Hartmann Flow of Two-Layered Fluids in Horizontal and Inclined Channels

Abstract

The effect of a transverse magnetic field on two-phase stratified flow in horizontal and inclined channels is studied. The lower heavier phase is assumed to be an electrical conductor (e.g., liquid metal), while the upper lighter phase is fully dielectric (e.g., gas). The flow is defined by prescribed flow rates in each phase, so the unknown frictional pressure gradient and location of the interface separating the phases (holdup) are found as part of the whole solution. It is shown that the solution of such a two-phase Hartmann flow is determined by four dimensionless parameters: the phases’ viscosity and flow-rate ratios, the inclination parameter, and the Hartmann number. The changes in velocity profiles, holdups, and pressure gradients with variations in the magnetic field and the phases’ flow-rate ratio are reported. The potential lubrication effect of the gas layer and pumping power reduction are found to be limited to low magnetic field strength. The effect of the magnetic field strength on the possibility of obtaining countercurrent flow and multiple flow states in concurrent upward and downward flows, and the associated flow characteristics, such as velocity profiles, back-flow phenomena, and pressure gradient, are explored. It is shown that increasing the magnetic field strength reduces the flow-rate range for which multiple solutions are obtained in concurrent flows and the flow-rate range where countercurrent flow is feasible.

1. Introduction

Flows of electrically conducting fluids in an electromagnetic field are found across a wide variety of phenomena, which are extensively investigated in the field of magnetohydrodynamics (MHD) and possess important technological applications (e.g., [1,2]). While much of the existing research on MHD focuses on single-phase flows, numerous practical challenges involve multiphase flow systems, particularly within the nuclear and petroleum industries, geophysics, and MHD power generation [3,4]. Among the applications are two-phase liquid metal magnetohydrodynamics (LMMHD) generators, which attracted attention in the industry due to their relatively simple structure, absence of moving parts, high-efficiency conversion, and reduced environmental pollution [5]. Alongside the liquid metal, another fluid phase (gas or liquid) is employed to convert thermal energy into kinematic energy. MHD flows can also transport weakly conducting fluids in microscale systems, e.g., in the microchannel networks, such as the microchannel networks of lab-on-a-chip devices [6,7], where the presence of a second non-conductive fluid enhances the mobility of the conducting fluid. Magnetic field-driven micropumps are in increasing demand due to their long-term reliability in generating flow, absence of moving parts, low power requirements, flow reversibility, and efficient mixing [8,9]. In all those applications, the system performance is dependent on the gas–liquid flow pattern in the channel, and the stability of the interface between the phases, which have been subject to experimental and numerical investigations (e.g., [10,11,12,13,14]).

Since the pioneering works of Hartmann and Lazarus in 1937 [15] the Poiseuille flow of an electrically conducting fluid in a transverse magnetic field (i.e., Hartmann flow) has been thoroughly studied in the literature (e.g., [16,17,18,19]). Hartmann [20] presented an analytical solution for the velocity profile of an isothermal laminar flow in an electrically insulated channel subjected to a wall-normal uniform magnetic field. It was demonstrated that the magnetic field flattens the velocity profile, a phenomenon often termed the ‘Hartmann effect’ in the literature. However, the presence of a second phase can drastically affect the velocity profile and the associated pressure gradient.

Due to the density difference between the phases, a stratified flow configuration is commonly encountered in both horizontal and inclined conduits. The part of the flow cross-section area occupied by the heavier fluid is referred to as the holdup, being an additional characteristic of the flow pattern. Shail [21] investigated the Hartmann flow of a conducting fluid between two horizontal insulating plates, with a layer of non-conducting fluid separating the top wall from the conducting fluid. It was discovered that the flow rate of the conducting fluid can be increased by approximately 30 percent for suitable ratios of the depths and viscosities of the two fluids used in an electromagnetic pump. Owen et al. [22] introduced two mechanistic models: a simple homogeneous model and a more sophisticated film flow model for computing the pressure drop in an MHD two-phase flow at high Hartmann numbers. Lohrasbi and Sahai [23] derived analytical solutions for the velocity and temperature profiles in a two-phase laminar steady MHD flow between two horizontal plates with one conducting phase. Malashetty and Umavathi [24], analytically investigated a similar system in an inclined channel, solving the nonlinear coupled momentum and energy equations in both phases. They found that, for a constant thickness of the layers, increasing the magnetic field has a dampening effect on the velocity of the conducting layer, akin to a single-phase flow. They also noted an increase in the velocity of the conducting fluid with a decrease in the thickness of the upper electrically non-conducting layer and/or with an increase in the channel inclination angle. A subsequent study by Umavathi et al. [25] addressed the magnetohydrodynamic Poiseuille–Couette flow and heat transfer of two immiscible fluids between inclined parallel plates. However, for isothermal flow, the effect of the different gravity driving force in the two layers is not considered in the model equations. Recently, Shah et al. [26] derived approximate analytical solutions for the velocity and temperature fields of unsteady MHD generalized Couette flows of two immiscible and electrically conducting fluids flowing between two horizontal electrically insulated plates subjected to an inclined magnetic field and an axial electric field. In all of these studies, the problem is solved for a pre-defined flow configuration (i.e., in situ holdup of the heavy layer and the pressure gradient).

An important issue in the context of two-phase flows is the occurrence of gravity-driven multiple solutions in inclined channels for a specified two-phase system and fixed operational conditions. It has been established in the two-phase flow literature [27,28,29,30] that there always exist two possible solutions for the holdup of steady countercurrent flow and up to three distinct steady-state solutions can be obtained within a limited range of the flow parameters in concurrent upward- and downward-inclined flows. The existence of multiple solutions for the holdup for some range of the superficial velocities in liquid–liquid flows was verified experimentally [28,29], where experimental verification of the existence of multiple solutions for the holdup within a certain range of superficial velocities in liquid–liquid flows has been conducted. However, the feasibility of achieving countercurrent flow in the presence of a magnetic field, and its impact on the potential for multiple stratified flow configurations in concurrent and countercurrent scenarios has not yet been explored.

In this study, we investigate the Hartmann flow of a conducting fluid between two parallel insulating plates, with a layer of non-conducting fluid separating the top wall from the conducting fluid. The presented analytical solution allows for the first time determination of the in situ holdup and the pressure gradient in horizontal or inclined channels for a specified two-phase system and fixed operational conditions. We identify the corresponding dimensionless input parameters that dictate the in situ stratified flow configuration. Furthermore, we utilize the solution to examine the impact of the presence of a second non-conductive layer and the intensity of the magnetic field on the flow characteristics in concurrent and countercurrent flows of the fluids. Special attention is given to the potential lubrication and pumping power saving achieved by introducing the gas phase, as well as to the operational conditions for which multiple solutions can be obtained for the stratified flow configuration in inclined flows.

2. Two-Phase Horizontal Flow

We consider isothermal, axial, steady and fully developed laminar stratified flow of two immiscible fluids between two infinite electrically insulating plates positioned at $z=0$ and $z=H$ (see Figure 1). Under these conditions, the pressure gradient, $G=dp/dx$, is the same in both layers and is constant (e.g., [31]). For horizontal flow, the channel inclination to the horizontal is β = 0. A plane and smooth interface between the phases is located at $z=h$. The lower layer (1) is occupied by an electrically conducting fluid (e.g., mercury), which is affected by a constant transverse magnetic field $\mathit{B}=B_0{\mathit{e}}_z$, while the upper lighter fluid is assumed to be non-conductive (e.g., air). Then, the x-component of the momentum equations in the two layers are as follows (e.g., Shail [21]):

$${\eta }_1\frac{d^2{u}_1\left(z\right)}{d{z}^2}-{\sigma }_1{B}_0^2{u}_1\left(z\right)=G\hspace{1em}\hspace{1em}0\le z\le h$$

(1)

$${ \eta }_2\frac{d^2{u}_2\left(z\right)}{d{z}^2}=G\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}h\le z\le \mathrm{H}$$

(2)

where ${\eta }_1,{\eta }_2$ are the fluids’ viscosities and ${\sigma }_1$ is the heavier fluid’s electric conductivity. The solutions for the velocity profiles are as follows:

$$u_1\left(\tilde{z}\right)=A_1\cosh \left(Ha \tilde{z}\right)+A_2\sinh \left(Ha \tilde{z}\right)-\frac{G}{{\sigma }_1{B}_0^2}\hspace{1em}0\le \tilde{z}\le \tilde{h}$$

(3)

$$u_2\left(\tilde{z}\right)=\frac{G{H}^2}{{ 2\eta }_2}{\tilde{z}}^2+B_{1}H\tilde{z}+B_2\hspace{1em}\hspace{1em}\tilde{h}\le \tilde{z}\le 1$$

