Nomenclature

$\mathcal A$

vertical aspect ratio (reduced height) of cavity

$\mathsf{A}$

arbitrary diffusivity tensor

$B$

Spalding's driving force for mass transfer, $\exp(-\varPhi )-1$

$b$

characteristic length, dimensional width of cavity

$C$

coefficient of Dirichlet term in Robin boundary condition

$c$

constant of integration

$c_{p}$

isobaric specific heat capacity

$D$

binary diffusivity

$d$

characteristic length of $V$ in §2.6.1

$E_K$

truncation error

$\hat{e}$

arbitrary unit vector

$\mathbf{f}$

body force term in Stokes equation

$\mathbf{g}$

gravitational acceleration, $-g\mbox{\boldmath$\hat{\jmath}$}$

Gr

(thermal) Grashof number, $g\beta\Delta T_*b^{3}/\nu^{2}$

$h$

specific enthalpy

$h_{fg}$

specific heat of vaporization

h.o.t.

higher order terms

$\hat{\imath}$

unit transverse vector

$I$

coefficient of Neumann term in Robin boundary condition

$i$

index for species in a multicomponent mixture

$\hat{\jmath}$

unit vertical vector

$\mathbf{j}_*$

mass flux of vapour relative to $\mathbf{u}_*$

$K$

number of evaluated terms in a truncated series

$\hat{k}$

unit spanwise vector

$k$

index for terms in a series

$Le_{A}$

vapour Lewis number, $\lambda/\rho c_{pA} \delta$

$M$

molar mass

$m$

reduced vapour mass fraction, $(m_*-m_{*r})/\Delta m_*$

$N$

buoyancy ratio, $\zeta\Delta m_*/\beta\Delta T_*$ number of species in a mixture in §2.6.1

$\hat{n}$

unit normal vector, positive outward from fluid phase

$\mathbf{n}_*$

absolute mass flux of vapour

$Nu$

Nusselt number, $-\mbox{\boldmath$\hat{\imath}$}\cdot\mathbf{e}_*b[1-\exp(-\varPhi _T)]/ \lambda\Delta T_*\varPhi _T$

$\mbox{$\mathcal P$}[\mathbf{v}]$

scalar defining the poloidal part of $\mathbf{v}$

$P_n^m$

function defined by (B.6)

$p$

reduced pressure,

$(p_*+\rho gy_*)/\rho g(\beta\Delta T_*+\zeta\Delta m_*)b\mbox{$\mathcal A$}$

Pr

Prandtl number, $\nu\rho c_{p}/\lambda$

$\mbox{\textit{Pr}}_r$

reference Prandtl number, $\nu\rho c_{pr}/\lambda$

$\mbox{\textit{Pr}}_I$

interdiffusion Prandtl number, $\nu\rho(c_{pA}-c_{pB})(1-m_{*r})/\lambda$

$\mathbf{r}$

reduced position vector, $\mathbf{r}_*/b$

$r$

reduced spherical coordinate, $\vert\mathbf{r}\vert$

Ra

(thermal) Rayleigh number, GrPr

$\mathit{Re}$

Reynolds number

$\mathcal S$

spanwise aspect ratio (reduced span) of cavity

$\mbox{$\mathcal S$}[\mathbf{v}]$

scalar defining the scaloidal part of $\mathbf{v}$

$S$

level surface of arbitrary scalar field

$s$

mesh stretching factor in §5.1.5

arbitrary scalar field in §2.6.1

Sc

Schmidt number, $\nu/D$

Sh

Sherwood number, $-\mbox{\boldmath$\hat{\imath}$}\cdot\mathbf{n}_*b/\rho D\varPhi$

$\mbox{$\mathcal T$}[\mathbf{v}]$

scalar defining the toroidal part of $\mathbf{v}$

$T$

reduced temperature, $(T_*-T_{*r})/\Delta T_*$

$\mathbf{u}$

reduced velocity, $\mathbf{u}_*\nu/g(\beta\Delta T_*+\zeta\Delta m_*)b^2$

$u$

reduced transverse component of velocity, $\mbox{\boldmath$\hat{\imath}$}\cdot\mathbf{u}$

