z r @z z ’ @z z z @z
(B.4)
(3) dyads:
P ¼i i P þi i P þi i P þi i P þi i P þi i P þi i P þii P þiiP ; (B.5)
r r rr r ’ r’ r z rz ’ r ’r ’ ’ ’’ ’ z ’z z r zr z ’ z’ z z zz
1 @ðrP Þ @P P’’
rP¼i rr zr
r r @r @z r
1 @ðrP Þ @P P 1 @ðrP Þ @P
þi r’ þ z’ þ ’r þi rz þ zz : (B.6)
’ r @r @z r z r @r @z
Then for strain tensor and strain velocity tensor we obtain, respectively:
1
D ruþðruÞtransp
2
¼i i @ur þi i 1 @u’ u’ þi i 1 @uz þ@ur
r r @r r ’ 2 @r r r z 2 @r @z
þi i 1 @u’u’ þi i ur þi i 1 @u’
’ r 2 @r r ’ ’ r ’ z 2 @z
þii 1 @uzþ@ur þi i 1 @u’þi i @uz; (B.7)
z r 2 @r @z z ’ 2 @z z z @z
D1 ruþðruÞtransp 1Iru
2 3
¼i i @ur 1Iru þi i 1 @u’u’ þi i 1 @uzþ@ur
r r @r 3 r ’ 2 @r r r z 2 @r @z
(B.8)
þi i 1 @u’ u’ þi i ur 1Iru þi i 1 @u’
’ r 2 @r r ’ ’ r 3 ’ z 2 @z
þi i 1 @uz þ@ur þi i 1 @u’ þi i @uz 1Iru :
z r 2 @r @z z ’ 2 @z z z @z 3
610 Elements of Tensor Calculus
Double-point Gibbs multiplication serves as operator widely used in hydrody-
namics. If, following the Gibbs notations,a;b;c;dare arbitrary vectors, thenab : cd
¼ðacÞðbdÞ. In particular, for unit vectors we have:
i i : i i ¼i iÞði i Þ¼d d ; (B.9)
j k l m j l k m jl km
and for two dyads we have:
! !
ð1Þ ð2Þ XX ð1Þ XX ð2Þ
D :D ¼ ijikD : ilimD
jk lm
j k l m
¼XXXXdd Dð1ÞDð2Þ¼XXDð1ÞDð2Þ; (B.10)
jl km jk lm jk jk
j k l m j k
or
2 2 2 2 2 2 2 2
2D:D¼2D þ2D þD þ4D þ4D þ4D ðruÞ : (B.11)
rr ’’ zz r’ rz z’ 3
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