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APPENDIX C VECTORS AND TENSORS C.1 Vectors The term scalar refers to a single real number used to describe the magnitude of a quantity. Pressure, temperature, and internal energy are all scalar. A vector is defined as an entity that possesses both magnitude and direction or as a directed line segment subject to the parallelogram law of addition. In three-dimensional space, a vector may be specified by three vector components. A unit vector is a vector whose magnitude is unity. A vector in a three-dimensional Cartesian coordinate system can be expressed as A = iAx + jAy + kAz (C.1) where i, j, and k are unit vectors in the x-, y-, and z-directions. The vector components in the x-, y-, and z-directions are Ax, Ay, and Az, respectively. A vector in a Cartesian coordinate system is shown in Fig. C.1. It can be seen that the magnitude of the vector is 2 2 A = Ax + Ay + Az2 (C.2) Figure C.1 Vector and its components. 1006 Transport Phenomena in Multiphase Systems and that its three components are the projections of the vector on the x, y, and zaxes. A vector can also be represented in matrix form: ⎡ Ax ⎤ A = ⎢ Ay ⎥ (C.3) ⎢⎥ ⎢ Az ⎥ ⎣⎦ The dot product (also referred to as the scalar or inner product) of two vectors A and B is a scalar; it is obtained by summation of the product of each of their corresponding components, i.e., A ⋅ B = Ax Bx + Ay By + Az Bz (C.4) The cross product (or vector product) of two vectors A and B is a vector, i.e., i jk A × B = Ax Bx Ay By Az Bz (C.5) = i Ay Bz − Az By + j ( Az Bx − Ax Bz ) + k Ax By − Ay Bx ( ) ( ) C.2 Operations with the ∇ Operator C.2.1 Cartesian Coordinate System An important vector for fluid mechanics and heat transfer is the ∇ operator, which is defined as ∂ ∂ ∂ ∇ =i + j +k (C.6) ∂x ∂y ∂z in a three-dimensional Cartesian coordinate system. It can be applied to a scalar function, φ , to obtain its gradient, ∂φ ∂φ ∂φ + j +k grad φ = ∇φ = i (C.7) ∂x ∂y ∂z which is a vector. The ∇ operator can also be applied to a vector function such as velocity, V = iu + jv + kw (C.8) to get its divergence: ∂u ∂v ∂w ++ (C.9) div V = ∇ ⋅ V = ∂x ∂y ∂z or its curl: ⎛ ∂w ∂v ⎞ ⎛ ∂u ∂w ⎞ ⎛ ∂v ∂u ⎞ curl V = ∇ × V = i ⎜ − ⎟ + j⎜ − (C.10) ⎟+k⎜ − ⎟ ⎝ ∂y ∂z ⎠ ⎝ ∂z ∂x ⎠ ⎝ ∂x ∂y ⎠ Appendix C Vectors and Tensors 1007 Another related operator that is very useful in fluid mechanics and heat transfer is the Laplacian operator, defined as ∂2 ∂2 ∂2 ∇2 = ∇ ⋅ ∇ = 2 + 2 + 2 (C.11) ∂x ∂y ∂z Application of the Laplace operator to a scalar function φ results in ∂ 2φ ∂ 2φ ∂ 2φ + + (C.12) ∂x 2 ∂y 2 ∂z 2 Forming the dot product of the velocity and the gradient of a scalar φ results in a scalar: ∂φ ∂φ ∂φ V ⋅ ∇φ = u +v +w (C.13) ∂x ∂y ∂z which is used to describe the advection of the property φ . The operation ∂ ⎛ ∂φ ⎞ ∂ ⎛ ∂φ ⎞ ∂ ⎛ ∂φ ⎞ ∇ ⋅ α∇φ = ⎜ α (C.14) ⎟ + ⎜α ⎟ + ⎜α ⎟ ∂x ⎝ ∂x ⎠ ∂y ⎝ ∂y ⎠ ∂z ⎝ ∂z ⎠ results in a scalar that describes the diffusion of the property φ . Application of the Laplace operator to a velocity vector, V, results in a vector: ∇ 2 V = i∇ 2u + j∇ 2 v + k∇ 2 w (C.15) ∇ 2φ = C.2.2 Cylindrical Coordinate System The cylindrical coordinate system (r, ϕ , z) shown in Fig. C.2 is related to Cartesian coordinates (x, y, z) by x = r cos ϕ y = r sin ϕ z = z (C.16) The velocity vector, V, in a cylindrical coordinate system has three components, i.e., V = k rVr + k ϕVϕ + k zVz (C.17) where k r , k ϕ and k z are the unit vectors in r-, ϕ -, and z-directions, respectively. The operations involving the ∇ operator and either the general scalar function φ or the vector V in a cylindrical coordinate system are summarized below: ∂φ 1 ∂φ ∂φ grad φ = ∇φ = k r + kϕ + kz (C.18) ∂r ∂z r ∂ϕ 1 ∂ (rVr ) 1 ∂Vϕ ∂Vz (C.19) div V = ∇ ⋅ V = + + r ∂r r ∂ϕ ∂z 1008 Transport Phenomena in Multiphase Systems Figure C.2 Relationship between cylindrical and Cartesian coordinates. ∇ 2φ = 1 ∂ ⎛ ∂φ ⎞ 1 ∂ 2φ ∂ 2φ + ⎜r ⎟ + r ∂r ⎝ ∂r ⎠ r 2 ∂ϕ 2 ∂z 2 ∂φ Vϕ ∂φ ∂φ + + Vz V ⋅ ∇φ = Vr r ∂ϕ ∂r ∂z (C.20) (C.21) (C.22) ∇ ⋅ α∇φ = 1 ∂ ⎛ ∂φ ⎞ 1 ∂ ⎛ α ∂φ ⎞ ∂ ⎛ ∂φ ⎞ ⎜ ⎟ + ⎜α ⎜ rα ⎟+ ⎟ r ∂r ⎝ ∂r ⎠ r ∂ϕ ⎝ r ∂ϕ ⎠ ∂z ⎝ ∂z ⎠ 2 2 ⎧ ∂ ⎡1 ∂ ⎪ ⎪ ⎤ 1 ∂ Vr 2 ∂Vϕ ∂ Vr ⎫ ∇2V = k r ⎨ ⎢ rVr ) ⎥ + 2 −2 + 2⎬ ( 2 r ∂ϕ ∂z ⎪ ⎦ r ∂ϕ ⎪ ∂r ⎣ r ∂r ⎩ ⎭ ⎧ ∂ 2V ∂ 2V ⎫ ⎪ ∂ ⎡1 ∂ +k ϕ ⎨ ⎢ ( rVϕ )⎤ + r12 ∂ϕ 2ϕ + r22 ∂Vr + ∂z 2ϕ ⎪ ⎬ ⎥ ∂ϕ ⎦ ⎪ ∂r ⎣ r ∂r ⎪ ⎩ ⎭ ⎧ ⎪ 1 ∂ ⎛ ∂Vz +k z ⎨ ⎜r ⎪ r ∂r ⎝ ∂r ⎩ 2 2 ⎫ ⎞ 1 ∂ Vz ∂ Vz ⎪ +2 + 2⎬ ⎟ 2 ∂z ⎪ ⎠ r ∂ϕ ⎭ (C.23) C.2.3 Spherical Coordinate System The spherical coordinate system (r, θ , ϕ ) shown in Fig. C.3 is related to the Cartesian coordinate system by x = r sin θ cos ϕ y = r sin θ sin ϕ z = r cos θ (C.24) A velocity vector, V, has three components, i.e., Appendix C Vectors and Tensors 1009 Figure C.3 Relationship between spherical and Cartesian coordinates. V = k rVr + kθ Vθ + k ϕVϕ (C.25) where k r , k θ , and k ϕ are the unit vectors in r, θ , and ϕ directions, respectively. The operations involving the ∇ operator and either the general scalar function φ or the vector V in a spherical coordinate system are summarized below: ∂φ 1 ∂φ 1 ∂φ (C.26) grad φ = ∇φ = k r + kθ + kϕ ∂r r ∂θ r sin θ ∂ϕ 1 ∂ (r 2Vr ) 1 ∂ (Vθ sin θ ) 1 ∂Vϕ div V = ∇ ⋅ V = 2 + + (C.27) ∂r r sin θ ∂θ r sin θ ∂ϕ r ∇ 2φ = 1 ∂ ⎛ 2 ∂φ ⎞ 1 ∂⎛ ∂φ ⎞ 1 ∂ 2φ r sin θ +2 +2 2 ⎜ ⎟ ⎜ ⎟ ∂θ ⎠ r sin θ ∂ϕ 2 r 2 ∂r ⎝ ∂r ⎠ r sin θ ∂θ ⎝ Vϕ ∂φ ∂φ Vθ ∂φ V ⋅ ∇φ = Vr + + ∂r r ∂θ r sin θ ∂ϕ 1 ∂⎛ ∂φ ⎞ 1 ∂⎡ ∂φ ⎤ ∇ ⋅ α∇φ = 2 ⎜ r 2α ⎟+ 2 ⎢( sin θ ) α ∂θ ⎥ ∂r ⎠ r sin θ ∂θ ⎣ r ∂r ⎝ ⎦ + 1 ∂ ⎛ ∂φ ⎞ ⎜α ⎟ 2 r sin θ ∂ϕ ⎝ ∂ϕ ⎠ 2 (C.28) (C.29) (C.30) 1010 Transport Phenomena in Multiphase Systems ⎧ 2V cot θ 2V 2 ∂V 2 ∂Vϕ ⎫ ∇ 2 V = k r ⎨∇ 2Vr − 2r − 2 θ − θ 2 −2 ⎬ r r ∂θ r r sin θ ∂ϕ ⎭ ⎩ ⎧ V 2 ∂V 2cos θ ∂Vϕ ⎫ +kθ ⎨∇ 2Vθ + 2 r − 2 θ − 2 2 ⎬ r ∂θ r sin θ r sin θ ∂ϕ ⎭ ⎩ Vϕ ⎧ 2 ∂Vr 2cos θ ∂Vθ ⎫ +k z ⎨∇ 2Vφ − 2 2 + 2 +2 2 ⎬ r sin θ r sin θ ∂ϕ r sin θ ∂ϕ ⎭ ⎩ (C.31) C.3 Tensors A tensor of rank n in the Cartesian coordinate system has 3n components. A scalar can be considered as a tensor of rank 0m because it has only one component. A vector is a tensor of rank 1 since it has 3 components. As was demonstrated in Section 1.3.1, the normal and shear stresses in a fluid can be described by a tensor of rank 2. The defining characteristics of a tensor is the manner in which its components transform under a rotation of the coordinate system where the components are defined. The transformation law for the components of tensor of a rank two is given as τ i′j = α ik α jlτ kl (C.32) where the prime denotes the tensor components in the rotated coordinate, and α ik and α jl represent, respectively, the cosines of the angles between the ith rotated axis and the kth original axis, and between the jth rotated axis and the lth original axis. It should be noted that it is not the tensor itself that transforms under this change in the reference coordinate system but, rather, the coordinates that describe the tensor. The application of the ∇ operator to each component of the velocity vector V = iu + jv + kw also yields a tensor of rank two: ⎡ ∂u ∂v ∂w ⎤ ⎢ ∂x ∂x ∂x ⎥ ⎢ ⎥ ⎢ ∂u ∂v ∂w ⎥ (C.33) ∇V = ⎢ ∂y ∂y ∂y ⎥ ⎢ ⎥ ⎢ ∂u ∂v ∂w ⎥ ⎢ ∂z ∂z ∂z ⎥ ⎣ ⎦ which can be used to determine the strain rate tensor. The dot product of a vector and a tensor of rank 2 is a vector. For example, the dot product of the ∇ operator and a stress tensor is Appendix C Vectors and Tensors 1011 ⎡ ∂τ xx ∂τ xy ∂τ xz ⎤ ⎡∂⎤ + + ⎢ ⎥ ⎢ ∂x ⎥ ∂y ∂z ⎥ ⎢ ∂x ⎢ ⎥ ⎡τ xx τ xy τ xz ⎤ ⎥ ⎢ ∂τ yx ∂τ yy ∂τ yz ⎥ ⎢ ∂ ⎥⎢ (C.34) ∇ ⋅ τ = ⎢ ⎥ ⎢τ yx τ yy τ yz ⎥ = ⎢ + + ⎥ ∂y ∂x ∂y ∂z ⎥ ⎢ ⎥ ⎢ ⎥ ⎣τ zx τ zy τ zz ⎦ ⎢ ⎢ ∂τ ⎥ ∂τ ⎢∂⎥ ⎢ zx + zy + ∂τ zz ⎥ ⎢ ∂z ⎥ ⎣⎦ ∂y ∂z ⎥ ⎢ ∂x ⎣ ⎦ The contraction of two tensors of rank two a and b is obtained by summing the products of the corresponding components from both tensors: a : b = axx bxx + axy bxy + axz bxz + a yx byx + a yy byy + a yz byz + azx bzx + azy bzy + azz bzz which can also be written as a : b = aij bij (C.35) (C.36) using the summation convention of tensors. According to the summation convention, the repetition of an index in a term denotes a summation with respect to that index over its range (i, j=x, y, z). The definitions and operations of the vectors and tensors reviewed here provide foundations for the governing equations for multiphase systems. Additional information about tensors and their associated operations can be found in a continuum mechanics textbook, such as Fung (1994). References Fung, Y.C., 1994, First Course in Continuum Mechanics, 3rd edition, Prentice Hall. 1012 Transport Phenomena in Multiphase Systems