(4)

where $\tilde{z}=z/H$, $\tilde{h}=h/H$ is the conductive (heavier) fluid holdup and Ha is the Hartmann number $Ha=B_{0}H\sqrt{\frac{{\sigma }_1}{{\eta }_1}}$. Applying no-slip boundary conditions at $\tilde{z}=0$ and $\tilde{z}=1$, and the continuity of velocities and viscous shear stresses at the fluids’ interface $\tilde{z}=\tilde{h}$ yields the following:

$$A_1=\frac{G}{{\sigma }_1{B}_0^2}, B_2=-\frac{G{H}^2}{{ 2\eta }_2}-B_{1}H$$

(5)

$$A_2=\frac{{GH}^2}{{ \eta }_2}\frac{-\frac{\left(1-\tilde{h}\right)}{2}-\frac{1}{H{a}^2{\eta }_{12}\left(1-\tilde{h}\right)}\left[\cosh \left(Ha\tilde{h}\right)-1\right]-\frac{1}{Ha}\sinh \left(Ha\tilde{h}\right)}{Ha{\eta }_{12}\cosh \left(Ha\tilde{h}\right)+\frac{1}{\left(1-\tilde{h}\right)}\sinh \left(Ha\tilde{h}\right)}$$

(6)

$$B_1=\frac{GH}{{\eta }_2}\left[-\tilde{h}+\frac{1}{Ha}sinh\left(Ha\tilde{h}\right)+Ha\frac{{\eta }_1{A}_2}{G{H}^2}\cosh \left(Ha\tilde{h}\right)\right]$$

(7)

The volumetric flow rates (per unit channel width) in the lower and upper layers are as follows:

$$Q_1=H{\int }_0^{\tilde{h}}{u}_1\left(\tilde{z}\right)d\tilde{z}=\frac{{GH}^3}{H{a}^3{\eta }_1}\sinh \left(Ha\tilde{h}\right)+\frac{A_{2}H}{Ha}\left[\cosh \left(Ha\tilde{h}\right)-1\right]-\frac{G{H}^3\tilde{h}}{H{a}^2{\eta }_1}$$

(8)

$$Q_2=H{\int }_{\tilde{h}}^1{u}_2\left(\tilde{z}\right)d\tilde{z}=\frac{G{ H}^3}{{ 6\eta }_2}\left(-2-{\tilde{h}}^3+3\tilde{h}\right)+\frac{B_1{H}^2}{2}\left(-1-{\tilde{h}}^2+2\tilde{h}\right)$$

(9)

As the left hand side (l.h.s) of both Equations (8) and (9) are proportional to G, the ratio of Q₁/Q₂ yields an implicit equation of calculation of the holdup of the conductive layer in the channel, $\tilde{h}=h/H$:

$$Q_{12}=\frac{Q_1}{Q_2}=\frac{U_{1s}}{U_{2s}}=\frac{1}{H{a}^3}\frac{\frac{1}{{\eta }_{12}}\sinh \left(Ha\tilde{h}\right)+H{a}^2\frac{A_2{ \eta }_2}{G{H}^2}\left[\cosh \left(Ha\tilde{h}\right)-1\right]-Ha\frac{\tilde{h}}{{\eta }_{12}}}{\frac{1}{6}\left(-2-{\tilde{h}}^3+3\tilde{h}\right)-\frac{B_1{ \eta }_2}{2GH}{\left(1-\tilde{h}\right)}^2}$$

(10)

where $U_{\mathrm{1,2}s}=Q_{\mathrm{1,2}}/H$ is the superficial velocity of the fluid in the channel cross-section. In view of Equations (6) and (7), the following expressions are introduced:

$$\frac{A_2{ \eta }_2}{G{H}^2}=\frac{-\frac{\left(1-\tilde{h}\right)}{2}-\frac{1}{H{a}^2{\eta }_{12}\left(1-\tilde{h}\right)}\left[\cosh \left(Ha\tilde{h}\right)-1\right]-\frac{1}{Ha}\sinh \left(Ha\tilde{h}\right)}{Ha{\eta }_{12}\cosh \left(Ha\tilde{h}\right)+\frac{1}{\left(1-\tilde{h}\right)}\sinh \left(Ha\tilde{h}\right)}$$

(11)

$$\frac{B_1{ \eta }_2}{GH}=-\tilde{h}+\frac{1}{Ha}sinh\left(Ha\tilde{h}\right)+Ha\frac{{\eta }_1{A}_2}{G{H}^2}\cosh \left(Ha\tilde{h}\right)$$

(12)

and ${\frac{A_2{\eta }_1}{G{H}^2}=\eta }_{12}\frac{A_2{\eta }_2}{G{H}^2}$.

The calculation of the flow-rate ratio for a specified holdup via Equation (10) (with (11) and (12)) is straightforward. However, in practice, the input flow rates of the fluids are known, whereas the location of the interface is unknown. Equation (10) indicates that $\tilde{h}=\tilde{h}\left(Q_{12},{\eta }_{12},Ha\right)$. In the limit of Ha → 0, Equation (10) converge to the expression obtained for $Q_{12}\left(\widetilde{h, }{\eta }_{12}\right)$ for a two-layer Poiseuille flow in the absence of a magnetic field (e.g., [32,33]).

Once Equation (10) is solved for the holdup, the corresponding pressure gradient $G$ can be determined. For example, using the solution for the holdup in Equation (9) yields the following:

$$G=\frac{dp}{dx}=\frac{{dp}_f}{dx}=\frac{U_{2s}}{\frac{{ H}^2}{{ 6\eta }_2}\left(-2-{\tilde{h}}^3+3\tilde{h}\right)-\frac{B_{1}H}{2G}{\left(1-\tilde{h}\right)}^2}$$

(13)

where B₁/G is independent of G (see Equation (7)). Obviously, in horizontal flow, the total pressure gradient is identical to the frictional pressure gradient. The corresponding dimensionless pressure gradient,$P_f^{\mathrm{2,0}}$ $P_f^{\mathrm{1,0}}$, can be obtained by normalizing $dp/dx$ with respect to the frictional pressure gradient of the single-phase flow of either the lighter or the heavier fluid obtained in the absence of a magnetic field, ${\left({dp}_f/dx\right)}_{2s,1s}^0=-12{\eta }_{\mathrm{2,1}}{U}_{2s,1s}$/H², respectively, whereby

$$P_f^{\mathrm{2,0}}=\frac{dp/dx}{{\left({dp}_f/dx\right)}_{2s}^0}=\frac{1}{2}\frac{1}{\left({\tilde{h}}^3-3\tilde{h}+2\right)+\frac{{{3\eta }_{2}B}_1}{GH}{(1-\widetilde{ h})}^2}$$

(14)

$$P_f^{\mathrm{1,0}}=\frac{dp/dx}{{\left({dp}_f/dx\right)}_{1s}^0}=\frac{P_f^{\mathrm{2,0}}}{Q_{12} {\eta }_{12}}$$

(15)

where the first superscript (1 or 2) represents the phase selected to scale the two-phase pressure gradient, and the second superscript (0) indicates that the pressure gradient in the absence of a magnetic field is used for the scaling (see also the nomenclature list). Note that $\frac{B_1{ \eta }_2}{GH}$ is also determined by $(Q_{12},{\eta }_{12},Ha)$ (see Equation (7)).

The dimensionless pressure gradient that would be obtained in the case of a single-phase (SP) flow of the conductive (heavier) fluid under the same magnetic field strength (i.e., the same Ha) is as follows:

$$P_{fs}^{\mathrm{1,0}}=\frac{{\left(dp/dx\right)}_{1s}}{{\left({dp}_f/dx\right)}_{1s}^0}=\frac{{Ha}^2}{12\left[1-\frac{2}{Ha}tgh\left(\frac{Ha}{2}\right)\right]}$$

(16)

Hence, when scaling the two-phase pressure gradient with respect to the SP flow of the heavy phase in the presence of the same magnetic field, the dimensionless pressure gradient is given by the following:

$$P_f^1=\frac{P_f^{\mathrm{1,0}}}{P_{fs}^1}=12\frac{{\tilde{G}}^{\mathrm{1,0}}}{{Ha}^2} \left[1-\frac{2}{Ha}tgh\left(\frac{Ha}{2}\right)\right]$$

(17)

The corresponding dimensionless pumping power (power factor) required for the given flow rates is as follows:

$$Po=P_f^1\frac{(U_{1s}+U_{2s})}{U_{1s}}=P_f^1(1+Q_{21})$$

(18)

Hence, the dimensionless pressure gradient (either $P_f^{\mathrm{2,0}}, P_f^{\mathrm{1,0}}$ or $P_f^1$) and the power factor are also determined in terms of $(Q_{12},{\eta }_{12},Ha)$.

