2.1. Domain and governing equations

A cross-sectional view of an AGHP (representatively made out of copper) is shown in figure 2(a). Note that a criterion on the transverse Bond numbers (see (2.40)) required to assume that all menisci have the same radius of curvature (at a given axial location in the adiabatic section), as delineated in figure 2, is provided below. Looking ahead, once we justify the slowly varying meniscus curvature assumption in § 2.3, for our purposes, we need only resolve a series of two-dimensional problems and some related integrals across the adiabatic section to compute the capillary limit. In the $ (r^*,\theta ,z^* )$ cylindrical coordinate system shown, $z^*$ is the streamwise coordinate, directed from the evaporator to the condenser, and $r^*$ and $\theta$ are the transverse ones, with positive $\theta$ anti-clockwise. The adiabatic section begins at $z^* = 0$ , where the evaporator one ends. There are $N_{\textit{g}}$ grooves (and lands); consequently, their (azimuthal) pitch is

(2.1) \begin{equation} \theta _{\textit{p}} = \frac {2\pi }{N_{\textit{g}}}. \end{equation}

The radius of the grooves is denoted by $R_{\textit{g}}^*$ . Observe that the vapour phase is continuous, but there are $N$ discrete liquid regions in the full problem. However, our domain includes only half of one of them; therefore, it is bounded by the dashed, red curves and has an azimuthal span of $\pi /N_{\textit{g}}$ . It extends over the length of the adiabatic section from $z^* = 0$ to $z^* = L_{\textit{a}}^*$ , where the condenser section begins. An enlarged view of its cross-section is shown in figure 2(b). The boundaries ( $\boldsymbol{F}$ , $\boldsymbol{G}$ and $\boldsymbol{H}$ ) at $\theta = 0$ , $\theta _{\textit{p}}/2$ become flow symmetry boundaries once the effect of the in-plane components of gravity are neglected, which we justify in § 2.3.

We assume constant density ( $\rho$ ) and viscosity ( $\mu$ ) and laminar, incompressible flow in the liquid and vapour phases. (The maximum Reynolds and Mach numbers in both phases for the range of operating conditions considered are provided in § 7 and are sufficiently low.) Then, for the steady-state conditions of interest, the continuity and Navier–Stokes equations are

(2.2) \begin{align} \boldsymbol{\nabla} ^* \boldsymbol{\cdot }\boldsymbol{v}_i^* &= 0, \end{align}

(2.3) \begin{align} \rho _i \boldsymbol{v}_i^* \boldsymbol{\cdot }\boldsymbol{\nabla} ^* \boldsymbol{v}_i^*&= - \boldsymbol{\nabla} ^* p_i^* + \mu _i \boldsymbol{\nabla} ^{*^2} \boldsymbol{v}_i^*+\rho _i \boldsymbol{g}, \end{align}

respectively, where the subscript $i$ denotes the liquid ( $\textit{l}$ ) or vapour ( ${v}$ ) phase and $\boldsymbol{g}$ is the gravity vector. Also, $\boldsymbol{v}^*_i = v_r^*\boldsymbol{\delta }_{r^*} +v_{\theta }^*\boldsymbol{\delta }_{\theta }+ v_z^*\boldsymbol{\delta }_{z^*}$ is the velocity vector, where $\boldsymbol{\delta }_{i}$ denotes a unit vector in the $i$ direction, and $p_i^*$ is pressure. The gravity vector is

(2.4) \begin{equation} \boldsymbol{g} = -g \cos (\phi) \sin \left (\theta \right )\boldsymbol{\delta }_{r^*} -g \cos (\phi) \cos \left (\theta \right )\boldsymbol{\delta }_{\theta } - g \sin (\phi) \boldsymbol{\delta }_{z^*}, \end{equation}

where $g$ is the standard acceleration of gravity, 9.81 m s $^{-2}$ . The ‘inventory’ or ‘fill ratio’ of working fluid in the heat pipe is assumed to be such that the triple-contact lines are pinned at the top of the grooves.

2.2. Boundary conditions

We define the location of the meniscus by the unknown vector function

(2.5) \begin{equation} \boldsymbol{L} = R_{{v}}^*\left (\theta ,z^*\right ) \boldsymbol{\delta }_{r^*} + z^*\boldsymbol{\delta }_{z^*}. \end{equation}

Then, the surfaces bounding the domain labelled in figure 2(b) are defined in table 1 (surface M would appear in a boundary perturbation with the meniscus Taylor expanded onto $r^* = R_{\textit{L}}^*$ ).

Table 1.

Domain boundaries.

Table 1. Long description

The table has 13 rows and 5 columns. The columns are labeled Surface, Phase(s), z∗, r∗, and theta. The rows are labeled with letters A to M. Each row provides specific values and conditions for the location of the meniscus and surfaces bounding the domain. Row 1: Surface A, Phase(s) Vapour, z∗ 0, r∗ 0 < r∗ < R∗_v for 0 < theta < theta_g/2, 0 < r∗ < R∗_L for theta_g/2 < theta < theta_p/2, theta 0 < theta < theta_p/2. Row 2: Surface B, Phase(s) Liquid, z∗ 0, r∗ R∗_v < r∗ < R∗_g, theta 0 < theta < theta_g/2. Row 3: Surface C, Phase(s) Vapour, z∗ L∗_a, r∗ 0 < r∗ < R∗_v for 0 < theta < theta_g/2, 0 < r∗ < R∗_L for theta_g/2 < theta < theta_p/2, theta 0 < theta < theta_p/2. Row 4: Surface D, Phase(s) Liquid, z∗ L∗_a, r∗ R∗_v < r∗ < R∗_g, theta 0 < theta < theta_g/2. Row 5: Surface E, Phase(s) Vapour, z∗ 0 < z∗ < L∗_a, r∗ 0, theta Not applicable. Row 6: Surface F, Phase(s) Vapour, z∗ 0 < z∗ < L∗_a, r∗ 0 < r∗ < R∗_v, theta 0. Row 7: Surface G, Phase(s) Vapour, z∗ 0 < z∗ < L∗_a, r∗ 0 < r∗ < R∗_L, theta theta_p/2. Row 8: Surface H, Phase(s) Liquid, z∗ 0 < z∗ < L∗_a, r∗ R∗_v < r∗ < R∗_g, theta 0. Row 9: Surface I, Phase(s) Solid-liquid, z∗ 0 < z∗ < L∗_a, r∗ R∗_L < r∗ < R∗_g, theta theta_g/2. Row 10: Surface J, Phase(s) Solid-vapor, z∗ 0 < z∗ < L∗_a, r∗ R∗_L, theta theta_p/2 < theta < theta_g/2. Row 11: Surface K, Phase(s) Solid-liquid, z∗ 0 < z∗ < L∗_a, r∗ R∗_g, theta 0 < theta < theta_g/2. Row 12: Surface L, Phase(s) Meniscus, z∗ 0 < z∗ < L∗_a, r∗ R∗_v, theta 0 < theta < theta_g/2. Row 13: Surface M, Phase(s) Meniscus, z∗ 0 < z∗ < L∗_a, r∗ R∗_L, theta 0 < theta < theta_g/2.

We define the pressures in the vapour and liquid phases at $z^*=0$ and $z^* = L_{\textit{a}}^*$ as per

(2.6) \begin{align} p_{{v}}^* &= p_{\textit{v,e}}^*\,\mathrm{on}\,\boldsymbol{A}, \end{align}

(2.7) \begin{align} p_{\textit{l}}^* &= p_{\textit{l,e}}^*\,\mathrm{on}\,\boldsymbol{B}, \end{align}

(2.8) \begin{align} p_{{v}}^* &= p_{\textit{v,c}}^*\,\mathrm{on}\,\boldsymbol{C}, \end{align}

(2.9) \begin{align} p_{\textit{l}}^* &= p_{\textit{l,c}}^*\,\mathrm{on}\,\boldsymbol{D}, \end{align}

where the subscripts e and c indicate that surfaces $\boldsymbol{A}$ and $\boldsymbol{B}$ and surfaces $\boldsymbol{C}$ and $\boldsymbol{D}$ are at the ends of the adiabatic section adjacent to the evaporator and condenser, respectively. The (streamwise) mass flow rates of working fluid in the liquid and vapour phases in the adiabatic section are equal in magnitude and thus satisfy

(2.10) \begin{equation} \int _{0}^{ \theta _{\textit{g}}/2 } \int _{0}^{R_{{v}}^*}v_{z^*,{v}}^*r^*\mathrm{d}r^*\mathrm{d}\theta + \int _{\theta _{\textit{g}}/2}^{{\theta _{\textit{p}}}/2} \int _{0}^{R_{\textit{L}}^*}v_{z^*,{v}}^*r^*\mathrm{d}r^*\mathrm{d}\theta = -\frac {1}{\tilde {\rho }}\int _{0}^{\theta _{\textit{g}}/2}\int _{R_{{v}}^*}^{R_{\textit{g}}^*}v_{z^*,\textit{l}}^*r^*\mathrm{d}r^*\mathrm{d}\theta , \end{equation}

where $\tilde {\rho } = \rho _{{v}}/\rho _{\textit{l}}$ .

Turning to the boundary conditions, the radial symmetry condition applies through line $\boldsymbol{E}$ passing through the centre of the domain at $r^* = 0$ as per

(2.11) \begin{align} v_{r^*,{v},r^*}^* &= v_{\theta ,{v},r^*}^* = v_{z*,{v},r^*}^*= 0\,\mathrm{on}\,\boldsymbol{E}, \end{align}

(2.12) \begin{align} p_{{v},r^*}^*& = 0\,\mathrm{on}\,\boldsymbol{E}, \end{align}

where the last variable in a subscript denotes a partial derivative. Mirror symmetry boundary conditions apply along surfaces $\boldsymbol{F}$ , $\boldsymbol{G}$ and $\boldsymbol{H}$ as per

(2.13) \begin{align} v_{r^*,{v},\theta }^*&=0,v_{\theta ,{v}}^*=0,v_{z^*,{v},\theta } ^*= p_{{v},\theta }^* = 0\,\mathrm{on}\,\boldsymbol{F}, \end{align}

(2.14) \begin{align} v_{r^*,{v},\theta }^*&=0,v_{\theta ,{v}}^*= 0,v_{z^*,{v},\theta }^*=0\,\mathrm{on}\,\boldsymbol{G}, \end{align}

(2.15) \begin{align} v_{r^*,\textit{l},\theta }^*&=0,v_{\theta ,\textit{l}}^*= 0,v_{z^*,\textit{l},\theta }^*=0\,\mathrm{on}\,\boldsymbol{H}. \end{align}

(This is justified below by showing that gravity effects transverse to the flow are small.) The no-slip and impermeability boundary conditions apply along both solid–liquid interfaces and the solid–vapour one as per

(2.16) \begin{align} \boldsymbol{v}_{\textit{l}}^* &= 0\,\mathrm{on}\,\boldsymbol{I}, \end{align}

(2.17) \begin{align} \boldsymbol{v}_{{v}}^* &=0\,\mathrm{on}\,\boldsymbol{J}, \end{align}

(2.18) \begin{align} \boldsymbol{v}_{\textit{l}}^* &=0\,\mathrm{on}\,\boldsymbol{K}. \end{align}

Turning to the boundary conditions along the meniscus, we define two (linearly independent) unit tangent vectors along it, $\hat {\boldsymbol{t}}_{\perp }$ in a (nominally) transverse (constant $z^*$ ) direction and $\hat {\boldsymbol{t}}_{\textit{s}}$ in a (nominally) streamwise (constant $\theta$ ) direction, and the corresponding unit normal pointing into the vapour phase ( $\hat {\boldsymbol{n}}$ ). They are

(2.19) \begin{align} \hat {\boldsymbol{t}}_{\perp } = \frac {R_{{v},\theta }^*/R_{{v}}^* \boldsymbol{\delta }_{r^*}+\boldsymbol{\delta }_{\theta }} {\sqrt {1+\big (R_{{v},\theta }^*/R_{{v}}^*\big )^2}} ;\, \hat {\boldsymbol{t}}_{\textit{s}} = \frac {R_{{v},z^*}^*\boldsymbol{\delta }_{r^*}+ \boldsymbol{\delta }_{z^*}} { \sqrt { 1+R_{{v},z^*}^{*^2} } };\, \hat {\boldsymbol{n}} = \frac { -\boldsymbol{\delta }_{r^*} + R_{{v},\theta }^*/R_{{v}}^*\boldsymbol{\delta }_{\theta } +R_{{v},z^*}^*\boldsymbol{\delta }_{z^*} } {\sqrt { 1+ \big (R_{{v},\theta }^{*}/R_{{v}}^*\big ) ^2 +R_{{v},z^*}^{*^2} } }. \end{align}

Relatedly, twice the mean curvature of the meniscus, which we denote by $\kappa ^*$ , equals $-\boldsymbol{\nabla }\boldsymbol{\cdot }\hat {\boldsymbol{n}}$ .

