1 Introduction

Nanofluid is the next-generation fluid that contains the predominant thermo-physical properties which empower it valuable in heat transport applications, including fusion power engines, cooling systems, heat exchangers, underground water cable systems, and many chemical processes (Esfe et al., 2014; Mashayekhi et al., 2018; Bazdar et al., 2020; Hassan et al., 2022). The new progressions in nanotechnology prompt the creation of a new group of liquids with advanced heat transport characteristics. Now the question comes to mind what is nanofluid? Nanofluid is the uniform and homogeneous combination of nanometer-scale-sized solid particles (oxide, silica, polymers, metals, non-metals, and carbon nanotubes) and conventional fluids (water, oils, alcohols, etc.) (Al-Kouz et al., 2021). In order to create improved heat transfer fluids with much greater thermal conductivities from an energy-saving and environmental cleansing point of view, a noble and novel technique that calls for the mixing of nanoparticles and the base fluid was devised. Nanoparticles possess high thermo-physical assets as linked to conventional liquids that accordingly upgrade the thermal transportation characteristics of the standard liquid. In 1995, the idea of suspension of nanoparticles was firstly coined by Choi and Eastman (Choi and Eastman, 1995) to root out the issues of Maxwell's work (Maxwell, 1873). The size range of nanoparticles is limited to 0 to 0.04 vol fraction because if the size of the nanoparticles surpasses this limit, sedimentation and clogging difficulties would occur. It is essential to specify here that nanofluids are not made by just embedding the nanoparticles in the base fluid but involve a specific chemical procedure to develop a homogenous mixture with effective and stable applications. Due to their potential implementation in the motion of dust-ladder air, fluid motion, gas-cooling strategies to improve heat transport, environmental degradation, powder innovation, geothermal power, nuclear reactor refrigeration, dust compendium, soil erosion, solid propellant nozzles, fossil fuel industry, crude oil sanitization, and so on, analyses of two-phase flowing in which solid sphere-shaped molecules are dispersed in base fluids are of great interest. Various specialists engaged in these technologies have worked on researching the momentum and heat transmission properties of a dusty liquid.

In laboratories, there are two techniques for preparing nanofluid: a one-step approach and a two-step method. In the scientific field, the two-step process for preparing nanofluid is commonly used. In the two steps method, primarily the raw material is converted to powder form through the physical or chemical procedure. Secondly, the powder is mixed with the regular fluid with highly extreme magnetic force, ultrasonic agitation, or homogenizing (Mukherjee and Paria, 2013). Nanofluid has improved characteristics such as increased heat capacitance, enhanced viscosity, higher thermal conductive, and advanced thermal transmit rate as linked to the conventional liquid. Using different nanofluids, Kavusi and Toghraie (Kavusi and Toghraie, 2017) created a two-dimensional numerical model to replicate the operation of a heat pipe. Through the use of the finite volume approach, the impact of various nanofluids at various concentrations and particle sizes on the functionality of heat pipes were also investigated. These important characteristics led Toghraie et al. (Toghraie et al., 2020) to examine the impact of heat fluxing profile in a concentric annulus utilizing H₂O-Al₂O₃ nanoliquid. By taking into account the radiative impact using single- and two-phase techniques with defined wall temperature circumstances and prescribed surface heat fluxing parameters, Mostafazadeh et al. (Mostafazadeh et al., 2019) analyzed heat transference and laminar flowing of a nanoliquid in a vertical pipe. Li et al. (Li et al., 2019) explored a non-Newtonian nanoliquid's two-phase mixed convection in a porousness h-shaped hole. They took four revolving cylinders and used the Boussinesq approximation to produce mixed convective within the enclosure. Najafabadi et al. (Najafabadi et al., 2022) discussed the mixed convection flow of nanofluid through a vertical channel. In another paper, Sarkar and Das (Sarkar and Das, 2022) discussed the heat transfer in a Cross nanofluid through a horizontal cylinder in the existence of magnetism force and chemical reaction. Ali et al. (Ali et al., 2022) reported the impact of Arrhenius activation energy in the electrical conductance flowing of nano liquid over a stretching cylinder. Some other important regarding applications of nanofluids can be seen in (Das et al., 2021; Ali et al., 2021; Ali et al., 2023; Jamshed et al., 2021; Najafabadi et al., 2022; Zangooee et al., 2022).

