Abstract
This study investigates the Darcy–Forchheimer magneto-hydrodynamic (MHD) flow of a ternary nanofluid through a convergent/divergent channel with stretching and shrinking walls. The fluid is assumed to be viscous, incompressible, and electrically conducting, allowing the Lorentz force to significantly influence the flow characteristics. Although related Jeffery-Hamel nanofluid models have previously been analyzed using numerical techniques such as RKF-45, ND Solve, bvp4c and other computational methods, the novelty of the present work lies in the incorporation of a Darcy–Forchheimer porous medium, ternary nanoparticles (Ag–Al2O3–TiO2), stretching/shrinking wall effects, and the application of the Improved Residual Power Series Method (IRPSM) to obtained accurate semi-analytic solutions. The governing nonlinear equations are derived through similarity transformations and solved using (IRPSM). The accuracy of the obtained solutions is verified through comparison with available results in the literature, showing excellent agreement. The effects of pertinent physical parameters on the velocity and temperature distributions are analyzed graphically, while the skin-friction coefficient and Nusselt number are also evaluated. The result reveals that the stretching/shrinking parameter and Reynold number produce similar effects on the velocity field in both converging/diverging channels. Increasing the magnetic parameter and channel opening angle suppresses the velocity profile, where the temperature profile is enhanced by stronger magnetic effects. The Darcy and Forchheimer parameters provide additional resistance to the flow and significantly influence the thermal behavior of the fluid. Furthermore, increasing the Reynold number reduces the temperature distribution. Compared with conventional numerical solvers, IRPSM offers an accurate semi-analytical framework that avoids discretization errors, fast convergence, and produces highly reliable solutions for strongly nonlinear flow and heat transfer problems. The ternary nanofluid exhibits superior heat transfer characteristic compared with conventional and hybrid nanofluids, demonstrating its potential for advanced thermal management and engineering applications involving porous MHD flows.
Keywords: Convergent/divergent-channels, Darcy–Forchheimermodel, Ternary nanofluids, IRPSM
Introduction
A ternary nanofluid consists of three distinct types of nanoparticles dispersed in a base fluid. The effective properties of ternary nanofluids, such as thermal conductivity, density, viscosity, and heat capacity, are determined by the volume fractions and individual characteristics of the three nanoparticles, as well as the base fluid. The advanced theory focuses on synergistic enhancement mechanisms such as Brownian motion and thermophoresis, cluster formation and percolation networks, interfacial layering, and electromagnetic and Lorentz forces that arise due to inter-particle interactions and collective behavior among the three nanoparticles. Ternary nanofluids are extensively applied in advanced cooling systems, solar collectors and energy storage devices, biomedical and chemical processes, magnetohydrodynamic and porous medium flows, and thermal management in aerospace and nuclear systems. Zayan et al. [1] investigated the rheological properties of a water-based ternary nanofluid. Humphrey et al. [2] discussed the applications and synthesis of ternary nanofluids in photovoltaic thermal systems and comparatively studied the energy and exergy performance of single and hybrid nanofluids. Vinutha et al. [3] numerically investigated ternary nanofluid flow in the presence of heat sources and sinks through convergent/divergent channels with pollutant concentration. Mohsin et al. [4] reported a numerical analysis of ternary nanofluids under MHD and porosity effects in convergent/divergent channels. Benaziza et al. [5] presented statistical and numerical results for MHD ternary nanofluid flow over stretching/shrinking walls. Gunay et al. [6] investigated a blood-based ternary nanofluid under velocity and thermal slip boundary conditions in convergent/divergent geometry. Malki et al. [7] examined AI-powered scrutiny of MHD ternary nanofluid flow through convergent/divergent channels. Ramesh et al. [8] investigated ternary nanofluid flow in the presence of heat sources and sinks through stretching/shrinking walls. Kezzar et al. [9, 10] discussed EMHD and polymer-based ternary hybrid nanofluids between convergent/divergent channels in the presence of thermal radiation. Batool et al. [11] studied blood-based tri-hybrid nanofluids between stretching/shrinking walls for drug delivery under couple stresses. Jahan et al. [12] examined the behavior of ternary hybrid nanofluids in microchannels with thermal radiation. Chebaane et al. [13] examined magnetized ternary nanofluid flow between convergent/divergent channels with thermal radiation. Various researchers have employed different methodologies to study ternary nanofluids in convergent/divergent channels. Mahariq et al. [14] used the Runge–Kutta method to investigate the thermal enhancement of ternary nanofluids in divergent/convergent channels. Lone et al. [15] studied dissipative and magnetized blood-based ternary hybrid nanofluids with porous media in convergent/divergent channels using the bvp4c approach. Ratha et al. [16] investigated ternary nanofluids with porosity parameters under dissipative and radiative heat effects using a fourth-order Runge–Kutta method. For further studies, see [4, 17–22].
