Appendix A Vectors, Tensors, and Their Notations

Transcription

1 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 = (a 1, a 2,...,a N ), where N is the dimension of the physical space on which a is considered, it is a vector, and a i (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, (α and β are scalars), (A.1) gives a vector. Sometimes, a i 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 secondorder 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(a 1 + a 2, b) = P(a 1, b) + P(a 2, b), P(a, b 1 + b 2 ) = P(a, b 1 ) + P(a, b 2 ), P(αa, b) = P(a,αb) = α P(a, b), (A.2) (A.3) (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 Transfer, DOI 1.17/ , C Springer-Verlag Berlin Heidelberg

2 212 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 continuously 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 consisted 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 = (x 1, x 2, x 3 ) is specified, the components a i s (i = 1, 2, 3) of vector field a and the components P ij 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 a i s and P ij 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 b i = 3 P ij a j, (i = 1, 2, 3). (A.6) j=1 According to the Einstein summation convention, we can eliminate the summation symbol to yield b i = P ij a j, (i = 1, 2, 3). (A.7) The Einstein summation convention is a rule of notation of binary-product operations 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 curvilinear coordinate systems.

3 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 = a i b i. (A.8) The dyadic of two vectors gives a second-order tensor P, P = ab = a i b j = P ij, (i, j = 1, 2, 3), (A.9) where, as usual, we do not distinguish a tensor P from its representative expression P ij, although the notation P ij 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 = T ij U ij. (A.1) The gradient of a scalar field f is a vector, and can be expressed as grad f = f = f x i, (i = 1, 2, 3). (A.11) The divergence of a vector field v is a scalar expressed as div v = v = v i x i. (A.12) The strain rate tensor 3 in fluid dynamics, ε, can be constructed by the dyadic of vectors and v as 4 ε = 1 [ ] v + ( v) T = 1 ( vi + v ) j = ε ij, (i, j = 1, 2, 3), (A.13) 2 2 x j x i where the superscript T denotes the transpose of matrix. Here, to simplify the notation further, we can indicate the differentiation with respect to x i by index i after an index denoting a component of vector or tensor with a comma separating the two indices. That is, grad f = f,i, div v = v i,i, ε = 1 ( ) 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, = ( / x 1, / x 2, / x 3 ).

4 214 Appendix A where since f is a scalar, no index appears before the comma before index i indicating the differentiation with respect to x i. 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 δ ij is a representation of the second-order identity tensor I, given by { 1 ifi = j, δ ij = otherwise. (A.15) The identity transformation from a vector a to a vector b can be written with the Kronecker delta as b = I a = δ ij a j = a i, (i = 1, 2, 3). (A.16) The Eddington epsilon ɛ ijk defined by 1 (i, j, k) = (1, 2, 3), (2, 3, 1), (3, 1, 2), ɛ ijk = 1 (i, j, k) = (3, 2, 1), (2, 1, 3), (1, 3, 2), i = j or j = k or k = i, (A.17) 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 = ɛ ijk a j b k = c i, (i = 1, 2, 3), (A.18) v k curl v = v = ɛ ijk = ɛ ijk v k, j, (i = 1, 2, 3). (A.19) x j Several relations involving δ ij and ɛ ijk are useful in manipulations of vectors and tensors: δ ij δ ij = 3, ɛ ijk ɛ ijk = 6, (A.2) ɛ ijk ɛ hjk = 2δ ih, (A.21) ɛ ijk ɛ mnk = δ im δ jn δ in δ jm. (A.22) A second-order tensor P is called symmetric, if P = P T,or P ij = P ji, (i, j = 1, 2, 3). (A.23) Clearly, the strain rate tensor ε and the Kronecker delta are the symmetric secondorder tensors.

