Page 294 - Intro to Tensor Calculus

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which can also be written in the alternative form

σ ij = −pδ ij + λ δ ij v k,k + µ (v i,j + v j,i ) (2.5.21)
∗
∗
involving the gradient of the velocity.
i
Upon transforming from a Cartesian coordinate system y ,i =1, 2, 3 to a more general system of
i
coordinates x ,i =1, 2, 3, we write
i
∂y ∂y j
σ mn = σ ij m n . (2.5.22)
∂x ∂x
Now using the divergence from equation (2.1.3) and substituting equation (2.5.21) into equation (2.5.22) we
obtain a more general expression for the constitutive equation. Performing the indicated substitutions there
results
i
∂y ∂y j
k
∗
∗
σ mn = −pδ ij + λ δ ij v + µ (v i,j + v j,i )
,k m n
∂x ∂x
σ mn = −pg + λ g v k + µ (v m,n + v n,m ).
∗
∗
mn mn ,k
i
Dropping the bar notation, the stress-velocity strain relationships in the general coordinates x ,i =1, 2, 3, is
ik
∗
σ mn = −pg mn + λ g mn g v i,k + µ (v m,n + v n,m ). (2.5.23)
∗
Summary
The basic equations which describe the motion of a Newtonian fluid are :
Continuity equation (Conservation of mass)
∂% i D%
~
+ %v =0, or + %∇· V =0 1 equation. (2.5.24)
∂t ,i Dt
i
i
Conservation of linear momentum σ ij + %b = %˙v , 3 equations
,j
~
DV
~
~
or in vector form % = %b + ∇· σ = %b −∇p + ∇·τ (2.5.25)
Dt
P 3 P 3 P 3 P 3
where σ = (−pδ ij + τ ij )ˆ i ˆ j and τ = j=1 ij ˆe i ˆe j are second order tensors. Conser-
τ
e
e
i=1 j=1 i=1
ji
vation of angular momentum σ ij = σ , (Reduces the set of equations (2.5.23) to 6 equations.) Rate of
deformation tensor (Velocity strain tensor)
1
D ij = (v i,j + v j,i ) , 6 equations. (2.5.26)
2
Constitutive equations
ik
∗
∗
σ mn = −pg mn + λ g mn g v i,k + µ (v m,n + v n,m ), 6 equations. (2.5.27)