Stability of conducting fluid flow between coaxial cylinders under thermal gradient and axial magnetic field

Abstract

Numerical simulations were performed to investigate the bifurcation in swirling flow between two coaxial vertical cylinders produced by the thermal gradient. The suppressed effects of an axial magnetic field on both vortex breakdown and fluid layers are analyzed. The governing Navier-Stokes, temperature, and potential equations are solved by using the finite-volume method. A conducting fluid is placed in the gap between two coaxial cylinders characterized by a small Prandtl number (Pr = 0.032). Three annular gaps were R = 0.7, 0.8, and 0.9 compared in terms of flow stability, and heat transfer rates. The combination of aspect ratio γ=1.5 and Reynolds number, Re=1500 is the detailed case in this study. In the hydrodynamic case, vortex breakdown takes place near the inner cylinder due to the increased pumping action of the Ekman boundary layer. In addition, the competition between buoyancy and viscous forces develops a fluid layered structure. It is shown that the onset of the oscillatory instability set in by increasing Reynolds number to the critical value. The results show that with an intensified magnetic field, the vortex breakdown disappears, the number of fluid layers will be reduced and the onset of the oscillatory instability will be retarded. Stability diagrams corresponding to the limits of transition from the multiple fluid layers to the one fluid layer are obtained.

1. Introduction

The external magnetic field is used by the tokamak powerful to confine and control the hot plasma of fusion fuels in a ring-shaped container called a ‘torus’. Many researchers are interested in this technology due to its enhanced performance compared with other approaches. The magnetic field keeps the hot plasma away from the machine walls. Two sets of magnetic coils-toroidal and poloidal create a complex 3D field to hold and shape plasma in the magnetic ‘cage’ The plasma is generated by sou blowing a small quantity of the fusion gazes (deuterium and tritium) into the container and breakdown it by using a high voltage to the neutral gas to form an electrically charged plasma, which can be controlled by the magnetic field.

A great plasma current is induced in the center as the primary of a transformer, which the plasma is the secondary winding. This current starts heating the plasma to fusion temperatures. Additional plasma heating is provided by neutral beam injection and microwave systems. Excess heat from the hot fusion plasma is exhausted using a magnetic ‘divertor’; streams of plasma are ‘diverted’ into special heat resistant plates at the top and bottom of the tokamak. The geometry of the coaxial shape plays an important role in the removal of excess heat. Many types of research are performed on this matter concerning different geometries, heat transfer, and magnetic fields. Several researchers, [1,2] are investigated a geometry consisting of two coaxial cylinders in which a liquid metal was placed in the annular gap while a helium gas was flowing inside the inner cylinder. They showed that in such a design, the interior gas can upgrade the heat transfer. Similar configurations were used by Kakarantzas et al. [3] and Mahfoud et al. [4] where a liquid metal is placed between two coaxial vertical cylinders. They showed that the damping of the magnetic field results in fluid deceleration and consequently flow stabilization.

Convection problems in the literature review confirmed that a magnetic field is used to stabilize the perturbation in the fluid motion and control the velocity field [5,6,7,8,9,10,11]. The influence of the geometry and the thermal gradient to generate small vortices with and without an applied magnetic field have been studied by [12,13]. In the hydrodynamic case, the increase of rotation rate accelerates the fluid and leads to the creation of a vortex breakdown bubble that occurred on the inner cylinder. The bottom disk in basic flow serves as a pump, drawing in fluid axially and driving it away in an outward spiral In closed coaxial cylinders, this fluid swirls along the annular gap and then spirals to the fixed top disk, then again turns into the axial direction towards the rotating bottom disk. The conservation of angular momentum causes an initial increase in swirl velocity, by the spiraling motion and so the creation of a concentrated vortex [11]. Effects of magnetic field on heat transfer using the nanofluid are also studied by many researchers [14,15]. Some researchers have detailed the position and length of a vortex under a magnetic field, as revealed in the studies of [16,17]. Laouari et al. [18] have recently studied the central position of the vortex and the limits of appearance and disappearance of the vortex breakdown in the different conductivity regimes. Then they have established stability diagrams corresponding to the transition from the vortex zone to the no- vortex zone for insulating and electrically conducting walls. Mahfoud et al. [19,20] investigate the behavior of vortex breakdown in vertical annuli under the axial magnetic field. The results obtained showed that the vortex breakdown is suppressed beyond the magnitude of the magnetic field exceeds a critical value and before the suppression of the electric vortex occurs.

In this paper, the first section presents forced convection without and with a magnetic field. The second section includes the mixed convection without and with magnetic effect. The first objective is to clarify the effects of the annular gaps on vortex breakdown (apparition, and suppression) in which three annular gaps are compared. The second objective is to identify the magnetic field intensity corresponding to eliminate the vortex bubble. The third objective is to clarify the temperature gradient role on the appearance of the bifurcation in form of fluid layers. Finally, the aim is to specify the magnetic field intensity corresponding to removing the fluid layers. Therefore, we present the stability limits correspond to the domain where the layering does not occur.

2. Flow configuration and model

A viscous conducting fluid characterized by a small Prandtl number (Pr = 0.032) rotates in the annular gap (R) between two coaxial vertical cylinders with height (H). The combined stabilizing action of the external magnetic field and thermal gradient are imposed in the vertical direction which is schematically plotted in Figure 1 Three annular gaps (R =0.7, 0.8 and 0.9) are examined. The case of aspect ratio, γ (H/R_o) = 1.5 and Reynolds number, Re=1500 are detailed. The annular gap is defined as (R = R_o −R_in), where R_in , R_o are the radius of the internal and outer cylinders, respectively and R_o =1.0 in all cases. A temperature difference (∆T) is axially imposed (the top disk is hotter than the bottom). The fluid and solid walls system are subjected to an external axial magnetic field, B (B₀ e_z), e_z is the unit vector in the z-direction. The lower disk rotates about the z-axis at an angular velocity Ω, which is supposed constant, while the upper disk is fixed. The magnetic Reynolds number Re_m = μ₀ σΩR_o² ≪ 1, that measure the ratio of induction to magnetic diffusion. In the specific problem, we can neglect the magnetic field induction by comparing it with the B₀. Also, the only effective force that remains is the electromagnetic force of Lorentz.

For simplification of the problem, some assumptions are given: (i) three fluids considered here are incompressible and Newtonian; (ii) properties of fluids are constants and appraised at the reference temperature; except the density, which is treated according to Boussinesq’s approximation, (iii) Joule heating, and viscous dissipation terms are neglected. Similarly, (iv) the container walls are electrically insulated, and (v) radiation heat transfer is often ignored in this calculation.

The magnetohydrodynamic equations are dimensionless by the following quantities: the lengths R_O, time 1/Ω, velocities ΩR_o, pressure ρ (ΩR_o) 2, temperature, Θ = (T − T₀)/∆T and electric potential, B₀ΩR_o². Also, the simulation considered the flow steady and laminar. So, the dimensionless equations can be defined as:

The induced electric current is given by the interaction of convective flow with the magnetic field, J = σ(E + VxB) hence, the electric charges become E = − ∇Φ. Where vector velocity V (v_r, v_θ, v_z), have the components in the radial (v_r), azimuthal (v_θ) and axial (v_z) direction, respectively. The conservation of the induced electric current ∇J = 0 gives the electric potential.

∇²Φ=∇.(VxB)

(4)

P, Θ, and Φ are the dimensionless pressure, temperature, and electric potential, respectively. Non-dimensional parameters are Reynolds number (Re = ΩR₀2/ν); the Hartmann number (Ha = B₀R_o; the Richardson number (Ri = βg∆T /Ω²R_o). Two other non-dimensional parameters control the comportment of the flow: the aspect ratio (γ=H/Ro) and the Prandtl number (Pr=ν/α). The symbols ρ, ν, β, α, and σ denote, respectively, the density, the kinematic viscosity, the thermal expansion, the thermal diffusivity, and electric conductivity. The Lorentz force is given by F_L = JXB, which are in the r, z, and θ directions, respectively:

(5)

with the current density, J is:

(6)

All simulations were performed starting from τ=0. The bottom disk starts its rotation with angular velocity Ω, while the top and side walls are stationary, respectively. No-slip boundary conditions on solid walls are affected (v_r= v_θ= v_z= 0), and which are also electrically insulated (∂Φ/∂n = 0). The boundary conditions are v_θ= r and Θ = −1/2 at a rotating cold bottom disk and v_θ= 0 and Θ = 1/2 at a stationary hot top disk. Owing to the geometrical symmetry, the periodicity conditions are V(r,θ,z) = V(r,θ + 2π, z).

The stream function ψ, which is specified as:

𝑣_𝑟 = 1/𝑟 (𝜕𝛹/𝜕𝑧), 𝑣_𝑧 = −1/𝑟(𝜕𝛹/𝜕𝑟)

(7)

The heat transfer and convection mode are concluded from the average Nusselt number which is calculated at the bottom disk as:

(8)

where

3. Numerical method and grid employed

The governing equations and boundary conditions of the system mentioned above are discretized by the finite volume method and solved by the tridiagonal matrix algorithm TDMA. SIMPLER scheme Patankar [21] was selected to solve the coupling between velocity and pressure. The central-difference approximation scheme is used for the diffusion and convective terms.

In this simulation, three staggered nonuniform meshes were applied to get better convergence and are listed in

Table 1. Grid sizes used in the numerical simulations.

. The influence imposed by the thinner Hartmann layers (~ 1/Ha), and annular gap lead adoptingted this grid for precision and time- saving. The details of check grid dependence have been examined using as a parameter for comparison the average Nusselt number, $\underset{\mathit{Nu}}{-}$ (see

Table 2. Grid independence test for the case of Re=1500 and γ=1.5 when Ri =1.0, and Ha =5.

).

4. Results and Discussion

First, the used numerical model has been validated by comparing the present numerical results with the numerical results of Mahfoud et al. [9] and Kakarantzas et al. [3] who investigated the flow and heat transfer of liquid metal which rotates in the annular of concentric cylinders under a magnetic field effect. Figure 2 shows the reproduction of the radial distribution of axial velocity in the middle of the domain, for Ha = 100 and the aspect ratio γ= 2.0 when the annular gap is R =0.4 and the Rayleigh number Ra=105. Although, there is a good agreement between the compared numerical results.

The second comparison is tested against the numerical results of Yu, Li & Thess [16], as shown in Figure 3. It shows the effects of the magnetic field on the central position of the vortex on the z-axis for the case when the top disk is electrically conducting. In this case, the aspect ratio, γ=1.5 when Ha = 0 and Ha=7 at different Reynolds numbers, which is in the range of 900 < Re < 1900.

Results of numerical simulations are presented for one aspect ratio, γ= 1.5, and three annular gaps (R = 0.7, 0.8, and 0.9). An electrically conducting fluid is placed in the gap between two coaxial cylinders characterized by a small Prandtl number (Pr = 0.032) corresponding to the PbLi 17 alloy, which is in the most appropriate position to ensure the fusion blankets The Richardson number analyzed here covers the range of 0 ≤ Ri ≤ 2.0. Multiple magnetic intensities characterized by Hartmann numbers are used to suppress both vortex breakdown and fluid layers.

The problem considered could be oscillatory or turbulent; the following section includes the justification. It is known that the typical sequence of evolution of a dynamic system towards chaos for increasing values of the control parameter consists of the following stages: transition to an oscillatory or periodic state; a quasi-periodic regime, and finally chaos (or turbulence). To detect the regime of flow, i.e., transient or steady-state, a series of numerical calculations are performed for each case. Figure 4 shows the temporal evolutions of axial and azimuthal velocities at the monitoring point (r = 0.5, z= 0.75). These simulations presented steady-state solutions obtained for the various case Re= 1500, R=0.9 and the ranges of controlling parameters: the Richardson number (Ri =0 and Ri=2.0) and the Prandtl number (Pr= 0.032), as shown in Figure 4(upper). The oscillatory aspect of the temporal evolutions of the axial of axial radial and azimuthal velocities at the same monitoring point (r = 0.5, z= 0.75) is shown in Figure 4(lower) for Re=2760, Ri=0. It is seen that the increase of Re enhances the fluid motion while the evolutions become time- dependent (oscillatory). Therefore, as the Reynolds number is increased, swirl strength increases, and hence the ability of waves to propagate against the flow increases [22]. The swirling flows will be steady and axisymmetric till the critical Reynolds numbers Re_cr ≈ 2585 when the oscillatory instability begins to set in for R=0.9 at γ= 1.5.

4.1. Annular gaps effect

The influence of the annular gaps on the flow pattern, the apparition, and suppression of vortex breakdown is also depicted in Figure 5 where four Reynolds numbers (Re=1500, 1750, 2000, 2250, and 2500) are compared for one aspect ratio γ=1.5 and three annular gaps R=0.7, 0.8, and 0.9, respectively. The first line of Figure 5 shows the swirling flow in the cases of R = 0.7, 0.8 and 0.9, they are represented schematically by the streamlines in meridional planes for Re= 1500. When the annular gap is increased (R =0.7, 0.8, and 0.9), the vortex appears at R=0.9 in which, the central position of the vortex on the z-axis is at z = 0.91 and on the r-axis is 0.19. The case of Re=1750 is shown in the second line of Figure 5, in which a small vortex breakdown appears at 0.8. Then the size of the vortex grows very quickly with the increasing annular gap to R=0.9. The z- dimensionless length is 0.17 when R=0.8 and 0.21 for R = 0.9. The size of the breakdown grows with the increase of Reynolds number to Re=2000 in both cases (R =0.8, 0.9). As clearly shown by the streamlines on the case of Re=2250 that a small vortex appears at R=0.7, on the other hand, the size of the vortex decreases for R = 0.9 and remains the same for R= 0.8. As the Reynolds number, Re of case R=0.7 increases (Re=2500), the size of the vortex increases and vice versa. The behavior in the case of R=0.9 is the opposite because the vortex size decreases with the increase of Re.

For the hydrodynamic case, the diagram presented in Figure 6 gives the stability limit in the (Re, γ) plane within which a vortex breakdown bubble occurred near to sidewall of the inner cylinder. The influence of the annular gaps on vortex breakdown zones in the (Re, γ) plane shows how the boundaries of the vortex breakdown shift. Three separate curves represent the limits of three-zone, i.e., the domains with and without vortex breakdowns could be observed. We observe that the decrease of the annular gap causes the decrease in the vortex breakdown zone so that the decrease in the values of the annular gaps causes boundaries to move towards the side of the lower aspect ratio.

4.2. Magnetic field effects on vortex breakdown

This section clarifies the effects of the magnetic field to control the central position of the vortex on the z-axis and r-axis, respectively. The second stage is to identify the most efficient Ha to eliminate the vortex bubble. The swirling flow in the annular gap produced by the rotating bottom disk under an axial magnetic can be divided into three regions, i.e., the core region, the Hartmann layer, and the side layer (Robert layer). The viscous forces can complete with magnetic forces in the Hartmann layer near to walls normal the applied magnetic field.

Thus, when the Hartmann number increases, the Ekman layer is progressively replaced by the Hartmann layer, which is near to walls normal to the magnetic field. These layers are perpendicular to the magnetic field, having a dimensionless thickness δ⊥ ≈ 1/Ha. Therefore, the intensified magnetic field results in Hartmann layer thickness. Where the walls are electrically insulating, the Hartmann layer near the disk has a similarity solution as in the case of the classical Ekman layer. The order of magnitude of the axial velocity is v_z~ Re/Ha3 [18], which results in the weakness of the axial velocity. Pao and Long [23] also that an external magnetic field that intensifies magnetic intensity results in boundary layer thickness and a decrease of the axial flow component.

Figure 7 shows streamlines for increasing Hartmann numbers (Ha=1, 5, 7 and 9) in three case of Re = 1300 (above), Re= 1500 (middle), and Re= 2000 (bellow) when R=0.9 and γ=1.5. For Re= 1300, the vortex appears at Ha=1 in which, the central position of the vortex on the z-axis is at z ≈ 0.84 and on the r-axis is 0.15. Then the size of the vortex grows with the increasing Ha to 5. The z- dimensionless length is 0.18 when Ha=1 and 0.11 for Ha = 5, and disappears for critical Hartmann number (Ha_cr ≈6.5). The case of Re=1500 is shown in the second line of Figure 7, in which the vortex breakdown size diminishes with increasing Hartmann numbers and finally disappears at Ha=9. For Re=2000, the vortex breakdown size rises with increasing Hartmann numbers and disappears for a value that exceeds Ha=9. Figure 7 show that the size of the breakdown grows with increasing Reynolds number. So, the central positions of the vortex on the z-axis increase with increasing Re and Ha, respectively.

Figure 8a,b shows the effects of the magnetic field on the central position of the vortex on the z-axis and r-axis, respectively in the case of γ=1.5 when the annular gap is R=0.9. Here, the central position is presented at different Reynolds numbers, which is in the range of 1100 ≤ Re ≤ 2500. Without a magnetic field, a vortex appears at Re = 1170 and disappears at Re = 2510. The central position on the z-axis is at z = 0.780 when the small vortex appears at Re = 1175 and rise to z = 1.157 until Re = 2500. In Figure 8a, the central positions of the vortex increase gradually with the increasing Hartmann number at a fixed Reynolds number. The effects of the magnetic field on the central positions of the vortex are much stronger for Ha=9 since the vortex appears close to Re = 1690 and disappears at Re = 2210. The central positions are at z= 1.029 and 1.14 for Re = 1700 and Re=2200, respectively. The results show that when the Hartmann number increase, the vortex breakdown is gradually suppressed by the magnetic field. In Figure 8b, the central position of the vortex on the r-axis under an axial uniform magnetic field is different from those in the central position on the z-axis. In this case, the distribution of the central position on the r-axis represents the shape of a semicircle As shown, their positions diminish gradually under the increasing effect of the magnetic field. When the Hartmann number increases, the central positions of the vortex on the r-axis drop at a fixed Reynolds number. Taking Re = 1800 for example, the central positions of the vortex are at r = 0.201, 0.189, 0.179 and 0.139 for Ha =0, 5, 7 and 9, respectively.

At Ha=0, 5, and 7 the central positions on the r-axis are divided into two parts, when Re ≤1800, the r-central position of vortex increase with the increasing Re, contrarily when Re ≥ 1800, they decrease with the increasing Re. Finally, for the case of R=0.9 at γ=1.5, the present results show that the increase of the Ha causes the increase of the z-central position of the vortex, but contrarily causes the decrease of the r-central position of the vortex breakdown. The situation can be interpreted as follows, the peak occurred in the region where the viscous and inertial forces are of the same magnitude. Moreover, Based upon experiments of Escudier [24], the breakdown region is characterised by a radius r, the corresponding Reynolds number is then (r/R₀)Re^1/2. The largest value of this quantity for the case of R=0.9 and γ=1.5 at Ha=0 is for Re=1800 when r/R_o ~ 0.202 and (r/R_o)Re^1/2 ~ 8.57. Figure 8c shows the relationships between Reynolds number of base flow and Reynolds number of breakdown region. The rotation rate of the breakdown region increase up to (r/R_o)Re^1/2~8.57, 8.02, 7.59, 5.93 for Ha =0, 5, 7 and 9, respectively and drops slowly when the Reynolds number is increased. In Figure 8d, the dimensionless lengths of the vortex for the case of R=0.9 at γ=1.5 are presented when the Reynolds number is in the range of 1100 to 2500 with the increment of 100. In Ha ≠0, the magnetic field gives a stronger influence on the length especially to the peak value, and this exhibition is more obvious than that in Ha=0. When Ha = 5 and Ha = 7, the lengths are ≈0,19 and 0,15 at the peak, respectively. For Ha = 7, the vortex appears close to Re = 1650, reaches the peak, 0.1014, at Re = 1700 and disappears at Re =2300.

Stability limits for Ha=0 and 5 at the annular gap R=0.9 are constructed on (Re, γ) plane for two cases as shown in Figure 9. There, curves represent the boundaries for the vortex zone and no-vortex zone and the limits between steady and unsteady zones. The boundaries shift shows that intensifying magnetic intensity contract the domain limits of the vortex breakdown. To obtain the impact of the Hartmann number on vortex breakdown zones, two selected magnitudes of Hartmann number, Ha=0 and 5 have been studied. The black curve with start symbols shows the boundaries vortex breakdown at Ha = 0. For Ha = 5, the vortex breakdown zone boundaries shown by the curve in red colour with circle symbols shrink and shift upward concerning the Ha=0 curve. Hence, this makes the increasing effects of the Hartmann number reduces the limits within which a vortex breakdown occurs and raises the transition to an unsteady regime [25].

4.3. Buoyancy effect on vortex breakdown and the fluid layers

To investigate the effects of thermal gradients on the fluid layers and the control of the vortex breakdown (i.e., location, or suppression of vortex breakdown), three annular gaps are compared (R=0.7, 0.8, and 0.9), which correspond to Re =1500 and γ= 1.5 Figure 10. The case (Ri =0) corresponds to a decoupling of the velocity and temperature fields, in which forced convection takes place.

The rotation of the bottom gives a centrifugal force to the fluid, and this force drives the fluid radially outward. As the fluid stoped by the outer cylinder sidewall it is turned upward, producing jets. The upward jets and downward flow result in the centrifugal flow structure, with an axial vortex near the inner cylinder axis. This behavior incurs a breakdown, with a single bubble. Figure 10 presents the superposed streamlines and isotherms in meridional planes for progressively increasing Richardson numbers. The results for R=0.9 show that when the Richardson number, Ri= 0.01 the vortex breakdown remains effective. So, by increasing to Ri =0.1, the vortex breakdown is suppressed. In this case, heat transfer convection is dominant. The maximum value of the stream function decrease with increasing Ri until 0.0064 at Ri =2.0. The streamlines when Ri =1.0 show a new region of counter- flow which grows with increasing Ri and then dominate the entire top section of the annular gap (plot first line of Figure 10).

The plots in Figure 10 for case R=0.8 show the decomposition in the counter-flow region, up to two-layered appear for Ri = 1.0. The stratified structure with two fluid layers is observed when Ri = 2.0. Note that, the separated zone is curved and the top layer grows with increasing Richardson. The isotherms plot when Ri =2.0 shows that conduction mode dominated the heat transfer, especially in the top region. Similarly, the buoyancy acts for case R=0.7 are stronger than those in cases R=0.9 and R=0.8 (plots in third line Figure 10). When the vertical temperature gradient is small (Ri=0.1), and the convection mode dominates heat flux, the vortex breakdown in the annular gap does not exist. In the range of Ri≥1, there is no big difference in the isothermal line distribution, but the maximum value of non-dimensional streamlines (Ψ_max) decreases as the increasing of Richardson number, and are 0.00763 and 0.00688 at Ri=0.1 and Ri=2, respectively, that indicates the flow is suppressed by the Buoyancy force. The process of increasing the Richardson number to Ri=1.0 induces two stratified layers. On the range of Ri considered, up to three-layered appear for Ri = 1.5 and Ri= 2.0 (Figure 10d-e).

Figure 11a shows the evolution of the number of fluid layers for the case of Re=1500 and for R=0.7, R=0.8, and R=0.9 cases, which are discussed above. The number of layers is the number of stratified recirculation zones in the meridional flow. For R=0.9, the flow is characterized by one concentrated vortex when Ri =0.01, this vortex undergoes breakdown, i.e., a stagnation point followed by a small recirculation zone near the inner cylinder wall. The flow contains a single layer, that occurs for Ri ≤0.7. It was found also that the number of layers, corresponding to R=0.9, grows with the increase of Richardson number and leads to a second layer to form beyond Ri ≥0.8. As to case R=0.8, a second layer flow structure is observed when Ri≥0.6. In case R=0.7, up to three layers appear for Ri ≥1.2.

Figure 11b compares the maximum hydrodynamic streamlines (Ψ) for the three annular gaps (R=0.7, R=0.8, and R=0.9, respectively) when Re=1500 and γ=1.5. The decreasing of the maximum value of hydrodynamic streamlines (Ψ) with increasing Ri for all three cases proves that the intensified buoyancy affects the number of recirculation zones formed by swirling flow.

4.4. Heat transfer

In this section, the relationship between the stratification of fluid and heat transfer will be clarified. However, the Nusselt number is analyzed in the remainder of this section for many parameters. The decreasing of the average Nusselt number with the Richardson number is presented in Figure 12, where three cases R=0.7, R=0.8, and R=0.9 are compared for Re=1500 and γ=1.5. Figure 12 shows that Nu Monotonically decreases with increasing Ri and approaches the reciprocal of the aspect ratio, i.e., 1/H (=0.66 at γ= 1.5) which means the value of the conduction limit.

In Figure 12, it is noticed that the maximum value of Nu is attained when a single layer fluid is established. For mixed convection, the influence of buoyancy force becomes stronger with the rising of the Richardson number, so the concurrence between viscous and buoyancy forces is increasingly important with increasing Ri. However, the lighter hot fluid close to the top hot disk sits on the top of the heavier cold fluid close to the bottom cold disk. The effects of natural convection continue to exist only near the heated top disk. In this case, the stable stratification of fluid opposes the flow produced by the bottom rotating disk, and so the net advective transport diminishes, and the value of Nu decreases with increasing Ri. Also, these fluid layers play the role of thermal insulation, since the number of layers influences the heat transfer. We conclude that the combination of Richardson number and annular gaps control the heat transfer by the presence or absence of fluid layering. Furthermore, it is seen that at a constant value of Ri the average Nusselt values become progressively grow as the annular gap is increased indicating that advective transport reinforces with increasing R, as observed from Figure 12. Consequently annular gap, R has an important influence on Nu in this case.

4.5. Magnetic effect on fluid layers

To investigate the effects of the axial magnetic field on the layering (i.e., apparition or suppression), taking, for example, the case of Re=1500, γ=1.5 and Ri =2.0 at three cases mentioned above (i.e., case R=0.7, R=0.8, and R=0.9). The hydrodynamic streamline plots in the case of R=0.9 at Ha=0 show a double layer, in which the bottom layer is the biggest Figure 13. The magnetic field in the vertical direction has a good suppressive effect on both vortex breakdown and fluid layers in which are shown in this case. As clearly shown by the streamlines on the first line of Figure 13, the clockwise recirculation top region diminishes in size and moves toward the sidewall when Ha = 10. On the contrary, the counterclockwise recirculation zone grows in size until it takes the entire top gap of the cylinder. Also, the maximum streamfunction decreases with increasing Ha and is 0.0064 and 0.0061 for Ha=0 and Ha = 10, respectively. The small toroidal vortex decreases in size and then disappears at Ha_cr= 20. The r-central position of the small toroidal region rises with increasing Ha and, on the contrary, the z- central diminishes slightly. The central positions are at z =1.08, 0.99, and 0.97 for Ha = 5, 10 and 15, respectively.

As for case R=0.8 shown in Figure 13, the streamline plots for Ha= 0, show three layers, in which the top layer becomes narrow. The streamlines for Ha=5 clearly show the two small toroidal vortices attached to the inner and outer wall, respectively. Another counterclockwise recirculation zone centered at z=0.82 for Ha=5 and decreases in size further and disappears at Ha = 22.

The effect of increasing Ha is distinctly seen in Figure 13 at case R=0.7. For Ha=0 and Ha=5, three layers are observed, but at Ha=10 the middle clockwise toroidal vortex divides into two cells and drives to the creation of two layers and stagnation-point [24] flow near to the inner wall. For Ha=20 the streamlines show a small toroidal vortex centered at z=075, then decreases in size and disappears at Ha_cr=25. For all cases the isotherms plots when Ri=1.0 show that conduction domine the heat transfer, especially in the top gap.

Figure 14 compares the magnetic field effect on the number of fluid layers for three cases (R=0.7, R=0.8, and R=0.9) when Re=1500 and Ri=2.0. The decrease in the curves as shown in Figure 14 indicates that increasing Ha has an important influence on the number of fluid layers formed. Therefore, the number of fluid layers decreases with increasing Ha for all three cases. For Ha=0 we have three layers for case R=0.7, three layers for R=0.8, and two layers for R=0.9.

For Ha=10 and 15, two layers is observed for R=0.8 and R=0.7, respectively. The critical Hartmann numbers, Ha_cr = 20, 22, and 25 correspondings to a single layer for cases R=0.7, 0.8 and 0.9, respectively. Consequently, the intensified magnetic reduces the number of resulting layers.

The diagram in the (Ha_cr–Ri) plane for all three cases when Re=1500 and γ=1.5 presented in Figure 15 gives the evolution of critical Hartmann number Ha_cr versus Ri. There are three separate curves, represent the limits of two-zone, i.e., the domains with and without stratification fluid layers. The blue curve with star symbols in Figure 15 represents the boundaries for the case of an annular gap, R=0.7. The red curve with a circle corresponding to case R=0.8. For case R=0.9, the threshold of transition is plotted by the black curve with square symbols. In all cases, we have seen that the increase of the Ri causes the increase of the Ha_cr. Also, increasing Ha removes the fluid layers at a constant value of Ri (the layering disappears after the amplitude of Ha goes beyond a critical value). The critical values (Ha_cr) for case R=0.7 are greater than those obtained in cases 0.8 and 0.9 for a fixed Richardson number.

5. Conclusion

Bifurcation and stability of mixed convection flow in the gap between two coaxial cylinders with a magnetic field have been numerically analyzed. The finite volume method has been used to capture the different vortex breakdowns in the isothermal case and the stratified layers under the temperature stratification condition. Three annular gaps were compared in terms of flow stability, and heat transfer rates for one Prandtl number, Pr=0.032 One configuration of Reynolds number and aspect ratio are detailed. The main results obtained are as follows:

• It was shown that the domain in which a vortex breakdown bubble occurred decreases with decreasing annular gaps, so that the decrease in the values of the annular gaps, causes boundaries to move towards the side of the lower aspect ratio.
• The present results show that the increase of the Hartmann number shrinks the vortex breakdown zone
• In R=0.9, increasing Richardson number to 0.1 resulted in the suppression of vortex breakdown in which one layer occupied the annular gap.
• The stratification layers increase with increasing Ri for both cases R=0.8 and 0.9, but for the case, R=0.7 more layers are developed.
• The fluid layers developed act to insulate the hot annular gap, so the average Nusselt number decrease with the increasing of stratified fluid layers.
• The number of stratified fluid layers increase with decreasing annular gap, R
• The intensified magnetic field results lead to a decrease in the number of fluid layers
• Finally, the transition from multiple fluid layers to one fluid layer evolves with increasing Ri.