5 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, velocity 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 ( 1 t 2 ρ v 2 + ρe ρ t + (ρv) =, (B.1) ρv + (ρvv + P) = ρb, (B.2) t ) [( ) ] ρ v 2 + ρe v + v P + q = ρb v + ρs, (B.3) where b is a body force exerted on the fluid per unit mass, 1 2 ρ v 2 + ρe (B.4) 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

6 216 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 = p I τ, (B.5) ( τ = 2με + μ b 2μ ) (ε : I)I, (B.6) 3 where μ is the viscosity coefficient, μ b is the bulk viscosity coefficient, 3 ε is the strain rate tensor defined by Eq. (A.13) in Appendix A, and the operator : means the contraction of two second-order tensors defined by Eq. (A.1) in Appendix A. Since the strain rate tensor ε and the identity tensor are symmetric, 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 temperature and pressure. (3) The thermodynamic relations hold among the pressure p, temperature T, internal energy e, and density ρ. This is the assumption of local equilibrium state. For the above four thermodynamic variables, there exist two independent thermodynamic relations. For example, if the fluid is an ideal gas, we have p = ρ RT, e = c v 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 (c v 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 thermodynamics 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.

7 Appendix B 217 ρ t + v ρ =, (B.9) 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 validations 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 equations, Eqs. (B.1), (B.2), and (B.3) with Eqs. (B.5), (B.6), and (B.7) may be called the set of Navier Stokes 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, t (B.1) 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 =, ϑ =, (B.11) Eq. (B.2) : f = v, φ = P, ϑ = b, (B.12) Eq. (B.3) : f = 1 v 2 + e, 2 φ = v P + q, ϑ = b v + S. (B.13) 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 + P in Eq. (B.2) is called the momentum flux density tensor, and ρ( 1 2 v 2 + e)v + v 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 2 and 21 in Chap Non-Newtonian fluids are excluded, of course. 7 The name Navier Stokes equations is often used to indicate the momentum conservation equations Eq. (B.2) with the stress tensor of Newtonian fluid (B.5) and(b.6) or its variations, ρ v = ρ(v )v p + τ + ρb, t and [ ] v ρ + (v )v = p + μ 2 v + (μ b + 13 ) t μ ( v) + ρb, for constant μ and μ b.

8 218 Appendix B per unit volume can be written as ρe t = (ρev) p v + τ : ε q + ρs. (B.14) 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): The momentum conservation equation (B.2): ρ t + ρv i x i =. (B.15) ρv i t + ρv iv j + P ij x j = ρb i, (i = 1, 2, 3). (B.16) The stress tensor of Newtonian fluid (B.5) and (B.6): P ij = pδ ij τ ij, ( vi τ ij = μ + v ) ( j + μ b 2μ x j x i 3 ) vk x k δ ij, (B.17) (B.18) where i, j = 1, 2, 3. The energy conservation equation (B.3): ( ) 1 t 2 ρv2 i + ρe + [( ) ] 1 x j 2 ρv2 i + ρe v j + v i P ij + q j = ρb j v j + ρs. (B.19) The heat flux based on the Fourier law (B.7): q j = λ T x j, (j = 1, 2, 3). (B.2)

9 Appendix C Supplements to Chapter 5 C.1 Generalized Stokes Theorem We here prove the generalized Stokes theorem by using the Gauss theorem: WdV = V S W nds, or n W..n dv = V S W..i n i ds, (C.1) 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. n T = n S T S S n S = n C n S h n C n B = n t C S B C 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

