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. Usually, fluid dynamics assumes that (1) The fluid is a Newtonian fluid in the sense that the stress tensor is given by the sum of the pressure p and the viscous stress tensor τ,2
P = pI − τ, (B.5) 2μ τ = 2με + μb − (ε : I)I, (B.6) 3
3 where μ is the viscosity coefficient, μb is the bulk viscosity coefficient, ε is the strain rate tensor defined by Eq. (A.13)inAppendix A, and the operator : means the contraction of two second-order tensors defined by Eq. (A.10)in Appendix A. Since the strain rate tensor ε and the identity tensor are symmet- ric, the viscous stress tensor τ is also symmetric. The viscosity coefficients are usually assumed as functions of temperature and pressure.4 (2) The heat flux obeys the Fourier law,
q =−λ∇T, (B.7)
where λ is the thermal conductivity coefficient and T is the temperature of fluid. The thermal conductivity coefficient is usually assumed as a function of tem- perature and pressure. (3) The thermodynamic relations hold among the pressure p, temperature T , inter- nal energy e, and density ρ. This is the assumption of local equilibrium state. For the above four thermodynamic variables, there exist two independent ther- modynamic relations. For example, if the fluid is an ideal gas, we have
p = ρ RT, e = cv T, (B.8)
where the first one is the (thermal) equation of state of ideal gas (R = k/m is the gas constant, k is the Boltzmann constant, and m is a mass of a molecule) and the second is the (caloric) equation of state of ideal gas (cv is the specific heat for constant volume per unit mass). If the gas is treated as an incompressible fluid, the density ρ is not a thermodynamic variable. Then, the first equation in Eq. (B.8) should be discarded and the definition of incompressible flows
1 In the incompressible fluid flows, we cannot assume any relation between ρ and other thermody- namics variables. Nevertheless, the conservation laws (B.1), (B.2), and (B.3) should be satisfied. 2 In many textbooks of fluid dynamics, the sign of stress tensor P is opposite to Eq. (B.5). 3 The bulk viscosity coefficient is sometimes called the second viscosity coefficient. 4 The viscosity coefficients and thermal conductivity coefficient of an ideal gas are functions of temperature. Appendix B 217 ∂ρ + v · ∇ρ = 0, (B.9) ∂t
should be used instead. At least for ideal gases, the above three statements are theoretically validated by the kinetic theory of gases in the limit that the Knudsen number goes to zero, if the nonlinearity is sufficiently weak.5 For liquids, although there are no theoretical vali- dations for Eqs. (B.5), (B.6), and (B.7), they are as a whole admitted and significant objections have never been raised against them.6 Thus, the system of equations in fluid dynamics is closed. In principle, we can solve it under appropriate boundary conditions and initial condition. The set of equa- tions, Eqs. (B.1), (B.2), and (B.3) with Eqs. (B.5), (B.6), and (B.7) may be called the set of NavierÐStokes equations.7 Equations (B.1), (B.2), and (B.3) are written in the so-called conservation law form, ∂ (ρ f ) =−∇ · (ρ f v + φ) + ρϑ, (B.10) ∂t where f and ϑ are vectors or scalars and φ is a vector or a tensor. In fact, Eqs. (B.1), (B.2), and (B.3) are recovered as follows:
Eq. (B.1) : f = 1, φ = 0,ϑ= 0, (B.11) Eq. (B.2) : f = v, φ = P,ϑ= b, (B.12) 1 Eq. (B.3) : f = v2 + e, φ = v · P + q,ϑ= b · v + S. (B.13) 2 In the above three equations, ρ f v+φ is very important for understanding the physics related to the interface: ρv in Eq. (B.1) is called the mass flux density vector, ρvv + ρ(1 |v|2 + )v + v · P in Eq. (B.2) is called the momentum flux density tensor, and 2 e P + q in Eq. (B.3) is called the energy flux density vector. In fluid dynamics, in addition to Eq. (B.3), there are several variations in the equation associated with the energy. For example, the equation of the internal energy
5 See Footnotes 20 and 21 in Chap. 2. 6 Non-Newtonian fluids are excluded, of course. 7 The name “NavierÐStokes equations” is often used to indicate the momentum conservation equa- tions Eq. (B.2) with the stress tensor of Newtonian fluid (B.5)and(B.6) or its variations, ∂v ρ =−ρ(v · ∇)v − ∇ p + ∇ · τ + ρb, ∂t and ∂v 2 1 ρ + (v · ∇)v =−∇ p + μ∇ v + μb + μ ∇ (∇ · v) + ρb, ∂t 3 for constant μ and μb. 218 Appendix B per unit volume can be written as
∂ρe =−∇ · (ρev) − p∇ · v + τ : ε − ∇ · q + ρS. (B.14) ∂t
B.2 Conservation Equations in Component Forms
As mentioned in Appendix A, actual numerical evaluations of vectors and tensors require the handling of their components. We therefore write down Eqs. (B.1), (B.2), and (B.3) and Eqs. (B.5), (B.6), and (B.7) in component forms with indices using the Einstein summation convention explained in Appendix A. The mass conservation equation (B.1): ∂ρ ∂ρv + i = 0. (B.15) ∂t ∂xi
The momentum conservation equation (B.2):
∂ρvi ∂ρvi v j + Pij + = ρbi ,(i = 1, 2, 3). (B.16) ∂t ∂x j
The stress tensor of Newtonian fluid (B.5) and (B.6):
Pij = pδij − τij, (B.17) ∂vi ∂vj 2μ ∂vk τij = μ + + μb − δij, (B.18) ∂x j ∂xi 3 ∂xk where i, j = 1, 2, 3. The energy conservation equation (B.3): ∂ ∂ 1ρv2 + ρ + 1ρv2 + ρ v + v + = ρ v + ρ . i e i e j i Pij q j b j j S ∂t 2 ∂x j 2 (B.19) The heat flux based on the Fourier law (B.7):
∂T q j =−λ ,(j = 1, 2, 3). (B.20) ∂x j Appendix C Supplements to Chapter 5
C.1 Generalized Stokes Theorem
We here prove the generalized Stokes theorem by using the Gauss theorem: . .
