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A Vectors, Tensors, and Their Operations

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A Vectors, Tensors, and Their Operations A Vectors, Tensors, and Their Operations A.1 Vectors and Tensors A spatial description of the fluid motion is a geometrical description, of which the essence is to ensure that relevant physical quantities are invariant under artificially introduced coordinate systems. This is realized by tensor analysis (cf. Aris 1962). Here we introduce the concept of tensors in an informal way, through some important examples in fluid mechanics. A.1.1 Scalars and Vectors Scalars and vectors are geometric entities independent of the choice of coor- dinate systems. This independence is a necessary condition for an entity to represent some physical quantity. A scalar, say the fluid pressure p or den- sity ρ, obviously has such independence. For a vector, say the fluid velocity u, although its three components (u1,u2,u3) depend on the chosen coordi- nates, say Cartesian coordinates with unit basis vectors (e1, e2, e3), as a single geometric entity the one-form of ei, u = u1e1 + u2e2 + u3e3 = uiei,i=1, 2, 3, (A.1) has to be independent of the basis vectors. Note that Einstein’s convention has been used in the last expression of (A.1): unless stated otherwise, a repeated index always implies summation over the dimension of the space. The inner (scalar) and cross (vector) products of two vectors are familiar. If θ is the angle between the directions of a and b, these operations give a · b = |a||b| cos θ, |a × b| = |a||b| sin θ. While the inner product is a projection operation, the cross-product produces a vector perpendicular to both a and b with magnitude equal to the area of the parallelogram spanned by a and b.Thusa×b determines a vectorial area 694 A Vectors, Tensors, and Their Operations with unit vector n normal to the (a, b) plane, whose direction follows from a to b by the right-hand rule. In particular, the inner and cross-products of Cartesian basis vectors satisfy e · e = δ i j ij , (A.2a,b) ei × ej = ek, i,j,k =1, 2, 3 and cycles where 1if i = j, δ = ij 0if i = j is the Kroneker symbol. The gradient operator ∂ ∇ = ei (A.3) ∂xi is also a vector, which as a single entity is invariant under any coordinate transformations. Thus, the pressure gradient ∇p is a vector; the divergence and curl of velocity, ∇·u = ϑ, ∇×u = ω, are the dilatation scalar and vorticity vector, respectively. A.1.2 Tensors Scalar and vectors can be considered as special tensors of rank (or order) zero and one respectively. In general, as the immediate extension of vectors, in an n-dimensional Euclidean space atensorT of rank m is a geometric entity independent of the choice of coordinate systems, and has nm components with respect to a given coordinate system. When n = 3, like the one-form (A.1) for m a vector, these 3 components Tij...k constitute the coefficients of an m-form of the given base vectors, i.e., T = Tij...keiej ...ek, i,j,...,k =1, 2, 3, and obey certain transformation rule to keep the invariance of T.1 The neces- sity of introducing tensors of ranks higher than one can be illustrated by the following simple example. Assume the fluid velocity at a spatial point x is u, and consider the velocity change at any neighboring point x +dx, see Fig. A.1. To the first-order of dx, there is du =(dx ·∇)u =dxju,j , (A.4a) where dxj (j =1, 2, 3) are Cartesian coefficients of dx. Here and throughout the book we have used a simple notation (·),j to indicate the derivative with 1 In many books a tensor is defined by requiring their components to obey this transformation rule, which is the inverse of that of the basis vector, to ensure the invariance of the tensor. A.1Vectors and Tensors 695 du u(x) x u(xϩdx) dx xϩdx Fig. A.1. A schematic interpretation of (A.4a) respect to xj; sometimes we also denote ∂/∂xi by ∂i.Sou has three directional derivatives ∂u u,j ≡ ,j=1, 2, 3, ∂xj each being a vector. Thus we may further expand them as u,j = ekuk,j,j,k=1, 2, 3. (A.4b) Then, just as p,j is the jth component of pressure gradient vector ∇p,the vector u,j is the jth component of a geometric entity, velocity gradient, defined as ∇u = eju,j = ejekuk,j,j,k=1, 2, 3, (A.5) where summation is to be taken twice, once for j and once for k. This is a two-form of ei (i =1, 2, 3), an example of tensors of rank 2, can also be directly obtained from (A.1) and (A.3). Note that in (A.5) the index j implies the