Along the meniscus, the impermeability condition as well as the continuity of velocities and the continuity of tangential stresses and normal stress apply as per

(2.20) \begin{align} \boldsymbol{v}_{{v}}^*\boldsymbol{\cdot }\hat {\boldsymbol{n}} &= 0\,\mathrm{on}\,\boldsymbol{L}, \end{align}

(2.21) \begin{align} \boldsymbol{v}_{{v}}^*&= \boldsymbol{v}_{\textit{l}}^*\,\mathrm{on}\,\boldsymbol{L}, \end{align}

(2.22) \begin{align} \left (p_{{v}}^*-p_{\textit{l}}^*\right )\hat {\boldsymbol{n}} + \hat {\boldsymbol{n}}\boldsymbol{\cdot }\left (\boldsymbol{\tau }_{\textit{l}}^* -\boldsymbol{\tau }_{{v}}^*\right ) &= \kappa ^*\,\sigma \,\hat {\boldsymbol{n}} \,\mathrm{on}\,\boldsymbol{L}, \end{align}

respectively (Landau & Lifshitz Reference Landau and Lifshitz1988). Here, $\boldsymbol{\tau }_i^*$ is the viscous stress tensor as per

(2.23) \begin{equation} \boldsymbol{\tau }_i^*= \mu _i \left [\boldsymbol{\nabla} ^*\boldsymbol{v}_i^*+ \left (\boldsymbol{\nabla} ^*\boldsymbol{v}_i^*\right )^{\mathrm{T}}\right ], \end{equation}

and $\sigma$ is surface tension. In expanded form, the impermeability condition becomes

(2.24) \begin{equation} -v_{r^*,{v}}^* + \frac {R_{{v},\theta }^*}{R_{{v}}^*}v_{\theta ,{v}}^* +R_{{v},z^*}^* v_{z^*,{v}}^* = 0\,\mathrm{on}\,\boldsymbol{L}, \end{equation}

and the continuity of velocities manifests itself as

(2.25) \begin{equation} v_{r^*,{v}}^* = v_{r^*,\textit{l}}^*;\,v_{\theta ,{v}}^*= v_{\theta ,\textit{l}}^*;\,v_{z^*,{v}}^*=v_{z^*,\textit{l}}^* \,\mathrm{on}\,\boldsymbol{L}. \end{equation}

Dotting the unit tangent vectors into the stress balance along the meniscus (2.22), the tangential stress balances in the transverse and streamwise directions become

(2.26) \begin{equation} \hat {\boldsymbol{t}}_{\perp }\boldsymbol{\cdot }\hat {\boldsymbol{n}}\boldsymbol{\cdot }\left (\boldsymbol{\tau }_{\textit{l}}^* -\boldsymbol{\tau }_{{v}}^*\right )=0,\, \hat {\boldsymbol{t}}_{\textit{s}}\boldsymbol{\cdot }\hat {\boldsymbol{n}}\boldsymbol{\cdot }\left (\boldsymbol{\tau }_{\textit{l}}^* -\boldsymbol{\tau }_{{v}}^*\right )=0\,\mathrm{on}\,\boldsymbol{L} , \end{equation}

respectively. Also, dotting the unit normal vector into the stress balance along the meniscus (2.22), the normal stress balance becomes

(2.27) \begin{equation} p_{{v}}^*-p_{\textit{l}}^* +\hat {\boldsymbol{n}}\boldsymbol{\cdot }\hat {\boldsymbol{n}}\boldsymbol{\cdot }\left (\boldsymbol{\tau }_{\textit{l}}^* -\boldsymbol{\tau }_{{v}}^*\right ) = \kappa ^*\,\sigma \,\mathrm{on}\,\boldsymbol{L}. \end{equation}

2.3. Slowly varying assumption and non-dimensionalisation

Normally, in an AGHP, the arc lengths spanning the bases of the grooves, menisci and tops of the lands are comparable, and of the same scale as the depth of the grooves ( $R_{\textit{g}}^*-R_{\textit{L}}^*$ ). Consequently, a key length scale is the groove pitch,

(2.28) \begin{equation} P^* = \theta _{\textit{p}}R_{\textit{L}}^*. \end{equation}

Also, the pitch of the grooves is normally small compared with the length of the adiabatic section, as is the radius of the lands, i.e. $P^*, R_{\textit{L}}^* \ll L_{\textit{a}}^*$ . We thus define the small parameter

(2.29) \begin{equation} \epsilon = \frac {P^*}{L_{\textit{a}}^*}, \end{equation}

where $\epsilon \ll 1$ . It represents the rate of change of meniscus deflection in the $z^*$ direction, which is of order $P^*$ , over the length of the adiabatic section $L_{\textit{a}}^*$ . The pressure difference across the meniscus (generally) decreases in the axial direction such that its curvature decreases with respect to $z^*$ . (In a gravity aided orientation ( $\phi = \pi /2$ , say), operating conditions where this is not true are discussed in § 7.) Relatedly, generally, the cross-sectional areas of the vapour and liquid phases are decreasing and increasing, respectively, with respect to $z^*$ . Since $\epsilon \ll 1$ , meniscus curvature is slowly varying (decreasing) in the axial direction.

To non-dimensionalise the streamwise velocities, we define

(2.30) \begin{equation} V_{\textit{s},{v}}^* = \frac { \Delta p_{\textit{cap,max}}^* P^*}{\mu _{{v}}}\epsilon ;\, V_{\textit{s},\textit{l}}^* = V_{\textit{s},{v}}^*\tilde {\mu }, \end{equation}

where $\tilde {\mu }$ is the ratio of the vapour-phase viscosity to the liquid-phase viscosity, i.e.

(2.31) \begin{equation} \tilde {\mu }=\frac {\mu _{{v}}}{\mu _{\textit{l}}}. \end{equation}

(Here, $\tilde {\rho }$ and $\tilde {\mu }$ are often small parameters but we do not exploit this.) The transverse terms in the continuity equation, (2.2), must balance the streamwise term due to the acceleration and deceleration of the vapour and liquid flows, respectively. The length scale over which both transverse velocities vary in the transverse plane is $P^*$ because gradients in radial velocities are highest within a pitch of the meniscus. Therefore, the continuity equations dictate that the scales of the (azimuthal and radial) transverse velocities are

(2.32) \begin{equation} V_{\perp ,{v}}^* = \epsilon V_{\textit{s},{v}}^*;\, V_{\perp ,\textit{l}}^* = \epsilon V_{\textit{s},\textit{l}}^*. \end{equation}

Physically, the slowly varying meniscus curvature assumption ( $\epsilon \ll 1)$ caused the transverse velocities to be an order higher in $\epsilon$ than the streamwise one.

We further introduce the following non-dimensional variables:

(2.33) \begin{align} \left (r,\theta ,z\right ) &=\left (\frac {r^*}{P^*},\theta ,\frac {z^*}{L_{\textit{a}}^*}\right )\!, \end{align}

(2.34) \begin{align} \boldsymbol{v}_i &= \overbrace {\frac {v_{r^*,i}^*}{V_{\perp ,i}^*}}^{v_{r,i}}\boldsymbol{{\delta }}_{r} +\overbrace {\frac {v_{\theta ,i}^*}{V_{\perp ,i}^*}}^{v_{\theta ,i}}\boldsymbol{{\delta }}_{\theta } +\overbrace {\frac {v_{z^*,i}^*}{V_{\textit{s},i}^*}}^{v_{z,i}}\boldsymbol{{\delta }}_{z}, \end{align}

(2.35) \begin{align} p_i &= \frac {p_i^*-p_{\textit{l,e}}^*}{ \Delta p_{\textit{cap,max}}^*}. \end{align}

The liquid-phase components of $\boldsymbol{v}_i$ are not scaled to vary between, nominally, 0 and 1 because the vapour-phase pressure drop in an AGHP is normally small compared with the liquid phase one to minimise the thermal resistance of the adiabatic section, which is not captured in (2.30). Furthermore, $R_{{v}}$ , $R_{\textit{L}}$ and $R_{\textit{g}}$ are their dimensional counterparts normalised by $P^*$ (such that $R_{\textit{L}} = 1/\theta _{\textit{p}}$ ) and $L_{i}$ is $L_{i}^*$ normalised by $L_{\textit{a}}^*$ . The (expanded) dimensionless forms of the continuity (2.2) and Navier–Stokes (2.3) equations are given in Appendix A, wherein

(2.36) \begin{equation} \textit{Re}_i = \frac {\rho _i V_{\textit{s},i}^*P^*}{\mu _i} \end{equation}

is the Reynolds number of phase $i$ .

We express the Navier–Stokes equations in the form provided in Appendix A as

(2.37) \begin{align} O(\epsilon ^3 \textit{Re}_i) &=-\frac {\partial p_i}{\partial r}-{\textit{Bo}_{\perp ,i}\sin (\theta )}+O(\epsilon ^2), \end{align}

(2.38) \begin{align} O(\epsilon ^3 \textit{Re}_i) &=-\frac {1}{r}\frac {\partial p_i}{\partial \theta } -{\textit{Bo}_{\perp ,i}\cos (\theta )}+O(\epsilon ^2), \end{align}

(2.39) \begin{align} O(\epsilon \textit{Re}_i) &=-\frac {\partial p_i}{\partial z} +\boldsymbol{\nabla} ^2_{\perp }v_{z,i}-{\textit{Bo}_{\parallel ,i}}+O(\epsilon ^2), \end{align}

where $\boldsymbol{\nabla} ^2_{\perp }$ is the Laplacian operator for the cross-plane ( $r,\theta$ ), and we have defined the transverse and streamwise Bond numbers as ( $i = v\mathrm{,l}$ )

(2.40) \begin{align} \textit{Bo}_{\perp ,i} &=\frac {\rho _i g P^*\cos (\phi)}{\Delta p_{\textit{cap,max}}^*}, \end{align}

(2.41) \begin{align} \textit{Bo}_{\parallel ,i} &= \frac {\rho _i g L_{\textit{a}}^*\sin (\phi)}{\Delta p_{\textit{cap,max}}^*}. \end{align}

We assume that the transverse Bond numbers are small, specifically,

(2.42) \begin{equation} \textit{Bo}_{\perp ,i} = O\big (\epsilon ^3 \textit{Re}_i,\epsilon \big ), \end{equation}

which is a condition always valid in the limit of the AGHP axis becoming parallel to the gravity vector ( $\phi =\pm 90^{\circ }$ ). The transverse-momentum equations (for any value of $\theta$ ) become

(2.43) \begin{align} \frac {\partial p_i}{\partial r}&=O(\epsilon ^3\textit{Re}_i,\epsilon ), \end{align}

(2.44) \begin{align} \frac {1}{r}\frac {\partial p_i}{\partial \theta }&=O(\epsilon ^3\textit{Re}_i,\epsilon ). \end{align}

A convenient expression for pressure follows as

(2.45) \begin{equation} p_i = -\textit{Bo}_{\parallel ,i}z + \xi _i\left (z\right )+O(\epsilon ^3\textit{Re}_i,\epsilon ), \end{equation}

where $\xi _i (z )$ is an unknown function, and the streamwise-momentum equation reduces to

(2.46) \begin{equation} 0=-\frac {\mathrm{d}\xi _i}{\mathrm{d}z}+\boldsymbol{\nabla} ^2_{\perp }v_{z,i}+O(\epsilon \textit{Re}_i,\epsilon ). \end{equation}

Notice that the streamwise and transverse Bond numbers are not independent but are related by $\textit{Bo}_{\parallel ,i} =\textit{Bo}_{\perp ,i} \tan (\phi )/\epsilon$ . This factor of $1/\epsilon$ means that, even if $\textit{Bo}_{\perp ,i}$ is small (gravity terms are negligible in the transverse-momentum equations), it is possible that $\textit{Bo}_{\parallel ,i}$ is not small (retain the gravity terms in the streamwise-momentum equation). (For example, if $\textit{Bo}_{\perp ,i}=O(\epsilon )$ then $\textit{Bo}_{\parallel ,i} = O(1)$ .) This is what we have done in the above, and it allows us to neglect the effect of gravity in the transverse direction and retain it in the streamwise direction in a consistent manner. Note, the streamwise gravity effects also vanish in the limit of a horizontal AGHP ( $\phi =0$ ). The reason we retain the gravity term in the streamwise direction (and $\xi _i$ is later converted back to $p_i$ ) is that it can be essential in the liquid phase for (non-horizontal) AGHPs.

Additionally, observe that the combination of the assumptions of slowly varying meniscus curvature ( $\epsilon \ll 1$ ) and $\textit{Bo}_{\perp ,i} = O (\epsilon ^3\textit{Re}_i,\epsilon )$ has decoupled the leading-order streamwise-momentum equation from the transverse ones and removed the inertial terms from the former, i.e. resulting in quasi-fully developed flow. Importantly, the form of the streamwise-momentum equation together with (2.45) justifies that the symmetry boundary conditions at $\theta =0,\theta _{\textit{p}}/2$ depicted in figure 2(b) are valid to $O(\epsilon )$ .

The dimensionless forms of the boundary conditions on surfaces $\boldsymbol{A}$ – $\boldsymbol{J}$ in table 1 are

(2.47) \begin{align} p_{{v}} &= p_{\textit{v,e}} \,\mathrm{on}\,\boldsymbol{A}, \end{align}

(2.48) \begin{align} p_{\textit{l}} &= 0\,\mathrm{on}\,\boldsymbol{B}, \end{align}

(2.49) \begin{align} p_{{v}} &= p_{\textit{v,c}}\,\mathrm{on}\,\boldsymbol{C}, \end{align}

(2.50) \begin{align} p_{\textit{l}} &= p_{\textit{l,c}}\,\mathrm{on}\,\boldsymbol{D}, \end{align}

(2.51) \begin{align} v_{r,{v},r} &= v_{\theta ,{v},r} = v_{z,{v},r} = p_{{v},r} = 0\,\mathrm{on}\,\boldsymbol{E}, \end{align}

(2.52) \begin{align} v_{r,{v},\theta }&=v_{\theta ,{v}}=v_{z,{v},\theta } = p_{{v},\theta } = 0\,\mathrm{on}\,\boldsymbol{F}, \end{align}

(2.53) \begin{align} v_{r,{v},\theta }&,v_{\theta ,{v}}= v_{z,{v},\theta }=0\,\mathrm{on}\,\boldsymbol{G}, \end{align}

(2.54) \begin{align} v_{r,\textit{l},\theta }&=v_{\theta ,\textit{l}}= v_{z,\textit{l},\theta }=p_{\textit{l},\theta } =0\,\mathrm{on}\,\boldsymbol{H}, \end{align}

(2.55) \begin{align} \boldsymbol{v}_{\textit{l}} &= 0\,\mathrm{on}\,\boldsymbol{I}, \end{align}

(2.56) \begin{align} \boldsymbol{v}_{{v}} &= 0\,\mathrm{on}\,\boldsymbol{J}, \end{align}

(2.57) \begin{align} \boldsymbol{v}_{\textit{l}} &=p_{\textit{l},r}=0\,\mathrm{on}\,\boldsymbol{K} . \end{align}

Along the meniscus, the unit tangent vectors and the unit normal one pointing into the vapour phase written in terms of dimensionless variables are