The flow of fluid and transfer of heat in porous media have conventional significant attention over the past several decades due to their wide ranging scientific, industrial, and environmental application. Porous material occur naturally in geological formation such as soil, rocks and aquifers, and are also engineered in industrial processes including packed bed reactors, catalytic converters, thermal insulation, fibrous materials, biomedical scaffolds, filtration systems, and energy storage devices. Guedri et al. [23] numerically investigated the thermal flow of ternary hybrid nanofluid in the presence of Darcy–Forchheimer through stretching sheet. Bharathi et al. [24] examined EMHD hyperbolic ternary hybrid nanofluid in wedge surface with thermal radiation in the presence of Darcy–Forchheimer. Alharbi et al. [25] computationally studied the Darcy ternary hybrid nanomaterials across the extending cylinder in the presence of induction effects. Gunay et al. [26] using bvp4c method examined the blood based ternary hybrid nanomaterials in the presence Darcy–Forchheimer and slip condition through porous medium. Babu et al. [27] presented the MHD dissipative ternary hybrid nanofluid through Darcy–Forchheimer permeable channel using numerical and neural network method. Many researchers explored various models such as [28–33]. For a more comprehensive background and resent advances in MHD nan nanofluid, hybrid and ternary nanofluids, porous stretching surfaces, and radiative Casson fluid models, the interested reader is referred to the study cited in ref ([34–46]).
Recent studies have shown growing interest in magnetohydrodynamic (MHD) nanofluid flow due to their significant application in thermal engineering, energy system, and industrial processes. Hybrid and ternary nanofluid have demonstrated superior heat transfer performance compared to conventional nanofluid because of the synergistic effects of multiple nanoparticles. Moreover, porous media, radiative heat transfer, and non-Newtonian fluid characteristic such as Casson behavior play an important role in controlling momentum and thermal transport in practical engineering systems. These developments highlight the importance of investigating the combined effects of magnetic field, porous resistance, thermal radiation, and complex nanoparticle mixtures to achieve enhanced thermal efficiency and better flow control. For more comprehensive background and recent advance in MHD nanofluids, hybrid and ternary and nanofluids, porous stretching surface, and radiative Casson fluid models, the interested reader is referred to the studies in ref [47, 48]
In the present study, the flow of a Darcy–Forchheimer MHD Casson ternary hybrid nanofluid through a convergent-divergent channel is investigated. The Darcy and Forchheimer terms account for porous medium resistance and inertial drag effects, respectively, while the Casson parameters characterizes the non-Newtonian behavior of the fluid.
Research gap
Despite the extensive literature on MHD Jeffery–Hamel flow in convergent/divergent channels, most existing studies have focused on simple fluids, or hybrid nanofluids and have primarily employed conventional numerical methods such as RKF-45, ND Solve and bvp4c. the combined effects of Darcy–Forchheimer porous media, ternary nanofluids (
, magnetic field, and stretching/shrinking wall conditions have received limited attention. Furthermore, the application of the Improved residual power series method (IRPSM) to such highly nonlinear flow and heat transfer problems has been thoroughly explored. Therefore, there exsista need for an accurate and efficient semi-analytical framework to investigate the thermal and flow characteristics of ternary nanofluids under these complex physical conditions. The present study fills this gap by developing and validating an IRPSM bases solution for Darcy–Forchheimer MHD Casson ternary hybrid nanofluid model and comparing its performance with existing methods.