Finally, upon scaling the velocity with respect to $U_{1s},$ the dimensionless velocity profiles are also obtained in terms of those three dimensionless parameters:

$${\tilde{u}}_1\left(\tilde{z}\right)=\frac{u_1\left(z\right)}{U_{1s}}={\tilde{A}}_1\left[\cosh \left(Ha\tilde{z}\right)-1\right]+{\tilde{A}}_2\sinh \left(Ha\tilde{z}\right)$$

(19)

$${\tilde{u}}_2\left(\tilde{z}\right)=\frac{u_2\left(z\right)}{U_{1s}}={\tilde{B}}_2\left({\tilde{z}}^2-1\right)+{\tilde{B}}_1\left(\tilde{z}-1\right)$$

(20)

with

$${\tilde{A}}_1=-\frac{12P_f^{\mathrm{1,0}}}{H{a}^2}; {\tilde{B}}_2=-6{\eta }_{12}{P}_f^{\mathrm{1,0}}$$

(21)

$${\tilde{A}}_2=-12{\eta }_{12}{P}_f^{\mathrm{1,0}}\frac{-\frac{\left(1-\tilde{h}\right)}{2}-\frac{1}{H{a}^2{\eta }_{12}\left(1-\tilde{h}\right)}\left[\cosh \left(Ha\tilde{h}\right)-1\right]-\frac{1}{Ha}\sinh \left(Ha\tilde{h}\right)}{Ha{\eta }_{12}\cosh \left(Ha\tilde{h}\right)+\frac{1}{\left(1-\tilde{h}\right)}\sinh \left(Ha\tilde{h}\right)}$$

(22)

$${\tilde{B}}_1=-12{\eta }_{12}{P}_f^{\mathrm{1,0}}\left[-\tilde{h}+\frac{1}{Ha}sinh\left(Ha\tilde{h}\right)+Ha\frac{{\eta }_1{A}_2}{G{H}^2}\cosh \left(Ha\tilde{h}\right)\right]$$

(23)

3. Two-Phase Inclined Flow

In the case of inclined flow, the momentum equations in the two layers are

$${\eta }_1\frac{d^2{u}_1\left(z\right)}{d{z}^2}-{\sigma }_1{B}_0^2{u}_1\left(z\right)=\frac{dp}{dx}-{\rho }_{1}gsin\beta =G_1\hspace{1em}\hspace{1em}0\le z\le h$$

(24)

$${\eta }_2\frac{d^2{u}_2\left(z\right)}{d{z}^2}=\frac{dp}{dx}-{\rho }_{2}gsin\beta =G_2\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}h\le z\le H$$

(25)

where $\beta$ is the angle of the channel downward inclination to the horizontal (x is in the downward direction) and ${\rho }_1,{\rho }_2$ are the fluids’ densities. The model assumptions are the same as those indicated in the case of the horizontal channel (see Section 2). The terms describing the gravity body force, which drives the flow along with the imposed pressure gradient, are added to the equations. The solution of (24) and (25) for the velocity profile that satisfies the no-slip boundary conditions at $\tilde{z}=0$ and $\tilde{z}=1$ is as follows:

$$u_1\left(\tilde{z}\right)=\frac{G_1}{{\sigma }_1{B}_0^2}\left[\cosh \left(Ha \tilde{z}\right)-1\right]+A_2\sinh \left(Ha \tilde{z}\right)\hspace{1em}\hspace{1em}\hspace{1em}0\le \tilde{z}\le \tilde{h}$$

(26)

$$u_2\left(\tilde{z}\right)=\frac{G_2}{{ 2\eta }_2}\left({\tilde{z}}^2-1\right)+B_1\left(\tilde{z}-1\right)\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\tilde{h}\le z\le 1$$

(27)

Imposing the boundary conditions of velocities and shear stress continuity at the interface yields the following:

$$B_{1}H=-\frac{G_2{H}^2\tilde{h}}{{\eta }_2}+\frac{G_1{H}^2}{{\eta }_2}\frac{1}{Ha}\sinh \left(Ha\tilde{h}\right)+A_2\mathit{Ha}{\eta }_{12}\cosh \left(Ha\tilde{h}\right)$$

(28)

$$A_2=\frac{-\frac{G_2{H}^2}{{ 2\eta }_2}\left(1-\tilde{h}\right)-\frac{G_1{H}^2}{H{a}^2{\eta }_1\left(1-\tilde{h}\right)}\left[\cosh \left(Ha\tilde{h}\right)-1\right]-\frac{G_1{H}^2}{Ha{\eta }_2}\sinh \left(Ha\tilde{h}\right)}{{\eta }_{12}\mathit{Ha}\cosh \left(Ha\tilde{h}\right)+\frac{1}{\left(1-\tilde{h}\right)}\sinh \left(Ha\tilde{h}\right)}$$

(29)

The volumetric fluxes in the lower and upper layers are as follows:

$$Q_1=U_{1s}H=H{\int }_0^{\tilde{h}}{u}_1\left(\tilde{z}\right)d\tilde{z}=\frac{{G_{1}H}^3}{H{a}^3{\eta }_1}\sinh \left(Ha\tilde{h}\right)+\frac{A_{2}H}{Ha}\left[\cosh \left(Ha\tilde{h}\right)-1\right]-\frac{G_1{H}^3\tilde{h}}{H{a}^2{\eta }_1}$$

(30)

$$Q_2=U_{2s}H=H{\int }_{\tilde{h}}^1{u}_2\left(\tilde{z}\right)d\tilde{z}=-{\left(1-\tilde{h}\right)}^2\left[\frac{G_2{ H}^3}{{ 6\eta }_2}\left(\tilde{h}+2\right)+\frac{B_1{H}^2}{2}\right]$$

(31)

Substituting $B_{1}H$ (Equation (28)) in Equation (31) and dividing Equations (30) and (31) by $U_{1s}$ yields the following:

$$1=-12\frac{{\tilde{G}}_1^{\mathrm{1,0}}}{H{a}^3}\sinh \left(Ha\tilde{h}\right)+\frac{\widetilde{A_2}}{Ha}\left[\cosh \left(Ha\tilde{h}\right)-1\right]+12\frac{{\tilde{G}}_1^{\mathrm{1,0}}\tilde{h}}{H{a}^2},$$

(32)

$$\begin{gathered} \frac{U_{2s}}{U_{1s}}=2{\left(1-\tilde{h}\right)}^2\left\{{\tilde{G}}_2^{\mathrm{1,0}}{\eta }_{12}\left(\tilde{h}+2\right)+\left[-3{\tilde{G}}_2^{\mathrm{1,0}}{\eta }_{12}\tilde{h}+\frac{{3\tilde{G}}_1^{\mathrm{1,0}}{\eta }_{12}}{Ha}\sinh \left(Ha\tilde{h}\right)- \\ \hspace{1em}\hspace{1em}\hspace{1em}-\frac{\widetilde{A_2}Ha{\eta }_{12}}{4}\cosh \left(Ha\tilde{h}\right)\right]\right\} \end{gathered}$$

(33)

where the following relations were introduced:

$$\frac{G_2}{U_{1s}{\eta }_2/H^2}=-\frac{12{\eta }_{12}{G}_2}{{\left({dp}_f/dx\right)}_{1s}^0}=-12{\eta }_{12}{\tilde{G}}_2^{\mathrm{1,0}}$$

(34)

$$\frac{G_1}{U_{1s}{\eta }_1/H^2}=-\frac{12G_1}{{\left({dp}_f/dx\right)}_{1s}^0}=-12{\tilde{G}}_1^{\mathrm{1,0}}$$

(35)

and ${\tilde{A}}_2$=$A_2/U_{1s}$, which by Equation (29) reads as follows:

$${\tilde{A}}_2=12\frac{\frac{{\tilde{G}}_2^{\mathrm{1,0}}{\eta }_{12}\left(1-\tilde{h}\right)}{2}+\frac{{\tilde{G}}_1^{\mathrm{1,0}}}{H{a}^2\left(1-\tilde{h}\right)}\left[\cosh \left(Ha\tilde{h}\right)-1\right]+\frac{{\tilde{G}}_1^{\mathrm{1,0}}{\eta }_{12}}{Ha}\sinh \left(Ha\tilde{h}\right)}{Ha{\eta }_{12}\cosh \left(Ha\tilde{h}\right)+\frac{1}{\left(1-\tilde{h}\right)}\sinh \left(Ha\tilde{h}\right)}$$

(36)

In principle, once $U_{1s}$,$U_{2s}$ are known, Equations (32) and (33) can be solved for the unknown holdup,$\tilde{h},$ and pressure gradient, $dp/dx,$ embedded both in ${\tilde{G}}_1^{\mathrm{1,0}}$ and ${\tilde{G}}_2^{\mathrm{1,0}}$. However, to reveal the dimensionless parameters of the solution in inclined flows, the solution form is further manipulated.

The total pressure gradient is a sum of the frictional and gravitational (hydrostatic) pressure gradients; hence:

$${\frac{dp}{dx}=\left(\frac{dp}{dx}\right)}_f+{\left(\frac{dp}{dx}\right)}_g={\left(\frac{dp}{dx}\right)}_f+\left[{\rho }_1\tilde{h}+{\rho }_2(1-\tilde{h})\right]gsin\beta$$

(37)

Therefore:

$$G_2=\frac{dp}{dx}-{\rho }_{2}gsin\beta ={\left(\frac{dp}{dx}\right)}_f+\widetilde{ h}({\rho }_1-{\rho }_2)gsin\beta$$

(38)

and

$${\tilde{G}}_2^{\mathrm{1,0}}=\frac{G_2}{{\left({dp}_f/dx\right)}_{1s}^0}=\left[{\tilde{P}}_f^{\mathrm{1,0}}+\widetilde{ h}{Y}^{\mathrm{1,0}}\right] ; Y^{\mathrm{1,0}}=\frac{({\rho }_1-{\rho }_2)gsin\beta }{{\left({dp}_f/dx\right)}_{1s}^0}$$

(39)

where $P_f^{\mathrm{1,0}}$ is the dimensionless frictional pressure gradient in the inclined two-phase flow and $Y^{\mathrm{1,0}}$ is the (a priori known) inclination parameter. Similarly,

$$G_1=\frac{dp}{dx}-{\rho }_{1}gsin\beta ={\left(\frac{dp}{dz}\right)}_f-\left(1-\tilde{h}\right)({\rho }_1-{\rho }_2)gsin\beta$$

(40)

and

$${\tilde{G}}_1^{\mathrm{1,0}}=\frac{G_1}{{\left({dp}_f/dx\right)}_{1s}^0}=\left[{\tilde{P}}_f^{\mathrm{1,0}}-\left(1-\tilde{h}\right)Y^{\mathrm{1,0}}\right]$$

(41)

Upon substituting (39) and (41) in Equations (32) and (33) (and in (36)), a solution for the holdup $\tilde{h}$ and the dimensionless frictional gradient $P_f^{\mathrm{1,0}}$ can be obtained in terms of the four (a priori known) dimensionless parameters (${\eta }_{12}$,$Q_{12},Ha, Y^{\mathrm{1,0}}$). The values of those parameters are a priori known for a specified two-flow system (i.e., channel geometry and inclination, fluid properties, and magnetic field intensity).