Dust granules and base fluid are combined to create the dusty fluid. As it has numerous uses in industry, engineering, and the field of medicine, including power technology, cooling nuclear reactors, recovering crude oil, the sedimentation process, treating wastewater, the formation of raindrops, environmental pollution, fluid-particle suspension past an elongated surface has received a great deal of research attention. In a variety of technological fields, the study of dusty fluids is of tremendous interest. Fluidization (flow of fluid through packed beds), flow in rocket tubes (when the metallic and carbon fuel particle is present together), the coalescing of small droplets (the process of formation of raindrops), extraction of matter from the fluid through a centrifugal separator, polymerization, fluid droplets sprays, purification of crude oil and cooling system are the areas having great technical importance. A noteworthy investigation of the complex plasma has been done by Fortov et al. (Fortov et al., 2005). They have reviewed the main types of experimental complex plasma, and many elementary operations, like the charging of grain in various regimes, the influence of change particles upon each other, and the exchange of momentum of various species are investigated. Time-dependent electro-hydro-dynamic flowing of a dusty liquid flowing on a stretchable plate has been discussed by Ramesh et al. (Ramesh et al., 2012). The authors also took the cause of non-uniformly heat generating in their analysis. In alkaline anion exchange membranes, Grew et al. (Grew and Chiu, 2010) employed the dusty fluid model to predict hydroxyl anion thermal conduction study. Dusty fluid flow with time-dependent viscosity and conductivity between two parallel plates has been inspected by Makinde and Chinyoka (Makinde and Chinyoka, 2010). 2-D hybrid dusty fluid flow through non-Darcian material under the effect of Cattaneo-Christov thermal fluxing has been assessed by Reddy et al. (Reddy et al., 2020). Some other exciting and significant industrial applications of dusty fluid flow and thermal radiation influence can be seen in (Radhika et al., 2020; Kaneez et al., 2020; Ahmadi et al., 2020; Dadsetani et al., 2020; Safaei et al., 2020; Abuldrazzaq et al., 2020; Abu-Hamdeh et al., 2021; Khosravi et al., 2021; Safaei et al., 2021; Reddy and Ferdows, 2021).

The flowing and thermal exchange of a power-law liquid over a stretchable plate is an impressive standard problematic in rheology, with many implementations in a variety of manufacturing fields; instances incorporate chemical manufacturing polymer fluids, the coolant of an endless metal surface in a freezing immersion, crystal growing, hot rolling, exotic lubricants, and suspension solutions (Megahed, 2012; Gulzar et al., 2020; Attar et al., 2022; Faghiri et al., 2022; Alali and Megahed, 2022). Non-Newtonian materials play a significant role in many mechanical and chemical processes as well as engineering implementations. These compounds have an important use in medications, surfactants, petroleum engineering, blood, and numerous other areas. Williamson fluid, which is a type of pseudoplastic fluid, is one of the non-Newtonian fluids. Williamson originally created the Williamson liquid in his pioneering study on pseudo-plastic materials flow. He developed a modeling approach to explain the flowing of shear-thinning liquids and tested this theory with experimentation. Several more investigators have applied the Williamson liquid since then. Keeping in view the characteristics of Williamson fluid, Hashim et al. (Hashim and Khan, 2018) probed the time-dependent combined convective flowing of Williamson nanofluid. The authors considered variable conductivity and magnetic field in their study. Impact of dual chemically reactive flowing along with the influence of active energy on Williamson nanoliquid has been evaluated by Hamid et al. (Hamid and Khan, 2018). In another paper, Hamid et al (Hamid et al., 2018) investigated the stagnating point flowing of Williamson liquid created by stretchable plate together with the impact of Ohmic heat. The phenomenon of thermal radiation and viscous dissipation in Williamson liquid flowing generated results in the stretchable wall has been reported by Megahed (Megahed, 2019). Reddy et al. (Reddy et al., 2022) discussed quantitatively the thermal radiation implications of Williamson nanoliquid flow alongside a stretched plate with porosity media while accounting for velocity and thermal slippage via the Keller box method. Other concerned and significant work on Williamson nanofluid can be found in (Hamid et al., 2019; Hamid et al., 2019; Hamid et al., 2019).