10 22 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 S S and the top and bottom base surfaces S T and S B. The unit normal vectors to these three surfaces are n S, n T, and n B, respectively. The unit normal and tangential vectors to the closed line C are defined as n C and t C, respectively. Now we substitute T n (ɛ nml T..m n l )intow of Eq. (C.1) to obtain (T n)dv = (T n) n S ds S V S S + (T n) n T ds T + (T n) n B ds B, (C.2) S T 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 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: S (T n) n = ɛ ijk T.. j n k n i = [n 1 (T..2 n 3 T..3 n 2 ) + n 2 (T..3 n 1 T..1 n 3 ) + n 3 (T..1 n 2 T..2 n 1 )] = [T..1 (n 2 n 3 n 3 n 2 ) + T..2 (n 3 n 1 n 1 n 3 ) + T..3 (n 1 n 2 n 2 n 1 )] =, (C.4) and that n T is equal to n and n B is equal to n. Therefore only the integration over the lateral surface of the column contributes to Eq. (C.2). Since n S is written as n C on the lateral surface, the left-hand side of Eq. (C.2) is rewritten as h (T n) n S ds, = (T n) n C dldh. (C.5) S S C With the use of n C = t C n and n t C =, the integrand of the left-hand side of Eq. (C.5) can be rewritten as ( ) (T n) n C = (T n) t C n = ɛ ijk T.. j n k ɛ imn tm C n n = ( ) δ jm δ kn δ jn δ km T.. j n k tm C n n = T.. j n k t C j n k T.. j n k tk C n j ( = T t C) ( ) (n n) (T n) t C n = T t C. (C.6) So we have the right-hand side of Eq. (C.2) as

11 Appendix C 221 (T n) n S ds, = S S h C T t C dldh. (C.7) Equating Eqs. (C.3) and (C.7), the Gauss theorem is rewritten as h S ( T) ndsdh = h C T t C dldh. (C.8) Equation (C.8) holds for arbitrary choice of h, and therefore we finally obtain ( T) nds = S C T t C dl. (C.9) 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 t = D 2 u, (D > ), (C.1) 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 = isgiven as u t= = ϕ(x), ( < x < ). (C.11) It is easily verified that solution of Eq. (C.1) is written as: u(x, t) = ( ) exp Dλ 2 t [A(λ) cos λx + B(λ) sin λx] dλ, (C.12) with coefficients A and B to be determined using initial condition (C.11). The substitution of the formal solution (C.12) into the initial condition (C.11) provides the Fourier Integral representation of ϕ(x): ϕ(x) = [A(λ) cos λx + B(λ) sin λx] dλ. (C.13) Coefficients A(λ) and B(λ) in Eq. (C.13) are obtained as A(λ) = 1 ϕ(ξ)cos λξdξ, B(λ) = 1 ϕ(ξ)sin λξdξ. 2π 2π (C.14)

12 222 Appendix C Substitution of relation (C.14) into Eq. (C.13) provides the following representation of u(x, t): u(x, t) = 1 ( ) dλ exp Dλ 2 t ϕ(ξ)cos λ(x ξ)dξ 2π = 1 ( ) ϕ(ξ)dξ exp Dλ 2 t cos λ(x ξ)dλ = 2π ] 1 ϕ(ξ) [ 2 π Dt exp (ξ x)2 dξ. 4Dt (C.15) Now we solve the heat conduction problem in a semi-infinite rod. The governing equation is Eq. (C.1). We consider that the initial temperature of the rod is uniform and the temperature of an end of a rod is set as. Then, the boundary and initial conditions are written as u(, t) =, (C.16) lim u(x, t) = 1, (C.17) x 1ifx >, u(x, ) = ϕ(x) = ifx =, 1 ifx <, (C.18) where all variables have been nondimensionalized. We require that ϕ(x) should be an odd function for Eq. (C.17) to be satisfied. This can be easily shown as follows. First we divide the solution (C.15) intotwo parts and rewrite as { 1 u(x, t) = 2 π Dt + ϕ( ξ)exp [ ϕ(ξ)exp [ ] (ξ x)2 dξ 4Dt ] } (ξ + x)2 dξ. (C.19) 4Dt Now we seek for the condition on which the solution satisfies Eq. (C.17) tobe imposed on ϕ(x). We have from Eq. (C.19), 1 u(, t) = 2 [ϕ(ξ) + ϕ( ξ)] exp ( ξ 2 ) dξ =. (C.2) π Dt 4Dt We find that Eq. (C.2) holds if and only if the relation ϕ( ξ) = ϕ(ξ), (C.21)