Novelty
The novelty of the present study lies in extending the classical Jeffery–Hamel flow model, which was previously limited to simple Newtonian or single-phase fluid configurations, to more generalized and physically realistic framework of Darcy–Forchheimer MHD Casson ternary hybrid nanofluids in convergent/divergent channels. In contrast to earlier study primly considered simplified fluid models and relied on standard numerical solvers such RKF-45, ND Solve, bvp4c, and related techniques, the present work introduces a new semi numerical analytic approach based on the Improved residual power series method (IRPSM). This method demonstrates superior accuracy, rapid convergency and computational efficiency when applied to a highly nonlinear system incorporating magnetic field effects, porous medium resistance, and ternary nanoparticles interactions. The combined influence of these physical mechanisms is examined within a unified convergent/divergent geometry.
Key novel contributions
Extension of the classical Jeffery–Hamel model to a Darcy–Forchheimer MHD Casson ternary hybrid nanofluid system in convergent/divergent channels.
Development and implementation of the Improved residual power series method (IRPSM) for solving the resulting strongly nonlinear governing equations.
Comparative validation of IRPSM against RKF-45, ND Solve and other existing numerical methods, demonstrating excellent agreement and fast convergence.
Comprehensive analysis of the effects of magnetic field, porous medium resistance, and ternary nanoparticles on velocity, temperature, skin friction and heat transfer characteristics.
Establishment of IRPSM as an efficient and reliable semi-analytic framework for complex nonlinear problems in fluid mechanics and mathematics.
Mathematical formulation
Let us consider a two-dimensional incompressible viscous Casson fluid through stretchable/shrinkable and convergent/divergent channels. The flow is assumed in source/sink at the intersection of the walls which have an angle 2α. The slippery velocity is expressed as 
where,s represents stretching/shrinking rate of the channels. The non-intersecting walls are assumed to be convergent if α < 0 and elsewhere α > 0 divergent. The cylindrical polar coordinates 
are used to investigate dynamic behavior through channels. The flow regime cane be describe in radial direction as 
. Furthermore, 
represents temperature, surface temperature, respectively. Figure 1 depicts the schematic representation of the current physical model.
Fig. 1.
Geometry of the problem
Equation of Continuity
 |
1 |
Equation of Momentum
 |
2 |
 |
3 |
Equation of Energy
 |
4 |
where 
are signified the Casson fluid factor, magnetic field strength, pressure term and heat source. Additionally, 
designates density of fluid, 
symbolized electrical conductivity, 
represents thermal conductivity while 
designates specific heat of ternary hybrid nanofluid (Table 1).
Table 1.
Thermo physical properties of nanoparticles with water as a base fluid
Boundary conditions
Following are the boundary conditions such as [44, 45]:
 |
5 |
It is assumed that, the flow is only in the radial direction so,
.
Transformed variables
 |
6 |
Thermo physical features of ternary nanofluid [46]
 |
7 |
 |
8 |
 |
9 |
 |
10 |
 |
11 |
The following equations are obtained when substitute the Eqs. (6–11) into Eqs. (1–5).
 |
12 |
 |
13 |
Boundary conditions
 |
14 |
The 
represents stretching/shrinking parameter, 
present heat source/sink parameter, 
represents Eckert number, 
are Reynolds number, 
are symbolized Prandtl number and 
shows porosity parameter. 
represented Forchheimer inertial drag, while 
is the magnetic. From the above equations it is concluded that, when α < 0 it means that, the channel is convergent otherwise divergent when α > 0.
Engineering quantities
Following are the skin friction 
and Nusselt number Nu such as:
 |
15 |
 |
16 |
 |
17 |
Discussion outcomes
In this section, we analyzed the velocity 
and temperature 
fields through significant emergent flow factors 
The mathematical investigation is performed with the help of semi-analytical method terms as Improved residual power series method (IRPSM).