In fact, Equation (32) can be used to obtain an explicit expression for $P_f^{\mathrm{1,0}}$ in terms of (${\eta }_{12},Ha,Y^{\mathrm{1,0}},\tilde{h})$, which can then be substituted into Equation (33) to obtain an implicit equation for the holdup. This step is important to explore the possibility of multiple solutions for prescribed values of the dimensionless parameters, since this becomes rather straightforward once a single algebraic equation for the holdup is obtained.

In view of Equation (37), the total pressure gradient in inclined flow (normalized with respect to the frictional pressure gradient of mercury flow in the absence of a magnetic field) is given by the following:

$$P^{\mathrm{1,0}}=\frac{dp/dx}{{\left({dp}_f/dx\right)}_{1s}^0}=P_f^{\mathrm{1,0}}+P_g^{\mathrm{1,0}}=P_f^{\mathrm{1,0}}+{\tilde{h}Y}^{\mathrm{1,0}}+\frac{{\rho }_2}{∆\rho }{Y}^{\mathrm{1,0}}$$

(42)

where the sum of the last two terms on the right hand side (r.h.s) represents the dimensionless hydrostatic pressure gradient, $P_g^{\mathrm{1,0}}.$ When normalized with respect to the frictional pressure gradient obtained for SP flow of the conductive layer under the same magnetic field,${\left({dp}_f/dx\right)}_{1s}$ (see Equation (16)), the dimensionless total pressure gradient is then $P^1=P^{\mathrm{1,0}}/P_{fs}^{\mathrm{1,0}}, { (P}_f^1$ and $P_g^1$ are the frictional and hydrostatic components). Note that the density ratio,${\rho }_{12}={{\rho }_1/\rho }_2$, is an additional dimensionless parameter that has to be prescribed in order to obtain the total pressure and power factors. However, in the case the lighter (non-conductive) fluid is a gas, the last term on the r.h.s of Equation (42) is negligible and practically $P^{\mathrm{1,0}}\approx {\tilde{G}}_2^{\mathrm{1,0}}$.

To examine the effect of the gas layer on the pressure gradient, its value compared to the total pressure gradient in SP flow of the conductive fluid which develops under the same Ha is of interest. It is given by the following:

$$P^{1T}=\frac{P^1}{1+\frac{{\rho }_2}{∆\rho }\frac{Y^{\mathrm{1,0}}}{P_{fs}^{\mathrm{1,0}}}}$$

(43)

and the corresponding power factor is $Po=P^{1T}(1+Q_{21})$. Once the holdup and pressure gradient have been obtained, the corresponding dimensionless velocity profile in the two layers are given by the following:

$${\tilde{u}}_1\left(\tilde{z}\right)=\frac{u_1\left(\tilde{z}\right)}{U_{1s}}={\tilde{A}}_1\left[\cosh \left(Ha\tilde{z}\right)-1\right]+{\tilde{A}}_2\sinh \left(Ha\tilde{z}\right)$$

(44)

$${\tilde{u}}_2\left(\tilde{z}\right)=\frac{u_2\left(\tilde{z}\right)}{U_{1s}}={\tilde{B}}_2\left({\tilde{z}}^2-1\right)+{\tilde{B}}_1\left(\tilde{z}-1\right)$$

(45)

The dimensionless coefficients are given by the following:

$${\tilde{A}}_1=-\frac{12{\tilde{G}}_1^{\mathrm{1,0}}}{H{a}^2}; {\tilde{B}}_2=\frac{G_2{H}^2}{{ 2\eta }_2{U}_{1s}}=-6{\eta }_{12}{\tilde{G}}_2^{\mathrm{1,0}}$$

(46)

${\tilde{A}}_2$ is given in Equation (36) and by using Equation (28):

$${\tilde{B}}_1=12\left[{\tilde{G}}_2^{\mathrm{1,0}}{\eta }_{12}\tilde{h}-\frac{{\tilde{G}}_1^{\mathrm{1,0}}{\eta }_{12}}{Ha}sinh\left(Ha\tilde{h}\right)\right]+{\tilde{A}}_{2}Ha{\eta }_{12}\cosh \left(Ha\tilde{h}\right)$$

(47)

4. Results and Discussion

The analytical solution presented in Section 2 and Section 3 enables determining the conductive liquid holdup and the dimensionless pressure gradient in MHD two-phase flow in terms of the set of dimensionless (${\eta }_{12},Q_{21},Ha,Y^{\mathrm{1,0}})$ parameters. The values of these parameters are determined by the channel geometry, fluid properties, and flow rates, and are independent of the in situ flow characteristics. Worth noting is that using the above analytical solution for the calculation of the holdups and velocity profiles appears to be non-trivial. Due to the hyperbolic functions involved, it is necessary to work with very large and very small numbers, which poses an accuracy problem for the usual calculations with 16, and even 32, decimal places. To obtain correct results we had to require 128 or even 256 decimal places, which was possible when the calculations were performed using Maple (https://www.maplesoft.com).

The analytical solution obtained is used to explore the effect of the presence of a second non-conductive (gas) layer and the intensity of the magnetic field on the flow characteristics in horizontal flows and in concurrent and countercurrent inclined flows of the fluids. With this aim, we selected two representative two-phase systems—mercury–air and sodium–argon. Their physical properties are provided in

Table 1. Properties of mercury (Hg) and air at room temperature.

Name	Notation	Mercury (Hg)	Air
Density	ρ	13,534 kg/m3	1.2 kg/m3
Dynamic viscosity	η	0.00149 kg/(m∙s)	1.8 × 10−5 kg/(m∙s)
Kinematic viscosity	ν = η/ρ	1.1 × 10−7 m2/s	1.8 × 10−5 m2/s
Electric conductivity	σ	106 1/(Ω∙m)
Surface tension	γ	0.4589 N/m

and

Table 2. Properties of liquid sodium (Na) and argon (Ar) at 400 °C.

Name	Notation	Sodium (Na)	Argon (Ar)
Density	ρ	856 kg/m3	0.713
Dynamic viscosity	η	0.000284 kg/(m∙s)	2.4 × 10−5
Kinematic viscosity	ν = η/ρ	3.32 × 10−7 m2/s	3.37 × 10−5
Electric conductivity	σ	4.52 × 106 1/(Ω∙m)
Surface tension	γ	0.161 N/m

, respectively. The viscosity ratio of the sodium–argon and the mercury–air systems are ${\eta }_{12}=11.83$ and ${\eta }_{12}=81.87$, respectively. The higher value of the latter results from the lower viscosity of sodium compared to mercury (and somewhat higher viscosity of argon compared to air). Worth noting is that, for the same $Ha$ number, the magnetic field intensity applied in the case of mercury flow is about five times larger than that in argon (e.g., B₀ = 0.01 T corresponds to Ha = 5.18 in mercury and to Ha = 25.23 in sodium).

4.1. Horizontal Channel

In a horizontal channel, $Y^{\mathrm{1,0}}=0,$ whereby, for specified fluids (i.e., specified ${\eta }_{12}$), the solutions for the holdup $\tilde{h}$ and the dimensionless pressure gradient $P_f^{\mathrm{1,0}}$ are a function of the phases’ flow-rate ratio, $Q_{12}$, and the Hartmann number, $\mathrm{H}\mathrm{a}$.

Figure 2 shows the effect of the magnetic field intensity variation of the holdup with the gas-to-liquid flow-rate ratio $Q_{21}=U_{2s}{/U}_{1s}$ for mercury–air and sodium–argon two-phase flows. Obviously, the liquid metal holdup decreases with the increase in the gas flow rate, and is higher in the mercury–air system of the larger viscosity ratio. The effect of the magnetic field on the holdup becomes significant for Ha > ~1. Due to the slowing down of the flow by the magnetic damping force, increasing the magnetic field intensity results in a higher holdup of the liquid metal and attenuates its decline with the increase in the gas flow rate. These trends are more pronounced for the same Ha in the more viscous mercury–air system.