components of ∇ and goes first, while the index k implies the components of u and goes after j. The order inverse in uk,j is only apparent, because this component of ∇u is merely an abbreviation of ∂uk/∂xj. Since ∇u is independent of the magnitude and direction of dx, in comparison with (A.4a) it is a more general description of the velocity variation at the neighborhood of a point. In physical and numerical experiments one always uses certain coordinate systems, and the direct output is always components of tensors. Using com- ponent operations is also convenient in deriving formulas, and sometimes only one typical component is sufficient. For example, for the velocity gradient we may just write down a representative component uk,j. But writing the final equations in coordinate-independent tensorial form enables seeing the physical objectivity of these equations clearly. Any tensor of rank 2 can be expressed by a square matrix, but not vise versa. Like any square matrix, the deformation tensor, or generally any tensor of rank 2, can always be decomposed into a symmetric part and an anti symmetric part, each being a tensor of rank 2 ∇u = D + Ω, (A.6a) 696 A Vectors, Tensors, and Their Operations where 1 D ≡ (u + u )=D (A.6b) ij 2 j,i i,j ji is called the strain-rate tensor, having six independent components, and 1 Ω ≡ (u − u )=−Ω (A.6c) ij 2 j,i i,j ij is the spin tensor or vorticity tensor, having three independent components. As illustrated by (A.6a), the addition (as well as differentiation and inte- gration) of tensors is the same as that of vectors. The multiplication of tensors, in general, results in a tensor of higher rank. A new operation of tensors is contraction by summing a pair of components with the same index (“dummy index”). This reduces the rank by two. The simplest example is the inner product of vectors: a · b = aibi or ∇·u = ϑ. Similarly, for two tensors of rank 2 there is Aij Bjk = Cik. A.1.3 Unit Tensor and Permutation Tensor The most fundamental symmetric tensor of rank 2 is the unit tensor I ≡ eiejδij (A.7) of which the components in any Cartesian coordinates are invariably the Kro- necker delta. The contraction of two unit tensors gives δij δjk = δik or I · I = I. Making contraction again (denoted by double dots) yields I : I = δii = n (dimension of the space). Clearly, the contraction of two tensors of rank 2 is the same as the multipli- cation of two matrices. With a few exceptions in Sect. 3.2, all tensors we need in this book but one will be of ranks no more than 2. Like (A.2b), the cross-product of vectors in three-dimensional space can be expressed by (a × b)i = ei · (ejaj × ekbk)=ajbkei · (ej × ek)=ijkajbk, where ijk are the Cartesian components of a tensor of rank 3 E ≡ eiejekijk. (A.8a) A.1Vectors and Tensors 697 To find these components, we use the familiar rule of vector product e · e e · e e · e δ δ δ i 1 i 2 i 3 i1 i2 i3 ≡ · × · · · ijk ei (ej ek)= ej e1 ej e2 ej e3 = δj1 δj2 δj3 ek · e1 ek · e2 ek · e3 δk1 δk2 δk3 (A.8b) 1if(ijk) = (123), (231), (312), = −1if(ijk) = (132), (213), (321), 0. otherwise. Therefore, these components in any Cartesian coordinates are invariably the permutation symbol ijk, completely anti symmetric under exchange of any pair of its three indices. For this reason, tensor E is called permutation tensor. It is used frequently whenever there is a cross-product. If more than one cross- products appear, one need the multiplication of two permutation tensors. In a three-dimensional space, from (A.8b) there is δil δim δin ijklmn = δjl δjm δjn . (A.9a) δkl δkm δkn Contraction with respect to k,nyields ijklmk = δilδjm − δimδjl. (A.9b) Continuing the contraction with respect to j, m then gives ijkljk =(3− 1)δil =2δil, (A.9c) and continuing again ijkijk = 3(3 − 1) = 6. (A.9d) One often needs to handle two-dimensional flow. If a and b are vectors in theflowplanesuchthata × b is along the third direction e3 normal to the plane, then in ijkajbk one of the i, j, k must be 3 due to (A.8b), with the other two varying between 1 and 2. In this case (A.9b) yields δjm δjn 3jk3lm = , j,k,m,n=1, 2. (A.10a) δkm δkn Contracting with respect to k,nyields 3jk3mk =(2− 1)δjm = δjm, (A.10b) and continuing again, 3jk3jk = 2(2 − 1) = 2. (A.10c) 698 A Vectors, Tensors, and Their Operations The particular importance of the permutation tensor in vorticity and vortex dynamics lies in the fact that ωi = ijkuk,j = ijkΩjk, (A.11a) due to (A.3) and (A.6).
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