Velocity profile 
In this section different physical parameters are discussed against velocity 
profile for both convergent/divergent 
channels. The behavior of alpha α against velocity 
for both convergent/divergent 
channels is examined in Fig. 2. Illustrate the influences of opening angle α versus velocity 
for both convergent/divergent 
walls. Physically, it happens that the variation in velocity with the opening angle α is attributed to change in channel geometry. An increase in α alters the flow area and wall confinement, thereby modifying the balance between inertial and viscous forces, which leads to either accelerating or decelerating of the fluid depending on the convergent or divergent nature of the channel. The investigation of Reynolds number 
compared to velocity 
in convergent/divergent 
walls are discussed in Fig. 3. The velocity 
of fluid enhanced due to increasing the Reynolds number 
for convergent α < 0 channel. Due to increase the inertial force, relatively weaker viscous resistance and decreasing the cross-sectional area increased the velocity of the fluid 
. And for divergent channels show revers behavior because the flow expansion dominates, increased resistance and reduced the fluid velocity through divergent channel. The investigation of opening angle α on 
is explored in Fig. 4. Fluid velocity 
diminished for opening angle α due to increase the pressure resistance and viscous dissipation. The investigation of magnetic parameter M against velocity 
for both 
channels are discussed in Fig. 5. The fluid velocity 
for both channels show decreasing behavior. Physically, it’s happened that an increases the magnetic parameter M strengthens the Lorentz force generated by the applied magnetic field. This force acts opposite to fluid motion, creating a resistive drag that suppresses the flow velocity 
. Consequently, the fluid experiences greater resistance, and the velocity profile 
decreases with increasing magnetic parameter. Figure 6. examine the influence of 
on 
. The fluid velocity increase by increasing the Forchheimer inertial drag parameter for convergent/divergent 
channels. Because enhances the inertial effects in the porous medium, which promotes fluid motion and increases the velocity in both channels.
Fig. 2.
Impact of alpha λ on 
Fig. 3.
Impact of Reynolds number 
on 
Fig. 4.
Impact of α on 
Fig. 5.
Impact of magnetic strength M on α
Fig. 6.
Impact of 
on 
Temperature profile 
Figure 7 illustrates the variations of Reynolds number 
against temperature profile 
for both extending/narrowing 
walls. The temperature 
of the fluid shows decreasing behavior when increasing the Reynolds number. Physically, it’s happened that enhances fluid convection and reduces the thermal boundary layer thickness, leading to a decrease in fluid temperature. Figure 8 examines the influence of magnetic parameter M on temperature profile 
for divergent/convergent 
channels. The temperature of the fluid increases due to increasing the applied magnetic field. Because magnetic field produce joule (ohmic) heating and suppresses the fluid motion enhance the thermal energy of the fluid.
Fig. 7.
Impact of Reynolds number 
on 
Fig. 8.
Impact of magnetic strength M on 
Figure 9. Demonstrate the behavior of nanoparticles on velocity 
and temperature 
. The graphical visualization shows that, for both profiles the ternary nanofluid curve lies lowest, the hybrid lies in between, and the single nanofluid show the highest for both 
and 
. The reason for fluid velocity is that by additions of multiple nanoparticles the increased effective viscosity and density the enhanced viscous resistance opposes fluid motion and suppresses the velocity field, resulting in a thicker momentum diffusion effect. For temperature profile the opposite trend is observed, where the ternary nanofluid attains the highest temperature, followed by the hybrid and single nanofluids. The inclusion of multiple nanoparticles significantly enhances the effective thermal conductivity of the fluid, enabling more efficient heat transport and greater thermal energy storage within the suspension. Furthermore, the reduction in fluid velocity weakens convective cooling, allowing more heat to accumulate in the fluid. Consequently, the thermal boundary layer becomes thicker, leading to higher temperature distributions for ternary nanofluids compared with hybrid and single nanofluid.
Fig. 9.
Impact of 
on 
and 
Validation
As shown in the table, the present numerical result with an excellent agreement with those obtained by other methods. While the IRPSM a semi-analytic method have a strong convergence approach and great accuracy compared to the other previously reported method.
The skin friction 
and Nusselt number 
for 
and 
are discussed in table (Tables 2, 3, 4 and 5) for different values of 
. The values obtained through 
and 
for ternary, hybrid nanofluids. When the values of 
rise then the drag force 
and heat transfer 
improved. Greater the porous medium opposes the fluid movement and exhibits the skin friction as a result and increased the surface drag force and improved the thermal boundary layer disruption. According to tables the achieved values through 
and 
are meaningfully high in case of ternary when associated with nanofluid and hybrid nanofluids in the presence of 
.
Table 2.