We discovered that the mixed convection Williamson dusty nanoliquid flowing produced by a stretchy plate in a porousness medium has never been investigated before. To close this gap, research is done on the 2-D mixed convection flowing of Williamson dusty nanofluid on a stretchy plate in the existence of a cross-cut magnetism force and radiative flowing. The goal of the current problem is to analyze the impact of thermal radiative fluxing on the electrical conductance flowing of Williamson's dusty nanofluid through a porous medium. The controlling PDEs are converted to non-linear ODEs using suitable similarity transform variables and then solved numerically. Plots are generated and thoroughly analyzed to provide insight into the ramifications of entered dimensionless parameters.

2 Mathematical formulation

Consider combined convective flowing of WDNF initiated by stretchable plate. The surface is expanded with the rapidity of $u_w(x)=ax$ , wherein $a$ is constant. The $x-$ axis is measured via the surface axis, whereas the $y-$ axis is measured in the path of flowing (Fig. 1(a)).

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Fig. 1a

Physical model and Coordinate system.

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In the present problem, we made the following assumptions.

1. In the perpendicular direction, a transverse magnetic field of intensity $B_0$ is applied.

2. It is presumed that the plate's surface is warmed by the temperature $T_w(x)=T_{\infty }+Ax$ in which $A$ being reference temperature and concentration is $C_w(x)=C_{\infty }+A^{\ast }x$ , where $A^{\ast }$ shows the source concentricity.

3. The temperature and concentricity at the ambient are $T_{\infty }$ and $C_{\infty }$ , correspondingly.

4. The nanofluid is non-Newtonian, laminar, electrically conducting, and incompressible.

5. For steady laminar flow, the Boussinesq approximation is used.

The principal equalities for impetus, energy, and mass along with the continuity equation are given as (Ramesh et al., 2012; Makinde and Chinyoka, 2010; Kaneez et al., 2020):

3 Results and discussion

The 4th order Runge-Kutta based shooting procedure is used to solve the coupled ordinary differential Eqs. (13–17) concerning the boundary conditions (18) and (19) (Fig. 1(b)).

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Fig. 1b

Flow Chart of 4th order Runge Kutta with shooting procedure.

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For computational outcomes, we considered $0\le We\le 0.4$ , $0\le M\le 1.5$ , $0\le K\le 1.5$ , $0\le {\beta }_1\le 1.5$ , $1\le \lambda \le 4$ , ${0\le \beta }_2\le 1.5$ , $0.72\le Pr\le 7$ , $0\le R_d\le 3$ , ${0.4\le N}_b\le 1$ , $0.4\le N_t\le 1$ . Except for the varying values indicated in the individual figures, these values are kept consistent throughout the study. Table 1 and Table 2 show a comparison of the skin friction coefficient and the Nusselt number, respectively, which were found to be in excellent agreement.

Moreover, Table 3 demonstrates the influence of different factors on the frictional force coefficient, Nusselt, and Sherwood numbers.