(2.58) \begin{equation} \hat {\boldsymbol{t}}_{\perp } = \frac {R_{{v},\theta }/R_{{v}} \boldsymbol{\delta }_{r}+\boldsymbol{\delta }_{\theta }} {\sqrt {1+\left (R_{{v},\theta }/R_{{v}}\right )^2}} ;\, \hat {\boldsymbol{t}}_{\textit{s}} = \frac {\epsilon R_{{v},z}\boldsymbol{\delta }_{r}+ \boldsymbol{\delta }_{z}} { \sqrt { 1+\epsilon ^2 R_{{v},z}^{2} } };\, \hat {\boldsymbol{n}} = \frac { -\boldsymbol{\delta }_{r} + R_{{v},\theta }/R_{{v}}\boldsymbol{\delta }_{\theta } +\epsilon R_{{v},z}\boldsymbol{\delta }_{z} } {\sqrt { 1 + \left (R_{{v},\theta }/R_{{v}}\right )^{2} +\epsilon ^2 R_{{v},z}^2 } }. \end{equation}

Moreover, the impermeability and continuity of velocity conditions along the meniscus read

(2.59) \begin{align} -v_{r,{v}} + \frac {R_{{v},\theta }}{R_{{v}}}v_{\theta ,{v}} +R_{v,z} v_{z,{v}} &= 0\,\mathrm{on}\,\boldsymbol{L}, \end{align}

(2.60) \begin{align} v_{r,{v}}&= \tilde {\mu }v_{r,\textit{l}};\,v_{\theta ,{v}}= \tilde {\mu }v_{\theta ,\textit{l}};\,v_{z,{v}}=\tilde {\mu }v_{z,\textit{l}} \,\mathrm{on}\,\boldsymbol{L}. \end{align}

Furthermore, the dimensionless analogues of the stress balances in the transverse tangential, streamwise tangential and normal directions (see (2.26) and (2.27)) are provided in Appendix B, where the modified capillary number therein and used below is

(2.61) \begin{equation} \widehat {\textit{Ca} }= \frac { \mu _{{v}} V_{\textit{s,v}}^* }{\epsilon \sigma }, \end{equation}

i.e. $\widehat {\textit{Ca} }=\Delta p_{\textit{cap,max}}^*P^*/\sigma$ , which is imposed to be $O (1 )$ .

The constraint that the mass flow rates of vapour and liquid are the same magnitude, at every $z$ , (2.10), in dimensionless form, is

(2.62) \begin{equation} \int _{0}^{ \theta _{\textit{g}}/{2} } \int _{0}^{R_{{v}}}v_{z,{v}}r\mathrm{d}r\mathrm{d}\theta + \int _{\theta _{\textit{g}}/{2}}^{{\theta _{\textit{p}}}/2} \int _{0}^{R_{\textit{L}}}v_{z,{v}}r\mathrm{d}r\mathrm{d}\theta =-\frac {\tilde {\mu }}{\tilde {\rho }} \int _{0}^{\theta _{\textit{g}}/2}\int _{R_{{v}}}^{R_{\textit{g}}}v_{z,\textit{l}}r\mathrm{d}r\mathrm{d}\theta . \end{equation}

The left side of (2.62) is the dimensionless volumetric flow rate of the vapour phase, i.e. $Q_{{v}} = Q_{{v}}^*/(V_{\textit{s,v}}^*P^{*2})$ , a positive quantity, and the double integral (in and of itself) on the right side of it is that of the liquid phase, i.e. $Q_{\textit{l}} = Q_{\textit{l}}^*/(V_{\textit{s,l}}^*P^{*2})$ , a negative quantity. Therefore,

(2.63) \begin{equation} Q_{{v}} +\frac {\tilde {\mu }}{\tilde {\rho }}Q_{\textit{l}} = 0. \end{equation}

4.1. Solution for $\theta _{\textit{m}}=0$

For the case $\theta _{\textit{m}}=0$ , where the radius of the meniscus is equal to that of the land (protrusion angle is zero), we append a superscript 0 to quantities and the streamwise-momentum equations, (4.2), are

(4.14) \begin{equation} \frac {1}{r}\frac {\partial }{\partial r} \left (r\frac {\partial V_{z,i}^{0}}{\partial r}\right ) +\frac {1}{r^2}\frac {\partial ^2 V_{z,i}^{0}}{\partial \theta ^2}= 1. \end{equation}

They are subject to boundary conditions (4.3)–(4.9) applied at $O (\theta _{\textit{m}}^{0} )$ and, along the meniscus, which has a radius $R_{\textit{L}}$ at this order as per surface $\boldsymbol{M}'$ in figure 2, (4.11) and (4.12) simplify to

(4.15) \begin{align} V_{z,{v}}^{0}&= \tilde {\mu } \tilde {\xi }^{0}\,V_{z,\textit{l}}^{0}\,\mathrm{on}\,\boldsymbol{M}', \end{align}

(4.16) \begin{align} \frac {\partial V_{z,{v}}^{0}}{\partial r}-\tilde {\xi }^{0}\,\frac {\partial V_{z,\textit{l}}^{0}}{\partial r} &=0\,\mathrm{on}\,\boldsymbol{M}'. \end{align}

We express the solutions for $V_{z,{v}}^{0}$ and $V_{z,\textit{l}}^{0}$ as sums of those for (one-dimensional) pressure- and/or gravity-driven flows through a no-slip tube and annulus, respectively, plus perturbation components denoted $\tilde {V}_{z,i}$ as per

(4.17) \begin{align} V_{z,{v}}^{0} &= \frac {r^2-R_{\textit{L}}^2}{4} +\tilde {V}_{z,{v}}^{0}, \end{align}

(4.18) \begin{align} V_{z,\textit{l}}^{0}&= \frac {r^2-R_{\textit{L}}^2}{4} +\frac {R_{\textit{L}}^2-R_{\textit{g}}^2}{4}\frac {\ln \left (r/R_{\textit{L}}\right )}{\ln \left (R_{\textit{g}}/R_{\textit{L}}\right )} +\tilde {V}_{z,\textit{l}}^{0}. \end{align}

The perturbation problems for $\tilde {V}_{z,{v}}^{0}$ and $\tilde {V}_{z,\textit{l}}^{0}$ are ${\nabla} ^2 \tilde {V}_{z,i}^{0} = 0$ subject to

(4.19) \begin{align} \tilde {V}_{z,{v},r}^{0}&= 0\,\mathrm{on}\,\boldsymbol{E}', \end{align}

(4.20) \begin{align} \tilde {V}_{z,{v},\theta }^{0}& = 0\,\mathrm{on}\,\boldsymbol{F}', \end{align}

(4.21) \begin{align} \tilde {V}_{z,{v},\theta }^{0}&=0\,\mathrm{on}\,\boldsymbol{G}', \end{align}

(4.22) \begin{align} \tilde {V}_{z,\textit{l},\theta }^{0}&=0\,\mathrm{on}\,\boldsymbol{H}', \end{align}

(4.23) \begin{align} \tilde {V}_{z,\textit{l}}^{0} &=-\frac {r^2-R_{\textit{L}}^2}{4} -\frac {R_{\textit{L}}^2-R_{\textit{g}}^2}{4}\frac {\ln \left (r/R_{\textit{L}}\right )}{\ln \left (R_{\textit{g}}/R_{\textit{L}}\right )}\,\mathrm{on}\,\boldsymbol{I}', \end{align}

(4.24) \begin{align} \tilde {V}_{z,{v}}^{0} &= 0\,\mathrm{on}\,\boldsymbol{J}', \end{align}

(4.25) \begin{align} \tilde {V}_{z,\textit{l}}^{0} &=0\,\mathrm{on}\,\boldsymbol{K}', \end{align}

and, as per (4.15) and (4.16), along the meniscus

(4.26) \begin{align} \tilde {V}_{z,{v}}^{0} &= \tilde {\mu }\tilde {\xi }^{0}\,\tilde {V}_{z,\textit{l}}^{0} \,\mathrm{on}\,\boldsymbol{M}', \end{align}

(4.27) \begin{align} \frac {\partial \tilde {V}_{z,{v}}^{0}}{\partial r} +\frac {R_{\textit{L}}}{2} -\tilde {\xi }^{0}\left [\frac {\partial \tilde {V}_{z,\textit{l}}^{0}}{\partial r}+\overbrace {\frac {R_{\textit{L}}}{2} +\frac {R_{\textit{L}}^2-R_{\textit{g}}^2}{4R_{\textit{L}}\ln \left (R_{\textit{g}}/R_{\textit{L}}\right )}}^{C_a}\right ]&=0\,\mathrm{on}\,\boldsymbol{M}'\,, \end{align}

where $C_a$ has been defined for later use.

Using separation of variables, the solution for $\tilde {V}_{z,{v}}^{0}$ takes the form

(4.28) \begin{equation} \tilde {V}_{z,{v}}^{0}= \left [A \ln (r) + B\right ]\left (C\theta +D\right ) + \left [E \sin \left (\alpha \theta \right )+F \cos \left (\alpha \theta \right )\right ] \left (\textit{Gr}^{-\alpha } + \textit{Hr}^{\alpha }\right ), \end{equation}

where $\alpha$ is the (finite) separation constant. We use (and reuse) the letters $A-$ $H$ to denote constants and their products as the results simplify. Applying the homogeneous boundary conditions given by (4.19)–(4.21) it follows that

(4.29) \begin{equation} \tilde {V}_{z,{v}}^{0} = \sum _{l=0}^{\infty }A_l r^{\alpha _l}\cos \left (\alpha _l \theta \right ), \end{equation}

where

(4.30) \begin{equation} \alpha _l = 2l\pi /\theta _{\textit{p}}\,\mathrm{for}\,l \in \mathbb{N}_0. \end{equation}

Next, since there are inhomogeneous boundary conditions on surfaces $\boldsymbol{I}'$ and $\boldsymbol{M}'$ on the $\tilde {V}_{z,\textit{l}}^{0}$ problem, we decompose it into

(4.31) \begin{equation} \tilde {V}_{z,\textit{l}}^{0} = \hat {V}_{z,\textit{l}}^{0} +\breve {V}_{z,\textit{l}}^{0}. \end{equation}

Here, $\hat {V}_{z,\textit{l}}^{0}$ accommodates the inhomogeneous boundary condition on surface $\boldsymbol{I}'$ as per (4.23) and $\breve {V}_{z,\textit{l}}^{0}$ contributes an infinite set of unknown constants appearing below in the triple-series equations that satisfy the three boundary conditions that apply at $r = R_{\textit{L}}$ on surfaces $\boldsymbol{J}'$ and $\boldsymbol{M}'$ . Via separation of variables, where the boundary conditions on constant $r$ surfaces are homogeneous, the solution for $\hat {V}_{z,\textit{l}}^{0}$ takes the form

(4.32) \begin{align} \hat {V}_{z,\textit{l}}^{0} = & \left [A \ln (r) + B\right ]\left (C\theta +D\right ) \nonumber \\ & + \left \{E\sin \left [\beta \ln (r)\right ]+F\cos \left [\beta \ln (r)\right ]\right \} \left [G \sinh \left (\beta \theta \right )+H \cosh \left (\beta \theta \right )\right ], \end{align}

where $\beta$ is the (finite) separation constant. We apply the homogeneous Dirichlet and Neumann boundary conditions in $r$ as per (4.25) and the homogeneous form of (4.27), i.e.

(4.33) \begin{equation} \partial \hat {V}_{z,\textit{l}}^{0}/\partial r =0 \mathrm{\,at\,} r=R_{\textit{L}}, \end{equation}

respectively, and the homogeneous one in $\theta$ given by (4.22)

(4.34) \begin{equation} \hat {V}_{z,\textit{l}}^{0} = \sum _{m=1}^{\infty } B_m \left \{ \sin \left [\beta _m\ln (r)\right ]-\tan \left [\beta _m \ln \left (R_{\textit{g}}\right )\right ] \cos \left [\beta _m\ln (r)\right ]\right \}\cosh \left (\beta _m \theta \right ), \end{equation}

where the eigenfunctions, subsequently denoted by $\phi _m$ , are enclosed in squiggly brackets and the eigenvalues ( $\beta _m$ ) are the positive roots of the eigencondition which takes the form

(4.35) \begin{equation} 1+\tan \left [\beta \ln \left (R_{\textit{g}}\right )\right ] \tan \left [\beta \ln \left (R_{\textit{L}}\right )\right ]=0. \end{equation}

It follows from this and a product-to-sum (trigonometric) identity that

(4.36) \begin{equation} \beta _m = \frac {\pi \left (2m-1\right )}{2\ln \left (R_{\textit{g}}/ R_{\textit{L}}\right )}\,\mathrm{for}\,m \in \mathbb{N}_1. \end{equation}

(This result and the fact that (4.34) is an odd function of $\beta _m$ causes the sum within it to be from $m=1$ to $m$ = $\infty$ .) The inhomogeneous boundary condition on surface $\boldsymbol{I}'$ takes the form

(4.37) \begin{align} &\sum _{m=1}^{\infty } B_m \left \{ \sin \left [\beta _m\ln (r)\right ]-\tan \left [\beta _m \ln \left (R_{\textit{g}}\right )\right ] \cos \left [\beta _m\ln (r)\right ]\right \}\cosh \left (\beta _m \theta _{\textit{g}}/2\right ) \nonumber \\ & \ =-\frac {r^2-R_{\textit{L}}^2}{4} -\frac {R_{\textit{L}}^2-R_{\textit{g}}^2}{4}\frac {\ln \left (r/R_{\textit{L}}\right )}{\ln \left (R_{\textit{g}}/R_{\textit{L}}\right )}. \end{align}

Upon casting the (regular) ordinary differential equation yielding the $r$ -dependent portion of the solution for $\hat {V}_{z,\textit{l}}$ in the requisite form, the Sturm–Liouville theorem dictates that the (simple) eigenfunctions are orthogonal with respect to the weighting function $1/r$ ; consequently,

(4.38) \begin{align} B_m = &\frac {1}{\left [\langle \phi _m,\phi _m\rangle \cosh \left (\beta _m \theta _{\textit{g}}/2\right )\right ]}\int _{R_{\textit{L}}}^{R_{\textit{g}}}\left [\frac {R_{\textit{L}}^2-r^2}{4r} -\frac {R_{\textit{L}}^2-R_{\textit{g}}^2}{4r}\frac {\ln \left (r/R_{\textit{L}}\right )}{\ln \left (R_{\textit{g}}/R_{\textit{L}}\right )} \right ] \times \end{align}

$\begin{align*} &\left \{ \sin \left [\beta _m\ln (r)\right ]-\tan \left [\beta _m \ln \left (R_{\textit{g}}\right )\right ] \cos \left [\beta _m\ln (r)\right ]\right \}\mathrm{d}r, \end{align*}$

where the inner product $\langle \phi _m,\phi _m\rangle$ is defined by

(4.39) \begin{equation} \langle \phi _m,\phi _m\rangle = \int _{R_{\textit{L}}}^{R_{\textit{g}}}\frac {\left \{ \sin \left [\beta _m\ln (r)\right ]-\tan \left [\beta _m \ln \left (R_{\textit{g}}\right )\right ] \cos \left [\beta _m\ln (r)\right ]\right \}^2}{r}\mathrm{d}r. \end{equation}

It follows that

(4.40) \begin{align} &\hat {V}_{z,\textit{l}}^{0} =\sum _{m=1}^{\infty } \hat {B}_m \sin \left [\beta _m \ln \left (r/R_{\textit{g}}\right )\right ] \cosh \left (\beta _m \theta \right )/\cosh \left (\beta _m\theta _{\textit{g}}/2\right ), \end{align}

where

(4.41) \begin{align} \hat {B}_m=&\frac {4\beta _m R_{\textit{g}}^2 \ln \big(\tilde {R}\,\big )-\left [2R_{\textit{L}}^2\beta _m^2\ln \big(\tilde {R}\,\big )+ \big (R_{\textit{g}}^2-R_{\textit{L}}^2\big ) \big (4+\beta _m^2\big)\right ](-1)^m}{2\beta _m^2\big (4+\beta _m^2\big )\left [\ln \big(\tilde {R}\,\big )\right ]^2 }, \end{align}

and $\tilde {R} = R_{\textit{L}}/R_{\textit{g}}$ .