Numerical computational values of 
for Re when α = 0.4, Ec = 3, Hs = 0.05, λ = 0.5
 |
Ternary
|
Hybrid
|
Nano
|
Hybrid versusternary (%) | Nano vsternary (%) | Hybrid versusnano (%) |
|---|---|---|---|---|---|---|
| 1 | − 1.87616 | − 1.99668 | − 1.93708 | 6.42 | 3.25 | 3.08 |
| 2 | − 1.85629 | − 1.97682 | − 1.91722 | 6.49 | 3.28 | 3.11 |
| 3 | − 1.83642 | − 1.95695 | − 1.91523 | 6.56 | 4.29 | 2.18 |
Table 3.
Numerical computational values of 
for φ when α = 0.4, Ec = 3, Hs = 0.05, λ = 0.5
| φ | Ternary
|
Hybrid
|
Nano
|
Hybrid-ternary(%) | Nano-ternary(%) | Hybrid-nano(%) |
|---|---|---|---|---|---|---|
| 0 | − 1.71324 | − 1.80729 | − 1.90133 | 5.49 | 10.98 | 5.20 |
| 1 | − 1.71126 | − 1.8053 | − 1.99934 | 5.50 | 16.84 | 10.75 |
| 2 | − 1.70927 | − 1.80331 | − 1.99914 | 5.50 | 16.99 | 10.86 |
Table 4.
Numerical computational values of 
for 
when α = 0.4, c = 3, Hs = 0.05, λ = 0.5
 |
Ternary
|
Hybrid
|
Nano
|
Hybrid-ternary(%) | Nano-ternary(%) | Hybrid-nano(%) |
|---|---|---|---|---|---|---|
| 0 | − 3.56391 | − 4.36052 | − 4.15817 | 5.49 | 10.98 | 5.22 |
| 1 | − 3.16266 | − 4.95962 | − 4.757612 | 5.50 | 16.86 | 10.91 |
| 2 | − 3.76153 | − 4.55884 | − 4.35716 | 5.50 | 16.97 | 10.92 |
Table 5.
Numerical computational values of 
for φ when α = 0.4, Ec = 3, Hs = 0.05, λ = 0.5
| φ | Ternary
|
Hybrid
|
Nano
|
Hybrid-ternary(%) | Nano-ternary(%) | Hybrid-nano(%) |
|---|---|---|---|---|---|---|
| 0 | − 0.017119 | − 0.197001 | − 0.1768759 | 1050.78 | 933.24 | 10.22 |
| 1 | − 0.07708 | − 0.156961 | − 0.1500148 | 103.63 | 94.62 | 4.43 |
| 2 | − 0.03704 | − 0.116921 | − 0.1500148 | 215.77 | 304.90 | 28.32 |
Table 6. presents a comparison of the present IRPSM results with the benchmark solutions available in the literature, including RKF-45, ND-Solve and Turkyilmazoglu results. The comparison demonstrates that the proposed IRPSM successfully captures the overall solution behavior and preserves the same physical trends as the benchmark methods. Although some quantitative differences are observed, the results confirm the effectiveness and applicability of IRPSM for solving the nonlinear Darcy–Forchheimer MHD Casson ternary hybrid nanofluid flow problem.
Table 6.
The comparison demonstrates consistency in the solution behavior, confirming the applicability of the proposed IRPSM approach
| α | λ | RKF-45[31] 
|
Turkyilmazoglu [32]
|
ND-solve [33]
|
Present results(IRPSM)
|
|---|---|---|---|---|---|
| 5 | − 1 | − 3.508103 | − 3.508103 | − 2.18315 | − 1.71404 |
| − 0.5 | − 2.173044 | − 2.173044 | − 1.567314 | − 1.78996 | |
| 0 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | |
| 0.5 | − 0.361846 | − 0.361846 | 0.378643 | − 1.28355 | |
| − 5 | 1 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |
| − 1 | − 4.652169 | − 4.652169 | − 3.56741 | − 0.853396 | |
| − 0 | − 2.833951 | − 2.833951 | − 1.34567 | − 0.42357 | |
| 1 | 0.0000000 | 0.0000000 | 0.000000 | 0.000000 |
Methodology
The (IRPSM) for solving 
order Initial Value Problems and Boundary Value Problems, about the associated conditions. This method expands a power series around the starting point 
To solve boundary value problems (BVPs), we begin with assumed initial condition values and then adjust them based on the boundary values. This approach refines the solution by blending the initial assumptions with the boundary conditions, resulting in greater accuracy.