Figure 3 shows the effect of the magnetic field intensity on the dimensionless frictional pressure gradient $P_f^{\mathrm{1,0}}$ (normalized with respect to the frictional pressure gradient of a single-phase liquid metal flow with the same $U_{1s}$, but without a magnetic field, see Equation (15)). As expected, increasing Ha results in a higher frictional pressure gradient, whereby the $P_f^{\mathrm{1,0}}$ factor reaches a value of ~1000 for Ha = 103.7 (corresponding to B₀ = 0.2 T and 0.041 T for mercury and sodium, respectively). It is well known that, in the absence of a magnetic field, the introduction of a gas flow to the flow of a viscous liquid can result in a lubrication effect (e.g., [31]), where $P_f^{\mathrm{1,0}}<1$ values are obtained for some range of low $Q_{21}$. The lubrication effect increases with increasing ${\eta }_{12}$ (approaching a value of 0.25 in the TP geometry for large ${\eta }_{12}$). For example, in mercury–air flow and Ha = 0, $P_f^{\mathrm{1,0}}$ can reach a value of ~0.35 (65% reduction of the pressure gradient) for $Q_{21}$ = 0.05, and $P_f^{\mathrm{1,0}}$ < 1 up to $Q_{21}$ < ~10 (see also Figure 4b below for low Ha, which shows the power reduction). Indeed, a reduction of the pressure gradient factor $P_f^{\mathrm{1,0}}$ by the gas flow at low $Q_{21}$ values is also obtained for low values of Ha < ~1. However, the figure shows that, in the examined range of Ha > 5, the presence of a gas layer further augments the pressure gradient factor,$P_f^{\mathrm{1,0}}$, in particular, when $Q_{21}>1$ and Ha is relatively low. The sensitivity of the $P_f^{\mathrm{1,0}}$ factor to the gas flow is higher for the less viscous sodium–argon system. In both systems and high Ha, the gas flow rate has a rather small effect on the $P_f^{\mathrm{1,0}}$ factor over a wide range of gas flow rates.

The lubrication effect of the gas flow is noticed over a wider range of Ha when examining the $P_f^1$ factor (Equation (17)), which is the dimensionless frictional pressure gradient normalized with respect to a single-phase liquid metal flow under the same magnetic field intensity. As shown in Figure 4a, the lubrication effect is more pronounced in the case of the more viscous mercury, where values of $P_f^1$ < 1 are obtained up to Ha~25 for $Q_{21}$ < 1. The adverse effect of the gas flow on the pressure gradient at $Q_{21}$ > ~10 is also reflected in the $P_f^1$ factor, which, for the same Ha, is higher for the sodium–argon system. The lubrication region is more visible in Figure 4b, where the pumping power factor, Po, is also shown for the mercury–air system for lower Ha (B₀ = 0.001 T, Ha = 0.5183). Values of Po = $P_f^1(1+Q_{21})<$1 indicate that pumping power saving can be achieved by introducing air flow to the conductive liquid flow. Note that for low $Q_{21}$ $\mathrm{P}\mathrm{o}~P_f^1$ (both approach 1 at $Q_{21}=0).$ As shown in Figure 4b, significant power reduction can be achieved for low Ha. For example, for Ha = 0.5183, the minimal value Po = 0.3586 is obtained by adding 3.5% air to the mercury flow. For such low magnetic field strength, the $P_f^1~P_f^{\mathrm{1,0}}$ and the value of both is almost the same as that obtained for Ha = 0. The potential for power saving by the air flow is obviously reduced with increasing the magnetic field strength (negligible for Ha > 25), and the minimum Po is shifted to lower $Q_{21}$ values.

Further insight into the effect of the gas flow on the holdup and pressure gradient can be obtained by examining the dimensionless velocity profiles (normalized by the conductive liquid superficial velocity, $U_{1s}$). Figure 5a demonstrates the change in the velocity profile in the mercury–air system when the gas flow rate is increased from $Q_{21}$ = 1 to 10 for a constant Ha (=25.92, B₀ = 0.05 T). As shown, in both cases, the air moves much faster than the mercury and its velocity profile is practically parabolic. The velocity gradients at the upper wall (hence the shear stress) are sharp and obviously increase with the gas flow rate. The velocity profiles in the mercury are depicted in Figure 5b for a wider range of $Q_{21}$. The magnetic field results in a flat velocity profile in the bulk of the mercury layer, with sharp velocity gradients at the wall and at the interface. As shown, except for very low $Q_{21}$, the maximum velocity is in the air layer, whereby, near the interface, the mercury is dragged by the air flow. The sharp velocity gradients at the interface indicate that high interfacial shear stresses are involved. Increasing the gas flow rate also results in a higher near-wall velocity gradient in the mercury. The increase in the wall shear stresses on both channel walls with the gas flow is reflected in the increased pressure drop factors, ${ P}_f^{\mathrm{1,0}}$ and $P_f^1$ (in particular for $Q_{21}>~10,$ Figure 3 and Figure 4).

Figure 6 shows that, with increasing the intensity of the magnetic field (for a fixed value of $Q_{21}=1$), even though the mercury holdup increases, the mercury velocity gradient near the wall and at the interface becomes steeper, resulting in an increased pressure gradient at higher Ha. The mercury velocity rapidly grows from the bottom wall and quickly attains an almost flat profile, like in the single-phase Hartmann flow. However, contrary to the latter, in the upper part of the mercury layer the velocity does not decay, but steeply grows towards the air velocity at the interface. The thickness of the gas layer decreases at increased applied magnetic fields, while its maximum velocity and interfacial shear stress increase. In the experimental study by Lu et al. [10], such a flow configuration was characterized as a gas jet. This configuration is likely to enhance mixing between the two phases, potentially causing the flow regime to gradually transition to a mixed flow.

The effect of $Q_{21}$ and Ha on the shape of the velocity profile can be deduced by examining the variation in the maximal velocity and the interfacial velocity. Figure 7a shows the variation in the maximal velocity in the velocity profile vs. $Q_{21}$ for various Ha (B₀ ranging from 0.001 to 0.1). The range of $Q_{21}$ where the maximal velocity is in the mercury layer is indicated by a dashed line, corresponding to the range of flow rates where the mercury flow drags the air at the interface. In this range of flow rates, the interfacial velocity is obviously lower than the maximal mercury velocity. However, as shown, this flow range is limited to low $Q_{21}$ and diminishes with increasing Ha. Examining the interfacial velocity in mercury–air flow vs. $Q_{21}$ for various values of Ha (see Figure 7b) shows that it increases with increasing $Q_{21}$, with a rather low sensitivity to the magnetic field strength.

4.2. Inclined Channel

In inclined channels, the holdup and (dimensionless) frictional pressure gradient ($P_f^{\mathrm{1,0}}$,$P_f^1)$ are determined by four dimensionless parameters (${\eta }_{12},Q_{21}, Ha,Y^{\mathrm{1,0}})$. To obtain the total (dimensionless) pressure gradient factors ($P^{\mathrm{1,0}}$, $P^1$, or $P^{1T}$; see Equations (42) and (43)) the hydrostatic pressure gradient contribution should be accounted for. It can be calculated based on the holdup; however, the density ratio, ${\rho }_{12}={{\rho }_1/\rho }_2$, is then an additional required parameter. Depending on the values set for $Q_{21}$ and $Y^{\mathrm{1,0}},$ concurrent up-flow, concurrent down-flow or countercurrent flow of the liquid metal and gas are considered. In the following, the effects of the system parameters on the flow characteristics are demonstrated by referring to the mercury–air system, thus fixing the values of ${\eta }_{12}$ = ($81.87$) and ${\rho }_{12}$ (=13,534).

4.2.1. Concurrent Upward Flow

According to the selected flow configuration and coordinates (Figure 1), concurrent upward flow corresponds to negative values of the superficial velocities, $U_{2s}{,U}_{1s}$ < 0; hence, $Q_{21}$ > 0 and $Y^{\mathrm{1,0}}>0$.

Figure 8a shows the effect of the magnetic field intensity on the variation in the holdup with the air-to-mercury flow-rate ratio for a fixed inclination parameter value of $Y^{\mathrm{1,0}}=2075.25$. The latter corresponds to a constant flow rate of the mercury ($U_{1s}=-0.005$ m/s) and a slight upward inclination of the channel, β = 0.2° (the channel height is 0.02 m). The variation in $Q_{21}$, in this case, actually corresponds to the change in the air flow rate. A comparison of Figure 8 with Figure 2 shows the drastic effect of the slight inclination on the holdup. Due to the retarding gravity force, the metal flow is slowed down. Consequently, even for very shallow channel inclination and low mercury flow rate, the mercury occupies most of the flow cross-section over a wide range of air flow rates ($\tilde{h}>0.9$ up to $Q_{21}=~$200 with low sensitivity to Ha). The holdup (and the hydrostatic pressure drop) steeply declines with the air flow rate for $Q_{21}$ > 2000. Then, for some range of high air flow rates, three different solutions for the holdup are obtained for the same air (and mercury) flow rates: the high holdup solution and two additional solutions of lower holdup values. An example of the triple solution is indicated by the three circles in Figure 8b. With increasing the magnetic field strength (i.e., higher Ha), the range of air flow rates where a triple solution is obtained diminishes and is shifted to higher air flow rates (see Figure 8a). For the tested parameter set, the triple solution is feasible up to Ha$~$52 (B_o$~$0.1 T). For even higher air flow rates, beyond the triple-solution region, only a single (low) holdup solution is obtained, and the sensitivity of its value to the magnetic field intensity is rather low. Figure 8b shows that, with increasing the mercury flow rate (i.e., reducing $Y^{\mathrm{1,0}}$), the range of gas flow rates where triple solutions are obtained is shifted to a lower $Q_{21}$, but becomes narrower. For $\left|U_{1s}\right|\ge 0.1 \mathrm{m}/\mathrm{s}$, only a single solution for the holdup is obtained even for low Ha (=0.518). For the same flow-rate ratio $(Q_{21}$) the holdup of the mercury layer increases with reducing the mercury flow rate (i.e., increasing $Y^{\mathrm{1,0}}$).