13 Appendix C 223 is satisfied; hence ϕ(ξ) should be an odd function. Substituting Eq. (C.18) in(c.19), and Setting σ 1 = (ξ x)/(2 Dt) and σ 2 = (ξ + x)/(2 Dt), the solution is given by 1 u(x, t) = 2 π Dt = 1 [ π χ { exp [ ( ) exp σ1 2 dσ 1 ] (ξ x)2 exp [ 4Dt χ ]} (ξ + x)2 dξ 4Dt ( ) ] exp σ2 2 dσ 2 = 2 π χ ( exp σ 2) dσ = erf(χ), (C.22) where χ = x 2 Dt. (C.23) Therefore the solution of the one-dimensional heat conduction equation (C.1) can be written by using only χ defined by (C.23). In Sect. 5.5, we use this χ to discuss the characteristic time of heat conduction. C.3 Abel s Integral Equation We shall seek the solution x of Abel s integral equation, f (t) = t x(η) dη (t η) ν ( <ν<1), (C.24) where f (t) is a given continuously differentiable function. Introducing a function φ(t) defined by φ(t) = t x(η)dη, (C.25) and using a formula π t sin νπ = η dξ (t ξ) 1 ν (ξ η) ν, (C.26)

14 224 Appendix C we can easily carry out the following integrations: π t t sin νπ φ(t) = dη η = = t t dξ ξ x(η) (t ξ) 1 ν (ξ η) ν dξ x(η) (t ξ) 1 ν (ξ η) ν dη f (ξ) dξ, (C.27) (t ξ) 1 ν where Eq. (C.24) has been used in the last equation of Eq. (C.27). Differentiating Eq. (C.27) with respect to t gives the solution of Eq. (C.24) as x(t) = sin νπ π d t dt f (ξ) dξ. (C.28) (t ξ) 1 ν

15 Index A Abel s integral equation, 27, 223 Adsorbed liquid film, 78 Antoine s equation, 8, 8 Association, 88 degree of, 88 B Berthelot equation, 138 BKW equation, 42 Boltzmann constant, 14, 2, 32, 113 Boltzmann equation, 1, 39 Bubble wall, 168, 2 Bulk viscosity, 57, 216 C Clausius Clapeyron equation, 127 Coefficient of thermal diffusivity, 174, 21, 221 Collision frequency, 4 Collision term, 39, 41 Complete-condensation condition, 44 Compression factor, 2, 116 Condensation coefficient, 4, 44, 53, 64 Condensation mass flux, 72 Conservation equation of energy at bubble wall, 171, 175, 21 for spherical bubble, 164 on interface, 161 Conservation equation of mass at bubble wall, 168, 21 for spherical bubble, 163 on interface, 158 Conservation equation of momentum at bubble wall, 17 for spherical bubble, 164 on interface, 16 Conservation equations, 25, 153, 215 Convective derivative, 149, 151 Convolution, 183, 24 Critical temperature, 112 Cut-off radius, 34, 113 D Density distribution function, 23 Diffuse-reflection condition, 43 Dirac delta function, 25, 182 E Einstein summation convention, 39, 85, 16, 212 Energy flux density vector, 217 Energy reflectance, 89 Equipartition theorem, 2 Error function, 1, 184, 25 Euler equations, 54, 55 Evaporation coefficient, 4, 44, 5, 64, 131 Evaporation into vacuum, 46, 132 F Flux balance on interface, 157 Fourier law, 165, 17, 216, 218 G Gauss divergence theorem, 26, 145, 147, 15, 151 Gaussian BGK Boltzmann equation, 5, 55, 77 Generalized Stokes theorem, 16, 221 Ghost effect, 54 Gibbs dividing surface (equimolor dividing surface), 129 Grad Boltzmann limit, 41 H Half-space problem, 55, 62 Hamilton s canonical equations of motion, 22 Hamiltonian, 22, 34, 37 Heat conduction,