Figs. 2(a) and 2(b) depict the influence of $\mathit{We}$ on quickness and energy profiles in both liquid and dust phases. The Weissenberg quantity is a measurable relationship between the fluid's recreation and a particular procedure time. The mounting values of the Weissenberg number cause an upsurge in the recreation on time of liquid molecules, which causes the viscosity to be more important and produces an impedance to a liquid flowing, resulting in a dwindle in liquid speed. The rapidity outline and related boundary layer thickening for the liquid phase reduce and the dust phase increases as the $\mathit{We}$ grows, however temperature boundary layer thickening increases for liquid and dust phases as $\mathit{We}$ increases.

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Fig. 2a

Variation of $f^{\text{'}}(\eta )$ and $F(\eta )$ for $\mathit{We}$ .

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Fig. 2b

Variation of $\theta \left(\eta \right)\mathrm{a}\mathrm{n}\mathrm{d}{\theta }_p(\eta )$ for $\mathit{We}$ .

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Figs. 3(a) and 3(b) demonstrate how the magnetic parameter affects the velocity and temperature profiles in together liquid and dusty phases. The graphs show that increasing the magnetic parameter diminishes the velocity field whilst increasing temperature outlines for together liquid and dusty phases. Because of the Lorentz force, which is the polar opposite of the flow when the magnetic field parameter is increased, which in turn upsurges the energy stored inside the fluid molecules and thus rises the temperature. This force tends to diminish the thickener of the quickness bounder-layer while growing the thickener of the thermal bounder-layer. As a result, the temperature profiles of a flow have risen. A similar effect is noticed for the variation of porosity parameter $K$ with rapidity and energy fields for together liquid and dusty phases, as shown in Figs. 4(a) and 4(b). This behavior is explained physically by the porous medium's inertial imparting resistance to the fluid flow mechanisms, causing the fluid to travel at a slower rate and at a lower velocity. Furthermore, as the number of dusty molecules in a liquid rose, a drag force increased, creating more impedance to a liquid flowing and, eventually, a reduction in the quickness field and a growth in an energy field. This effect is due to the physical explanation which considers the augmented pores to be the reason behind this.

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Fig. 3a

Variation of $f^{\text{'}}(\eta )$ and $F(\eta )$ for $M$ .

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Fig. 3b

Variation of $\theta \left(\eta \right)$ and ${\theta }_p(\eta )$ for $M$ .

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Fig. 4a

Variation of $f^{\text{'}}(\eta )$ and $F(\eta )$ for $K$ .

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Fig. 4b

Variation of $\theta \left(\eta \right)$ and ${\theta }_p(\eta )$ for $K$ .

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Figs. 5(a) and 5(b) portray the impact of momentum dust variable on quickness and energy outlines for liquid and dusty phases. It's worth noting that when ${\beta }_1$ increases, the rapidity of a liquid phase decreases while the speed of a dusty phase increases, as illustrated in Fig. 5(a). Since the fluid-particle phase contact is so strong, the particle phase exerts an opposing force on a liquid phase up to the molecule's rapidity and approaches a liquid swiftness. Increasing the ${\beta }_1$ value improves the temperature outline of the liquid and dusty stages, as shown in Fig. 5(b). This is consistent with the common fact that contact among a liquid and molecules stages improves flow thermal conductance. Furthermore, an increase in ${\beta }_1$ causes the internal opponent power to the flowing to developing until the rapidity of a nanofluid phase is reached by the dust phase.

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Fig. 5a

Variation of $f^{\text{'}}(\eta )$ and $F(\eta )$ for ${\beta }_1$ .

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Fig. 5b

Variation of $\theta \left(\eta \right)$ and ${\theta }_p(\eta )$ for ${\beta }_1$ .

