Fluid Dynamics I - Fall 2017 Tensor Algebra and Calculus for Fluid Dynamics

Fluid dynamics quantities and equations are naturally described in terms of tensors. We’ll make precise later what makes something a tensor, but for now, it suﬃces that scalars are zeroth order tensors (rank 0 tensors), vectors are ﬁrst order tensors (rank 1 tensors), and square matrices may be thought of as second-order tensors (rank two tensors). The notes that follow consider concepts and operations with tensors expressed in terms of Cartesian coordinates (x, y, z) = (x1, x2, x3). T The standard Cartesian basic vectors are ei, i = 1, 2, 3 where for example, e1 = (1, 0, 0) .A vector v may be represented in terms of this basis as the linear combination

v = v1e1 + v2e2 + v3e3. (1)

The scalars v1, v2, v3 are the coordinates of v with respect to the standard Cartesian basis. The expression for v above can also be written

3 X v = vjej = vjej (2) j=1

The second term in this equation uses Σ to indicate summation over the indicated range of the index j. The third term uses the summation convention (or Einstein convention) that when an index is repeated in a term, summation over that index is implied and the range of the index has to be inferred from context. Summation convention is concise, but if you are ever confused, just resort to the use of explicit Σ symbols.

In an equation, e.g., 3 X ai = bi + cij j=1 j is a dummy index used for the summation, while i is a free index. The same free indices must occur in every term of an equation.

Deﬁnition: The Kronecker delta is: 1, if i = j δ = i, j = 1, 2, 3. ij 0, if i 6= j

Deﬁnition:A permutation of a list of integers is a listing of these integers in another order, e.g. 312 instead of 123. A transposition is a permutation in which two adjacent integers are interchanged, e.g. 312 → 132. An even permutation is a permutation that can be achieved in an even number of transpositions. An odd permutation is a permutation that can be achieved in an odd number of transpositions.

Deﬁnition: The Permutation Symbol is:   0, if and two of i, j, k are equal ijk = 1, when ijk is an even permutation of 123  −1, when ijk is an odd permutation of 123

1 Remark: Note that transposing two indices in ijk just changes its sign, e.g., jik = −ijk.

Proposition: δii = 3, δijδij = 3, ijkijk = 6.

If x is a vector with coordinates xi, then

∂xi = δij. ∂xj

An important relationship between the Kronecker delta and the permutation symbol is

Proposition:

ijkpqk = δip δjq − δjp δiq . |{z} |{z} |{z} |{z} first second inner outer Vector operations such as dot product and cross product are deﬁned in terms of the basis vectors

ei · ej = δij ei × ej = ijkek. Using these deﬁnitions we ﬁnd that

u · v = uiei · vjej = uivjδij = uivi and u × v = ijkuivjek. A second order tensor T represents a linear function that takes a vector as input and gives a vector as output. It can be represented as a linear combination of the nine second order tensors: T eiej (outer product) i = 1, 2, 3, j = 1, 2, 3. The same quantity is sometimes written in dyadic form T eiej = eiej = ei ⊗ ej. Note that, for example, 1 0 0 1 e1e3 = 0 0 0 1 = 0 0 0 0 0 0 0

Expressed in terms of this basis, T = Tijeiej = Tijei ⊗ ej. The numbers Tij are the components of the tensor T with respect to the Cartesian basis eiej, i = 1, 2, 3, j = 1, 2, 3.

There are three basic operations on 1st amd 2nd order tensors. The inner or double inner products (·, :) are contractions and reduce the rank of tensors while the tensor product or outer product (⊗) increases the rank of tensors. Let u, v be vectors and S, T be second order tensors. Then we have

u · v = uiei · vjej = uivjδij = uivi (rank 0) u · T = uiei · Tjkejek = uiTjkei · ejek = uiTjkδijek = uiTikek (rank 1) T · u = Tjkejek · uiei = uiTjkejek · ei = uiTjkδikej = uiTjiej (rank 1) S · T = Tijeiej · Sklekel = TijSkleiδjkel = TijSjleiel (rank 2) u ⊗ v = uivjeiej (rank 2) u ⊗ T = uiei ⊗ Tjkejek = uiTjkeiejek (rank 3) T : S = TijSij (rank 0)

2 Remark: The tensors e = e ⊗ e form a unit basis for second rank tensors with respect to the ij i j inner product :. 1 0 0 0 0 0 0 0 0 0 0 0 , 1 0 0 , ..., 0 0 0 . 0 0 0 0 0 0 0 0 1 The set of 3 × 3 second rank tensors is a nine-dimensional vector space.