16 226 Index Heat equation, 178, 194, 22, 221 Heat flux, 29 Heaviside function, 179 Helmholtz free energy, 153 Hertz Knudsen Langmuir formula, 66, 132 H-theorem, 42, 56 I Ideal gas, 2, 41 Incident shock wave, 79 Incompressible fluid, 216 Interface, 3 Interface velocity, 145 Interferometer, 89 Intermolecular force, 21 Internal degrees of freedom, 51, 56, 74 Intramolecular force, 21 Inverse Laplace transform, 181 K Kelvin equation, 111, 127, 169 Kinetic boundary condition, 3, 43, 5, 53, 57 Kinetic theory, 38 Kinetic theory of gases, 25, 38 Knudsen layer, 3, 55, 61 Knudsen layer analysis, 61 Knudsen number, 2, 55 L Laplace equation, 126 Laplace transform, 179, 23 Latent heat, 171, 21 Leap-frog scheme, 33, 113 Lennard-Jones potential, 31, 113 Liouville equation, 24 Liquid film, 79 Liquid temperature at bubble wall, 188 gradient at bubble wall, 189 of bubble interior, 186 Liquid velocity at bubble wall, 165, 169 Local equilibrium, 2, 28, 38, 55, 59 Local Maxwellian, 41 Loschmidt constant, 2 M Mach number, 82 Mass flux across the interface, 45, 47, 53, 65, 158 of molecules spontaneously evaporating, 47, 72, 131 Mass flux density vector, 217 Mass fraction, 87 Maxwell distribution function, 4 Mean collision frequency, 4 41 Mean free path, 3, 4 42, 57, 116 Mean free time, 14, 4 Molecular dynamics, 31, 46 Molecular gas dynamics, 25 Momentum flux density tensor, 217 Monatomic molecule, 21, 39, 72 Moving boundary problem, 176 N Nanodroplet, 112 Navier Stokes equations, 1, 54 55, 217 Net mass flux of condensation, 78 Net mass flux of evaporation, 76 Newton s equation of motion, 21 Newtonian fluid, 216, 218 Noncondensable gas, 87 Nonequilibrium state, 2, 46, 78, 131 NVE simulation, 31, 113 P Partition function, 74 Periodic boundary condition, 35, 113 Permanently absorbed liquid film, 93 Phase space, 23, 25 Polyatomic molecule, 5, 55, 73 Prandtl number, 42, 57, 192 Pressure of bubble interior, 196 R Ratio of specific heats, 56, 75 Rayleigh Plesset equation, 172 Reflected shock wave, 79 Reflection mass flux, 72 S S expansion, 58 Saturated vapor density, 8, 44, 72 Saturated vapor pressure, 8, 116 Schrage formula, 66 Second harmonics, 18 Shock tube, 6, 78 Shock wave, 6, 13, 78 Slip coefficient, 62, 63 Solvability condition, 59 Sound resonance, 17 Sound resonance method, 16 Standing wave, 18 State equation of real gas, 138 Stress tensor, 28, 29 Surface entropy, 152 Surface of tension, 129 Surface tension, 116, 124, 159

17 Index 227 T Temperature discontinuity at bubble wall, 158, 171 Temporal transition phenomenon, 12 Temporarily adsorbed liquid film, 94 Thermal diffusion-controlled condensation, 13 Thickness of interface, 2, 46, 129 Tolman equation, 111, 13 Tolman length, 129 Total curvature, 147 Transition layer, 2, 35, 116 Transition time, 11 Triple point temperature, 36, 5, 112, 138 U Unit normal of interface, 145 V Vapor pressure, 116 Vapor temperature at bubble wall, 195 gradient at bubble wall, 195 of bubble exterior, 194 outside thermal boundary layer, 19 with uniform interior, 26 Vapor velocity at bubble wall, 165, 169 Vapor liquid interface, 143 Velocity distribution function, 39, 48, 73, 77 Velocity field of bubble interior, 197 Velocity scaling, 47, 114, 133 Volterra integral equation of the second kind, 9, 8