 |
18 |
whereas the initial conditions are:
 |
19 |
In Eqs. (18–19), k are known function and 
are constant respectively, where the unknown function are k to be determined. The 
power series can be written as.
 |
20 |
For the determination of 
and 
putt the value of m = 0 in Eq. (18) these equation are reduce
 |
21 |
Comparing with 
initial conditions and putting μ = 0 in Eq. (21) we get:
 |
22 |
Now take the m = 1 for calculation of 
and 
finding the 
derivative of Eq. (18) and put μ = 0 compare with 
initial condition.
 |
23 |
Take m = 2 for calculation of 
and 
finding the 
derivative of Eq. (18) and put μ = 0 compare with 
initial condition.
 |
24 |
Similarly for the 
and 
putt n = m - 1 and finding the 
derivatives compare with the 
initial conditions.
 |
25 |
According to the boundary value problems, initially we started considering the values of the unknown initial conditions. Later, we determine these values using the boundary conditions. Different values of 
and 
for n = m,m + 1.
 |
26 |
The 
truncated series for 
put n = 0,1,2,3...m - 1.
 |
27 |
Plug the values of 
and 
in Eq. (26) we get
 |
28 |
For the residuals of Eq. (18).
 |
29 |
For the 
residual function using Eq. (21) in Eq. (20) gives the following function.
 |
30 |
Differentiating Eq. (21) for μ = 0 on both sides
 |
31 |
Finding the values of 
and 
the 
truncated series are:
 |
32 |
This process can be repeated until the solution achieves the desired accuracy.
Implementation of IRPSM
In order to implement the IRPSM to solved the Eqs. (8)–(10). We define the boundary conditions as:
 |
33 |
Here α and β are missing conditions with the help of boundary condition are determined later on. Assume the series solution as:
 |
34 |
For i = 0,1,2 we have:
 |
35 |
The residual function are define as:
 |
36 |
After evaluations through MATHEMATICA 11.0 while taking and using boundary conditions we get the approximate solutions as:
 |
37 |
 |
 |
38 |
 |
39 |
 |
40 |
 |
41 |
 |
42 |
 |
43 |
 |
44 |
 |
45 |
 |
46 |
Final remarks
The ternary hybrid nanoparticles are investigated through convergent/divergent channels. The nonlinear system of ODEs is cracked by IRPSM through MATHEMATICA-11. The following key points are listed below as:
The velocity field
for stretching/shrinking parameter λ, and Reynolds number 
have similar effect for both diverging/converging 
channel.For the magnetic field parameter M and opening angle α, the fluid velocity
diminished, and due to the inertial drag (Forchheimer) parameter 
the velocity profile 
have enhanced behavior for increasing the numerical values of 
.High magnitude of Reynolds number
thermal performance of fluid 
show decreasing behavior.With increasing the energy transmission
, when values of magnetic field strength M enhanced.The nanomaterials
have essential rule in the especially in the heat transfer analyses with MHD flow problems.
Acknowledgements
Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2026R8). Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia
List of symbols
Polar coordinates
- P
Pressure
Reynolds number
- v
Kinematic viscosity
Temperature of wall
- α
Channel angle
Thermal conductivity of nanofluids
Magnetic field
- s
Stretching/shrinking rate
Dynamic viscosity
Fluid density
Inertial drag Forchheimer
Specific heat
- σ
Electrical conductivity
- \Pr
Prandtl number
- T
Temperature
- M
Magnetic parameter
Radioactive heat flux
Absorption coefficient
- λ
Stretching/Shrinking parameter
- k
Thermal conductivity
Heat source / sink
Nanoparticles volume fraction
- Ec
Eckert number
Author contributions
Weiguo Lai: formal analysis, Mohsin Ul Haq: investigations, software, Ji Li: funding, Zareen A Khan: writing orignal drafting, Subhan Ullah: supervision, edeting orignal drafting.
Funding
This research is supported by the High-Level Talent of Shandong Xiehe University, Jinan, China.
Data availability
All data generated or analyzed during this study are included in this published article.
Declarations
Ethics approval and consent to participate
Not applicable.
Consent for publication
Not applicable.
Competing interests
The authors declare no competing interests.
Footnotes
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Contributor Information
Zareen A. Khan, Email: zakhan@pnu.edu.sa
Subhan Ullah, Email: subhan.bcst@gmail.com.
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