The possibility of obtaining multiple holdup solutions is typical to gravity-dominated two-phase systems, where the body force is of the order of the frictional pressure gradient (or higher), and the forces acting on the fluids can be balanced in more than one flow configuration. A demonstration of the velocity profiles associated with the three different solutions for the holdup is shown in Figure 9. The velocity profile in the air flow cross-section is almost parabolic, whereby the air flow practically determines the values of the pressure gradient and the interfacial shear stress. In the low holdup solution, the gravity force acting on the thin mercury layer is low, and the pressure gradient and air shear at the interface are capable of carrying upward the flow over the entire mercury layer. In the high holdup solution, the mercury experiences the highest counter-flow body force, whereby the pressure gradient and the air interfacial shear are insufficient to carry the entire mercury layer upward. Consequently, a back-flow of the mercury is observed near the lower channel wall, and the wall shear direction is reversed. As shown in the figure, the back-flow is reduced in the middle holdup solution, which results in a lower reversed wall shear.

Back-flow of the heavier fluid is a source of instability of the stratified flow configuration, as it introduces a large disturbance at the inlet, where the fluids are introduced into the channel. The effect of the magnetic field strength on the velocity profiles of the high and middle holdup solutions is demonstrated in Figure 10 (for $Y^{\mathrm{1,0}}=2075.25)$. As shown, due to the magnetic field damping force, increasing Ha reduces the back-flow in both flow configurations, although the mercury holdup corresponding to the high holdup solution increases with Ha, and, therefore, is associated with a higher backward gravity force. For the examined parameters, triple-holdup solutions are obtained up to Ha~18 (B₀ = 0.035 T). For higher Ha, a single (relatively high) holdup solution is obtained, yet with a significant region of mercury back-flow (e.g., Ha = 51.83 in Figure 10b). The effect of the air flow rate on the velocity profile in the mercury layer for Ha = 51.84 is demonstrated in Figure 10c,d. For this magnetic field intensity, a single solution for the holdup is obtained for any $Q_{21}$. As shown, although the mercury holdup is reduced with increasing the air flow rate, the back-flow intensity increases. Note that the same dimensionless velocity profiles would be obtained for lower superficial velocities of the conductive layer and the gas, provided the value of $Y^{\mathrm{1,0}}$ is the same (e.g., by referring to a smaller channel size and/or inclination, and/or density difference), and $Q_{21}$ and ${\eta }_{12}$ are unchanged.

The effect of the magnetic field strength and the air flow rate on the pressure gradient is examined in light of Figure 11. As expected, the frictional pressure gradient factor $P_f^{\mathrm{1,0}}$ (scaled with the frictional pressure gradient of single-phase mercury flow for Ha = 0) increases with Ha (see Figure 11a). Introducing the air flow further increases the frictional pressure gradient. In fact, at high $Q_{21}$ (beyond the triple-solution region, corresponding to the loop in the pressure gradient curves) the increase in the frictional pressure gradient is dominated by the gas flow and is insensitive to the magnetic field intensity. Examining the frictional pressure gradient factor $P_f^1$ (i.e., scaled with the frictional pressure gradient of mercury flow under the same Ha, Figure 11b) reveals that the lubrication effect, obtained in the horizontal channel at low air flow rates (Figure 7b), is lost at even such a slight upward inclination of the channel. The variation in the hydrostatic pressure gradient factor, $P_g^1$ (see Equation (42)), is shown in Figure 11c. Obviously, for a constant Ha, it follows the trend of the holdup variation with $Q_{21}$ (shown in Figure 7). With increasing Ha, the hydrostatic pressure gradient factor decreases, and values < 1 are reached at high Ha and $Q_{21}$.

From the perspective of reducing the pressure gradient by the air flow the values of the total pressure gradient factor, $P^{1T}$ (i.e., $dp/dx$ normalized with respect to the total pressure gradient of single-phase mercury flow with the same magnetic field, Equation (43)), should be examined. Figure 11d shows that, at low $Q_{21}$, introducing the air flow practically does not affect the total pressure gradient. However, in the triple-solution region and in its vicinity, the steep reduction in the mercury holdup (and thus the hydrostatic pressure gradient), results in a pronounced reduction in the total pressure gradient compared to the mercury single-phase up-flow (e.g., by a factor of 0.122 at low Ha, and by a factor of 0.285 for B₀ = 0.1 T, Ha = 51.83). The potential for pressure drop reduction in this region is reduced with the increase in the magnetic field strength.

The trends of the variation in the holdup with the channel upward inclination are similar to those obtained upon changing the mercury flow rate (Figure 8b). Obviously, an increase in $Y^{\mathrm{1,0}}$ can be affected by reducing the flow rate of the conductive phase (mercury), by increasing the channel upward inclination, or by increasing the channel size. Figure 12a shows that, for the same Ha and maintaining the same flow rates of both phases (i.e., same $Q_{21})$, the holdup of the conductive layer is higher at a steeper upward channel inclination. Also, with increasing β, the triple-solution region is shifted to higher air flow rates. In this region, owing to the lower values of the low and middle holdup solutions at steeper inclinations, smaller values of the total pressure-gradient factor, $P^{1T}$, can be obtained (e.g., 0.00425 for β = 5° in Figure 12b). Obviously, the same range of $Y^{\mathrm{1,0}}$ examined in Figure 12a can be obtained for a lower superficial velocity of the conductive layer in a smaller channel size, in which case the $Q_{21}$ range of the triple-solution region corresponds to lower gas dimensional velocity.

The concurrent down-flow corresponds to $Y^{\mathrm{1,0}}<0 {U_{1s},U}_{2s}$ > 0 ($Q_{21}$ > 0). In countercurrent flow, a net downward flow of conductive heavier fluid is considered, namely $U_{1s}$ > 0 and $Y^{\mathrm{1,0}}<0$, while the light fluid flows upward ($U_{2s}$ < 0); hence, $Q_{21}$ < 0. Countercurrent flow is a basic configuration in many heat and mass transfer systems.

4.2.2. Countercurrent and Concurrent Downward Flows

Figure 13 demonstrates the effect of the magnetic field intensity on the countercurrent flow characteristics of mercury and air for a given inclination parameter value of $Y^{\mathrm{1,0}}=-259.1$. The latter corresponds to a constant flow rate of the mercury ($U_{1s}=1$ m/s) and a channel inclination of β = 5° (the channel height is 0.02 m). The variation in $Q_{21}$ can then be attributed to variation in $U_{2s}$. As shown in the figure, countercurrent flow can be established for a limited range of sufficiently low (upward) air flow ($Q_{21}<0)$ and magnetic field strength. In this range, two distinct solutions for the holdup (and the corresponding flow characteristics) are obtained for fixed flow rates of the air and mercury, which merge to a single solution at the flooding point, beyond which countercurrent flow is not feasible. The countercurrent flow region diminishes with increasing Ha. For Ha > 53.7, countercurrent flow of mercury and air is not feasible (see also Figure 15b below). It is worth noting that the possibility of establishing the two high and low holdup configurations in the countercurrent region was demonstrated experimentally by Ullmann et al. [28] for Ha = 0. The flow configuration realized actually depends on the resistance at the heavy phase outlet. The high holdup configuration is obtained by increasing the resistance at the outlet while maintaining the same flow rates.

Figure 13b,c show the variation in frictional pressure gradient factors $P_f^{\mathrm{1,0}}$, $P_f^1$ with $Q_{21}$ and Ha. In the countercurrent flow region, the positive values of the frictional pressure gradient factors are associated with the low-holdup configuration, indicating that the frictional pressure gradient is dominated by the downward mercury flow. Negative values of these factors are obtained (usually, but not exclusively, in the high holdup configuration) for low Ha (e.g., Ha = 0.518 and 5.18 in Figure 13b), which indicates that the frictional pressure gradient is dominated by the upward gas flow. With increasing Ha, the difference between the $P_f^{\mathrm{1,0}}$ and $P_f^1$ values associated with the low and high holdup configurations diminishes (both $P_f^{\mathrm{1,0}}$ values are positive). With high Ha values, the frictional pressure gradients become close to that obtained in single-phase mercury flow with the same Ha (i.e., the $P_f^1$ values are close to 1). The corresponding $P_f^{\mathrm{1,0}}$ values at high Ha are in the concurrent flow outside the range of Figure 13b. The downward flow of the mercury is obviously assisted by the hydrostatic pressure, ${\left(dp/dx\right)}_g>0$ ($P_g^1$ < 0, Figure 13c, and, when ${-P}_g^1$ > $P_f^1$, no pump is needed to drive the mercury flow.