Separation of variables (for homogeneous boundary conditions on surfaces of constant  $\theta$ ) dictates that $\breve {V}_{z,\textit{l}}^{0}$ takes the form of $\tilde {V}_{\textit{z,v}}^{0}$ specified in (4.28), but with the eigenvalues denoted by $\gamma$ . Thus,

(4.42) \begin{equation} \breve {V}_{z,\textit{l}}^{0}= \left [A \ln (r) +B\right ]\left (C\theta +D\right ) + \left [E \sin \left (\gamma \theta \right )+F \cos \left (\gamma \theta \right )\right ] \left (\textit{Gr}^{-\gamma } + \textit{Hr}^{\gamma }\right )\!. \end{equation}

Applying the homogeneous Neumann boundary condition on surface $\boldsymbol{H}'$ and the homogeneous Dirichlet ones on surfaces $\boldsymbol{I}'$ and $\boldsymbol{K}'$

(4.43) \begin{equation} \breve {V}_{z,\textit{l}}^{0}= \sum _{n=0}^{\infty } C_n \big (r^{\gamma _n} - R_{\textit{g}}^{2\gamma _n}r^{-\gamma _n}\big ) \cos \left (\gamma _n \theta \right ), \end{equation}

where the eigencondition imposes that

(4.44) \begin{equation} \gamma _n = \frac {\pi \left (2n+1\right )}{\theta _{\textit{g}}} \quad \mathrm{for} \quad n \in \mathbb{N}_0. \end{equation}

The boundary conditions at $r=R_{\textit{L}}$ close the coupled $V_{z,{v}}^{0}$ and $V_{z,\textit{l}}^{0}$ problems as per velocity continuity along $\boldsymbol{M}'$ (4.26), no-slip along $\boldsymbol{J}'$ (4.24) and shear stress continuity along $\boldsymbol{M}'$ (4.27). These constitute triple-series equations with $\tilde {\xi }^{0}$ -dependent forcings on the right sides of (4.26) and (4.27) that we solve to compute $A_l$ and $C_n$ in (4.29) and (4.43), respectively. We avoid solving for unbounded coefficients (and therefore avoid any unbounded terms in the boundary conditions) by defining new coefficients

(4.45) \begin{align} \hat {A}_l &=A_l R_{\textit{L}}^{\alpha _l}, \end{align}

(4.46) \begin{align} \hat {C}_n &= C_n R_{\textit{g}}^{2\gamma _n}R_{\textit{L}}^{-\gamma _n}, \end{align}

such that the boundary conditions (4.26), (4.24), (4.27) become

(4.47) \begin{align} \sum _{l=0}^{\infty }\hat {A}_l \cos \left (\alpha _l \theta \right )- \tilde {\mu } \tilde {\xi }^{0}\sum _{n=0}^{\infty } \hat {C}_n \big (\tilde {R}^{2\gamma _n}-1\big ) \cos \left (\gamma _n \theta \right )=&\tilde {\mu }\tilde {\xi }^{0}\sum _{m=1}^{\infty }\frac { \hat {B}_m (-1)^m\cosh \left (\beta _m \theta \right )}{\cosh \left (\beta _m\theta _{\textit{g}}/2\right )}\nonumber \\ &\mathrm{on}\,\boldsymbol{M}', \end{align}

(4.48) \begin{align} \sum _{l=0}^{\infty }\hat {A}_l \cos \left (\alpha _l \theta \right )=&0\,\mathrm{on}\,\boldsymbol{J}', \end{align}

(4.49) \begin{align} \sum _{l=0}^{\infty }\hat {A}_l \frac {\alpha _l}{R_{\textit{L}}}\cos \left (\alpha _l \theta \right )-\tilde {\xi }^{0}\sum _{n=0}^{\infty }\hat {C}_n\frac {\gamma _n}{R_{\textit{L}}} \big (\tilde {R}^{2\gamma _n}+1\big )\cos \left (\gamma _n \theta \right )=& \frac {R_{\textit{L}}}{2}\big (\tilde {\xi }^{0}-1\big )\nonumber \\ &-\frac {\tilde {\xi }^{0}\big (R_{\textit{L}}^2-R_{\textit{g}}^2\big )}{4R_{\textit{L}}\ln \big(\tilde {R}\,\big )}\,\mathrm{on}\,\boldsymbol{M}'. \end{align}

Discretising the triple-series equations in $\theta$ to resolve $\hat {A}_l$ and $\hat {C}_n$ produces a near-singular coefficient matrix. Consequently, we multiply the boundary conditions on fluid velocity, i.e. (4.47) and (4.48), by the eigenfunction $\cos (\alpha _r \theta )$ and that imposing continuity of shear stress, i.e. (4.49), by the eigenfunction $\cos (\gamma _s \theta )$ and integrate them over their ranges of validity in $\theta$ . (Here, $\alpha _r$ and $\gamma _{s}$ obey (4.30) and (4.44), respectively, with the new indices.) A truncated form of the triple-series equations becomes

(4.50) \begin{align} &\sum _{l=0}^{N-1}\hat {A}_ld(l,r) -\tilde {\mu }\tilde {\xi }^{0}\sum _{n=0}^{N-1}\hat {C}_n \big (\tilde {R}^{2\gamma _n}-1\big )e\left (n,r\right )=\tilde {\mu }\tilde {\xi }^{0} \sum _{m=1}^{2N} \hat {B}_m (-1)^mf(m,r), \end{align}

(4.51) \begin{align} &\sum _{l=0}^{N-1} \hat {A}_l k(l,r)=0, \end{align}

(4.52) \begin{align} &\sum _{l=0}^{N-1}\hat {A}_l\frac {\alpha _l}{R_{\textit{L}}}g (l,s) -\tilde {\xi }^{0}\sum _{n=0}^{N-1} \hat {C}_n \frac {\gamma _n}{R_{\textit{L}}}\big (\tilde {R}^{2\gamma _n}+1\big )h (n,s)=\left [ \frac {R_{\textit{L}}}{2}\big (\tilde {\xi }^{0}-1\big ) -\frac {\tilde {\xi }^{0}\big (R_{\textit{L}}^2-R_{\textit{g}}^2\big )}{4R_{\textit{L}}\ln \big(\tilde {R}\,\big )}\right ]j(s), \end{align}

where the $2N$ upper limit on the sum on the right side of (4.50) is arbitrarily large. The new functions $d (l,r )$ to $k (l,r )$ are provided in integral and integrated forms in Supplementary Information S1 is available at https://doi.org10.1017/jfm.2026.11806.

We define $\hat {\boldsymbol{A}} = (\hat {A}_0, \hat {A}_1, \ldots , \hat {A}_{N-1})^{\mathrm{T}} \in \mathbb{R}^{N}$ and $\hat {\boldsymbol{C}} = (\hat {C}_0, \hat {C}_1, \ldots , \hat {C}_{N-1})^{\mathrm{T}} \in \mathbb{R}^{N}$ . Then, $\boldsymbol{M}\boldsymbol{a} = \boldsymbol{f}$ , where $\boldsymbol{a} = (\hat {\boldsymbol{A}},\hat {\boldsymbol{C}})^{\mathrm{T}}$ and the left sides of the triple-series equations determine $\boldsymbol{M} \in \mathbb{R}^{2N}\times \mathbb{R}^{2N}$ and their right sides the forcing $\boldsymbol{f} \in \mathbb{R}^{2N}$ . The first $N$ rows of $\boldsymbol{M}$ and $\boldsymbol{f}$ correspond to the sum of (4.50) and (4.51) for $r$ spanning from 0 to $N-1$ . The subsequent $N$ rows of it correspond to (4.52) for $s$ spanning from 0 to $N-1$ .

Matching the magnitudes of the mass flow rates in the two phases, as per (4.13), resolves $\tilde {\xi }^{0}$ via iteration on it as per

(4.53) \begin{equation} \tilde {\xi }^{0} = -\frac {\tilde {\rho }\tilde {Q}_{{v}}^{0}}{\tilde {\mu }\tilde {Q}_{\textit{l}}^{0}}. \end{equation}

Here,

(4.54) \begin{align} \tilde {Q}_{{v}}^{0} &= \int _0^{\frac {\theta _{\textit{p}}}{2}} \int _0^{R_{\textit{L}}}V_{z,{v}}^{0}\left (r,\theta \right )r \mathrm{d}r\mathrm{d}\theta, \end{align}

(4.55) \begin{align} \tilde {Q}_{\textit{l}}^{0} &= \int _{0}^{\frac {\theta _{\textit{g}}}{2}} \int _{R_{\textit{L}}}^{R_{\textit{g}}}V_{z,\textit{l}}^{0}\left (r,\theta \right )r \mathrm{d}r\mathrm{d}\theta . \end{align}

Evaluating the integrals, we find that

(4.56) \begin{align} &\tilde {Q}_{{v}}^{{0}} = \frac {\hat {A}_0 R_{\textit{L}}^2\theta _{\textit{p}}}{4} -\frac {R_{\textit{L}}^4 \theta _{\textit{p}}}{32}, \end{align}

(4.57) \begin{align} &\tilde {Q}_{\textit{l}}^{{0}} = \overbrace { \frac {\theta _{\textit{g}}}{32} \left [R_{\textit{L}}^{4}-R_{\textit{g}}^{4} -\frac {\big (R_{\textit{g}}^{2}-R_{\textit{l}}^{2}\big)^2}{\ln \big(\tilde {R}\,\big )}\right ]}^{K_1}+ \overbrace {\sum _{m=1}^{2N} \frac {-\hat {B}_m\left [2R_{\textit{L}}^{2}(-1)^m+\beta _mR_{\textit{g}}^2\right ]\tanh \left (\beta _m \theta _{\textit{g}}/2\right )}{\beta _m \left (4+\beta _m^2\right ) }}^{K_2}\nonumber \\ &\,+\overbrace { \sum _{n=0}^{N-1}\frac {\hat {C}_n\left [R_{\textit{L}}^{{2}} \tilde {R}^{2\gamma _n}\left (2-\gamma _n\right ) +2R_{\textit{g}}^{{2}}\tilde {R}^{{\gamma _n}}\gamma _n -R_{\textit{L}}^{{2}}\left (2+\gamma _n\right ) \right ]\sin \left (\gamma _n\theta _{\textit{g}}/2\right )}{\gamma _n\left (\gamma _n^2-4\right )}}^{K_3}, \end{align}

where $K_1$ , $K_2$ and $K_3$ are defined for later use. The convergence criterion for a sufficiently large value of $N$ at which to truncate all sums is determined from the behaviour of $\tilde {\xi }^{0}$ as a function of $N$ for the ‘baseline’ set of AGHP parameters specified in § 7. Since the change in the magnitude of $\tilde {\xi }^{0}$ from $N=900$ to $N=1000$ is 2.57 $\times$ 10 $^{-9}$ times its value at $N=900$ , we conservatively set $N=1000$ . Further validation of the solutions of the triple-series equations follows at the end of the next section. As per our Data Availability Statement, Matlab code for computing $\tilde {Q}_{{v}}^{0}$ and $\tilde {Q}_{\textit{l}}^{0}$ when the working fluid, operating temperature and geometry and orientation of an AGHP are prescribed is available. The capillary limit then follows from (6.20).

4.2. Solution for arbitrary $\theta _{\textit{m}}$

We numerically resolve the problem from § 4 for arbitrary values of $\theta _{\textit{m}}$ . To facilitate our solution in MATLAB, we define the velocities and volumetric flow rates slightly differently as per Supplementary Information S2 but revert back to the same variables henceforth. The end result is an array of $\tilde {\xi }$ , $\tilde {Q}_{{v}}$ and $\tilde {Q}_{\textit{l}}$ as a function of $\theta _{\textit{m}}$ (or as per (3.12) and (3.13) $\Delta p$ ) for a prescribed set of parameters. Finally, we outline a boundary perturbation as an alternative method to resolve the effects of non-zero (but small) $\theta _{\textit{m}}$ in the Supplementary Information, section S3.

Figure 4.

