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Appendix a Vectors, Tensors, and Their Notations Appendix A Vectors, Tensors, and Their Notations A.1 Scalar, Vector, and Tensor A physical quantity appears in this book is a scalar, vector, or tensor. A quantity expressed by a number, such as mass, volume, temperature, is a scalar. If a quantity a is expressed as a one-dimensional array of numbers, a = (a1, a2,...,aN ), where N is the dimension of the physical space on which a is considered, it is a vector, and ai (i = 1, 2,...,N) is called the ith component of vector a. The number N is three in this book. Displacement, velocity, and force are vectors. If a and b are vectors, a linear combination αa + βb,(αand β are scalars), (A.1) gives a vector. Sometimes, ai is used as a representative of vector a. A tensor is a multi-dimensional array of numbers. If P is an n-dimensional array of numbers, it is called an nth-order tensor. A scalar and a vector may be called a zeroth-order tensor and a first-order tensor, respectively. An example of a second- order tensor is a stress tensor in continuum mechanics. A second-order tensor can be expressed by a matrix. It is important to notice that a tensor is a multilinear functional of tensors.1 For example, a second-order tensor P(a, b), which is a functional of two first-order tensors (vectors) a and b, satisfies the bilinearity relations P(a1 + a2, b) = P(a1, b) + P(a2, b), (A.2) P(a, b1 + b2) = P(a, b1) + P(a, b2), (A.3) P(αa, b) = P(a,αb) = α P(a, b), (A.4) where α is a scalar. Remember that a stress tensor gives a stress if two unit vectors are specified, one determines the normal direction of a surface and another does the component of the stress acting on the surface. 1 A functional is a mapping of some functions to a number. S. Fujikawa et al., Vapor-Liquid Interfaces, Bubbles and Droplets, Heat and Mass 211 Transfer, DOI 10.1007/978-3-642-18038-5, C Springer-Verlag Berlin Heidelberg 2011 212 Appendix A If a tensor (scalar or vector) is a function of time t and position x,itiscalled a tensor field (scalar field or vector field). Usually, they are assumed to be contin- uously differentiable with respect to t and x. In fluid dynamics, a pressure field, velocity field, and stress field are, respectively, a scalar field, vector field, and tensor field, and all of them are continuously differentiable. Thus, the basic conservation laws of physics can be expressed in the forms of partial differential equations con- sisted of scalars, vectors, and tensors. It is essential for physics and its applications to engineering that the basic conservation laws expressed by scalars, vectors, and tensors are unchanged under the rotation of coordinate system with a fixed origin and the Galilean transformation. A.2 Einstein Summation Convention After a Cartesian coordinate system x = (x1, x2, x3) is specified, the components ai ’s (i = 1, 2, 3) of vector field a and the components Pij’s (i, j = 1, 2, 3) of a second-order tensor field P are determined.2 The expressions of mathematically complicated equations can often be made compact by using symbols like a and P instead of ai ’s and Pij’s. However, the numerical evaluations of vectors and tensors require the handling of their components. In the following, we summarize notations of some binary-product operations of vectors and tensors presented in component forms. The inner product (or scalar product) of a second-order tensor P and a vector a gives a vector b, i.e., b = P · a, (A.5) and this can be written as 3 bi = Pija j ,(i = 1, 2, 3). (A.6) j=1 According to the Einstein summation convention, we can eliminate the summation symbol to yield bi = Pija j ,(i = 1, 2, 3). (A.7) The Einstein summation convention is a rule of notation of binary-product oper- ations of vectors and tensors in a single term, which states that if a same index (subscript) appears twice in a single term, then the summation is taken from one to 2 The representations of vectors and tensors in component forms are possible in arbitrary curvilin- ear coordinate systems. Appendix A 213 three for the index in the term. The index is called a dummy index. Hereafter, we use the Einstein summation convention. The scalar product of two vectors, a and b, gives a scalar α, α = a · b = ai bi . (A.8) The dyadic of