The difference between the flow characteristics associated with the two-holdup solutions in countercurrent flow can be elucidated by examining the velocity profiles. Figure 14 shows the effect of Ha on the velocity profiles of the two holdup solutions obtained for a downward flow of mercury ($U_{1s}=1$ m/s) and a low upward gas flow ($U_{2s}=-0.05 \mathrm{m}/\mathrm{s},Q_{21}=-0.05$). The velocity profiles in the lower holdup solution (Figure 14a) show that the downward flow of the mercury drags the air downward near the interface (i.e., air back-flow region). With increasing Ha, the mercury holdup increases and its (downward) velocity is reduced, resulting in lower air back-flow near the interface. Figure 14b shows the velocity profiles in the corresponding upper holdup solution. For the low $Q_{21}$ considered, the mercury occupies most of the channel, and the air flows in a very thin layer. Here, the air flow drags the mercury upward near the interface (i.e., mercury back-flow region). Such velocity profiles are associated with negative values of $P_f^{\mathrm{1,0}}$, $P_f^1$ (see Figure 13b,c). As shown, the mercury back-flow in the upper holdup solution diminishes with increasing Ha (for Ha = 25.92, the mercury flows downward over the entire layer). As shown in Figure 13a, for higher upward air flow rates, the thickness of the air layer in the high holdup solution is much larger. Yet, for low Ha, the mercury back-flow phenomenon near the interface is sustained over a wide range of negative $Q_{21}$ corresponding to $P_f^{\mathrm{1,0}}$, $P_f^1$ < 0 (Figure 13b,c).

The variation in the holdup with the magnetic field intensity in concurrent down-flow of mercury and air is elucidated in view of Figure 15 ($Y^{\mathrm{1,0}}=-259.1$, corresponding to $U_{1s}=1$ m/s, β=5°, H = 0.02 m). Figure 15a implies that, apparently, a single solution for the holdup is obtained for specified Ha and $Q_{21}$. As expected, the holdup increases with Ha and is reduced by increasing the air flow rate. However, the enlargement of the region of high holdups and low $Q_{21}$ (Figure 15b) shows that two additional solutions of higher holdups can be obtained in the range of Ha < 53.92. This range of Ha is slightly higher than the maximal Ha value for obtaining countercurrent flow (Ha~53.7). However, independently of the magnetic field intensity, for the tested inclination parameter value $(Y^{\mathrm{1,0}}=-259.1)$, already, only the low holdup solution branch persists in the concurrent flow for $Q_{21}>0.006$. Note that the same inclination parameter can be obtained for lower mercury superficial velocity, lower channel inclination, or smaller channel height.

The frictional pressure gradient factor $P_f^1$ is shown in Figure 15c. For low Ha, and (even low) $Q_{21}$ values, which are already outside the triple-solution region (see Figure 15a), the low mercury holdup solution is associated with higher frictional pressure gradients compared to single-phase mercury flow under the same Ha. The effect of the airflow on the frictional pressure gradient diminishes with increasing Ha. Enlargement of the low $Q_{21}$ (triple solution) region is shown in Figure 15c, where $P_f^1$ values associated with the two additional high holdup solutions at low $Q_{21}$ are shown (e.g., for Ha = 25.92 and 51.84). The two high holdup solutions in the triple-solution regions correspond to $P_f^1~1$(only slightly lower than the frictional pressure gradients of single-phase mercury under the same Ha) and are lower than that obtained for the low holdup solution. For sufficiently high Ha, only a single holdup solution is obtained in the entire concurrent down-flow region, for which the dimensionless frictional pressure gradient values are only slightly lower than 1 at low $Q_{21}$, due to the air lubrication effect. Practically, for large Ha, the frictional pressure gradient factor,$P_f^1,$ in the concurrent region, is of the order of 1 in a wide range of $Q_{21}$. For example, for $Q_{21}$ = 200, the frictional pressure gradient is only about 5% larger than the single-phase mercury flow. Obviously, it then keeps increasing with further increase in the air flow rate.

The difference between the velocity profiles of the three holdup solutions in concurrent down-flow is demonstrated in Figure 16. As shown, the velocity profile of the lower holdup solution (Figure 16a) is similar to that of the lower solution in countercurrent flow (Figure 14a). The difference is that here the net flow of air is downward, while there is a region of upward (back-flow) of the air near the upper channel surface. The steep mercury velocity gradient at the wall at low Ha is responsible for the large frictional pressure gradient associated with the lower holdup solution discussed above (with reference to Figure 15c). With increasing Ha, the holdup of the mercury increases, its velocity is reduced, and the velocity profile is flatter and becomes more similar to that observed in single-phase mercury flow under the same Ha. The back-flow in the thinner air layer diminishes, and eventually, for sufficiently large Ha, the flow in the entire gas layer is downward, and only one solution exists. A similar effect of Ha on the mercury velocity profile is observed in the middle and high holdup solutions (the latter is not shown), as both correspond to high mercury holdup and the velocity profiles in the mercury layer are similar. The main difference in the velocity profile of the middle and high holdup configurations is in the air layer (shown in Figure 16c,d). While in the upper solution, the flow in the entire air layer is downward for all Ha; in the middle holdup solution, back-flow of the air is still obtained. The increase in the mercury holdup with Ha results in a higher (positive) hydrostatic pressure gradient, which facilitates the downward flow of the heavier conductive liquid. Due to the opposite sign of the hydrostatic and frictional pressure gradient in concurrent down-flow, and also in countercurrent flow, the interpretation of the total pressure gradient value is more complicated than in concurrent up-flow. For example, for a specified single-phase flow rate of mercury in the particular channel size considered, the hydrostatic (dimensional) pressure gradient can be larger than the (negative) frictional pressure gradient. Hence, the total pressure gradient can attain positive values, indicating that the specified mercury flowrate is entirely driven by gravity. In fact, a restriction (valve) at the channel outlet is required to maintain the specified flow rate.

To analyze the effect of the air flow and the magnetic field intensity on the total pressure gradient, in concurrent down-flow and countercurrent flow, the values of $dp/dx$ are normalized with respect to the hydrostatic pressure gradient in single-phase mercury flow, i.e., ${\pi }_T=\left(\frac{dp}{dx}\right)/\left({\rho }_{1}gsin\beta \right).$ The obtained ${\pi }_T$ values are shown in Figure 17. The effect of the air flow rate on the total pressure gradient is demonstrated in Figure 17a, where the ${\pi }_T$ values are depicted vs. $Q_{21}$. Positive values are obtained in the countercurrent flow region (as long as the magnitude of the magnetic field is low enough to enable countercurrent flow), indicating that the specified mercury downward flowrate is also entirely driven by gravity in the presence of air upward gas flow. Figure 17b shows the ${\pi }_T$ values when presented vs. the mercury holdup. The range of holdups that correspond to countercurrent flow extends from a holdup of almost 1 (marked by a square) to the holdup for which ${\pi }_T=0$(marked by a diamond). For this holdup, the frictional pressure gradient for the air–mercury system with the specified mercury flow rate (i.e., specified ${ Y}^{\mathrm{1,0}}$) is exactly balanced by the hydrostatic pressure gradient. This point corresponds to the lowest holdup solution obtained for $Q_{21}=0$(see Figure 13a), where the air is circulating in the channel with a zero net flow. The high holdup configuration (indicated by the squares) at $Q_{21}=0$ also corresponds to gas circulation in the channel, however, with ${\pi }_T>0$(see Figure 17a). As shown in Figure 17b, the holdup range of countercurrent flow diminishes with increasing Ha. Countercurrent flow is feasible for magnetic field intensities below a critical value, for which the ${\pi }_T$ value for single-phase mercury flow (i.e., holdup = 1) is 0 (i.e., the frictional pressure gradient for the single-phase mercury flow is balanced by the hydrostatic pressure gradient). For a higher Ha, the single-phase dimensionless pressure gradient is <1, and a pump is needed to drive the mercury flow, whereby countercurrent flow of air becomes unfeasible.

The negative values of ${\pi }_T$ in Figure 17 correspond to concurrent down-flow, and in the triple-solution region they correspond to the lower holdup solution. The figure shows that these ${\pi }_T$ values become more negative as the magnitude of the magnetic field strength and/or the downward air flow are increased, indicating that higher pumping power is needed to drive the concurrent downward mercury–air flow. For Ha values for which triple solutions in concurrent down-flow are feasible, the ${\pi }_T$ values of the two high holdup solutions (which are close to 1, and cannot be distinguished in Figure 17b) are positive, and are only slightly lower than the ${\pi }_T$ value of single-phase mercury flow for the same Ha. This indicates that in these configurations the mercury–air down-flow is driven by gravity. It is worth noting that the correspondence of ${\pi }_T=0$ and the low holdup solution of countercurrent flow at $Q_{21}=0$ is valid in cases of ${\rho }_2/∆\rho \ll 1$. Otherwise, values ${\pi }_T>0$ can also be obtained in the low holdup solution at $Q_{21}=0$ as well as in concurrent flow ($Q_{21}>0$).