Remark: There is no analog of the cross product for tensors other than two ﬁrst rank ones.

Deﬁnition: A tensor T is symmetric if Tij = Tji and anti-symmetric (or skew-symmetric) if Tij = −Tji.

Deﬁnition: A tensor is isotropic if its components do not change for any rotation of the coordinate system.

Remark: All rank 0 tensors are isotropic. No rank 1 tensors are isotropic.

Proposition: The only isotropic tensors of rank 2 and 4 are of the form

αδij rank 2 αδijδpq + β(δipδjq + δiqδjp) + γ(δipδjq − δiqδjp) rank 4.

The permutation symbol ijk is useful for expressing the determinant of a 3 × 3 matrix A:

det(A) = ijka1ia2ja3k = (a1 · (a2 × a3))

th where ai is the i column of A.

Derivatives of Tensors: For tensors which are functions of spatial location x and/or time t, we can deﬁne derivatives. For now, we restrict ourselves to Cartesian coordinate systems.

Definition: The del or nabla operator is ∂ ∇ = ei = ei∂xi = ei∂i. ∂xi Definition: The gradient of a tensor field A of any rank is

∇A = ∇ ⊗ A.

Note that ∇A is one rank higher than A. For example, if φ(x) is a scalar ﬁeld,

∂ ∇φ = ek φ = (∂kφ)ek. ∂xk

∂φ It is sometimes convenient to write this as ∂x. If u(x) is a vector ﬁeld, then

∇u(x) = (ei∂i)(ujej) = (∂iuj)eiej.

3 Definition: The divergence of a tensor field A of any rank is divA = ∇ · A. Note that divA is one rank lower than A. For example, if u(x) is a vector field and T (x) is a second-order tensor field, then

div(u) = (ei∂i) · (ujej) = (∂iuj)ei · ej = (∂iuj)δij = ∂iui and ∂Tik div(T ) = (ei∂i) · Tjkejek = (∂iTjk)δijek = (∂iTik)ek = ek. ∂xi Deﬁnition: The curl of a vector ﬁeld u(x) is curl(u) = ∇×u = ∂mem×unen = ∂munem×en = ∂munijk(em)i(en)jek = ∂munijkδmiδnjek = ∂iujijkek.

Deﬁnition: A vector ﬁeld u(x) is irrotational if ∇ × u(x) = 0 for all x.

Deﬁnition: The Laplacian operator is the composition of the divergence and the gradient operators ∆ = ∇2 = ∇ · ∇. For a scalar ﬁeld φ(x),

∆φ = ∇ · ∇φ = (ei∂i) · (∂jφej) = (∂i∂jφ)ei · ej = ∂i∂iφ = ∂iiφ. For a vector ﬁeld u(x),

∆u = (∂iei) · (∂jej)(ukek) = (∂i)(∂j)δij(ukek) = ∂i∂iukek. th 2 T It is the vector with k component equal to ∂i uk, the Laplacian of uk. ∆u = (∆u1, ∆u2, ∆u3) .

Proposition: Let φ be a scalar ﬁeld and u be a vector ﬁeld. Then ∇ · (φu) = ∇φ · u + φ∇ · u, ∇ × (∇φ) = 0, ∇ · (∇ × u) = 0, ∇ × (∇ × u) = ∇(∇ · u) − ∆u.

The standard divergence theorem from calculus applies to vector ﬁelds. The theorem is

Theorem: The divergence theorem states that if n is the outward unit normal vector to the surface S enclosing the volume V , and u is a vector (that is, a ﬁrst rank tensor), then Z Z n · udS = ∇ · udV. S V This theorem can be extended to second rank tensor ﬁelds:

Theorem: The (general) divergence theorem states that if n is the outward unit normal vector to the surface S enclosing the volume V , and T is a second rank tensor, then Z Z n · T dS = ∇ · T dV. S V Note that the order matters in the general divergence theorem.