Page 293 - Intro to Tensor Calculus

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tensor components are

∂u ∂v ∂w ∂u ∂v
∗
τ xx =(λ +2µ ) + λ ∗ + τ yx = τ xy =µ ∗ +
∗
∂x ∂y ∂z ∂y ∂x
∂v ∂u ∂w ∂w ∂u
∗
∗
τ yy =(λ +2µ ) + λ ∗ + τ zx = τ xz =µ ∗ +
∂y ∂x ∂z ∂x ∂z

∂w ∂u ∂v ∂v ∂w
∗
τ zz =(λ +2µ ) + λ ∗ + τ zy = τ yz =µ ∗ +
∗
∂z ∂x ∂y ∂z ∂y
In cylindrical form, with velocity components v r ,v θ ,v z , the viscous stess tensor components are
∂v r
∗
τ rr =2µ ∗ + λ ∇· V ~
∂r ∗ 1 ∂v r ∂v θ v θ
τ θr = τ rθ =µ + −
1 ∂v θ v r r ∂θ ∂r r
∗
τ θθ =2µ ∗ + + λ ∇· ~ V
r ∂θ r ∂v r ∂v z
τ rz = τ zr =µ ∗ +
∂v z ∂z ∂r
∗
τ zz =2µ ∗ + λ ∇· V ~
∂z 1 ∂v z ∂v θ
τ zθ = τ θz =µ ∗ +
1 ∂ 1 ∂v θ ∂v z r ∂θ ∂z
~
where ∇· V = (rv r )+ +
r ∂r r ∂θ ∂z
In spherical coordinates, with velocity components v ρ ,v θ ,v φ , the viscous stress tensor components have the
form
∂v ρ
∗
τ ρρ =2µ ∗ + λ ∇· V ~
∂ρ ∂ v θ 1 ∂v ρ
τ ρθ = τ θρ =µ ∗ ρ +
1 ∂v θ v ρ ~ ∂ρ ρ ρ ∂θ
∗
τ θθ =2µ ∗ + + λ ∇· V
ρ ∂θ ρ 1 ∂v r ∂ v θ
τ φρ = τ ρφ =µ ∗ + ρ
1 v θ cot θ ρ sin θ ∂φ ∂ρ ρ
∗
τ φφ =2µ ∗ ∂v φ + v ρ + + λ ∇· V ~
ρ sin θ ∂φ ρ ρ sin θ ∂ v φ 1 ∂v θ
τ θφ = τ φθ =µ ∗ +
1 ∂ 1 ∂ 1 ∂v φ ρ ∂θ sin θ ρ sin θ ∂φ
~
2
where ∇· V = ρ v ρ + (sin θv θ )+
2
ρ ∂ρ ρ sin θ ∂θ ρ sin θ ∂φ
Note that the viscous stress tensor is a linear function of the rate of deformation tensor D ij . Such a
fluid is called a Newtonian fluid. In cases where the viscous stress tensor is a nonlinear function of D ij the
fluid is called non-Newtonian.
Definition: (Newtonian Fluid) If the viscous stress tensor τ ij
is expressible as a linear function of the rate of deformation tensor
D ij , the fluid is called a Newtonian fluid. Otherwise, the fluid is
called a non-Newtonian fluid.
Important note: Do not assume an arbitrary form for the constitutive equations unless there is ex-
perimental evidence to support your assumption. A constitutive equation is a very important step in the
modeling processes as it describes the material you are working with. One cannot arbitrarily assign a form
to the viscous stress and expect the mathematical equations to describe the correct fluid behavior. The form
of the viscous stress is an important part of the modeling process and by assigning different forms to the
viscous stress tensor then various types of materials can be modeled. We restrict our study in these notes
to Newtonian fluids.
In Cartesian coordinates the rate of deformation-stress constitutive equations for a Newtonian fluid can
be written as
∗ ∗
σ ij = −pδ ij + λ δ ij D kk +2µ D ij (2.5.20)