The effect of channel inclination on the holdup and the total pressure gradient factor, ${\pi }_T$, in both countercurrent and concurrent downward flow is illustrated in Figure 18. The range of β = 0.5° to 10° corresponds to $Y^{\mathrm{1,0}}=$ −25.94 to −516.2. Decreasing the channel inclination reduces the range of airflow rates for countercurrent flow (Figure 18a). With decreasing channel inclination, the thickness of the mercury layer associated with the high holdup solution decreases, while thicker mercury layers are associated with the low holdup solution. Conversely, at shallower inclinations, the range where a triple-holdup solution is obtained in concurrent downward flow extends over a wider range of downward airflow rates (refer to Figure 18b, which depicts an enlargement of the high holdup region at low $Q_{21}$; the corresponding low holdup solution is shown in Figure 18a). While the total pressure gradient factor (${\pi }_T$) of the high and middle solutions is positive (values close to 1 for all the examined inclinations, Figure 18d), the ${\pi }_T$ value of the low holdup solution becomes more negative at shallower channel inclinations (see Figure 18c). This indicates that a pump is required to propel the concurrent downward mercury–air flow over most of the range of $Q_{21}>0$, where the low holdup is the only solution of the flow equations.

The influence of channel inclination on the flow characteristics illustrated in Figure 18 can be used to anticipate the effect of the mercury flow rate or the density of the conductive fluid. This is evident from the fact that the same variation in $Y^{\mathrm{1,0}}$ can be achieved by adjusting the mercury flow rate while keeping β constant, or by altering the density of the conductive layer (${\rho }_2$ is significantly less than ${\rho }_1$ and has minimal impact on the density difference). It is noteworthy that increasing the viscosity of the lighter layer (i.e., reducing ${\eta }_{12}$) has a similar effect to channel inclination (though not depicted), as it diminishes the countercurrent flow region and marginally expands the range of $Q_{21}$ for triple solutions in concurrent down-flow. However, for higher ${\eta }_2$, the value of the single (low) holdup solution in concurrent down-flow (at a specified $Q_{21}$) is reduced. On the other hand, the impact of altering ${\eta }_1$ is more complex, as it also affects the values of Ha and $Y^{\mathrm{1,0}}$, and thus necessitates specific examination of the particular two-phase system of interest.

4.3. Numerical Calculations

As already mentioned above, the hyperbolic functions involved in the analytical solutions pose an accuracy problem for usual calculations with 16, and even 32, decimal places. To obtain correct results, the calculations were carried out in Maple with 128 or 256 decimal places. For example, consider the mercury–air system with $H=0.2 \mathrm{m}, B_0=0.1 \mathrm{T}$. The holdup noticeably decreases below 1 for $Q_{21}=100$ and larger. The correct holdup values are calculated using 256 floating point digits, which yields $h=0.962, 0.919, 0.829,$ and $0.645$ for $Q_{121}={10}^2, {10}^3, {10}^4$, and ${10}^5$, respectively. The inaccurate results calculated by Maple using 128 digits are $0.594, 0.590, 0.585,$ and $0.579$, respectively.

An alternative method of calculation of the holdups and the associated velocity profiles and pressure gradients is a numerical solution of the flow, Equations (1), (2), (24), and (25). The numerical schemes do not involve evaluation of the hyperbolic functions, and therefore can be carried out with a usual floating-point precision. We obtained numerical solutions using the finite difference and Chebyshev collocation methods. The Chebyshev collocation method is the same as that presented in [33], and the finite difference method applies central differences on an arbitrary stretched grid. The holdups and pressure drops are calculated using the superposition principle and the secant method as described in [34,35].

For the above example, using the finite difference method for the above parameters and at all values of $Q_{21}$ considered, the holdup converges to the value obtained by the analytical solution within the fourth decimal place with 200 uniformly distributed grid points, and within the fifth place with 700 points. Trying to improve the convergence, we applied the $tanh$ stretching,

$$x\leftarrow 0.5+0.5\frac{tanh\left[s\left(x-0.5\right)\right]}{tanh\left(0.5s\right)},$$

(48)

near the boundaries and the interface. However, with the stretching parameter gradually increased up to $s=3$, we did not observe any noticeable improvement in the convergence.

The Chebyshev collocation method exhibits much faster, spectral convergence. Thus, for the same test cases, convergence within four decimal places is reached with the truncated series of 50 polynomials, while beyond 90 polynomials eight decimal places are readily converged. Obviously, the analytical solution is important for the validation of the numerical results and for assuring that all possible solutions are considered in inclined flows, where multiple steady-state configurations are feasible. An example of the holdup convergence study and comparison with the analytic solution is presented in Appendix A.

It is emphasized that, for example, the numerical study of stability of these flows is preferably conducted by using the Chebyshev collocation method, as demonstrated in [32,33]. At the same time, consideration of two-phase flows in bounded rectangular ducts or circular pipes (see [34,35]) will require lower-order methods. In these cases, the present results on convergence of the finite difference method will help estimate the computational resources required.

5. Concluding Remarks

We analyzed the characteristics of two-phase laminar stratified flows in horizontal and inclined channels under the effect of a transverse magnetic field, when the lower, heavier liquid is electrically conducting while the upper fluid is an electric insulator. The simplified channel geometry of two infinite parallel plates is considered, allowing for analytical solutions, and thus facilitating the analysis of system parameters’ effects on flow characteristics.

The flow is defined by prescribed flow rates of the fluids so that the pressure gradient needed to drive the flow, as well as the conducting layer holdup, is obtained together with the solution of the whole problem. We focused on two main issues:

(a)

The effect of the magnetic field on the liquid metal holdup and the pressure gradient driving the flow for prescribed flow rates of the fluids.

(b)

The effect of the magnetic field on the multiple states (i.e., different holdups, pressure gradients, etc., for the same operational conditions), which, under certain conditions, may exist in inclined stratified two-phase flows.

The main conclusions are as follows:

(1)

In horizontal flow, there exists a single state whose holdup increases with the increase in magnetic field strength. This increase is explained by the slowing down of the flow by the magnetic damping force. This effect is also valid in concurrent upward and downward flows for operational conditions that correspond to a single holdup solution of the flow governing equations.

(2)

In inclined concurrent flows, the high-density difference between the liquid metal and the gas results in high sensitivity of the flow configuration to the channel inclination, which allows for multiple states (three) for fixed flow rates under low-to-moderate magnetic field intensities.

(3)

In the triple-solution region, growth of the magnetic field strength leads to an increase in the lower and upper holdup values, but to a decrease in the holdup of the middle solution.

(4)

The triple-solution regions are associated with the possibility of a back-flow region (downward) of the heavy phase near the lower wall in upward flow, and a back-flow region (upward) of the gas near the upper wall. However, back-flow is not exclusively associated with multiple solutions and can also be obtained under conditions where a single solution exists.

(5)

Back-flow is a source of instability of the stratified flow configuration, as it introduces a large disturbance at the inlet, where the fluids are introduced into the channel. It can result in the formation of liquid metal slugs in an upward flow or entrainment of the gas into the liquid in a downward flow.

(6)

The magnetic field weakens the back-flow, and the velocity profile of the liquid metal flattens. Consequently, the ranges of flow rates where multiple solutions exist narrow, and then disappear in strong magnetic fields. In this respect, a magnetic field stabilizes the single holdup flow configuration.

(7)

In countercurrent flow, two distinct states exist up to a certain value of the phase’s flow-rate ratio, beyond which countercurrent flow is impossible. In most cases, in the state of the smaller holdup, the heavier liquid metal drags the lighter one near the interface, while in the other state of a larger holdup, the thinner gas layer drags the liquid metal near the interface.

(8)

With the increase in magnetic field strength, the countercurrent region diminishes, and beyond a certain Hartmann number, countercurrent flow is not feasible.

(9)

Compared to the classical single-phase Poiseuille flow, the magnetic field damping force increases the pressure gradient needed to reach the same flow rate. In liquid metal–gas horizontal stratified flows, the addition of a gas layer may yield a lubrication effect so that, compared to the single-phase Hartmann flow, the same flow rate of the metal liquid can be reached with a smaller pressure gradient.

(10)

In horizontal flows, the lubrication effect increases with increasing the liquid–metal/gas viscosity ratio and can reach about 75% pressure gradient reduction with high viscosity liquid metal under low magnetic field strength. The maximal lubrication effect is reached by adding a few percent gas flow rate to that of the liquid metal flow rate, and therefore can also be associated with pumping power reduction. However, the lubrication effect diminishes with increasing the Hartmann number (e.g., for mercury–air flow, it becomes insignificant for Ha > 25).

(11)

In upward concurrent flow, the introduction of gas flow does not result in a lubrication effect and, in fact, increases the frictional pressure gradient. However, as the hydrostatic pressure gradient is reduced in the presence of a gas layer, the total pressure gradient can be significantly reduced. Indeed, a reduction in the total pressure gradient compared to the single-phase flow of the liquid metal is obtained in the triple-solution region and its vicinity. As this region is associated with high gas/liquid flow-rate ratios, power reduction may not be obtained. In any case, the potential for a pressure drop reduction in this region is reduced with the increase in the magnetic field strength.

(12)

In concurrent downward flow, the addition of the gas layer also has a small effect on the frictional pressure gradient in the triple-solution region. However, as the driving force of the hydrostatic pressure is reduced in the presence of a gas layer, there is no benefit to adding gas to the flow of the liquid metal.

(13)

In countercurrent flows, the frictional pressure gradient is higher in the state of the smaller holdup. In both states, it is larger than in single-phase mercury flow under the same magnetic field strength.

The analytical solution has been used to validate the results obtained by the numerical solution of the flow equations, which is intended to be used for studying the effect of the magnetic field on the stability of the laminar stratified flow pattern in horizontal and inclined flows.