∇·WdV = W · ndS, or ∂n W..ndV = W..i ni dS, (C.1) V S V S where W (W..n) is a tensorial quantity of any order. The Gauss theorem turns the surface integral of W over a closed surface S which is enclosing a volume V into the volume integral of a derivative of W (the divergence) over the interior of S, i.e., over the volume of V . We assume here that the surface S where the integration is evaluated is the plane surface as shown in Fig. C.1, for simplicity. Although this choice of the integral surface is rather special, the following discussion is also valid for general integral surfaces by considering the integration over an infinitesimal area element on the tangential surface at a point of contact.
nT = n ST
SS n h nS = nC S
nC C C t SB nB = – n
Fig. C.1 A volume considered in the proof of the generalized Stokes theorem
We draw a smooth closed line C on the plane surface S, and construct a column perpendicular to its base whose peripheral edge is C, as shown in Fig. C.1.The
219 220 Appendix C height of this column is h. The unit normal vector to the plane surface S is denoted as n. The column is enclosed by the lateral closed surface SS and the top and bottom base surfaces ST and S B. The unit normal vectors to these three surfaces are nS, nT , and nB, respectively. The unit normal and tangential vectors to the closed line C are defined as nC and tC , respectively. Now we substitute T × n (nmlT..mnl )intoW of Eq. (C.1) to obtain . ∇·(T × n)dV = (T × n) · nSdSS V SS + (T × n) · nT dST + (T × n) · nBdSB , (C.2) ST S B where T is a tensor of any order. The integrand of the left-hand side of Eq. (C.2) can be rewritten as . h ∇·(T × n)dV = (∇×T) · ndSdh, (C.3) V 0 S by using the fact that n is constant since the surface S is plane. Now we rewrite the right-hand side of Eq. (C.2). Notice that the following holds:
(T × n) · n = ijkT.. j nkni = [n1(T..2n3 − T..3n2) + n2(T..3n1 − T..1n3) + n3(T..1n2 − T..2n1)] = [T..1(n2n3 − n3n2) + T..2(n3n1 − n1n3) + T..3(n1n2 − n2n1)] = 0, (C.4) and that nT is equal to n and nB is equal to −n. Therefore only the integration over the lateral surface of the column contributes to Eq. (C.2). Since nS is written as nC on the lateral surface, the left-hand side of Eq. (C.2) is rewritten as . . h (T × n) · nSdS, = (T × n) · nC dldh. (C.5) SS 0 C
With the use of nC = tC × n and n · tC = 0, the integrand of the left-hand side of Eq. (C.5) can be rewritten as ! " ( × ) · C = ( × ) · C × = C T n n T n t n ijkT.. j nk imntm nn # $ C C C = δ δ − δ δ T.. n t n = T.. n t n − T.. n t n ! jm kn" jn km j k m! n " j k j k j k k j = T · tC (n · n) − (T · n) tC · n = T · tC . (C.6)
So we have the right-hand side of Eq. (C.2)as Appendix C 221 . . h (T × n) · nSdS, = T · tC dldh. (C.7) SS 0 C
Equating Eqs. (C.3) and (C.7), the Gauss theorem is rewritten as . . h h (∇×T) · ndSdh = T · tC dldh. (C.8) 0 S 0 C
Equation (C.8) holds for arbitrary choice of h, and therefore we finally obtain . (∇×T) · ndS = T · tC dl. (C.9) S C
C.2 Characteristic Time of Heat Conduction
We discuss the characteristic time of heat conduction by considering the simplest case, i.e., heat conduction in a uniform rod. Temperature u at position x in a uniform rod is governed by one-dimensional heat conduction equation:
∂u ∂2u = D ,(D > 0), (C.10) ∂t ∂x2 where D is the coefficient of thermal diffusivity. We first investigate the case that the rod is infinitely long; hence the domain of definition is −∞ < x < ∞. Suppose that temperature distribution at t = 0isgiven as
u|t=0 = ϕ(x), (−∞ < x < ∞). (C.11)
It is easily verified that solution of Eq. (C.10) is written as: