Pressure-Driven Instability Characteristics and Stability Analysis of Magnetohydrodynamic (MHD) Flow through a Rotating Curved Square Duct with Hall and Ion-Slip Currents

1  Introduction

Fluid flow with Magneto-hydrodynamics (MHD) finds applications across various industries, including geophysics, astrophysics, engineering, and industrial sectors. Practical uses of MHD encompass MHD pumps, generators, flow meters, thermal insulators, metallurgy, metal dispersion, bearings, and geothermal energy extraction. Moreover, MHD technology is utilized in ship propulsion, cooling of electronic equipment, flow control, magnetic filtration and separation, jet printer propulsion, and flight control for rockets and hypersonic vehicles. MHD power generation is also being explored for space applications though on a smaller scale. Ion-slip effects are caused by the simultaneous movement of ions and neutral particles in the fluid, whereas the Hall Effect is the result of the interactions between charged particles and magnetic fields. The situation becomes more difficult when Hall and ion-slip effects are included in the analysis. Moreover, rotational fluids are highly important due to their presence in various natural phenomena and technological applications influenced by the Coriolis force. Many scientific fields incorporate significant and necessary properties of these fluids. Considering these effects, it is imperative to investigate the isothermal fluid flow in a revolving curved duct. This research is key to understanding the complex dynamics of such flows and discovering new ways to enhance their performance.

As a pioneering researcher, Ref. [1] demonstrated that there are two counter-rotating vortices in the flow in a curved pipe. He continued his investigation into the behavior of viscous, incompressible fluid flow in the presence of a constant pressure gradient and discovered that the flow is governed by a parameter Dn=Gd3μν2dL known as the Dean number. The identities of μ, υ, G, L, and d of the pipe are given in the nomenclature section. An exact solution for the laminar flow of extremely conductive liquids in rectangular or circular pipes in the presence of a constant oblique magnetic field was derived by [2]. Except for pressure, all the physical parameters in his findings depended on the transverse coordinate. After the 50 s, there was a growing interest among applied physicists and engineers in studying the flow of electrically conductive fluids, driven by the importance of controlled fusion research and advancements in space technology. Numerous researchers explored different flow scenarios, examining various configurations and the effects of both conductive and non-conductive walls. Notable contributions came from [3–6] among others. A study on the flow of conductive fluids in curved pipes at low to intermediate Hartmann numbers was conducted by [7]. Their results showed that the axial velocity contours shifted toward the outer wall in the presence of a strong magnetic field, increasing fluid compression there. Additionally, the Lorentz forces modify the flow patterns. The effect of the induced magnetic field on MHD flow in rectangular ducts under no-slip conditions has been thoroughly investigated by several researchers, such as [8,9]. An accurate numerical approach, the spectral method, to solve the magneto-fluid-mechanical problem in a square duct involving heat transfer was applied by [10]. They had to solve the energy equations and the N-S equations to calculate the pressure drop and Nu for both low and high Hartmann numbers, including those when M = 1000. A numerical study on flow through a rectangular duct was performed by [11]. They studied details of steady and unsteady solution and analyzed the heat transfer for this flow.

A solution framework for thermal flows in CSD was developed by [12]. The objective of this methodology was to furnish dependable and perceptive attributes, augmenting the basic comprehension of spiraling flow and its thermal dynamics. The dual effects of gyration and curvature of the thermal flow in CRD was described, in detail, by [13] and investigated the heat transfer between two different heating walls. The flow behaviors of transient fluid flow through a CRD with strong positive and negative rotation, along with a high-pressure gradient was also demonstrated by [14]. Their findings reveal that secondary flow greatly enhances convective heat transfer. An experimental visualization study on secondary flow in a curved square duct with rotating walls under an imposed azimuthal pressure gradient was examined by [15]. The observed patterns agree well with numerical predictions, leading to a classification map in the Taylor-Dean number plate. Very recently, A numerical study for investigating the flow structure and progressing the Dean vortices with CHT through a curved channel was successfully performed by [16]. However, all these studies are performed in a bending geometry without magnetic effects.

An investigation on the MHD fluid flow considering a magnetic field in a rectangular duct was performed by [17,18]. The numerical solution to acquire the effects of Dn and curvature, considering a wide range of M was presented by [19]. In another study, the MHD flow in a rotating curved pipe was analyzed by [20], examining how rotation, magnetic fields, and curvature influenced the flow, revealing the presence of four vortex structures in the system. All the studies explored the impact of Magnetic parameter but overlooked hall currents, thus neglecting the simultaneous effect of both m and M in the application of Ohm’s law. Consequently, the Hall effect reduces effective conductivity and generates a fluid velocity component. The numerical solutions to analyze the heat transfer in a rectangular duct counting the Hall term and variable viscosity were performed by [21]. Unsteady flow between parallel plates was examined by [22,23], incorporating the effects of Hall current, Joule heating, and viscous dissipation.

All the studies referenced earlier explored the impact of hall parameters but neglected the ion slip currents, failing to account for the blended effect of both hall and ion slip currents. However, under a strong magnetic force (M), both hall (m) and ion-slip (α) become considerable. These effects have various applications across fields such as power generation, transformers, magnetometers, etc. To the authors’ knowledge, no previous research has examined the blended effects of m and α on MHD flow in a curved channel, which is the primary focus of this study. There have been many studies on the effects of hall and ion-slip currents through different geometries. The hybrid nanofluid flow over a photocatalytic surface was examined in [24] using collocation method to solve the ordinary differential equations. They discovered that the heat transfer coefficient is progressed, while the mass transfer is prolonged. The effects of hall and ion slip currents on the flow of MHD Williamson nanofluid over a stretching sheet with various parameters was investigated by [25]. They found that both primary and secondary profiles decrease if Williamson parameter increases. They also found that the temperature and concentration profiles increase with the increase of thermal radiation parameters. The combined impacts of Hall current, Soret-Dofour, magneto hydrodynamics and non-Darcy on chemically reactive micropolar fluid flow over a stretching sheet were examined by [26]. The effects of stratification, Hall and Ion slip in a high-prosperity medium were numerically tested in [27] and it was found that the velocity profile increases with increasing values of hall parameters. The combined effects of hall and ion-slip currents and the heat transfer coefficient on the peristaltic transport of a nanofluid flow with water as the base fluid and copper as the nanoparticle, were studied by [28]. The results showed that increasing the hall parameter or introducing copper nanoparticles boosts fluid velocity at the channel center while reducing it near the walls. The steady MHD flow and heat transfer characteristics of a Casson nanofluid over a stretching and rotating disk subject to a spatially varying magnetic field and internal heat generation was investigated by [29]. Their results showed that strengthening the magnetic field or increasing the heat source intensity leads to a pronounced thickening of the thermal boundary layer, whereas the velocity field near the disk is reduced due to magnetic damping effects. Moreover, higher nanoparticle concentrations lessen the hall parameter’s influence on both the pressure gradient and temperature. Furthermore, numerous studies have explored the behavior of different fluid models in diverse flow geometries. For instance, an investigation performed by [30] on micropolar fluid flow in a rectangular channel while accounting for Hall and Ion slip current effects. The characteristics of the Couette flow between two parallel plates in a rotating frame were examined by [31], revealing several noteworthy influences of Hall and ion slip currents. Additionally, a theoretical study of buoyancy-induced flow in a vertical microchannel incorporating both Hall and ion slip effects was provided in [32]. The influence of Hall and ion slip currents on rotating flow over an exponentially accelerated plate and on flow through a porous medium was further analyzed by [33,34], respectively.

Many previous studies have primarily focused on analyzing the impact of several dimensionless parameters on velocity, temperature, and concentration profiles. Although numerous studies have examined MHD flow in straight and simple geometries, only limited attention has been given to pressure-driven instability in a rotating curved duct under the simultaneous influence of hall and ion-slip currents. Existing works typically analyze either (i) isothermal MHD flows without Hall/ion-slip effects, or (ii) Hall or ion-slip induced modifications in simple channels, but do not address their combined impact on the onset of instability, bifurcation structure, and transition to chaos in curved duct systems. This gap limits our understanding of how electromagnetic parameters may be used to control or suppress chaotic flow in a practical curved duct rotating system. The present work addresses these deficiencies by performing a comprehensive stability and time-evolution analysis of MHD flow in a rotating curved square duct, explicitly incorporating both Hall and ion-slip currents. We provide new insights into how these MHD effects reshape the steady solution structure, shift critical bifurcation points, stabilize previously unstable regimes, and suppress chaotic behavior at high magnetic interaction parameters. To the best of the authors’ knowledge, this is the first detailed investigation that couples Hall and ion-slip physics with high-Dean number instability in rotating curved ducts, offering a novel contribution to the field and supplying the missing theoretical foundation required for advanced control of electrically conducting flows in engineering applications.

2  Mathematical Formulations

2.1 Basic Governing Equations

In this study, the fluid is assumed to be fully developed, viscous, MHD, and incompressible 2D flow with constant thermophysical properties through a curved square duct (CSD). The computational domain along with the necessary notations is exhibited in Fig. 1. The flow is in the axial direction, and the duct rotates with a uniform angular velocity about the vertical axis. The magnetic field is imposed in the z′-direction. The Hall and ion-slip effects are incorporated through the generalized Ohm’s law for partially ionized plasmas. Body-force contributions other than Lorentz and centrifugal forces are neglected. The lower and upper walls of the duct are assumed to be heated and cold, respectively, and the two vertical sidewalls remain insulated. The fluid (water) is treated as isothermal with constant physical properties, and the no-slip condition is applied at all solid boundaries. The fluid flows uniformly in the z-direction.

Figure 1: Working system (a) Geometrical shape (b) Cross-sectional view

When a fluid flows with uniform magnetic field ‘B’ and the entire system rotates with constant angular velocity ΩT about the y-axis, the continuity and the momentum equations for two-dimensional incompressible steady fluid flow in the cylindrical coordinate (r~,θ~,y~) take the following form:

Continuity equation:

∇~.a~_+u~r~=0.(1)

Momentum equations:

(∇~_u~_).a~_−w~2r~=−1ρ∂p~∂r~+ν[∇~_2−1r~2]u~+2(Ω→×q→)+1ρ(J→×B→),(2)

(∇~_v~_).a~_=−1ρ∂p~∂y~+ν∇~_2v~,(3)

(∇~_w~).a~_+u~w~r~=−1ρ1r~∂p~∂θ~+ν[∇~_2−1r~2]w~+2(Ω→×q→)+1ρ(J→×B→),(4)

here,

a~_=u~i^+v~j^,∇~_=∂∂r~i^+∂∂y~j^;∇~_2=∂2∂r~2+1r~∂∂r~+∂2∂y~2,

where ‘r~’ represents the radial variable, ‘θ~’ is corresponds to the circumferential angle and ‘y~’ is the vertical variable. We transform the cylindrical coordinate to rectangular (x,y,z), by using the transformations r~=L+xd, y~=hy, Lq~=zd.

The generalized Ohm’s law for introducing the Hall and ion-slip currents is defined as follows

J→=σμe(q→∧B→)−mB0(J→∧B→)+mαB02((J→∧B→)∧B→).(5)

ωeτe=m, the hall factor, where ωe: cyclotron frequency and τe: electron collision time, where B→: magnetic field vectors, q→: the fluid velocity vector, J→: the current density vector, σ: the conductivity of the fluid, μe: magnetic permeability, α: Ion-slip parameter. B0: the component of B→ on the direction of the flow. Based on these assumptions, the continuity and momentum equations in dimensional variables are provided below

Continuity equation:

∇~.a~_+u~r~=0.(6)

Momentum equations:

(∇~_u~_).a~_−w~2r~=−1ρ∂p~∂r~+ν[∇~_2−1r~2]u~+C[mw~−(1+mα)u~]+2w~Ω0,(7)

(∇~_v~_).a~_=−1ρ∂p~∂y~+ν∇~_2v~,(8)

(∇~_w~).a~_+u~w~r~=−1ρ1r~∂p~∂θ~+ν[∇~_2−1r~2]w~−C[mu~−(1+mα)w~]−2u~Ω0,(9)

here, C=αμeB02(1+mα)2+m2 and Ω=(0,−Ω0,0), r~=L+x~, p~ is the dimensional pressure, the symbol ρ refer to the density and the symbol υ refer to the kinematic viscosity. Now we introduce non-dimensional variables.

u=u′U0;v=v′U0;w=2δU0w′;x=(x′d−1δ);y~=y′d;z=z′d;t=U0dt′;δ=dL;P=P′ρU02.}(10)

We introduce the stream function in the x- and y-directions as:

u=1b∂ψ∂y;v=−1b∂ψ∂x,where b=1+δx(11)

Then the derived dimensionless equations for w and ψ are as follows:

−∂(w,ψ)∂(x,y)+δ∂w∂x+Dn+b[Δ2−d2]w−d∂ψ∂yw−δTr∂ψ∂y−E[2δm∂ψ∂y+b(1+mα)w]=0,(12)

−∂(Δ2ψ,ψ)∂(x,y)+d[∂ψ∂y(2Δ2ψ−3d∂ψ∂x+∂2ψ∂x2)−∂ψ∂x∂2ψ∂x∂y]−d[3δ∂2ψ∂x2−3δd∂ψ∂x]−2δ∂∂xΔ2ψ+bw∂ψ∂y+bΔ22ψ+b22Tr∂ψ∂y−bE[(1+mα)∂2ψ∂y2−m2δb∂w∂y]=0,(13)

here, Δ2≡∂2∂x2+∂2∂y2; ∂(f,g)∂(x,y)=∂f∂x∂g∂y−∂f∂y∂g∂x; d=δ1+δx; and E=M(1+mα)2+m2.

The non-dimensional parameters Dn, Tr and M, used in Eqs. (12) and (13), are defined as Dn=Gd3μυ2dL; Tr=2d22δδυΩ0; and M=d2σμeB02ρυΩ0.

In the present study, the parameters δ, m, Tr and Pr are fixed while M and Dn vary.

The functional boundary conditions are

w(±1,y)=w(x,±1)=ψ(±1,y)=ψ(x,±1)=∂ψ∂x(±1,y)=∂ψ∂y(x,±1)=0.(14)

2.2 The Stability Equations for MHD Flow

To assess the stability of the flow, we initiate perturbations in two dimensions on the steady flow and observe the development of these perturbations, i.e., we put

w(x,y,z)=w¯(x,y,z)+w^(x,y,z)ψ(x,y,z)=ψ¯(x,y,z)+ψ^(x,y,z) }(15)

where w¯ and ψ¯ are the steady solutions of the axial velocity, and secondary flow, respectively, and w^, and ψ^ are their perturbations. Substituting Eq. (15) into Eqs. (12) and (13) and disregarding the quadratic terms of the perturbation, the equations for w^ and ψ^ are derived as

bΔ2w^+δ∂w^∂x−dδw^−∂(w¯,ψ^)∂(x,y)−∂(w^,ψ¯)∂(x,y)−d[∂ψ¯∂yw^+∂ψ^∂yw¯]−δTr∂ψ¯∂y−E[2δm∂ψ^∂y−b(1+mα)w^]=b∂w^∂t,(16)

(Δ2−d∂∂x)∂ψ^∂t=−1b[∂(Δ2ψ¯,ψ^))∂(x,y)+∂(Δ2ψ^,ψ¯)∂(x,y)]+∂ψ¯∂y{2dδΔ2ψ^−3δd3∂ψ^∂x+1δd2∂2ψ^∂x2}+∂ψ^∂y{2δd2Δ2ψ¯−3δd3∂ψ¯∂x+1δd2∂2ψ¯∂x2}−1δd2{∂ψ¯∂x∂2ψ^∂x∂y+∂ψ^∂x∂2ψ¯∂x∂y}−2d∂∂xΔ2ψ^+3d2∂2ψ∂x2−3d3∂ψ^∂x+{w¯∂w^∂y+w^∂w¯∂y}+Δ22ψ^+bTr2∂w^∂y−E{(1+mα)∂2ψ^∂y2−m2δb∂w^∂y}.(17)

Eqs. (16) and (17) are the basic equations for the linear stability analysis. The boundary conditions for w^ and ψ^ are the same as those of w and ψ, respectively.

2.3 Resistance Coefficient

The alternative name of resistant coefficient λ is called the hydraulic resistance coefficient and denoting the quantity of the flow state, generally used in fluids engineering defined as

P1∗−P2∗Δz∗=λdh∗12ρ⟨w∗⟩2,(18)

where ⟨⟩ positions for the mean over the cross-section of the duct and dh∗ is the hydraulic diameter, defined by dh∗=4(2d×2dl)/(4d+4dl) and ⟨ω∗⟩ is the stream-wise main velocity deliberated by

⟨w∗⟩=ν42δd∫−11dx∫−11w(x,y,t)dy.(19)

Since P1∗−P2∗Δz∗=G, λ is linked to the mean non-dimensional axial velocity ⟨w⟩=2δd⟨w∗⟩/υ as

λ=42δDn⟨w⟩2.(20)

Eq. (20) will be used to find the resistance coefficient of the flow evolution by numerical calculations.

2.4 Flux through the Duct

The flux, which is representative of the total flow in the curved channel, is calculated by the formula

Q′=∫−dd∫−ddw′dx′dy′=VdQ,(21)

where Q is dimensional heat-flux and w′¯ is the mean axial velocity defined by

w′¯=Qν4d.(22)

Converting the dimensional Eq. (22) into non-dimensional form, the flux Q is formulated as

Q=∫−11∫−11wdxdy.(23)

3  Numerical Calculations

3.1 Method of Numerical Calculation

This study relies on numerical computations, and to obtain these numerical solutions, the spectral method [35] is employed as a fundamental technique. The key objective of this technique is to utilize polynomial function expansions, where the variables are represented as a series of functions composed of the Chebyshev polynomials. The expansion functions are denoted as φn(x) and ψn(x) are described as follows:

φn(x)=(1−x2)Cn(x)ψn(x)=(1−x2)2Cn(x)},(24)

where, Cn(x)=cos⁡(ncos−1⁡(x)) is the nth order Chebyshev polynomial; w(x,y,t) and ψ(x,y,t) are expressed in terms of φn(x) and ψn(x) as

w(x,y,t)=∑m=0M∑n=0Nwmn(t)φm(x)φn(y)ψ(x,y,t)=∑m=0M∑n=0Nψmn(t)ψm(x)ψn(y)},(25)

where M and N are the truncation numbers in the x- and y-directions, respectively; and wmn and ψmn are the coefficients of expansion. To obtain a steady solution w¯(x,y) and ψ¯(x,y) the expansion series (25) are then substituted into the basic Eqs. (12) and (13), and the collocation method is applied following [35]. As a result, a set of nonlinear algebraic equations for wmn and ψmn are obtained. The collocation points (xi,yj) are taken as

xi=cos⁡[π(1−1M+2)]; yj=cos⁡[π(1−jN+2)](26)

where i=1,⋯,M+1 and j=1,⋯,N+1.

The convergence is assured by taking suitably small εp(εp<10−10) defined as

εp=∑m=0M∑n=0N[(wmn(p+1)−wmnp)2+(ψmn(p+1)−ψmnp)2].(27)

For sufficient accuracy of the solutions, we take M = 20 and N = 20.

3.2 Time-Evolution Calculations

For time-evolution calculation, Crank-Nicolson as well as Adams-Bashforth method is applied [35] in Eqs. (12) and (13), and consequently the equations are obtained as

F(t)w(t+Δt)=Dn+{(1+2δx)F(t)−dδ−Eb(1+mα)−2δΔtx+δ∂∂x}w(t)−[δTr+E2δm+dw(t)]∂ψ(t)∂y+∂(ψ,w)∂(x,y),(28)

F(t)Δ2ψ(t+Δt)=b{(F(t)+Δ2)Δ2−2dΔt∂δx−2d∂δxΔ2+3d2∂2δx2−3d3∂δx}ψ(t)−Eb(1+mα)∂2ψ∂y2+Em2δb2∂w∂y+2δΔt∂ψ∂x+d{∂ψ∂y(∂2ψ∂x2+2Δ2ψ−3d∂ψ∂x)−∂ψ∂x∂2ψ∂x∂y}−∂(Δ2ψ,ψ)∂(x,y)+bw∂w∂y.(29)

3.3 Mesh Sensitivity Test

In this section, the accuracy of the code is validated by Mash efficiency. Previous researchers such as [36,37] determined grid efficiency by calculating the minimum variation of the primary axial velocity for different grid sizes M × N. In this study, the accurate grid efficiency is confirmed by calculating the slight variation in relative errors of the primary axial velocity for different gird sizes. The present relative error is calculated using a formula

Ep=|Currentvalue−Previousvalue|Currentvalue×100%

It should be mentioned here that we measured primary velocity and the present relative errors to determine the suitable truncation number for changing the truncation number and fixed parameters Dn = 500, Pr = 7.0, Tr = 0, M = 0.5, m = 0.01, α = 0.01 and δ = 0.1. Our findings were compared with those of [37], who used a CSD similar to our model, M × N which represents grid size, remained the same in each scenario. Results showed that numerical solutions approximated by M = 20 and N = 20 were accurate (marked in Bold letters) and matched well with the findings obtained by [37] as shown in Table 1.

4  Results and Discussion

4.1 Solution Structure without MHD

The study explores branching structure of steady solution (SS) for MHD fluid flow in a CSD considering several non-dimensional parameters including M, m, α, Dn and Tr. This investigation is depicted through graphical representations with M varying from 0.5 to 50.0 for 0 < Dn ≤ 6500. The study specifically focuses on SS which is determined based on Dn while other factors such as M = 0, m = 0, α = 0 remain constant for Tr = 50 and curvature δ = 0.1.

The SS curve, symbolized by a → b → c → d, is represented by a solid red line as depicted in Fig. 2a. The asymmetric curve originates at point a (Dn = 10.0) and proceeds to point b (Dn ≈ 1333.399) where λ decreases as Dn increases. The branch then alters its direction once more and ascends to point c (Dn ≈ 606.357). Afterward, the branch alters its direction again, progressing to a higher Dn, eventually arriving at point d (Dn ≈ 6500) without any additional changes in direction. It should be noted that point b is very sharp the point c is a smooth curve (see Fig. 2b,c). It is worth noting that [37] also obtained similar type of SS in their study. In Fig. 3, the secondary and axial flow patterns of the SS are displayed. The branch can be divided into three parts based on the structure of the secondary flow patterns. The first part (a to b) consists of symmetric 2-vortex solutions, and the second part (b to c) consists of asymmetric and symmetric 2- and 4-vortex solutions, while the third part (c to d) only symmetric 4-vortex solution. As Dn increases, the axial velocity becomes stronger and shifts towards the outer side of the duct.

Figure 2: (a) SS for M = 0, m = 0, α = 0, Tr = 50, Gr = 0 and δ = 0.1, (b) Zoomed graph of point b (c) Zoomed graph of point c

Figure 3: Vortex pattern for Dn = 503, 1333, 1250, 611, 606.35, 1000 and 2000

4.2 Solution Structure with MHD

To analyze the SS structure considering the influence of the dimensionless M, the SS is acquired through the employment of the arc-length technique and illustrated graphically. To assess the impact of M, we have graphed the SS structure within the confines of 10 ≤ Dn ≤ 6500 and 0.5 ≤ M ≤ 50.0 maintaining the consistency of other flow variables as m = 0.01 and α = 0.01. In this study, we assume a fixed rotation of the CSD with Tr = 50 and Pr = 7.0. The hall and ion-slip parameters are both set to 0.01. The magnetic parameter (M) is arbitrarily chosen as 0.5, 2.5, 5.5, 7.5, 10.5, 12.0, 32.6175, 32.6178, 32.63, 33.0, 35.0, 40.0, 45.0 and 50.0. The chosen range of parameters M allows us to capture the transition in the SS structure that arises from the interplay between duct geometry, curvature effects and MHD forces.

Fig. 4a displays the solution structure of the steady solution (SS) for the above-mentioned values of M. However, due to the proximity of the curves, they are not clearly distinguishable. Therefore, Fig. 4b,c provides a closer look at the smaller and larger estimates of both M and Dn, respectively. Different colors and line patterns distinguish each SS. The purple-colored solid line (lower solid line) in Fig. 4b represents the steady solution for M = 0.5 which is like the SS without MHD effect presented in Fig. 2a. However, there are slight changes in the position of turning points b (Dn = 801.277), c (Dn = 647.785), reflecting the geometric effects of curvature, which naturally introduce limit-point behavior. Fig. 4d,e represents the zoomed figures at points b and c, respectively, where it is observed that the curve turns sharply at point b and smoothly at point c like that in Fig. 2b,c. The green-colored solid line in Fig. 4b represents the SS for M = 2.5 which is similar in structure to the previous SS with turnings but with higher position and slightly different positions at points b (Dn = 1007.566) and c (Dn = 823.7389). Increasing the value of M does not affect the structure of the SS significantly but the positions of the points b and c gradually shift to larger Dn and the length of bc gradually increases. Physically this happens because the Lorentz force suppress curvature-induced secondary motion, requiring a higher Dean number (strong curvature-driven inertia) to produce the same geometric turning behavior. Consequently, the distance between points b and c (the length bifurcation segment) increases with M. This geometric trend continues, where the asymmetry of the SS reaches the upper limit of the computational domain. At M = 32.6178 and beyond, shown in Fig. 4c, the SS curve no longer exhibits a turning at high Dn; instead λ monotonically approaches zero. This transition indicates that the magnetic field becomes sufficiently strong to dominate over the curvature geometry, eliminating the multiple steady states that arise from the geometric bifurcation mechanisms. In Fig. 4a–c, we observed that λ increases gradually as M increases which leads to a decrease in the flux value Q. The study shows that M has a decelerating effect on the flux value. This decelerating effect is consistent with the physical damping imposed by the Lorentz force and reflects the increasing geometric confinement of the axial flow when MHD forces oppose curvature-driven momentum transport.

Figure 4: (a) Steady graph for M = 0.5, 2.5, 5.5, 7.5, 10.5, 12.0, 32.0, 33.0, 35.0, 40.0, 45.0 and 50.0. Zoomed graph of (a): (b) for 0.5 ≤ M ≤ 32.6178; (c) for 32.63 ≤ M ≤50.0; (d) Zoomed at point b; (e) Zoomed at point c

The bifurcation zone represented by the region of bc, moves to larger values of Dn as M increases. The extent of the bifurcation zone also increases as M increases, and this is demonstrated in Fig. 5, where the normal and bifurcation zones are illustrated in blue and red-colors, respectively and their numerical values are presented in Table 2. Fig. 5 indicates that the bifurcation zone becomes weaker as M is increased, demonstrating that the geometric bifurcation becomes less pronounced when magnetic forces counteract curvature-induced inertial effects.

Figure 5: Schematic diagram of bifurcation zone for 0 < Dn ≤ 6500 and 0.5 ≤ M ≤ 32.63

To investigate the structure of SS, it is necessary to analyze the pattern variation of secondary and axial velocity profiles. Based on Fig. 6a–c, it is observed that for 0.5 ≤ M ≤ 32.6175 the structure of each SS is almost identical. Each SS starts with two relatively weak vortex solutions followed by two additional vortices, called Dean vortices, appear around point b. The remaining part of the SS consists of 4-vortex solutions. This multi-vortex structure reflects the geometrical amplification of centrifugal effects in curved ducts. Conversely, all SSs in 32.6178 ≤ M ≤ 50.00 are composed of only 2-vortex solution as shown in Fig. 6d,e. Therefore, in both cases, M appears to increase the extent of 2-vortex solution in a SS. The magnetic field suppresses the curvature-induced secondary flow so strongly that the geometry is unable to support the development of Dean vortices. This corresponds exactly to the disappearance of turning points in Fig. 4c, confirming that the geometric bifurcation mechanism collapses under strong MHD damping. Moreover, increasing the value of M is found to reduce the intensity of the axial velocity and shift towards the center of the duct. This implies that the axial velocity can completely be controlled by increasing the value of M.

Figure 6: Impacts of magnetic parameter (M) on flow pattern. (a) M = 0.5 (b) M = 5.5 (c) M = 32.6175 (d) M = 32.6178 (e) M = 50.0

4.3 Linear Stability Analysis

In this section, we examine the linear stability of the SS for eight values of M, e.g., M = 0.5, 2.5, 5.5, 7.5, 10.5, 12.0, 32.6175 and 32.63. The purpose is to determine which parts of the branches are stable or unstable using numerical and graphical methods. The stable region of the SSs is represented by a solid black line in Fig. 7a,b. Note that the x-axis of the SSs for M = 32.6175 and M = 32.63 is specified in the top label of Fig. 7a.

Figure 7: Schematic diagram of linear stability analysis. (a) For 0.5 ≤ M ≤ 32.63; (b) For M = 32.63, 50.0

Tables 3–7 provide the eigenvalues with the largest real and imaginary component of the first eigenvalue denoted by σr and σi, respectively, where bold letters signify the linearly stable solutions. We investigated the linear stability of the SS for M = 0.5, 12.0, 32.6175 and found that the stable regions of these three solutions exist within certain limits of Dn with the other portion of the corresponding solution being linearly unstable. Specifically, the stable regions are found for 0 < Dn ≤ 801.2768, 0 < Dn ≤ 2239.3544 and 0 < Dn ≤ 6499.50, respectively. It is noteworthy that linearly stable region increases gradually as M increases, and if M = 32.63 or above, all solutions become linearly stable, and the unstable region disappears for 0 < Dn ≤ 6500. Therefore, the critical value of M is determined as Mc = 32.63.

5  Investigation of Unsteady Behavior

5.1 Unsteady Solution for M = 0.5

Time-evolution calculations for m = 0.01, α = 0.01 and 0.5 ≤ M ≤ 50 are numerically carried out and displayed in the t-λ plane where the requisite phase trajectory (PT) is drawn to justify the periodicity/multi-periodicity/chaotic behavior. We explored the oscillating behavior for M = 0.5, 2.5, 5.5, 7.5, 12.0 and 32.63. For brevity, we only present the US with M = 0.5, M = 12.0 and 32.63 but all USs are presented in a graph at the end of this section. To investigate the oscillating behavior of λ with transition stages for M = 0.5 and Dn = 250, 801.27, 802, 1400, 1415, 2135, 2140, 2310, 2320, 2710, 2720, 3850, 3860 and 6500. Because the curved duct geometry inherently produces centrifugal imbalance, small variations in Dn alter the strength of secondary flow and shift the system between steady-state, multi-periodic, and chaotic states. Fig. 8a,b shows that the flow remains steady state with two asymmetric vortices for 250 ≤ Dn ≤ 801.27. However, at Dn = 802, the flow suddenly becomes multi-periodic as seen in Fig. 9a. The PT diagram, as depicted in Fig. 9b, shows that the flow is multi-periodic because the orbits of the trajectory do not intersect each other. The oscillating flow at Dn = 802 consists of asymmetric 3- and 4-vortex solutions for a single period in 8.70 ≤ t ≤ 16.30 is shown in Fig. 9c, demonstrating that the geometry promotes additional vortex pairs when inertial effects overcome magnetic damping. For Dn = 1400, the time dependent solutions, presented in Fig. 10a,c, show a multi-periodic flow with an asymmetric 2-vortex solution as verified by PT and displayed in Fig. 10b. This indicates that while inertial forces are stronger, they are not yet sufficient to support additional vortex splitting imposed by the curved duct geometry.

Figure 8: Results for M = 0.5, Dn = 250 and 801.27 (a) US (b) Vortex pattern for Dn = 250 and Dn = 801.27

Figure 9: Results for M = 0.5, Dn = 802 (a) US (b) PT (c) Vortex pattern

Figure 10: Results for M = 0.5, Dn = 1400 (a) US (b) PT (c) Vortex pattern

Again, the multi-periodic flow turns into steady-state flow for Dn = 1415 which continues up to Dn = 2135 and the 2-vortex steady-state flow at Dn = 1415 turns into 4-vortex for Dn = 2135 (Fig. 11a,b). If Dn is slightly increased, for example Dn = 2140, the flow changes from a steady-state to multi-periodic flow arising only 4-vortex as depicted in Fig. 12a,c. To verify the flow characteristics, Fig. 12b confirms that the flow is multi-periodic for Dn = 2140. For 2140 ≤ Dn ≤ 2310, the flow remains multi-periodic and unaltered as shown in Fig. 13a even though Fig. 13a appears to be a periodic flow. The phase trajectory, as shown in Fig. 13b, signifies that the flow is multi-periodic. Moreover, the multi-periodic flow consists of 3- and 4-vortex solutions for a single period of 16.50 ≤ t ≤ 17.11 as demonstrated in Fig. 13c. The multi-periodic flow transforms again into a steady-state nature if Dn is raised to 2320 and this remains unchanged until Dn = 2710 as shown in Fig. 14a. The flow structure presented in Fig. 14b reveals that the steady-state flow comprises a 2-vortex solution in both cases.

Figure 11: Results for M = 0.5, Dn = 1415 and 2135 (a) US (b) Vortex pattern

Figure 12: Results for M = 0.5 and Dn = 2140 (a) US (b) PT (c) Vortex pattern

Figure 13: Results for M = 0.5, Dn = 2310 (a) Unsteady solution (b) PT (c) Vortex pattern

Figure 14: Results for M = 0.5, Dn = 2320, 2710 (a) US (b) Vortex pattern

To assess the oscillating behavior, we employ Dn = 2720 and Dn = 3850. The flow remains multi-periodic with asymmetric 2-vortex solutions throughout the entire interval 2720 ≤ Dn ≤ 3850 as shown in Fig. 15a,c. The graph presented in Fig. 15a shows an US for Dn = 3850 which appears to have a multi-periodic pattern. However, the PT depicted in Fig. 15b confirms that the flow is chaotic which is referred to as transitional chaos [37]. As Dn is increased, the flow loses multi-periodicity and enters an unsteady oscillatory regime with 2- to 4-vortex solutions until it reaches at Dn = 3860 where a chaotic state occurs and continues up to Dn = 6500 as shown in Fig. 16a–c, reflecting the strong destabilizing influence of curvature at very high inertia. The range of 1400 < Dn < 1415, 2135 < Dn < 2140, 2310 < Dn < 2320, 2710 < Dn < 2720 and 3850 < Dn <3860 exhibits five transition stages from multi-periodic to steady-state, steady-state to multi-periodic, multi-periodic to steady-state, steady-state to multi-periodic and multi-periodic to chaotic. Based on the afore-mentioned discussion, it has been discovered that there are specific range of Dn that corresponds to different types of solutions for M = 0.5. The steady-state solution exists for 0 < Dn ≤ 801.27, 1415 ≤ Dn ≤ 2135 and 2320 ≤ Dn ≤ 2710. The multi-periodic solution is found for 802 ≤ Dn ≤ 1400, 2140 ≤ Dn ≤ 2310 and 2720 ≤ Dn ≤ 3850. Finally, the chaotic solution exists for 3860 ≤ Dn ≤ 6500. These transitions are governed by the curved geometry, which amplifies inertial effects and produces vortex interactions highly sensitive to the Dean number. Furthermore, an analysis of the linear stability of the SS for M = 0.5 indicates that only a small section of this branch is linearly stable, and it exists for 0 ≤ Dn ≤ 801.27. The steady-state solution for 1415 ≤ Dn ≤ 2135 and 2320 ≤ Dn ≤ 2710 is not linearly stable. As a result, the findings from oscillating behavior and the linear stability analysis are consistent.

Figure 15: Results for M = 0.5, Dn = 2720 and 3850 (a) US (b) PT (c) Vortex pattern

Figure 16: Results for M = 0.5, Dn = 3860 and 6500 (a) US (b) PT (c) Vortex pattern

5.2 Unsteady Solution for M = 12.0

To investigate the flow characteristics for M = 12.0, we conducted a study of the flow characteristics with the transition stages. A steady-state flow with a 2-vortex solution is found for Dn ≤ 2239 as shown in Fig. 17a,b. As Dn is increased to 2250, the steady-state flow is transformed into a multi-periodic flow as depicted in Fig. 18a which continued until Dn = 3650. A clear understanding of the flow at Dn = 2250 and Dn = 3650 is obtained from the PT as depicted Fig. 18b, which indicates that the flow is multi-periodic at both values of Dn. The flow pattern consists of 2- and 4-vortex solution for Dn = 2250, and 2- and 3-vortex solution for Dn = 3650 as presented in Fig. 18c,d, respectively. With increasing Dn, the axial velocity shifted towards the outer side of the duct as observed in Fig. 18b. For Dn = 3665, the multi-periodic flow turned into a steady-state with a 2-vortex solution which remained approximately similar up to Dn = 5350 except that the number of vortices increased to a 4-vortex solution (see Fig. 19a). As Dn is increased from 3665 to 5350, the axial flow shifted to the outer side of the duct and became stronger (see Fig. 19a,b). The flow then lost the steadiness stage and transformed into a multi-periodic stage for higher values of Dn (5370 ≤ Dn ≤ 6500) as depicted in Fig. 20 with a 4-vortex solution at Dn = 5370 and a 2-vortex at Dn = 6500. The transitional stages from steady-state to multi-periodic, multi-periodic to steady-state and steady-state to multi-periodic occurred in the regimes 2240 ≤ Dn < 2250, 3650 < Dn < 3665 and 5350 < Dn < 5370, respectively.

Figure 17: Results for M = 12.0, Dn = 250 and 2239 (a) US (b) Vortex pattern

Figure 18: Results for M = 12.0, Dn = 2250 and 3650. (a) US (b) PT (c) Vortex pattern for Dn = 2250 (d) Vortex pattern for Dn = 3650

Figure 19: Results for M = 12.0, Dn = 3665 and 5360 (a) US (b) Vortex pattern

Figure 20: Results for M = 12.0, Dn = 5370 and 6500 (a) US (b) Vortex pattern

As discussed above, the relationship between linear stability analysis and time evaluation calculations have been built up for M = 12.0. It was found that the regions of the steady-state solution are 0 < Dn ≤ 2239 and 3665 ≤ Dn ≤ 5350 whereas the regions of the multi-periodic solution are obtained for 2250 ≤ Dn ≤ 3650 and 5370 ≤ Dn < 6500. Notably, no region of chaotic solution is identified for M = 12.0. It is worth mentioning that the branches for M = 33.0 and above contain only steady-state solution in the working domain 0 < Dn ≤ 6500. The analysis of the linear stability for M = 12.0 demonstrated that only a small section of the branch, i.e., the region for 0 < Dn ≤ 2239 constitutes a linearly stable SS whereas the other parts of the branch are linearly unstable. Furthermore, the steady-state solution for 3665 ≤ Dn ≤ 5350 is not linearly stable. Thus, the results from the oscillating behavior calculations align with those from the linear stability analysis. Furthermore, it is noteworthy that the linear stability region of the branch increases with the increase of M.

5.3 Unsteady Solution for M = 32.63

In this section. we have examined how the magnetic parameter M = 32.63 changes over time for four randomly selected values for Dn = 255, 3000, 5000 and 6500. Our observations as presented in Fig. 21a,b reveal a steady-state solution. Based on these findings, it can be asserted that any US for M = 32.63 and 0 < Dn ≤ 6500 converge to a steady-state solution characterized by only 2 vortices.

Figure 21: Results for M = 32.63, Dn = 255, 3000, 5000 and 6500 (a) US (b) Vortex pattern

5.4 Schematic Plot of Unsteady Solutions

The summarization of the unsteady solutions for various values of M is depicted in Fig. 22. It depicts how the region of the steady-state solution increases while the regions of the periodic and chaotic solutions gradually diminish as M increases in the working domain. It is evident that the largest number of variations in the time dependent solutions is observed at M = 0.5 and in the region of small Dn. The largest area of the chaotic solution is found at M = 0.5 as illustrated in Fig. 22. However, the number of transitions in the USs gradually decreases and approaches that of a single steady-state solution as M increases. When M = 32.63 or higher, all the existing time dependent solutions transform into a single steady-state with a 2-vortex solution. As M increases, the Lorentz force acts to suppress the curvature-induced secondary flow, thereby weakening the geometric mechanisms responsible for vortex splitting and oscillatory instabilities. Consequently, the areas corresponding to periodic and chaotic solutions contract, while the region of steady-state solutions expands across the Dn-M domain. The number of transitions in the unsteady solutions decreases because the magnetic field damps the geometric asymmetry that normally destabilizes the flow.

Figure 22: Schematic plot of unsteady solutions for 0.5 ≤ M ≤ 50.0 and 0 < Dn ≤ 6500

5.5 Effects of Hall Parameter

The paragraph describes the effects of hall parameter m (0.1 ≤ m ≤ 20) on the SS of the isothermal flow through a CSD for Dn = 500, M = 0.1 and α = 0.1. Fig. 23a shows that as m increases, the flux value Q of the fluid increases rapidly for 0 ≤ m ≤ 5, after which the rate of increase gradually weakens and becomes nearly negligible beyond m = 10. This behavior is strongly linked to the geometry of the curved duct: for small m, the Hall current reduces the effective Lorentz force acting on the field, allowing the curvature-driven secondary flow to strength. This enhances the cross-sectional circulation and increases the flux. However, once m becomes sufficiently large, the electromagnetic influence saturates, and the geometric curvature can no longer generate additional momentum redistribution. Thus, the SS curve flattens for m>10. Fig. 23b shows that the SS consists of only a symmetric 2-vortex solution throughout the entire range of m. The curved duct geometry, at this moderate Dean number, supports only a single pair of counter-rotating vortices, and variations in the Hall parameter are not strong enough to alter the geometric vortex pattern. Consequently, both the primary and secondary velocity fields remain nearly unchanged with increasing m, indicating that the Hall effect primally modifies the electromagnetic resistance without significantly affecting the curved-induced vortex geometry.

Figure 23: (a) Steady plot for 0.1 ≤ m ≤ 20 (b) Vortex pattern for m = 2.0, 5.0, 15.0 and 20.0

5.6 Effects of Ion-Slip Parameter

This section discusses the impact of the ion-slip parameter on the SS of the isothermal flow through CSD for Dn = 500, M = 0.1 and m = 0.1. Fig. 24a illustrates that as the value of ion-slip increases, the flux also increases gradually. This suggests that the ion-slip parameter has a positive effect on the acceleration of the flux. Meanwhile, Fig. 24b depicts that the flow has 2-vortex solution and there is no significant effect of the ion-slip parameter on the axial and secondary velocities for all values of α.

Figure 24: (a) Steady plot for 0.1 ≤ α ≤ 20; (b) Vortex pattern for α = 2.0, 5.0, 15.0 and 20.0

6  Validation of the Result

We consider the experimental results for the rotating bent square duct to verify our numerical studies. The experimental results are shown in the secondary flow pattern (SFP) of each of the left-hand graphics of Fig. 25, while the numerical-based studies are shown in the right-hand graphics of each group. Our numerical studies in the right pair are well-aligned with a visually striking representation of the flow patterns, namely a 4-vortex flow experimented by [38–40], as shown in Fig. 25a,b,d in the left pair, respectively. The results in the right part of Fig. 25c are accurate because [36] investigated a 2-vortex secondary flow in their experiments for moderate curvature (δ=0.03). The results of the published experiments confirmed the conclusions for the stationary instances.

Figure 25: Experimental findings against numerical data: On the left: laboratory-based experimental findings by (a) [38], (b) [39], (c) [36], and (d) [40]; on the right: numerical findings by the authors

So far, no experimental works have been completed on the MHD flow in a rotating bending duct. Our simulation results have been verified using the experimental vortex structure for our square channel flow with rotation by [15]. Our vortex structure was in line with the rotational effects on the duct flow, both positive and negative, as shown in Figs. 26 and 27, respectively. A fair agreement is found between our numerical output in the right of each pair of Figs. 26 and 27 and the 2- and 4-vortex structures for the positive rotation (Tr=150) and the negative rotation (Tr=−150) experimentally investigated by [15] in the left of each pair of Figs. 26 and 27. In contrast to [15], who examined the channel’s rotational flow behavior for the three walls alone, we have extended the rotation to the entire system, not just the three walls. The outer wall rotates around the center of curvature. As per [15], the centrifugal effect was regarded as same.

Figure 26: Analysis of rotating curved square duct flow at Tr=150: comparison between experimental and numerical results. (a,b) Experimental result (left) by [15] and the authors’ numerical result (right)

Figure 27: Analysis of rotating curved square duct flow at Tr=−150: Comparison between experimental and numerical results. (a,b) Experimental result by [15] (left) and numerical result by the authors (right)

7  Conclusions

The present study investigates the characteristics of MHD fluid flow through a rotating curved square duct, considering hall and ion-slip currents for the Dean number 0 < Dn ≤ 6500 and magnetic parameter 2.5 ≤ M ≤ 50.0. To observe the impact of M in the fluid flow, the steady solution (SS) is first calculated without imposing the magnetic field, and then the magnetic field is applied, considering hall and ion-slip currents. Unsteady solutions (US) are also calculated for various values of M and Dn. Specific findings of the current study show that-

•   The magnetic parameter (M) tends to decelerate the flux value, i.e., the value of the resistance coefficient gradually increases with an increase in M.

•   The bifurcation zone on the SS gradually increases with an increase of M. The bifurcation zone gradually moves toward larger Dn, and the SS approaches symmetry as the magnetic field is intensified.

•   The steady solution (SS) consists of 2- and 4-vortex solutions for 2.5 ≤ M < 32.63, a 2-vortex solution for M ≥ 32.63. M enhances the extent of the 2-vortex solution of an SS. The intensity of axial velocity decreases as M increases and shifts to the central position of the duct.

•   Linearly stability analysis shows that the stability region gradually increases with an increase of M.

•   Unsteady flow transition undergoes in the scenario ‘steady-state→multi-periodic→steady-state→multi-periodic→steady-state→multi-periodic→chaotic’, ‘steady-state→multi-periodic→steady-state→multi-periodic’, ‘steady-state→multi-periodic’, ‘steady-state’ for M = 0.5, 12.0, 32.0, and 32.63, respectively. The transient flow consists of asymmetric 2- to 4-vortex solutions.

•   M reduces the tendency of frequent transition, i.e., the chaotic and multi-periodicity gradually converted to steady-state nature with the increase of M.

•   The hall parameter accelerates the flux, but the secondary flow and axial flow remain minimally impacted with the increase of m.

•   The ion-slip parameter gradually accelerates the flux value, but no significant impacts are seen on the secondary flow and axial flow distribution as α is increased.

Future Perspectives

Since this paper is based on spectral-based numerical study on isothermal and Magnetohydrodynamic (MHD) fluid flow through a rotating curved square duct, our future intention is to extend the work for non-isothermal flow with Nanofluid or Hybrid nanofluid, taking into account MHD effects, which have remarkable applications in engineering and industry. The problem can also be extended for investigating nonlinear characteristics and convective heat transfer for MHD flow in curved channels for triangular or elliptical cross-sectional geometry.

Acknowledgement: None.

Funding Statement: Rabindra Nath Mondal (R. N. Mondal), one of the authors, would like to gratefully acknowledge the financial support from Jagannath University, Dhaka to conduct this research work (Ref. No. JnU/Research/Pro-38/2024/281, Dated: 09 April 2025), URL: www.jnu.ac.bd.

Author Contributions: The authors confirm contribution to the paper as follows: Study conception and design: Ratan Kumar Chanda and Rabindra Nath Mondal; data collection: Ratan Kumar Chanda; analysis and interpretation of results: Ratan Kumar Chanda and Rakesh Bhowmick; draft manuscript preparation: Ratan Kumar Chanda and Rabindra Nath Mondal; review and editing: Rakesh Bhowmick and Giulio Lorenzini; visualization, validation and supervision: Rabindra Nath Mondal and Giulio Lorenzini. All authors reviewed the results and approved the final version of the manuscript.

Availability of Data and Materials: The authors confirm that the data supporting the findings of this study are available from the corresponding authors upon request.

Ethics Approval: Not applicable.

Conflicts of Interest: The authors declare no conflicts of interest to report regarding the present study.

Nomenclature

References

1. Dean WR XVI. Note on the motion of fluid in a curved pipe. Lond Edinb Dublin Philos Mag J Sci. 1927;4(20):208–23. doi:10.1080/14786440708564324. [Google Scholar] [CrossRef]

2. Hartmann J, Lazarus F. Hg-dynamics II. Math-Fys Medd. 1937;15(7):1–45. [Google Scholar]

3. Gold RR. Magnetohydrodynamic pipe flow. Part 1. J Fluid Mech. 1962;13(4):505–12. doi:10.1017/s0022112062000889. [Google Scholar] [CrossRef]

4. Grinberg GA. On steady flow of a conducting fluid in a rectangular tube with two nonconducting walls, and two conducting ones parallel to an external magnetic field. J Appl Math Mech. 1961;25(6):1536–50. doi:10.1016/0021-8928(62)90133-8. [Google Scholar] [CrossRef]

5. Grinberg GA. On some types of flow of a conducting fluid in pipes of rectangular cross-section, placed in a magnetic field. J Appl Math Mech. 1962;26(1):106–17. doi:10.1016/0021-8928(62)90105-3. [Google Scholar] [CrossRef]

6. Hunt JCR. Magnetohydrodynamic flow in rectangular ducts. J Fluid Mech. 1965;21(4):577–90. doi:10.1017/s0022112065000344. [Google Scholar] [CrossRef]

7. Issacci F, Ghoniem NM, Catton I. Magnetohydrodynamic flow in a curved pipe. Phys Fluids. 1988;31(1):65–71. doi:10.1063/1.866578. [Google Scholar] [CrossRef]

8. Barrett KE. Duct flow with a transverse magnetic field at high hartmann numbers. Int J Numer Meth Eng. 2001;50(8):1893–906. doi:10.1002/nme.101. [Google Scholar] [CrossRef]

9. Carabineanu A, Lungu E. Pseudospectral method for MHD pipe flow. Int J Numer Meth Eng. 2006;68(2):173–91. doi:10.1002/nme.1706. [Google Scholar] [CrossRef]

10. Al-Khawaja MJ, Selmi M. Highly accurate solutions of a laminar square duct flow in a transverse magnetic field with heat transfer using spectral method. J Heat Transf. 2006;128(4):413–7. doi:10.1115/1.2177289. [Google Scholar] [CrossRef]

11. Mondal RN, Bhowmick S, Lorenzini G, Adhikari SC. A computational modeling on flow bifurcation and energy distribution through a loosely bent rectangular duct with Vortex structure. Front Heat Mass Transf. 2025;23(1):249–78. doi:10.32604/fhmt.2024.057990. [Google Scholar] [CrossRef]

12. Hasan M, Mondal R, Lorenzini G. Centrifugal instability with convective heat transfer through a tightly coiled square duct. Math Model Eng Probl. 2019;6(3):397–408. doi:10.18280/mmep.060311. [Google Scholar] [CrossRef]

13. Chanda RK, Hasan MS, Alam MM, Mondal RN. Taylor-heat flux effect on fluid flow and heat transfer in a curved rectangular duct with rotation. Int J Appl Comput Math. 2021;7(4):146. doi:10.1007/s40819-021-00986-8. [Google Scholar] [CrossRef]

14. Chanda RK, Hasan MS, Alam MM, Mondal RN. Hydrothermal behavior of transient fluid flow and heat transfer through a rotating curved rectangular duct with natural and forced convection. Math Model Eng Probl. 2020;7(4):501–14. doi:10.18280/mmep.070401. [Google Scholar] [CrossRef]

15. Yamamoto K, Wu X, Nozaki K, Hayamizu Y. Visualization of Taylor-Dean flow in a curved duct of square cross-section. Fluid Dyn Res. 2006;38(1):1–18. doi:10.1016/j.fluiddyn.2005.09.002. [Google Scholar] [CrossRef]

16. Hussen S, Chanda RK, Alam MS, Mondal RN. Flow instability and development of Dean vortices in a bending rectangular channel with bottom wall heating and cooling from the ceiling. Int J Thermofluids. 2025;26(1):101049. doi:10.1016/j.ijft.2024.101049. [Google Scholar] [CrossRef]

17. Moreau R, Smolentsev S, Cuevas S. MHD flow in an insulating rectangular duct under a non-uniform magnetic field. PMC Phys B. 2010;3(1):3. doi:10.1186/1754-0429-3-3. [Google Scholar] [CrossRef]

18. Dousset V. Numerical simulations of MHD flows past obstacles in a duct under externally applied magnetic field [dissertation]. Coventry, UK: Coventry University; 2009. [Google Scholar]

19. Hoque M, Anika NN, Alam MM. Magnetic effects on direct numerical solution of fluid flow through a curved pipe with circular cross section. Euro J Sci Res. 2013;103(3):343–61. [Google Scholar]

20. Hoque MM, Alam MM. A numerical study of MHD laminar flow in a rotating curved pipe with circular cross section. Open J Fluid Dyn. 2015;5(2):121–7. doi:10.4236/ojfd.2015.52014. [Google Scholar] [CrossRef]

21. Sayed-Ahmed ME, Ali Attia H. MHD flow and heat transfer in a rectangular duct with temperature dependent viscosity and hall effect. Int Commun Heat Mass Transf. 2000;27(8):1177–87. doi:10.1016/S0735-1933(00)00204-9. [Google Scholar] [CrossRef]

22. Afikuzzaman M, Ferdows M, Alam MM. Unsteady MHD casson fluid flow through a parallel plate with hall current. Procedia Eng. 2015;105:287–93. doi:10.1016/j.proeng.2015.05.111. [Google Scholar] [CrossRef]

23. Kabir MF, Alam MM. The unsteady Casson fluid flow through parallel plates with hall current, joule heating and viscous dissipation. AMSE J. 2015;84(1):1–22. [Google Scholar]

24. Yusuf A, Bhatti MM. Coupled nonthermal plasma and photocatalytic surface reactions in mass-based ZnO-Au/water hybrid nanofluid flow. Int J Heat Fluid Flow. 2026;117(10):110087. doi:10.1016/j.ijheatfluidflow.2025.110087. [Google Scholar] [CrossRef]

25. Alamirew WD, Zergaw GA, Gorfie EH. Effects of hall, ion slip, viscous dissipation and nonlinear thermal radiation on MHD Williamson nanofluid flow past a stretching sheet. Int J Thermofluids. 2024;22(6):100646. doi:10.1016/j.ijft.2024.100646. [Google Scholar] [CrossRef]

26. Laltesh K, Atar S, Vimal KJ, Sharma K. MHD micropolar fluid flow with hall current over a permeable stretching sheet under the impact of dufour-soret and chemical reaction. Int J Thermofluids. 2025;26(1):1–12. doi:10.1016/j.ijft.2024.101042. [Google Scholar] [CrossRef]

27. Hossain MD, Ali ME, Samad MA, Alam MM, Hafez MG. Numerical investigation of ion-slip current and stratification on MHD flow through high porosity medium with soret and dufour effects in a turning scheme. Partial Differ Equ Appl Math. 2025;14(1):101158. doi:10.1016/j.padiff.2025.101158. [Google Scholar] [CrossRef]

28. Nowar K, Halouani B. Combined effects of hall and ion-slip currents and heat transfer on peristaltic transport of a copper-water nanofluid. AIP Adv. 2025;15(2):025307. doi:10.1063/5.0246334. [Google Scholar] [CrossRef]

29. Anwar T, Raza Q, Mushtaq T, Ali B, Hemati E. MHD casson nanofluid flow over a stretching rotating disk with heat source and spatially varying magnetic field. Int J Thermofluids. 2025;30(1):101422. doi:10.1016/j.ijft.2025.101422. [Google Scholar] [CrossRef]

30. Srnivasacharya D, Shiferaw M. MHD flow of micropolar fluid in a rectangular duct with hall and ion slip effects. J Braz Soc Mech Sci Eng. 2008;30(4):331–8. doi:10.1590/s1678-58782008000400007. [Google Scholar] [CrossRef]

31. Jha BK, Apere CA. Combined effect of hall and ion-slip currents on unsteady MHD couette flows in a rotating system. J Phys Soc Jpn. 2010;79(10):104401. doi:10.1143/jpsj.79.104401. [Google Scholar] [CrossRef]

32. Jha BK, Malgwi PB. Combined effects of Hall and ion-slip current on MHD free convection flow in a vertical micro-channel. SN Appl Sci. 2019;1(10):1163. doi:10.1007/s42452-019-1164-2. [Google Scholar] [CrossRef]

33. Veera Krishna M, Ameer Ahamad N, Chamkha AJ. Hall and ion slip effects on unsteady MHD free convective rotating flow through a saturated porous medium over an exponential accelerated plate. Alex Eng J. 2020;59(2):565–77. doi:10.1016/j.aej.2020.01.043. [Google Scholar] [CrossRef]

34. Krishna MV, Chamkha AJ. Hall and ion slip effects on MHD rotating flow of elastico-viscous fluid through porous medium. Int Commun Heat Mass Transf. 2020;113(1):104494. doi:10.1016/j.icheatmasstransfer.2020.104494. [Google Scholar] [CrossRef]

35. Gottlieb D, Orszag SA. Numerical analysis of spectral methods. Philadelphia, PA, USA: Society for Industrial and Applied Mathematics; 1977. doi:10.1137/1.9781611970425. [Google Scholar] [CrossRef]

36. Wang L, Yang T. Periodic oscillation in curved duct flows. Phys D Nonlinear Phenom. 2005;200(3–4):296–302. doi:10.1016/j.physd.2004.11.003. [Google Scholar] [CrossRef]

37. Mondal RN, Kaga Y, Hyakutake T, Yanase S. Bifurcation diagram for two-dimensional steady flow and unsteady solutions in a curved square duct. Fluid Dyn Res. 2007;39(5):413–46. doi:10.1016/j.fluiddyn.2006.10.001. [Google Scholar] [CrossRef]

38. Bara B, Nandakumar K, Masliyah JH. An experimental and numerical study of the dean problem: flow development towards two-dimensional multiple solutions. J Fluid Mech. 1992;244:339–76. doi:10.1017/s0022112092003100. [Google Scholar] [CrossRef]

39. Mees PAJ, Nandakumar K, Masliyah JH. Instability and transitions of flow in a curved square duct: the development of two pairs of dean vortices. J Fluid Mech. 1996;314:227–46. doi:10.1017/s0022112096000298. [Google Scholar] [CrossRef]

40. Chandratilleke TT. Secondary flow characteristics and convective heat transfer in a curved rectangular duct with external heating. In: Proceedings of the 5th World Conference on Experimental Heat Transfer, Fluid Mechanics and Thermodynamics; 2001 Sep 24–28; Thessaloniki, Greece. 36 p. [Google Scholar]