$\mathcal V$

term of an inner matched asymptotic expansion

$V$

volume enclosed by level surface, $S$

$\mathbf{v}$

arbitrary vector field

$v$

reduced vertical component of velocity, $\mbox{\boldmath$\hat{\jmath}$}\cdot\mathbf{u}$

$w$

reduced spanwise component of velocity, $\mbox{\boldmath$\hat{k}$}\cdot\mathbf{u}$

$x$

reduced transverse coordinate, $\mbox{\boldmath$\hat{\imath}$}\cdot\mathbf{r}$

$y$

reduced vertical coordinate, $\mbox{\boldmath$\hat{\jmath}$}\cdot\mathbf{r}$

$z$

reduced spanwise coordinate, $\mbox{\boldmath$\hat{k}$}\cdot\mathbf{r}$

Greek symbols

$\beta$

thermal coefficient of cubic expansion

$\gamma_{ij}$

component of metric tensor

$\varDelta$

inner gauge function in a matched asymptotic expansion

$\Delta$

characteristic difference

$\delta$

outer gauge function in a matched asymptotic expansion

$\zeta$

vapour coefficient of cubic expansion

stretched coordinate for region near front wall in §7.4.2

$\eta$

spherical coordinate: colatitude, $\arctan(\sqrt{x^2+z^2}/y)$

$\theta$

spherical coordinate: colatitude, $\arctan(\sqrt{x^2+y^2}/z)$

$\lambda$

thermal conductivity

$\varLambda$

latent heat factor, $h_{fg}/c_{pA}\Delta T_*$

$\mu$

(dynamic) viscosity

$\nu$

kinematic viscosity

$\xi$

stretched coordinate for region near hot wall

$\rho$

density

$\sigma$

function defined by (4.29), appearing in (4.27)

$\upsilon$

spherical coordinate: azimuth, $\arctan(y/z)$

$\varPhi$

mass transfer rate factor, $\ln[(1-m_{*r})/(1-m_{*r}-\Delta m_*)]$

$\varPhi _T$

thermal mass transfer rate factor, $\varPhi (\mbox{\textit{Pr}}_r+\mbox{\textit{Pr}}_I)/\mbox{\textit{Sc}}$

$\phi$

spherical coordinate: azimuth, $\arctan(x/y)$

$\varphi$

test function for variational form of Navier-Stokes equation

$\psi$

vertical component of solenoidal vector potential for $\mathbf{u}_{\perp}$

Stokes's stream-function in §8.2

$\varOmega$

rotational speed

$\Omega$

domain, with boundary $\partial\Omega$

Superscripts

$\hat{}$

unit magnitude

$\bar{}$

mean

$\tilde{}$

transformed

$\Vert$

in a domain with $\mbox{$\mathcal S$}\rightarrow\infty$

$=$

in a domain with $\mbox{$\mathcal S$}\rightarrow 0$

$\Box$

in a domain with rectangular horizontal section

$\circ$

in a domain with circular horizontal section

in a domain with elliptic horizontal section

$i$

contravariant component

$j$

contravariant component

$(P)$

poloidal part

$(S)$

scaloidal part

$(T)$

toroidal part

Subscripts

$,ij\ldots$

covariant derivative with respect to $x^i, x^j,\ldots$

$*$

dimensional

$\infty$

fully developed

$\perp$

restricted to a horizontal plane

$\odot$

restricted to a plane of constant $z$

$0$

at the wall $x=0$

$1$

at the wall $x=1$

$A$

species A, the vapour

$B$

species B, the gas

$f$

forced

$i$

Cartesian tensor component

$(i)$

species index in a multicomponent mixture

$j$

Cartesian tensor component

$m$

pertaining to mass transfer or composition

$n$

natural

$r$

at the reference state

$T$

thermal