two vectors gives a second-order tensor P, P = ab = ai b j = Pij,(i, j = 1, 2, 3), (A.9) where, as usual, we do not distinguish a tensor P from its representative expression Pij, although the notation Pij is often used as the (i, j)th component of tensor P. The scalar product (or contraction) of two tensors T and U, denoted by T : U,gives a scalar α, α = T : U = TijUij. (A.10) The gradient of a scalar field f is a vector, and can be expressed as ∂ f grad f = ∇ f = ,(i = 1, 2, 3). (A.11) ∂xi The divergence of a vector field v is a scalar expressed as ∂v div v = ∇ · v = i . (A.12) ∂xi The strain rate tensor3 in fluid dynamics, ε, can be constructed by the dyadic of vectors ∇ and v as4 ∂v ∂v 1 T 1 i j ε = ∇v + (∇v) = + = εij,(i, j = 1, 2, 3), (A.13) 2 2 ∂x j ∂xi where the superscript T denotes the transpose of matrix. Here, to simplify the notation further, we can indicate the differentiation with respect to xi by index i after an index denoting a component of vector or tensor with a comma separating the two indices. That is, 1 # $ grad f = f,i , div v = vi,i , ε = vi, j + v j,i , (A.14) 2 3 It is sometimes called the rate-of-strain tensor or rate-of-deformation tensor. 4 Precisely, ∇ is not a vector because it is not an array of numbers but an array of differential operators, ∇ = (∂/∂x1,∂/∂x2,∂/∂x3). 214 Appendix A where since f is a scalar, no index appears before the comma before index i indi- cating the differentiation with respect to xi . The Einstein summation convention is also applied to this type of simplified notation as shown in the second equation in Eq. (A.14). The Kronecker delta δij is a representation of the second-order identity tensor I, given by 1ifi = j, δij = (A.15) 0 otherwise. The identity transformation from a vector a to a vector b can be written with the Kronecker delta as b = I · a = δija j = ai ,(i = 1, 2, 3). (A.16) The Eddington epsilon ijk defined by ⎧ ⎪ ( , , ) = ( , , ), ( , , ), ( , , ), ⎨⎪ 1 i j k 1 2 3 2 3 1 3 1 2 = − ( , , ) = ( , , ), ( , , ), ( , , ), ijk ⎪ 1 i j k 3 2 1 2 1 3 1 3 2 (A.17) ⎩⎪ 0 i = j or j = k or k = i, is the third-order alternating unit tensor. A vector product of two vectors and curl operation to vector field v can be expressed as c = a × b = ijka j bk = ci ,(i = 1, 2, 3), (A.18) ∂vk curl v = ∇ × v = ijk = ijkvk, j ,(i = 1, 2, 3). (A.19) ∂x j Several relations involving δij and ijk are useful in manipulations of vectors and tensors: δijδij = 3,ijkijk = 6, (A.20) ijkhjk = 2δih, (A.21) ijkmnk = δimδ jn − δinδ jm. (A.22) A second-order tensor P is called symmetric, if P = P T ,or Pij = Pji,(i, j = 1, 2, 3). (A.23) Clearly, the strain rate tensor ε and the Kronecker delta are the symmetric second- order tensors. Appendix B Equations in Fluid Dynamics B.1 Conservation Equations Let the macroscopic variables be defined everywhere in a space filled with a fluid, and let them be continuously differentiable functions of time t and position x.The macroscopic variables that should be defined at this stage are the density ρ, veloc- ity v, internal energy per unit mass e, stress tensor P, and heat flux q. Then, the conservation equations of mass, momentum and energy of the fluid, in general, are respectively written as ∂ρ + ∇ · (ρv) = 0, (B.1) ∂t ∂ρv + ∇ · (ρvv + P) = ρb, (B.2) ∂t ∂ 1 1 ρv2 + ρe + ∇ · ρv2 + ρe v + v · P + q ∂t 2 2 = ρb · v + ρS, (B.3) where b is a body force exerted on the fluid per unit mass, 1 ρv2 + ρe (B.4) 2 is the total energy of the fluid per unit volume, and S is a heat generated in the fluid per unit mass and per unit time. The body force b and heat generation S are independent of the motion of fluid and prescribed by some other rules. Equations (B.1), (B.2), and (B.3) are the most fundamental equations in fluid dynamics, and can be derived, for example, by considering the conservation of mass, momentum, and energy in a volume element in the physical space without specifying the explicit forms of P and q. Furthermore, the relation between the density ρ and the internal energy e is not necessary for the derivation of Eqs. (B.1), 215 216 Appendix B (B.2), and (B.3).1 Clearly, the number of unknown variables in Eqs. (B.1), (B.2), and (B.3) exceeds the number of Eqs. (B.1), (B.2), and (B.3), and therefore we have to add some equations.
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