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Appendix: Fields of Vectors and Tensors Appendix: Fields of Vectors and Tensors A.1 Vectors and Tensors A spatial description of the flow field is nothing but 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 coordinate systems. This independence is a necessary condition for an entity to represent some physical quantity. A scalar, say the fluid pressure p or density ρ, obviously has such independence. For a vector, say the fluid velocity u, although its three components (u1, u2, u3) depend on the chosen coordinates, say Cartesian coordinates with unit basis vectors (e1, e2, e3), as a single geometric entity the one-form of ei , u = u1e1 + u2e2 + u3e3 = ui ei , (A.1.1) has to be independent of the basis vectors. Note that Einstein’s convention has been used in the last expression of (A.1.1): 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 a and b, the inner product generates a scalar a · b =|a||b| cos θ, (A.1.2) © Springer-Verlag Berlin Heidelberg 2015 J.-Z. Wu et al., Vortical Flows, DOI 10.1007/978-3-662-47061-9 405 406 Appendix: Fields of Vectors and Tensors which is a projection operation and changing θ to −θ does not alter the result. In contrast, the cross product of a and b generates a vector a × b = n(|a||b| sin θ), (A.1.3) where n is a unit vector perpendicular to both a and b, and |a||b| sin θ is the area of the parallelogram spanned by a and b. Thus a × b determines a vectorial area.The positive direction of n follows the right-hand rule from a to b. Since changing θ to −θ alters the sign of the result, we have a × b =−b × a. In particular, the inner and cross products of Cartesian basis vectors satisfy ei · e j = δij, (A.1.4a) ei × e j = ek, i, j, k = 1, 2, 3 and cycles, (A.1.4b) where 1ifi = j, δ = (A.1.5) ij 0if i = j is the Kroneker symbol. The gradient operator ∂ ∇=ei (A.1.6) ∂xi is also a vector, which as a single entity is invariant under any coordinate trans- formations. Thus, the pressure gradient ∇ p is a vector; the divergence and curl of velocity, ∇·u = ϑ, ∇×u = ω, (A.1.7) are the scalar dilatation and vector vorticity, respectively. If we shift to another coordinates system, the components of u must vary according to certain rule during the coordinate transformation to ensure the invariance of vector ( ) ( ) u. For example, consider two sets of Cartesian coordinates: S ei and S ei .Let · = . ei e j cij Then the basis vectors of S can be expressed in terms of those of S and vise versa: = , e j cijei (A.1.8a) = . ei cije j (A.1.8b) Since · = δ = · = δ , e j ek jk cijclkei el cijclk il Appendix: Fields of Vectors and Tensors 407 we have cljclk = δ jk, (A.1.9) i.e., the matrices C and C T are orthogonal: C T C = I . Then, we can expend any vector u in S and S. Because u is independent of coordinates, there must be = = = . u ei ui e j u j cije j ui Therefore, using (A.1.8b) we find = . u j cijui (A.1.10) { } { } { } Comparing (A.1.8a) and (A.1.10) indicates that as ei are transformed to ei , ui { } are transformed to ui by the same transformation rule as that of the basis vectors. Note that coordinate transformations can be either continuous or discrete. In the former cij vary continuously as a set of parameters, say the rotation angles of the basis vectors; while in the latter the direction of one of the basis vectors is reversed or one changes from right-handed to left-handed coordinate systems, like viewing the vectors through a mirror, and cij vary discretely. A true vector (or polar vector) is invariant under both continuous and discrete transformations, for example position vector x, velocity u, force F, and electric field E, etc. The force includes pressure force −∇ p, so the vector operator ∇ is also a true vector. On the other hand, a pseudo vector (or axial vector) is invariant only under con- tinuous transformation but changes sign under discrete transformation. This happens once a cross product is involved. For example, in (A.1.4b) the unit vector n of the area, perpendicular to both a and b, is an axial vector. The reason is that the positive direction of n has no objective definition but is just chosen by convention: if we turn a into b through the angle θ<180◦, then we require (a, b, n) form a right-hand triad. The direction of n will be reversed if we change the convention to the left-hand one, since b×a =−a×b. In other words, an axial vector is associated with a rotation, like the axis in which a screw with a right-hand thread will advance. Examples include angular velocity , torque x × F, and magnetic field B; after all the vorticity field ω =∇×u that concerns us most in this book. Note however the curl of a pseudo vector is a true vector. Moreover, the inner product of a true vector and a pseudo vector, say c · (a × b),mustbeapseudo scalar that changes sign as we change from right-handed to left-handed coordinate systems. A.1.2 Tensors A physical field often involves geometric entities more complicated than vectors. For example, let the fluid velocity at a spatial point x be u, and consider the velocity change at any neighboring point x + dx. To the first order of dx, there is 408 Appendix: Fields of Vectors and Tensors du = (dx ·∇)u = dxj u, j , where dxj ( j = 1, 2, 3) are Cartesian coefficients of dx.Herewehaveusedasimple notation (·), j to indicate the derivative with respect to x j .Sou has three directional derivatives ∂u u, j ≡ , j = 1, 2, 3, ∂x j each being a vector. Thus we may further expand them as u, j = ekuk, j , j = 1, 2, 3. Now, just as p, j is the jth component of pressure gradient vector ∇ p, the vector u, j can also be viewed as the jth component of a geometric entity called deformation tensor, which is defined as the operator ∇ directly acting to u and equals the sum of each e j times u, j . Consequently, we obtain ∇u = e j u, j = e j ekuk, j , (A.1.11) which can also be directly obtained from (A.1.1) and (A.1.6). In this geometric entity the basis vectors appear twice, and hence the summation is to be taken twice, once for j and once for k; so we call this entity a second-rank tensor. Note that in (A.1.11) 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, 1 because this component of ∇u is merely an abbreviation of ∂uk/∂x j . Since ∇u is independent of the magnitude and direction of dx, in comparison with (1.1.3)itisa more general description of the velocity variation at the neighborhood of a point and will appear in many later analyses. As another example of tensor of second rank, assume on an arbitrary surface element dS = ndS there is a pressure p > 0, where n is the unit normal vector of the surface. Then by definition p must act as a normal pressing force on the element, i.e., a surface stress t p =−np or {tpi }=−{ni p} per unit area. But if on dS there is a viscous force tv, it is generically not along n but pointing to another direction, yet still depending linearly on the orientation of n. Thus, in general we may write 2 {tvi }={n j Vji} with {Vji} being the 3 coefficients. Since tv and n are both geometric entities independent of the choice of coordinates, so must be V = e j ei Vji; it is called a tensor of second rank. Later we will see that the relation {tvi }={n j Vji} has a vector form tv = n · V as the “contraction” or inner product of n and V. Scalars and vectors can be considered as special tensors of rank (or order) zero and one respectively. In general, as the immediate extension of vectors, a tensor T of rank n is a geometric entity independent of the choice of a class of coordinate systems, and has 3n components with respect to a given coordinate in three-dimensional space. 1 The order of j, k is a matter of convention. A single convention has to be followed throughout the whole analysis. Appendix: Fields of Vectors and Tensors 409 n Like the one-form (A.1.1) for a vector, these 3 components Tij...k constitute the coefficients of an n-form of the given base vectors, i.e., T = Tij...kei e j ...ek. The transformation rule of vector components can be extended to higher-rank tensor.
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