Row 1: ( $\theta _{\textit{m}}=0$ ) Dimensional velocities in both phases from the series solution when $N$ = 5000, 10 000 and 15 000 and local analysis in the vicinity of the triple-contact line vs. (angular) distance from it ( $\theta - \theta _{\textit{g}}/2$ ) for baseline case. (b,c) $|\hat {A}_l|$ ( $|\hat {C}_n|$ ) vs. $l$ (and $n$ ) for values of it up to 5000, 10 000 and 15 000 and red lines corresponding $O(1/l)$ (and $O(1/n)$ ).

6.1. Evaporator and condenser

Always, as per the above, in the evaporator and condenser, we invoke the usual assumption of a no-slip boundary condition on the vapour phase along a meniscus of radius of $R_{\textit{L}}^*$ . Then, as per the Darcy friction factor for laminar flow through a tube (64/ $\textit{Re}_{\textit{D,v}}$ , where $\textit{Re}_{\textit{D,v}}$ is the Reynolds number based on the diameter of the vapour phase, i.e. $2\rho _{{v}}\overline {v}_{z,{v}}^{*}R_{\textit{L}}^*/\mu _{{v}}$ , where $\overline {v}_{z,{v}}^{*}$ is the mean velocity of the vapour phase) (Bird, Stewart & Lightfoot 2007)

(6.1) \begin{align} \frac {\mathrm{d}p_{{v}}^{*}}{\mathrm{d}z^*}=&- \frac {8\mu _{{v}}q^*_{\textit{t}}} {\pi \rho _{{v}} R_{\textit{L}}^{*^4}h_{\textit{lv}}} \left (\frac {z^*}{L_{\textit{e}}^*}+1\right )- \rho _{{v}}g\sin (\phi)\,\mathrm{for}\,-L_{\textit{e}}^* \leqslant z^* \leqslant 0\,\mathrm{(evaporator)}, \end{align}

(6.2) \begin{align} \frac {\mathrm{d}p_{{v}}^{*}}{\mathrm{d}z^*}=&- \frac {8\mu _{{v}}q^*_{\textit{t}}} {\pi \rho _{{v}} R_{\textit{L}}^{*^4}h_{\textit{lv}}} \left ( 1-\frac {z^*-L_{\textit{a}}^*}{L_{\textit{c}}^*} \right )- \rho _{{v}}g\sin (\phi)\,\mathrm{for}\,L_{\textit{a}}^* \leqslant z^* \leqslant L_{\textit{a}}^*+ L_{\textit{c}}^*\,\mathrm{(condenser)}. \end{align}

Here, we use $q_{\textit{t}}^*$ , the heat load on the evaporator, rather than $q_{\textit{cap}}^*$ because the results are valid for $q_{\textit{t}}^* \leqslant q_{\textit{cap}}^*$ .) Relatedly,

(6.3) \begin{align} p_{{v}}^{*} &= p_{\textit{sat}}^*\big(T_{\textit{o}}^*\big) - \frac {8\mu _{{v}}q^*_{\textit{t}}} {\pi \rho _{{v}} R_{\textit{L}}^{*^4}h_{\textit{lv}}}\left (\frac {z^{*^2}}{2 L_{\textit{e}}^*}+z^* +\frac {L_{\textit{e}}^{*}}{2} \right )-\rho _{{v}}g\sin (\phi)\left (L_{\textit{e}}^*+z^*\right )\,\mathrm{for}\,-L_{\textit{e}}^* \leqslant z^* \leqslant 0\nonumber \\ &\mathrm{(evaporator)}, \end{align}

where $p_{\textit{sat}}^* (T_{\textit{o}}^* )$ is the saturation pressure of the working fluid at the operating temperature of the AGHP ( $T_{\textit{o}}^*$ ). Also, the pressure differences driving the vapour flow through the evaporator and condenser are

(6.4) \begin{align} \Delta p_{\textit{v,e}}^{*} &=\frac {4\mu _{{v}}q^*_{\textit{t}}L_{\textit{e}}^*} {\pi \rho _{{v}} R_{\textit{L}}^{*^4}h_{\textit{lv}}} +\rho _{{v}} g \sin (\phi)L_{\textit{e}}^*, \end{align}

(6.5) \begin{align} \Delta p_{\textit{v,c}}^{*} &=\frac {4\mu _{{v}}q^*_{\textit{t}}L_{\textit{c}}^*} {\pi \rho _{{v}} R_{\textit{L}}^{*^4}h_{\textit{lv}}} +\rho _{{v}} g \sin (\phi)L_{\textit{c}}^*, \end{align}

respectively. The latter expressions resolve the second and sixth terms on the right side of (1.4) and are based on the mean of the linearly increasing (decreasing) mass flow rate of vapour in the evaporator (condenser). Relatedly, the vapour pressure at the beginning of the adiabatic section ( $z^* =0$ ) follows as

(6.6) \begin{equation} p_{\textit{v,e}}^{*} = p_{\textit{sat}}^*\big(T_{\textit{o}}^*\big)- \frac {4\mu _{{v}}q^*_{\textit{t}}L_{\textit{e}}^*} {\pi \rho _{{v}} R_{\textit{L}}^{*^4}h_{\textit{lv}}} -\rho _{{v}} g \sin (\phi)L_{\textit{e}}^*. \end{equation}

In the adiabatic section, where the flow rate of vapour is at its maximum, for cases A and B,

(6.7) \begin{equation} \Delta p_{\textit{v,a}}^{*} =\frac {8\mu _{{v}}q^*_{\textit{t}}L_{\textit{a}}^*} {\pi \rho _{{v}} R_{\textit{L}}^{*^4}h_{\textit{lv}}} +\rho _{{v}} g \sin (\phi)L_{\textit{a}}^*. \end{equation}

6.2. Case A

We model the vapour phase in the adiabatic section by analogy to the foregoing treatment of it in the evaporator and condenser; hence,

(6.8) \begin{equation} \frac {\mathrm{d}p_{{v}}^{*^{\mathrm{A}}}}{\mathrm{d}z^*}=- \frac {8\mu _{{v}}q^*_{\textit{t}} } {\pi \rho _{{v}} R_{\textit{L}}^{*^4}h_{\textit{lv}}}- \rho _{{v}}g\sin (\phi) \quad \mathrm{for}\quad 0 \leqslant z^* \leqslant L_{\textit{a}}^*\,\mathrm{(adiabatic\,section)}. \end{equation}

The analogues of (6.3) for the vapour pressure in the adiabatic and condenser sections are

(6.9) \begin{align} p_{{v}}^{*^{\mathrm{A}}} &= p_{\textit{sat}}^*\big(T_{\textit{o}}^*\big) - \frac {8\mu _{{v}}q^*_{\textit{t}}} {\pi \rho _{{v}} R_{\textit{L}}^{*^4}h_{\textit{lv}}}\left (\frac {L_{\textit{e}}^{*}}{2}+z^* \right )- \rho _{{v}}g\sin (\phi)\left (L_{\textit{e}}^*+z^*\right )\quad \mathrm{for}\quad 0 \leqslant z^* \leqslant L_{\textit{a}}^*\nonumber \\ &\mathrm{(adiabatic\,section)}, \end{align}

(6.10) \begin{align} p_{{v}}^{*^{\mathrm{A}}} &= p_{\textit{sat}}^*\big(T_{\textit{o}}^*\big) - \frac {8\mu _{{v}}q^*_{\textit{t}}} {\pi \rho _{{v}} R_{\textit{L}}^{*^4}h_{\textit{lv}}}\left [\frac {L_{\textit{e}}^{*}}{2}+L_{\textit{a}}^*+\frac {\left (z^*-L_{\textit{a}}^*\right ) \left (L_{\textit{a}}^*+2L_{\textit{c}}^*-z^*\right )}{2L_{\textit{c}}^*} \right ] \nonumber \\ &\quad -\rho _{{v}}g\sin (\phi)\left (L_{\textit{e}}^*+z^*\right ) \quad \mathrm{for} \quad 0 \leqslant z^* \leqslant L_{\textit{a}}^*+L_{\textit{c}}^* \,\mathrm{(condenser)}. \end{align}

Relatedly, the total pressure drop across the vapour phase is

(6.11) \begin{equation} \Delta p_{\textit{v,t}}^{*^{\mathrm{A}}} =\frac {8\mu _{{v}}q^*_{\textit{t}}L_{\textit{eff}}^*} {\pi \rho _{{v}} R_{\textit{L}}^{*^4}h_{\textit{lv}}} +\rho _{{v}} g \sin (\phi)L_{\textit{t}}^*, \end{equation}

where the effective length of the AGHP, $L_{\textit{eff}}^*$ , is $L_{\textit{e}}^*/2+L_{\textit{a}}^*+L_{\textit{c}}^*/2$ and the total length of it, $L_{\textit{t}}^*$ , is $L_{\textit{e}}^*+L_{\textit{a}}^*+L_{\textit{c}}^*$ .

We, to be consistent with the literature, apply Darcy’s law to the flow of liquid through the grooves (albeit necessary for, say, a sintered wick, but not an axially grooved one); therefore,

(6.12) \begin{equation} \frac {\mathrm{d}p_{\textit{l}}^{*^{\mathrm{A}}}}{\mathrm{d}z^*} =\frac {\mu _{\textit{l}}\dot {m}^*}{K\rho _{\textit{l}}A_{{w}}^*} -\rho _{\textit{l}}g\sin (\phi), \end{equation}

where $K$ is the permeability of the wick and $A_{{w}}^*$ is its (entire) cross-sectional area, i.e.

(6.13) \begin{equation} A_{{w}}^*=\pi \big ( R_{\textit{g}}^{*^2}-R_{\textit{L}}^{*^2}\big ). \end{equation}

In the standard approach, the liquid flows through a (no-slip) duct in the shape of an annular sector; hence, the results by Sparrow, Chen & Jonsson (Reference Sparrow, Chen and Jonsson1964) for the (Fanning) friction factor–Reynolds number product, $ (f_{\textit{F}}\textit{Re} )_{\textit{Sparrow}}$ , apply. The permeability of the wick becomes

(6.14) \begin{equation} K=\frac {D_{\textit{h}}^2\phi _{\textit{l}}}{2\left (f_{\textit{F}}\textit{Re}\right )_{\textit{Sparrow}}}, \end{equation}

where $\phi _{\textit{l}}$ is the liquid fraction in the porous wick, i.e. $N\theta _{\textit{g}}/ (2\pi )$ , and $D_{\textit{h}}$ is the hydraulic diameter of the annular sector, i.e.

(6.15) \begin{equation} D_{\textit{h}} = \frac {2\theta _{\textit{g}}\big (R_{\textit{g}}^{*^2}-R_{\textit{L}}^{*^2}\big)} {2\big(R_{\textit{g}}^*-R_{\textit{L}}^*\big)+\theta _{\textit{g}} \big (R_{\textit{L}}^*+R_{\textit{g}}^*\big )}. \end{equation}

The pressure distribution in the liquid phase becomes

(6.16) \begin{align} p_{\textit{l}}^{*^{\mathrm{A}}} &= p_{\textit{sat}}^*\big(T_{\textit{o}}^*\big)- \Delta p_{\textit{v,t}}^{*^{\mathrm{A}}}+ \frac {\mu _{\textit{l}}q^{*}_{\textit{t}}}{K\rho _{\textit{l}}A_{{w}}^* h_{\textit{lv}}}\left ( \frac {z^{*^2}}{2L_{\textit{e}}^*}+z^*-\frac {L_{\textit{c}}^*}{2} -L_{\textit{a}}^*\right ) +\rho _{\textit{l}}g\sin (\phi)\big(L_{\textit{a}}^*+L_{\textit{c}}^*-z^* \big)\,\mathrm{for} \nonumber \\ &-L_{\textit{e}}^* \leqslant z^* \leqslant 0\,\mathrm{(evaporator)} ,\end{align}

(6.17) \begin{align} p_{\textit{l}}^{*^{\mathrm{A}}} &= p_{\textit{sat}}^*\big(T_{\textit{o}}^*\big)- \Delta p_{\textit{v,t}}^{*^{\mathrm{A}}}+ \frac {\mu _{\textit{l}}q^{*}_{\textit{t}}}{K\rho _{\textit{l}}A_{{w}}^* h_{\textit{lv}}} \left (z^*-\frac {L_{\textit{c}}^*}{2} -L_{\textit{a}}^*\right ) +\rho _{\textit{l}}g\sin (\phi)\big(L_{\textit{a}}^*+L_{\textit{c}}^*-z^* \big)\,\mathrm{for}\nonumber \\ &0 \leqslant z^* \leqslant L_{\textit{a}}^*\,\mathrm{(adiabatic\,section)} ,\end{align}

(6.18) \begin{align} p_{\textit{l}}^{*^{\mathrm{A}}} &= p_{\textit{sat}}^*\big(T_{\textit{o}}^*\big)- \Delta p_{\textit{v,t}}^{*^{\mathrm{A}}}- \frac {\mu _{\textit{l}}q^{*}_{\textit{t}}}{K\rho _{\textit{l}}A_{{w}}^* h_{\textit{lv}}}\frac {\left ( L_{\textit{a}}^*+L_{\textit{c}}^*-z^*\right )^2}{2L_{\textit{c}}^*}+\rho _{\textit{l}}g\sin (\phi)\big(L_{\textit{a}}^*+L_{\textit{c}}^*-z^* \big)\,\mathrm{for}\nonumber \\ &\,L_{\textit{a}}^* \leqslant z^* \leqslant L_{\textit{a}}^*+L_{\textit{c}}^*\,\mathrm{(condenser)}. \end{align}

The total pressure drop in the liquid phase follows as

(6.19) \begin{equation} \Delta p _{\textit{l}}^{*^{\mathrm{A}}} = \frac {\mu _{\textit{l}} q^{*^{\mathrm{A}}}_{\textit{t}} L_{\textit{eff}}^*}{K\rho _{\textit{l}}A_{{w}}^* h_{\textit{lv}}}-\rho _{\textit{l}}g\sin (\phi)L_{\textit{t}}^*. \end{equation}

