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Analysis of chemical, ion slip, and thermal radiation effects on an unsteady magnetohydrodynamic dusty fluid flow with heat and mass transfer through a porous media between parallel plates
Bulletin of the National Research Centre volumeÂ 47, ArticleÂ number:Â 49 (2023)
Abstract
Background
Investigating the dusty fluids flow attracted substantial attention in latest years because of their widespread utility within several industrial applications, including polymer technology, gas cooling systems, combustion, petroleum industry, and transport processes. The current article is dedicated to inspect the flow with mass transfer and heat of an unsteady Magnetohydrodynamic dusty fluid. Consequently, impacts of chemical and thermal radiation, Joule, ion slip, Hall, and viscous dissipation toward heat and mass transferring and fluid flows are provided.
Results
Numerical solution of the controlling partial differential equations was performed. The temperature, velocity, and distribution of concentration for the particle and fluid phases were inspected under the influences of various physical parameters and their discussion was supplemented with diagrams.
Conclusions
The findings specify that these parameters have a significantly governed the solutions. The thermal radiation denotes efficiency enhancement of temperature distribution.
Background
The dusty fluids' heat transfer and flow studies are extremely useful in improving the operation of many applications in several industries and engineering problems, including dust collection, atmospheric fallout, powder technology, nuclear reactor, and paint spraying (Islam et al. 2022). The dusty fluid flow under the simplifying assumptions was formulated by (Saffman 1962). Hence, development and research are used to enhance the flow and heat transfer of dusty fluids attributes by using various models and situations have been previously documented (Michael and Miller 1966; Peddieson 1976; Meena et al. 2022; Safwa et al. 2022; Dey and Chutia 2022, Tavousi et al. 2023). Recently, many investigators studied the flows of particle fluids under different conditions are carried out. Attia et al. (2014) examined the fluid flow of the unsteady dusty with heat transfer in the existence of uniform injection and suction and porosity impact through parallel plates. The mathematical solutions for dualphase dusty boundarylayer nonNewtonian fluid flow through the semiinfinite vertical heated surface are obtained (Hamid et al. 2022). Mahanthesh et al. (2019a, b) investigated the nonlinear convective transport influence on nonNewtonian Dusty fluids flow through a stretched sheet. The influence of thermal radiation of squeezed dusty fluid flow with heat transfer between parallel plates was examined by Abbas et al. (2020). Chandrawat et al. (2022) inspected numerically the unsteady flow of two unmixable micropolar and dusty fluids throughout a horizontal plate. Several researchers were interested in latest developments regarding dusty nanofluids flow under different categories and conditions are highlighted with many researchers (Mishra et al. 2022; Dey and Chutia 2020; Gireesha et al. 2018). Several researchers have considered magnetohydrodynamics (MHD) flow of Newtonian/nonNewtonian fluids due to its substantial potentiality in scientific and industrial concerns (Jha and Apere 2013; Dey and Chutia 2020; Rafiq et al. 2022) The Hall influences are significant for the MHD applications. It is toward the conducting fluid is an ionized gas with and strong magnetic field (Abbas and Sayed 2017). Because the reduction in conductivity is normal to the magnetic field as a result of the electrons and ions free spiraling about the force's magnetic lines before collisions and a current is stimulated normal to both magnetic and electric fields. Bafakeeh et al. (2022) examined the Hall influences regarding the hydromagnetic oscillatory unsteady flow for nonNewtonian secondgrade fluid. The analytical solution for an unsteady dusty fluid flow throughout a circular pipe having an Ion Slip effect was studied by Abdeen et al. (2013). Alkot and Abbas (2017) reported the mathematical procedure of blood flow throughout catheterized arteries having overlapping stenosis under the hall effect. The exponential heat source and hall effects on unsteady dusty nanofluid with timedependent velocity were studied in Hussein (2023), Mahanthesh et al. (2019a, b). The impact of chemical reactions has very important because it has many industrial applications in chemical engineering, food processing, oil emulsions, and geothermal reservoirs.
Muthuraj et al. (2016) inspected the peristaltic motion of dusty fluid having mass transfer and heat through a horizontally straight channel under a chemical reaction effect. The MHD stagnation point of nanofluid Casson flow with heat transfer toward a stretched sheet under activation energy and chemical reaction effect was investigated (Khan et al. 2018). Kumar et al. (2021) explored the chemical reaction and heat source impact on the MHD flow of the blood in bifurcated permeable arteries. The chemical reaction influence on the peristaltic electroosmotic flow of Jeffrey nanofluid was investigated in the existence of the Brownian effect (Rafiq et al. 2022). Also, there are numerous researchers who investigated the problems pf the dusty flow by considering the effect of thermal radiation (Abozaid et al. 2021; Mahanthesh et al. 2021; Ghadikolaei et al. 2018; Rafiq et al. 2022).