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Figs. 6(a) and 6(b) illustrate the variation of $\lambda$ on rapidity and energy outlines of liquid and dusty stages. It has been observed that the liquid and dusty stage velocities boosted with growth $\lambda$ and the liquid and dusty phase temperatures decreased. Because of greater $\lambda$ , buoyancy force takes precedence over viscous force. Following this, the buoyancy force parameter improves liquid flowing, which manages to boost fluid velocity and the corresponding boundary layer thickening. Further, this might happen when high buoyant values obtained enable the liquid to waste thermal and kinetic energy from dusty particle collisions, leading both numbers to drop at the top edge. Moreover, the mixed convection parameters combine the flow and heat fields. This represents the ability of the liquid to expand thermally. This is the underlying concept of liquid-in-glass thermometers. A rise in temperature causes the liquid to expand, causing it to ascend the glass. Figs. 7(a) and 7(b) exhibit the effect of ${\beta }_2$ on the rapidity and temperature outlines for together liquid and dusty phases. Fig. 7(a) shows that as ${\beta }_2$ value increases, the rapidity of the fluid and dusty phases decline. On the other hand, the temperature of the fluid phase tends to decrease as the ${\beta }_2$ increases and the temperature of the dusty phase rise as demonstrated in Fig. 7(b). Result the dusty elements generate frictional force in the liquid, slowing it down. This might be owing to enhanced contact between the nanoparticle phase and the dust particle phase, which allows for more conduction heat transfer in the dusty phase. Also, up until the dust phase reaches the fluid phase's rapidity, the fluid phase and flow are opposed internally as the liquid particles interface parameter increases. Moreover, this may happen because the liquid and particle phases interact strongly, leading to the particle phase developing an opposing force to the liquid phase until the particle speed reaches the liquid rapidity. Figs. 8(a) and 8(b) show the effect of $\mathit{Pr}$ on the rapidity and temperature outlines of both the liquid and dust phases. The relationship between momentum and thermal conductivity is expressed in Prandtl number. The increase in $\mathit{Pr}$ has a greater effect on the decrease in thermal bounder-layer thickener than on the impetus bounder-layer thickener. In a heat transport situation, the Prandtl number is utilized to govern the relative impetus and thickening of the temperature bounder-layer. The rate of thermal diffusivity slows as $\mathit{Pr}$ rises, resulting in a decline in thermal bounder- layer thickener. Physically, the fluid with a lesser Prandtl quantity has greater thermal conductance and a bigger thermal bounder-layer structure, allowing heat to permeate from the plate more quickly than the liquid with a greater Prandtl quantity, which has a thinner boundary layer.

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Fig. 6a

Variations of $f^{\text{'}}(\eta )$ and $F(\eta )$ for $\lambda$ .

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Fig. 6b

Variations of $\theta \left(\eta \right){\theta }_p(\eta )$ for $\lambda$ .

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Fig. 7a

Variations of $f^{\text{'}}(\eta )$ and $F(\eta )$ for ${\beta }_2$ .

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Fig. 7b

Variation of $\theta \left(\eta \right)$ and ${\theta }_p(\eta )$ for ${\beta }_2$ .

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Fig. 8a

Variation of $f^{\text{'}}(\eta )$ and $F(\eta )$ for $\mathit{Pr}$ .

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Fig. 8b

Variation of $\theta \left(\eta \right)$ and ${\theta }_p(\eta )$ for $\mathit{Pr}$ .

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Figs. 9(a) and 9(b) are plotted to visualize the appearance of $R_d$ on quickness and temperature outlines of liquid and dusty phases. Interestingly, the quickness and energy fields of both liquid and dusty stages rise with growing amounts of $R_d$ . The following behaviour can be explained physically by the fact that large warming rates of solely viscous nanofluids can also be predicted using the thermal radiation phenomenon. Another possible reason could be because the Stefan–Boltzmann value dominates, and the mean absorbing factor has less influence in the Rosseland approximation. Also, higher radiation parameter values ​​increase the heating of the fluid, improving the energy outline and the thickener of the thermal bounder-layer between the liquid and dusty stages. Radiation enhances a conductive effect and tends to expand the thermal bounder-layer. This will improve the rapidity and temperature at any point away from the sheet metal surface. The excess radiation upsurges the turbulence of the fluid particles' movement, including dust particles as well, which surges internal collisions and increases the stored energy, raising the fluid's velocity and improving temperatures.