The capillary limit becomes

(6.20) \begin{equation} q^{*^{\mathrm{A}}}_{\textit{cap}} = \frac {\Delta p_{\textit{cap,max}}^*-\left (\rho _{{v}}-\rho _{\textit{l}}\right ) g \sin (\phi)L_{\textit{t}}^*}{8 \mu _{{v}}L_{\textit{eff}}^*/\big(\pi \rho _{{v}} R_{\textit{L}}^{*^4}h_{\textit{lv}}\big )+ \mu _{\textit{l}} L_{\textit{eff}}^*/\big (K\rho _{\textit{l}}A_{{w}}^* h_{\textit{lv}}\big )}. \end{equation}

Also, it follows from (2.45), (4.13) and the foregoing results that the (constant) ratio of the frictional component of the liquid-to-vapour-phase pressure gradients is

(6.21) \begin{equation} \tilde {\xi }^A = -\frac {\pi \tilde {\rho }R_{\textit{L}}^{*^4}}{8K A_{{w}}^*\tilde {\mu }}. \end{equation}

Finally, the shear stress on the vapour phase (along the meniscus and land) is

(6.22) \begin{align} \tau _{rz,{v}}^{*^{\mathrm{A}}} &= -\frac {4 \mu _{{v}}q_{\textit{t}}^{*}} { \pi \rho _{{v}} R_{\textit{L}}^{*^3}h_{\textit{lv}}}\left (\frac {z^*}{L_{\textit{e}}^*}+1\right )\,\mathrm{on}\,\boldsymbol{M}\, \mathrm{and}\,\boldsymbol{J}\,\mathrm{for}\,-L_{\textit{e}}^*\leqslant z^*\leqslant 0^* \,\mathrm{(evaporator)}, \end{align}

(6.23) \begin{align} \tau _{rz,{v}}^{*^{\mathrm{A}}} &= -\frac {4 \mu _{{v}}q_{\textit{t}}^{*}} { \pi \rho _{{v}} R_{\textit{L}}^{*^3}h_{\textit{lv}}}\,\mathrm{on}\,\boldsymbol{M}\, \mathrm{and}\,\boldsymbol{J}\,\mathrm{for}\,0\leqslant z^*\leqslant L_{\textit{a}}^*\,\mathrm{(adiabatic\, section)}, \end{align}

(6.24) \begin{align} \tau _{rz,{v}}^{*^{\mathrm{A}}} &= -\frac {4 \mu _{{v}}q_{\textit{t}}^{*}} { \pi \rho _{{v}} R_{\textit{L}}^{*^3}h_{\textit{lv}}} \left (1-\frac {z^*-L_{\textit{a}}^*}{L_{\textit{c}}^*}\right )\,\mathrm{on}\,\boldsymbol{M}\, \mathrm{and}\,\boldsymbol{J}\,\mathrm{for}\,L_{\textit{a}}^*\leqslant z^*\leqslant L_{\textit{a}}^*+L_{\textit{c}}^*\,\mathrm{(condenser)}, \end{align}

and that on the liquid phase along the meniscus follows from the Sparrow et al. (Reference Sparrow, Chen and Jonsson1964) results.

6.3. Case B

Here, we resolve the velocity and pressure fields in the liquid under the assumption discussed in § 1.2 that the $z^*$ -dependent shear stress exerted by the vapour phase on the liquid phase corresponds to case A, i.e. there is one-way coupling of the flows. The streamwise-momentum equation governing the flow of the liquid is

(6.25) \begin{equation} \frac {1}{r^*}\frac {\partial }{\partial r^*} \left (r^*\frac {\partial v_{z,\textit{l}}^{*^{\textit{B}}}}{\partial r^*}\right ) +\frac {1}{r^{*^2}}\frac {\partial ^2 v_{z,\textit{l}}^{*^{\textit{B}}}}{\partial \theta ^2}=\frac {1}{\mu _{\textit{l}}} \frac {\mathrm{d} {p}_{\textit{l}}^{*^{\textit{B}}}}{\mathrm{d}z^*} +\frac {\rho _{\textit{l}}g\sin (\phi)}{\mu _{\textit{l}}}. \end{equation}

It is subject to homogeneous boundary conditions on surfaces $\boldsymbol{H}'$ , $\boldsymbol{I}'$ and $\boldsymbol{K}'$ as per

(6.26) \begin{align} \frac {\partial v_{z,\textit{l}}^{*^{\textit{B}}}}{\partial \theta } &= 0\,\mathrm{on}\,\boldsymbol{H}', \end{align}

(6.27) \begin{align} v_{z,\textit{l}}^{*^{\textit{B}}}&=0\,\mathrm{on}\,\boldsymbol{I}', \end{align}

(6.28) \begin{align} v_{z,\textit{l}}^{*^{\textit{B}}}&=0\,\mathrm{on}\,\boldsymbol{K}', \end{align}

respectively. Additionally, noting that the magnitude of the local mass flow rate of both phases is given by (1.6), the inhomogeneous boundary condition on surface $\boldsymbol{M}'$ manifests itself as

(6.29) \begin{align} \frac {\partial v_{z,\textit{l}}^{*^{\textit{B}}}}{\partial r^*} &=- \frac {{4}q^*_{\textit{t}}\tilde {\mu }}{\pi \rho _{{v}} R_{\textit{L}}^{*^3} h_{\textit{lv}}}\left (\frac {z^*}{L_{\textit{e}}^*}+1\right )\,\mathrm{on}\,\boldsymbol{M}'\,\mathrm{for}\,-L_{\textit{e}}^* \leqslant z^* \leqslant 0\,\mathrm{(evaporator)}, \end{align}

(6.30) \begin{align} \frac {\partial v_{z,\textit{l}}^{*^{\textit{B}}}}{\partial r^*} &=- \frac {{4}q^*_{\textit{t}}\tilde {\mu }}{\pi \rho _{{v}} R_{\textit{L}}^{*^3} h_{\textit{lv}}}\,\mathrm{on}\,\boldsymbol{M}'\,\mathrm{for}\,0 \leqslant z^* \leqslant L_{\textit{a}}^*\,\mathrm{(adiabatic\,section)}, \end{align}

(6.31) \begin{align} \frac {\partial v_{z,\textit{l}}^{*^{\textit{B}}}}{\partial r^*} &=- \frac {{4}q^*_{\textit{t}}\tilde {\mu }}{\pi \rho _{{v}} R_{\textit{L}}^{*^3} h_{\textit{lv}}}\left ( 1-\frac {z^*-L_{\textit{a}}^*}{L_{\textit{c}}^*} \right ) \,\mathrm{on}\,\boldsymbol{M}'\,\mathrm{for}\,L_{\textit{a}}^* \leqslant z^* \leqslant L_{\textit{a}}^*+L_{\textit{c}}^*\,\mathrm{(condenser)}. \end{align}

This problem is, in part, analogous to that for ${V}_{z,\textit{l}}^{0}$ in § 4.1. One distinction between them is that the Poisson equation governing ${V}_{z,\textit{l}}^{0}$ (4.14) has a forcing of unity rather than that specified by (6.25). Conversely, a commonality between them is that boundary conditions (6.26), (6.28) and the homogeneous form of (6.29)–(6.31) for the present $v_{z,\textit{l}}^{*^{\textit{B}}}$ problem are analogous to the homogeneous boundary conditions on subproblem $\hat {V}_{z,\textit{l}}^{0}$ within the solution for ${V}_{z,\textit{l}}^{0}$ . Moreover, boundary condition (6.27) is modified in the same manner as boundary condition (4.23) on surface $\boldsymbol{I}'$ due the manner in which superposed solutions are used to resolve both problems. Therefore, it follows by analogy with (4.18), that up to the $\hat {V}_{z,\textit{l}}^{0}$ component of $\tilde {V}_{z,\textit{l}}^{0}$ in it provided by (4.40), we have

(6.32) \begin{align} v_{z,\textit{l}}^{*^{\textit{B}}} &= \left [\frac {1}{\mu _{\textit{l}}}\frac {\mathrm{d}{p}_{\textit{l}}^{*^{\textit{B}}}}{\mathrm{d}z^*} +\frac {\rho _{\textit{l}}g\sin (\phi)}{\mu _{\textit{l}}}\right ]\left \{ \frac {r^{*^2}-R_{\textit{L}}^{*^2}}{4} -\frac {R_{\textit{L}}^{*^2}-R_{\textit{g}}^{*^2}}{4}\frac {\ln \left (r^*/R_{\textit{L}}^*\right )}{\ln \big(\tilde {R}\,\big )} \right .\nonumber \\ &\, \left .+\sum _{m=1}^{\infty }\frac {\hat {B}_m^* \sin \left [\beta _m \ln \big (r^*/R_{\textit{g}}^*\big )\right ]\cosh \left (\beta _m \theta \right ) }{\cosh \left (\beta _m \theta _{\textit{g}}/2\right )}\right \} +\check {v}_{z,\textit{l}}^{*^{\textit{B}}}\left (r^*,\theta \right )\!, \end{align}

where $\beta _m$ is given by (4.36) and $\hat {B}_m^* = P^{*^2}\hat {B}_m$ , where $\hat {B}_m$ is given by (4.41).

The $\check {v}_{z,\textit{l}}^{*^{\textit{B}}}$ problem follows as

(6.33) \begin{equation} \frac {1}{r^*}\frac {\partial }{\partial r^*} \left (r^*\frac {\partial \check {v}_{z,\textit{l}}^{*^{\textit{B}}}}{\partial r^*}\right ) +\frac {1}{r^{*^2}}\frac {\partial ^2 \check {v}_{z,\textit{l}}^{*^{\textit{B}}}}{\partial \theta ^2}=0, \end{equation}

subject to

(6.34) \begin{align} \frac {\partial \check {v}_{z,\textit{l}}^{*^{\textit{B}}}}{\partial \theta } &= 0\,\mathrm{on}\,\boldsymbol{H}', \end{align}

(6.35) \begin{align} \check {v}_{z,\textit{l}}^{*^{\textit{B}}} &=0 \,\mathrm{on}\,\boldsymbol{I}', \end{align}

(6.36) \begin{align} \check {v}_{z,\textit{l}}^{*^{\textit{B}}} &= 0\,\mathrm{on}\,\boldsymbol{K}', \end{align}

(6.37) \begin{align} \frac {\partial \check {v}_{z,\textit{l}}^{*^{\textit{B}}}}{\partial r^*} &= \overbrace {- \frac {{4}q^*_{\textit{t}}\tilde {\mu }}{\pi \rho _{{v}} R_{\textit{L}}^{*^3} h_{\textit{lv}}}\left (\frac {z^*}{L_{\textit{e}}^*}+1\right )}^{f_{\textit{e}}\left (z^*\right )}-C_a^*\left [\frac {1}{\mu _{\textit{l}}}\frac {\mathrm{d}{p}_{\textit{l}}^{*^{\textit{B}}}}{\mathrm{d}z^*} +\frac {\rho _{\textit{l}}g\sin (\phi)}{\mu _{\textit{l}}}\right ]\,\mathrm{on}\,\boldsymbol{M}'\nonumber \\[3pt] &\,\mathrm{for}\,-L_{\textit{e}}^* \leqslant z^* \leqslant 0\,\mathrm{(evaporator)} ,\end{align}

(6.38) \begin{align} \frac {\partial \check {v}_{z,\textit{l}}^{*^{\textit{B}}}}{\partial r^*} &= \overbrace {- \frac {{4}q^*_{\textit{t}}\tilde {\mu }}{\pi \rho _{{v}} R_{\textit{L}}^{*^3} h_{\textit{lv}}}}^{f_{\textit{a}}}-C_a^*\left [\frac {1}{\mu _{\textit{l}}}\frac {\mathrm{d}{p}_{\textit{l}}^{*^{\textit{B}}}}{\mathrm{d}z^*} +\frac {\rho _{\textit{l}}g\sin (\phi)}{\mu _{\textit{l}}}\right ]\,\mathrm{on}\,\boldsymbol{M}'\nonumber \\[3pt] &\,\mathrm{for}\,0 \leqslant z^* \leqslant L_{\textit{a}}^*\,\mathrm{(adiabatic\,section)} ,\end{align}

(6.39) \begin{align} \frac {\partial \check {v}_{z,\textit{l}}^{*^{\textit{B}}}}{\partial r^*} &= \overbrace {- \frac {{4}q^*_{\textit{t}}\tilde {\mu }}{\pi \rho _{{v}} R_{\textit{L}}^{*^3} h_{\textit{lv}}}\left ( 1-\frac {z^*-L_{\textit{a}}^*}{L_{\textit{c}}^*} \right )}^{f_{\textit{c}}\left (z^*\right )}-C_a^*\left [\frac {1}{\mu _{\textit{l}}}\frac {\mathrm{d}{p}_{\textit{l}}^{*^{\textit{B}}}}{\mathrm{d}z^*} +\frac {\rho _{\textit{l}}g\sin (\phi)}{\mu _{\textit{l}}}\right ] \,\mathrm{on}\,\boldsymbol{M}'\nonumber \\[3pt] &\,\mathrm{for}\,L_{\textit{a}}^* \leqslant z^* \leqslant L_{\textit{a}}^*+L_{\textit{c}}^*\,\mathrm{(condenser)}, \end{align}

where $f_{{i}} (z^* )$ (where $i$ is ‘e’ or ‘c’) and $f_{\textit{a}}$ are defined for later use, and $C_a^*$ is $P^{*}C_a$ , where $C_a$ is defined by (4.27). This problem is the same as that for $\breve {V}_{z,\textit{l}}^{0}$ in § 4.1 up to the non-homogeneous boundary conditions on $\boldsymbol{M}'$ . It thus follows from (4.43) that

(6.40) \begin{equation} \check {v}_{z,\textit{l}}^{*^{\textit{B}}}=\left .\frac {\partial \check {v}_{z,\textit{l}}^{*^{\textit{B}}}}{\partial r^*}\right |_{r^*=R_{\textit{L}}^*} \sum _{n=0}^{\infty } {F}_n^* \Big (r^{*^{\gamma _n}} - R_{\textit{g}}^{*^{2\gamma _n}}r^{*^{-\gamma _n}}\Big ) \cos \left (\gamma _n \theta \right ), \end{equation}

where the eigenvalues ( $\gamma _n$ ) remain as given by (4.44), and evaluation of the non-homogeneous boundary conditions yields ${F}_n^*$ as

(6.41) \begin{equation} {F}_n^* = \frac {4(-1)^n} {\big(R_{\textit{L}}^{*^{\gamma _n-1}}+R_{\textit{g}}^{*^{2\gamma _n}} R_{\textit{L}}^{*^{-\gamma _n-1}}\big )\theta _{\textit{g}}\gamma _n^2}. \end{equation}

Integrating the velocity profile given by (6.32) over the cross-sectional area of the domain, the corresponding volumetric flow rate is

(6.42) \begin{equation} {Q}_{\textit{l},i}^{*^{\textit{B}}} = \int _0^{\theta _{\textit{g}}/2} \int _{R_{\textit{L}}^*}^{R_{\textit{g}}^*}{v}_{z,\textit{l}}^{*^{\textit{B}}} r^* \mathrm{d} r^* \mathrm{d}\theta . \end{equation}

It follows that

(6.43) \begin{equation} {Q}_{\textit{l},i}^{*^{\textit{B}}} = \left [\frac {1}{\mu _{\textit{l}}}\frac {\mathrm{d}{p}_{\textit{l}}^{*^{\textit{B}}}}{\mathrm{d}z^*} +\frac {\rho _{\textit{l}}g\sin (\phi)}{\mu _{\textit{l}}}\right ]\left (K_1^*+K_2^*\right ) +\left .\frac {\partial \check {v}_{z,\textit{l}}^{*^{\textit{B}}}}{\partial r^*}\right |_{r^*=R_{\textit{L}}^*}K_4^*, \end{equation}

where ‘i’ denotes any section of the AGHP, $K_1^* = P^{*^4}K_1$ , $K_2^*=P^{*^4}K_2$ and $K_4^*$ follows from $K_3$ when $C_n$ is replaced by $F_n^*$ and the result is multiplied by $P^{*^3}$ as per

(6.44) \begin{equation} K_4^* =\sum _{n=0}^{N-1} \frac {4(-1)^nR_{\textit{g}}^{*^3}\left [2\gamma _n\tilde {R}^{\gamma _n} +\tilde {R}^{3\gamma _n+2}\left ( 2-\gamma _n\right )-\tilde {R}^{2}\left (2+\gamma _n\right )\right ]}{\theta _g\gamma _n^3 \big(\gamma _n^2-4\big )\big (\tilde {R}^{2\gamma _n-1}+\tilde {R}^{-1}\big )}. \end{equation}

The pressure gradient in the liquid phase follows as

(6.45) \begin{equation} \frac {\mathrm{d}{p}_{\textit{l}}^{*^{\textit{B}}}}{\mathrm{d}z^*}=\frac {\left [{Q}_{\textit{l},i}^{*^{\textit{B}}}-f_i\left (z^*\right )K_4^*\right ] \mu _{\textit{l}}} {K_1^*+K_2^*-C_a^*K_4^*}-\rho _{\textit{l}}g\sin (\phi). \end{equation}

The magnitudes of $\bar {Q}_{\textit{l},i}^{*^{\textit{B}}}$ and $f_i (z^* )$ evaluated at $r^*=R_{\textit{L}}^*$ are linearly increasing (decreasing) with $z^*$ in the evaporator (condenser). Therefore, noting that the cross-sectional area of the liquid portion of the domain is $1/ (2N_g )$ times that of all of the liquid in the AGHP in specifying the former, their mean values correspond to the middle of these sections as per

(6.46) \begin{align} \bar {Q}_{\textit{l},i}^{*^{\textit{B}}}& = -\frac {q^{*^{\textit{B}}}_{\textit{t}}}{4N_{\textit{g}}\rho _{\textit{l}}h_{\textit{lv}}}, \end{align}

(6.47) \begin{align} \overline {f_i\left (z^*\right ) }&=- \frac { {2}q^{*^{\textit{B}}}_{\textit{t}} \tilde {\mu }}{\pi \rho _{{v}} R_{\textit{L}}^{*^3} h_{\textit{lv}} }, \end{align}

from which the mean value of the pressure gradient in these sections follows. Correspondingly,

(6.48) \begin{equation} \Delta p_{\textit{l},i}^{*^{\textit{B}}} = \frac { {2}q^{*}_{\textit{t}}\tilde {\mu }K_4^*/ \big (\pi \rho _{{v}}R_{\textit{L}}^{*^3}h_{\textit{lv}}\big )-q^{*}_{\textit{t}}/\left (4N_{\textit{g}}\rho _{\textit{l}}h_{\textit{lv}} \right ) } {K_1^*+K_2^*-C_a^*K_4^*}\mu _{\textit{l}}L_i^*-\rho _{\textit{l}}g\sin (\phi) L_i^*, \end{equation}

which resolves the first and fifth terms on the right side of (1.4). To apply this expression in the adiabatic section, the relevant lengths are $2L_{\textit{a}}^*$ and $L_{\textit{a}}^*$ in the first and second terms on its right side, respectively. The capillary limit becomes

(6.49) \begin{equation} q^{*^{\textit{B}}}_{\textit{cap}} = \frac {\Delta p_{\textit{cap,max}}^*-\left (\rho _{{v}}-\rho _{\textit{l}}\right ) g \sin (\phi)L_{\textit{t}}^*}{\left [ \frac {8\mu _{{v}}} {\pi \rho _{{v}} R_{\textit{L}}^{*^4}h_{\textit{lv}}} + \frac {{4}\tilde {\mu }K_4^*/ \big(\pi \rho _{{v}}R_{\textit{L}}^{*^3}h_{\textit{lv}}\big)-1/\big(2N_{\textit{g}}\rho _{\textit{l}}h_{\textit{lv}}\big) } {K_1^*+K_2^*-C_a^*K_4^*}\mu _{\textit{l}}\right ] L_{\textit{eff}}^*}. \end{equation}

The pressure distributions in the vapour phase are the same as those for case A as per (6.3), (6.9) and (6.10). Those in the liquid phase follow from (6.45) as

(6.50) \begin{align} p^*_{\textit{l}} &= p^*_{\textit{sat}}\big(T_{\textit{o}}^*\big)- \Delta p_{\textit{v,t}}^*+\frac {{4}\tilde {\mu }K_4^*/ \big (\pi \rho _{{v}}R_{\textit{L}}^{*^3}h_{\textit{lv}}\big )-1/\big(2N_{\textit{g}}\rho _{\textit{l}}h_{\textit{lv}}\big) } {K_1^*+K_2^*-C_a^*K_4^*}\mu _{\textit{l}}q_{\textit{t}}^{*} \left (\frac {z^{*^2}}{2L_{\textit{e}}^*}+z^*-\frac {L_{\textit{c}}^*}{2}-L_{\textit{a}}^*\right ) \nonumber \\[3pt] &+\rho _{\textit{l}}g\sin (\phi)\big(L_{\textit{a}}^*+L_{\textit{c}}^*-z^* \big) \ \mathrm{for} \ -L_{\textit{e}}^* \leqslant z^* \leqslant 0\,\mathrm{(evaporator)} ,\end{align}

(6.51) \begin{align} p^*_{\textit{l}} &= p^*_{\textit{sat}}\big(T_{\textit{o}}^*\big)- \Delta p_{\textit{v,t}}^*+\frac {{4}\tilde {\mu }K_4^*/ \big(\pi \rho _{{v}}R_{\textit{L}}^{*^3}h_{\textit{lv}}\big)-1/\big(2N_{\textit{g}}\rho _{\textit{l}}h_{\textit{lv}}\big) } {K_1^*+K_2^*-C_a^*K_4^*}\mu _{\textit{l}}q_{\textit{t}}^{*} \left (z^*-\frac {L_{\textit{c}}^*}{2}-L_{\textit{a}}^*\right ) \nonumber \\[3pt] &+\rho _{\textit{l}}g\sin (\phi)\big(L_{\textit{a}}^*+L_{\textit{c}}^*-z^* \big)\ \mathrm{for}\ 0 \leqslant z^* \leqslant L_{\textit{a}}^*\,\mathrm{(adiabatic\,section)} ,\end{align}

(6.52) \begin{align} p^*_{\textit{l}} &= p^*_{\textit{sat}}\big(T_{\textit{o}}^*\big)- \Delta p_{\textit{v,t}}^*-\frac {{4}\tilde {\mu }K_4^*/ \big(\pi \rho _{{v}}R_{\textit{L}}^{*^3}h_{\textit{lv}}\big)-1/\big(2N_{\textit{g}}\rho _{\textit{l}}h_{\textit{lv}}\big) } {K_1^*+K_2^*-C_a^*K_4^*}\mu _{\textit{l}}q_{\textit{t}}^{*} \frac {\left ( L_{\textit{a}}^*+L_{\textit{c}}^*-z^*\right )^2}{2L_{\textit{c}}^*} \nonumber \\ &+\rho _{\textit{l}}g\sin (\phi)\big(L_{\textit{a}}^*+L_{\textit{c}}^*-z^* \big)\ \mathrm{for}\ L_{\textit{a}}^* \leqslant z^* \leqslant L_{\textit{a}}^*+ L_{\textit{c}}^*\,\mathrm{(condenser)}. \end{align}

It may also be shown that the (constant) value of $\tilde {\xi }^{\textit{B}}$ is

(6.53) \begin{equation} \tilde {\xi }^{\textit{B}} = \frac {\pi \tilde {\rho } R_{\textit{L}}^{*^4}/\left (16N_{\textit{g}}\tilde {\mu }\right )- K_4^*R_{\textit{L}}^*/{2}}{K_1^*+K_2^*-C_a^*K_4^*}. \end{equation}

Finally, the velocity of the liquid along the meniscus in the adiabatic section is

(6.54) \begin{align} v_{z^*,\textit{l}}^{*^{\textit{B}}} &= \left [\frac {1}{\mu _{\textit{l}}}\frac {\mathrm{d}{p}_{\textit{l}}^{*^{\textit{B}}}}{\mathrm{d}z^*} +\frac {\rho _{\textit{l}}g\sin (\phi)}{\mu _{\textit{l}}}\right ] \sum _{m=1}^{\infty }\frac {\hat {B}_m^* \sin \left [\beta _m \ln \left (r^*/R_{\textit{g}}^*\right )\right ]\cosh \left (\beta _m \theta \right )} {\cosh \left (\beta _m \theta _{\textit{g}}/2\right )} \nonumber \\[3pt] &+ \left \{f_a-C_a^*\left [\frac {1}{\mu _{\textit{l}}}\frac {\mathrm{d}{p}_{\textit{l}}^{*^{\textit{B}}}}{\mathrm{d}z^*} +\frac {\rho _{\textit{l}}g\sin (\phi)}{\mu _{\textit{l}}}\right ]\right \} \sum _{n=0}^{\infty } {F}_n^* \left (R_{\textit{L}}^{*^{\gamma _n}} - R_{\textit{g}}^{*^{2\gamma _n}}R_{\textit{L}}^{*^{-\gamma _n}}\right ) \cos \left (\gamma _n \theta \right )\,\mathrm{on}\,\boldsymbol{M}'. \end{align}

6.4. Case C $_0$

When the protrusion angle is zero ( $\theta _{\textit{m}}=0$ ), $\tilde {Q}_i^{0}$ are known from § 4.1 for prescribed values of $\theta _{\textit{g}}$ , $\theta _{\textit{p}}$ , $R_{\textit{L}}$ , $R_{\textit{g}}$ $\tilde {\mu }$ and $\tilde {\rho }$ . The corresponding dimensional pressure gradients in the adiabatic section are

(6.55) \begin{align} \frac {\mathrm{d}p_{{v}}^{*^{0}}}{\mathrm{d}z^*} & =\frac {q_{\textit{t}}^*\mu _{{v}}} {\rho _{{v}}h_{\textit{lv}}P^{*^4}\tilde {Q}_{{v}}^{0}}-\rho _{{v}} g \sin (\phi)\,\mathrm{for}\,0 \leqslant z^* \leqslant L_{\textit{a}}^*\,\mathrm{(adiabatic\,section)}, \end{align}

(6.56) \begin{align} \frac {\mathrm{d}p_{\textit{l}}^{*^{0}}}{\mathrm{d}z^*} & =-\frac {q_{\textit{t}}^*\mu _{\textit{l}}} {\rho _{\textit{l}}h_{\textit{lv}}P^{*^4}\tilde {Q}_{\textit{l}}^{0}}-\rho _{\textit{l}} g \sin (\phi)\,\mathrm{for}\,0 \leqslant z^* \leqslant L_{\textit{a}}^*\,\mathrm{(adiabatic\,section)}. \end{align}

Equation (6.3) remains valid for the pressure distribution in the vapour phase in the evaporator and those in the adiabatic section and condenser become

(6.57) \begin{align} p_{{v}}^{*^{\textit{C}_0}} &= p_{\textit{sat}}^*\big(T_{\textit{o}}^*\big) - \frac {4\mu _{{v}}q^*_{\textit{t}}L_{\textit{e}}^*} {\pi \rho _{{v}} R_{\textit{L}}^{*^4}h_{\textit{lv}}}+\frac {q_{\textit{t}}^*\mu _{{v}}} {2N_{\textit{g}}\rho _{{v}}h_{\textit{lv}}P^{*^4}\tilde {Q}_{{v}}^{0}}z^*- \rho _{{v}}g\sin (\phi)\left (L_{\textit{e}}^*+z^*\right ) \,\mathrm{for}\,0 \leqslant z^* \leqslant L_{\textit{a}}^*\nonumber \\ &\mathrm{(adiabatic\,section)} ,\end{align}