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Fig. 9a

Variation of $f^{\text{'}}(\eta )$ and $F(\eta )$ for $R_d$ .

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Fig. 9b

Variations of $\theta \left(\eta \right)$ and ${\theta }_p(\eta )$ for $R_d$ .

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Figs. 10(a) and 10(b) elucidate the variation of $N_b$ on rapidity and temperature of the liquid and dusty phases. It can be seen that rapidity and temperature outlines for both liquid and dusty phases enhance with increasing values of $N_b$ . A high Brownian motion parameter value represents that nanoparticles and base fluid particles collide frequently. As a result, there is an enhancement in heat transference qualities within the system, resulting in enhanced temperature and velocity fields. Figs. 11(a) and 11(b) describe the variation of rapidity and temperature profiles for increasing values of $N_t$ for together liquid and dusty phases. The thermophoretic force between the particles grows as the temperature difference in the system increases, causing the dusty nanofluid to heat up and the temperature of the nanofluid and dusty nanofluid to rise. Furthermore, this force can move suspended particles from a warm to a cold state, boosting the nanoparticle's strength in the fluid.

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Fig. 10a

Variations of $f^{\text{'}}(\eta )$ and $F(\eta )$ for $N_b$ .

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Fig. 10b

Variation of $\theta \left(\eta \right)$ and ${\theta }_p(\eta )$ for $N_b$ .

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Fig. 11a

Variation of $f^{\text{'}}(\eta )$ and $F(\eta )$ for $N_t$ .

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Fig. 11b

Variation of $\theta \left(\eta \right)\mathrm{a}\mathrm{n}\mathrm{d}{\theta }_p(\eta )$ for $N_t$ .

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4 Conclusion

The flowing of a Williamson-dusty nanoliquid across a stretched plate has been studied using magnetohydrodynamic (MHD) mixed convection. The similarity solutions are achieved by applying the Runge Kutta Fehlberg technique and Maple software's algebraic code. The following are significant findings:

• The increment in $\mathit{We}$ diminishes the velocity of the nanofluid while the dusty phase improved.

• The temperature of both the nanoliquid and the dusty phase are enlarged with the upsurge in $\mathit{We}$ .

• A rise in $M$ increased the fluid temperature while decreasing the fluid velocity.

• The influence of $K$ on the temperature field of the dust phase is more profound than the rapidity of the dusty phase.

• The improvement of ${\beta }_1$ cause an augmentation in the quickness and energy of the dust phase.

• ${\beta }_2$ has an opposing influence on the temperature field of both fluid and dust phases.

• Increasing $\lambda$ has a significant impact on the temperature field of both phases.

• Velocity and thermal fields show the same behavior of an improvement with the growing amounts of $R_d$ .

• An expansion in the $N_t$ leads to a growth in both the temperature and velocity of both phases.

• Higher values of ${\beta }_1$ and ${\beta }_2$ were shown to cause a drop in the heat transfer coefficient.

In the future, the existing method might be used to a number of physical and technical obstacles (Jamshed and Aziz, 2018; Jamshed, 2021; Jamshed and Nisar, 2021; Jamshed et al., 2021; Jamshed, 2021; Jamshed et al., 2022; Islam et al., 2022; Pasha et al., 2022; Rasool et al., 2022; Rasool et al., 2022; Rasool et al., 2022).

CRediT authorship contribution statement

Imran Ullah: Conceptualization. Farhad Ali: Formal analysis. Sharena Mohamad Isa: Methodology. Saqib Murtaza: Software. Wasim Jamshed: Validation, Investigation. Mohamed R. Eid: Investigation, Validation, Writing - original draft, Writing - review & editing. Ayesha Amjad: Writing – review & editing.