(6.58) \begin{align} p_{{v}}^{*^{\textit{C}_0}} &= p_{\textit{sat}}^*\big(T_{\textit{o}}^*\big) - \frac {4\mu _{{v}}q^*_{\textit{t}}L_{\textit{e}}^*} {\pi \rho _{{v}} R_{\textit{L}}^{*^4}h_{\textit{lv}}}+\frac {q_{\textit{t}}^*\mu _{{v}}L_{\textit{a}}^*} {2N_{\textit{g}}\rho _{{v}}h_{\textit{lv}}P^{*^4}\tilde {Q}_{{v}}^{0}} - \frac {8\mu _{{v}}q^*_{\textit{t}}} {\pi \rho _{{v}} R_{\textit{L}}^{*^4}h_{\textit{lv}}}\left [\frac {\left (z^*-L_{\textit{a}}^*\right ) \left (L_{\textit{a}}^*+2L_{\textit{c}}^*-z^*\right )}{2L_{\textit{c}}^*} \right ] \nonumber \\[3pt]&- \rho _{{v}}g\sin (\phi)\left (L_{\textit{e}}^*+z^*\right ) \,\mathrm{for}\,L_{\textit{a}}^* \leqslant z^* \leqslant L_{\textit{a}}^*+L_{\textit{c}}^*\,\mathrm{(condenser)}. \end{align}

Relatedly, the total pressure drop across the vapour phase is

(6.59) \begin{equation} \Delta p_{\textit{v,t}}^{*^{\textit{C}_0}} =\frac {4\mu _{{v}}q^*_{\textit{t}}\left (L_{\textit{e}}^*+L_{\textit{c}}^*\right )} {\pi \rho _{{v}} R_{\textit{L}}^{*^4}h_{\textit{lv}}} -\frac {q_{\textit{t}}^*\mu _{{v}}L_{\textit{a}}^*} {2N_{\textit{g}}\rho _{{v}}h_{\textit{lv}}P^{*^4}\tilde {Q}_{{v}}^{0}} +\rho _{{v}} g \sin (\phi)L_{\textit{t}}^*, \end{equation}

In the liquid phase

(6.60) \begin{align} p_{\textit{l}}^{*^{\textit{C}_0}} &= p_{\textit{sat}}^*\big(T_{\textit{o}}^*\big)- \Delta p_{\textit{v,t}}^{*^{\textit{C}_0}}- \frac {\mu _{\textit{l}}q^{*}_{\textit{t}}L_{\textit{c}}^*}{2K\rho _{\textit{l}}A_{{w}}^* h_{\textit{lv}}} +\frac {q_{\textit{t}}^*\mu _{\textit{l}}L_{\textit{a}}^*} {2N_{\textit{g}}\rho _{\textit{l}}h_{\textit{lv}}P^{*^4}\tilde {Q}_{\textit{l}}^{0}}+ \frac {\mu _{\textit{l}}q^{*}_{\textit{t}}}{K\rho _{\textit{l}}A_{{w}}^* h_{\textit{lv}}}\left (\frac {z^{*^2}}{2L_{\textit{e}}^*}+z^*\right )\nonumber \\[3pt] &+\rho _{\textit{l}}g\sin (\phi)\big(L_{\textit{a}}^*+L_{\textit{c}}^*-z^* \big)\,\mathrm{for} -L_{\textit{e}}^* \leqslant z^* \leqslant 0\,\mathrm{(evaporator)} ,\end{align}

(6.61) \begin{align} p_{\textit{l}}^{*^{\textit{C}_0}} &= p_{\textit{sat}}^*\big(T_{\textit{o}}^*\big)- \Delta p_{\textit{v,t}}^{*^{\textit{C}_0}}- \frac {\mu _{\textit{l}}q^{*}_{\textit{t}}L_{\textit{c}}^*}{2K\rho _{\textit{l}}A_{{w}}^* h_{\textit{lv}}} -\frac {q_{\textit{t}}^*\mu _{\textit{l}}\left (z^*-L_{\textit{a}}^*\right )} {2N_{\textit{g}}\rho _{\textit{l}}h_{\textit{lv}}P^{*^4}\tilde {Q}_{\textit{l}}^{0}} +\rho _{\textit{l}}g\sin (\phi)\big(L_{\textit{a}}^*+L_{\textit{c}}^*-z^* \big)\,\mathrm{for}\nonumber \\[3pt] &0 \leqslant z^* \leqslant L_{\textit{a}}^*\,\mathrm{(adiabatic\,section)} ,\end{align}

and (6.18) remains valid in the condenser section using case C $_0$ parameters. Relatedly, the total pressure drop (from the condenser to evaporator end caps) across the liquid phase is

(6.62) \begin{equation} \Delta p_{\textit{l,t}}^{*^{\textit{C}_0}} =\frac {q_{\textit{t}}^*\mu _{\textit{l}}\left (L_{\textit{e}}^* +L_{\textit{c}}^*\right )}{2K\rho _{\textit{l}}A_{{w}}^*h_{\textit{lv}}} -\frac {q_{\textit{t}}^*\mu _{{v}}L_{\textit{a}}^*} {2N_{\textit{g}}\rho _{{v}}h_{\textit{lv}}P^{*^4}\tilde {Q}_{{v}}^{0}} -\rho _{\textit{l}} g \sin (\phi)L_{\textit{t}}^*. \end{equation}

The capillary limit becomes

(6.63) \begin{equation} q_{\textit{cap}}^{*^{\textit{C}_0}}= \frac {\Delta p_{\textit{cap,max}}^*-\left (\rho _{{v}}-\rho _{\textit{l}}\right ) g \sin (\phi)L_{\textit{t}}^*}{\frac {4\mu _{{v}}\left (L_{\textit{e}}^* +L_{\textit{c}}^*\right )}{\pi \rho _{{v}} R_{\textit{L}}^{*^4}h_{\textit{lv}}}+ \frac {\mu _{\textit{l}}\left (L_{\textit{e}}^* +L_{\textit{c}}^*\right )}{2K\rho _{\textit{l}}A_{{w}}^*h_{\textit{lv}}} -\frac {\mu _{{v}}L_{\textit{a}}^*}{2N_{\textit{g}}\rho _{{v}}h_{\textit{lv}}P^{*^4} \tilde {Q}_{{v}}^{0}}-\frac {\mu _{\textit{l}}L_{\textit{a}}^*}{2N_{\textit{g}}\rho _{\textit{l}}h_{\textit{lv}}P^{*^4} \tilde {Q}_{\textit{l}}^{0}}}. \end{equation}

The velocity fields in the vapour and liquid phases in the adiabatic section are

(6.64) \begin{align} v_{z,{v}}^{*^{0}} &=\frac {q_{\textit{t}}^{*}}{2N_{\textit{g}}h_{\textit{lv}}\rho _{{v}} P^{*^2}\tilde {Q}_{{v}}^{0}}\left [ \frac {r^2-R_{\textit{L}}^2}{4} + \sum _{l=0}^{\infty }\hat {A}_l \left (\frac {r}{R_{\textit{L}}}\right )^{\alpha _l}\cos \left (\alpha _l \theta \right ) \right ]\,\mathrm{(adiabatic\,section)}, \end{align}

(6.65) \begin{align} v_{z,\textit{l}}^{*^{0}} &=\frac {-q_{\textit{t}}^{*}}{2N_{\textit{g}}h_{\textit{lv}}\rho _{\textit{l}} P^{*^2}\tilde {Q}_{\textit{l}}^{0}}\left \{\frac {r^2-R_{\textit{L}}^2}{4} -\frac {R_{\textit{L}}^2-R_{\textit{g}}^2}{4}\frac {\ln \left (r/R_{\textit{L}}\right )}{\ln \big(\tilde {R}\,\big )} +\sum _{m=1}^{\infty }\frac { \hat {B}_m \sin \big [\beta _m \ln \big (r/R_{\textit{g}}\big )\big ] \cosh \big(\beta _m \theta \big )}{\cosh \big (\beta _m\theta _{\textit{g}}/2\big )}\right . \nonumber \\[3pt] &\left .+\sum _{n=0}^{\infty }\hat {C}_n\left [\left (\frac {r}{R_{\textit{g}}}\tilde {R}\right )^{\gamma _n}- \left (\frac {R_{\textit{L}}}{r}\right )^{\gamma _n} \right ]\cos \left (\gamma _n \theta \right ) \right \}\!, \end{align}

respectively. The velocity along the meniscus and land in the adiabatic section is

(6.66) \begin{equation} v_{z}^{*^{0}} = \frac {q_{\textit{t}}^{*^{\textit{C}_0}}}{h_{\textit{lv}}\rho _{{v}} P^{*^2}\tilde {Q}_{{v}}^{0}} \sum _{l=0}^{\infty }\hat {A}_l \cos \left (\alpha _l \theta \right )\,\mathrm{on}\,\boldsymbol{M}\, \mathrm{and}\,\boldsymbol{J}\,\mathrm{for}\,0\leqslant z^*\leqslant L_{\textit{a}}^*\,\mathrm{(adiabatic\, section)}. \end{equation}

Equivalently, along the meniscus

(6.67) \begin{equation} v_{z}^{*^{0}} =\frac {-q_{\textit{t}}^{*^{\textit{C}_0}}}{h_{\textit{lv}}\rho _{\textit{l}} P^{*^2}\tilde {Q}_{\textit{l}}^{0}}\left \{ \sum _{m=1}^{\infty }\frac { \hat {B}_m (-1)^m \cosh \left (\beta _m \theta \right )}{\cosh \left (\beta _m\theta _{\textit{g}}/2\right )}+ \sum _{n=0}^{\infty }\hat {C}_n\left (\tilde {R}^{2\gamma _n}- 1 \right )\cos \left (\gamma _n \theta \right ) \right \}\!. \end{equation}

The shear stress on the vapour phase along the meniscus and land in the adiabatic section is

(6.68) \begin{equation} \tau _{rz,{v}}^{*^{0}}=\frac {\mu _{{v}}q_{\textit{t}}^{*^{\textit{C}_0}}}{h_{\textit{lv}}\rho _{{v}}P^{*^3} \tilde {Q}_{{v}}^{0}}\left [\frac {R_{\textit{L}}}{2}+\sum _{l=0}^{\infty } \frac {\hat {A}_l \alpha _l}{ R_{\textit{L}}} \cos \left (\alpha _l \theta \right ) \right ]\!. \end{equation}

Equivalently,

(6.69) \begin{align} \tau _{rz,{v}}^{*^{0}} =\frac {\mu _{\textit{l}}q_{\textit{t}}^{*^{\textit{C}_0}}}{h_{\textit{lv}}\rho _{\textit{l}}P^{*^3} \tilde {Q}_{\textit{l}}^{0}} \left \{\frac {R_{\textit{L}}}{2} - \frac {R_{\textit{L}}^2 -R_{\textit{g}}^2}{4R_{\textit{L}}\ln \big(\tilde {R}\,\big )}+\sum _{n=0}^{\infty }\hat {C}_n \gamma _n \left [\frac {\tilde {R}^{2\gamma _n}}{R_{\textit{L}}}+\frac {1}{R_{\textit{L}}}\right ] \cos \left (\gamma _n\theta \right )\right \}\!. \end{align}

Equations (6.22) and (6.24) remain valid in the evaporator and condenser, respectively.

6.5. Case C

Turning to case $C$ , where $\theta _{\textit{m}}$ is not restricted in value, it follows from (2.45) and the definition of $\tilde {Q}_i$ that

(6.70) \begin{equation} \frac {\mathrm{d}\left (\Delta {p}\right )}{\mathrm{d}z} =\Delta \textit{Bo}_{\parallel }+ \frac {Q_{{v}}}{\tilde {Q}_{{v}}} - \frac {Q_{\textit{l}}}{\tilde {Q}_{\textit{l}}}, \end{equation}

where

(6.71) \begin{align} \Delta p &= p_{{v}}-p_{\textit{l}}, \end{align}

(6.72) \begin{align} \Delta \textit{Bo}_{\parallel } &= \textit{Bo}_{\parallel ,\textit{l}} -\textit{Bo}_{\parallel ,{v}}, \end{align}

or, from (2.63),

(6.73) \begin{equation} \frac {\mathrm{d}\left (\Delta {p}\right )}{\mathrm{d}z} =\Delta \textit{Bo}_{\parallel }+ Q_{{v}} \left ( \frac {1}{\tilde {Q}_{{v}}}+\frac {\tilde {\rho }}{\tilde {\mu }\tilde {Q}_{\textit{l}}} \right )\!. \end{equation}

We use (4.13) to express $\tilde {Q}_{\textit{l}}$ in terms of $\tilde {Q}_{{v}}$ and integrate over the length of the adiabatic section. Imposing the limits of integration as per (2.47)–(2.50)

(6.74) \begin{equation} \int _{{p_{{{v}},{\mathrm{e}}}}}^{{p_{{{v}},{\mathrm{c}}}} - {p_{{\textit{l}},{\mathrm{c}}}}} {\frac {{{\mathrm{d}}\left ( {\Delta {p}} \right )}}{{\Delta {\mathrm{B}}{{\mathrm{o}}_\parallel } + Q_{{v}}(1 - {{\tilde \xi }})/\tilde Q_{{v}}}}} = 1, \end{equation}

where $\tilde {\xi }$ and $\tilde {Q}_{\mathrm v}$ as functions of $\Delta p$ are from an array computed as described in § 4.2, and

(6.75) \begin{align} p_{\textit{v,e}} &=1- \frac {8N_{\textit{g}}Q_{{v}}L_{\textit{e}}}{\pi R_{\textit{L}}^{4}}- \frac {N_{\textit{g}}\tilde {\rho }Q_{{v}}L_{\textit{e}}}{\tilde {\mu }A_{{w}}} +\Delta \textit{Bo}_{\parallel }L_{\textit{e}}, \end{align}

(6.76) \begin{align} p_{\textit{v,c}}-p_{\textit{l,c}} &= \frac {8N_{\textit{g}}Q_{{v}}L_{\textit{c}}}{\pi R_{\textit{L}}^{4}} +\frac {N_{\textit{g}}\tilde {\rho }Q_{{v}}L_{\textit{c}}}{\tilde {\mu }A_{{w}}} -\Delta \textit{Bo}_{\parallel }L_{\textit{c}}, \end{align}

where $A_{{w}} = A_{{w}}^*K/P^{*^4}$ . We numerically evaluate the integral in (6.74) when using fzero to find $Q_{{v}}$ and thus the capillary limit (fzero is a MATLAB function used to find the roots of nonlinear functions. The convergence tolerance is set to the default value based on machine precision, i.e. $2^{-52}$ ).