In the current study, we are motivated to examine the flow with mass transfer and heat of an unsteady MHD dusty fluid under thermal radiation and chemical reaction effects. Viscous dissipation and Joule, ion slip, Hall, and heat generation is considered. The controlling nonlinear equations for the particles and fluid are solved mathematically by the finite difference technique. The influences of suitable nondimensional parameters on the temperature, concentration, and velocity profiles are considered and analyzed with the assistance of diagrams.
The dusty fluid flow throughout horizontal porous plates was considered. Figure 1 shows a diagram of physical geometry. While the lower plate is immobile, the upper one moves in the xdirection with an unvarying velocity \({\mathbf{u}}_{{\varvec{o}}}\). A uniform injection and suction through the plates are considered in the ydirection with constant velocity vo. Both plates are regarded as nonconductive electrically and preserved within two fixed temperatures and the species concentration.
At the lower wall \({\mathbf{T}}_{{\varvec{o}}}\),\({\mathbf{C}}_{{\varvec{o}}}\), \({\mathbf{T}}_{1}\) and \({\mathbf{C}}_{1}\) for the upper wall where \({\mathbf{T}}_{1}>{\mathbf{T}}_{{\varvec{o}}}\). In this study, a uniform magnetic field with strength \({\mathbf{B}=(0,0,\mathbf{B}}_{{\varvec{o}}})\) is applied perpendicularly toward the plates (positive ydirection). Additionally, the lack of Hall current is related to the magnetic field incidence, which likely influences fluid flow motion. The thermal equation takes into account radiation, Joule and viscous dissipations, and heat generation are taken into account in the species equation. Additionally, the flow is assumed to be noncompressible, and the two phases have constant densities, and there exists a chemical reaction in the mixture.
According to the aforementioned assumptions the basics equations are (Abbas et al. 2020).
Fluid phase.
Particle phase:
where \(\rho\), \({\rho }_{p}\) are the fluid and particles densities, v denotes the fluid velocity vector, \(\mathbf{v}=u(y,t){\varvec{i}}+vo{\varvec{j}}+w(y,t){\varvec{k}}\), \(\mathbf{p}\) the pressure gradient, \(\mu\) denotes fluid viscosity. \(KN\left(v{v}_{p}\right)\) is the force resulted from the relative motion among dust particles and fluid, whereas K represents the Stokeâ€™s drag constant which is \(\mathrm{K}=6\mathrm{\pi \mu ra}\) in case of spherical particles having radius a, N denotes the dust particles' number density per unit volume of the fluid, \({v}_{p}\) is the dust particles' velocity vector, and \({K}_{D}\) is the Darcy permeability. T denotes the fluid temperature, \({\mathrm{T}}_{p}\) is the dust temperature, \({c}_{p}\) denotes the fluid specific heat capacity at fixed volume, \({k}_{f}\) denotes the fluid thermal conductivity, \({\gamma }_{T}\) denotes the temperature relaxation time, and c_{s} denotes the particles specific heat capacity. \(\varphi\) represents the viscous dissipation, \({q}_{r}\) denotes the radiative heat flux, and Q is heat source constant. C denotes the concentration, \({D}_{B}\) denotes the diffusion coefficient, and \({k}_{c}\) is the chemical reaction parameter. J accounts for the current density, and B accounts for total magnetic induction vector. When the ion slip and Hall terms are maintained, the current density J from the general Ohm's law can be obtained by Attia et al. (2015):
where Ïƒ represents the electric conductivity and \({B}_{i}\mathrm{ and}\) Î² denote the ion slip parameter and Hall factor. Lorentz force can be represented as
where \(\mathrm{Be}=\mathrm{\sigma \beta }{B}_{o}\) denotes the Hall parameter. The heat flux according to the Rosseland approximation is manifested by:
where Ïƒâ€‰âˆ—â€‰and kâ€‰âˆ—â€‰account for the Stefanâ€“Boltzmann constant and the mean absorption coefficient, respectively. Furthermore, \({T}^{4}\) could be written using the Taylor series about \({T}_{o}\) as:
By substituting Eqs. (6â€“9) into Eqs. (1â€“5) and after some arrangements yield:
Fluid phase:
pick the following proper dimensionless forms
The nondimensional forms after dropping the hats are:
Particle phase
where \({m}_{p}\) accounts for the dust particles' average mass. The initial and boundary conditions can be obtained by:
At \(t \le 0\):
At \(t > 0\):
At \(t > 0\):
Fluid phase:
Particle phase:
where \({R}_{e}=\rho {U}_{0 }h/\mu\) is the Reynolds number, \({Ha}^{2}=\sigma {B}_{0}^{2}{h}^{2}/\mu\) denotes the Hartmann number squared, \(S={v}_{0 }/{U}_{0}\) denotes the suction parameter, \({E}_{c}={{U}_{0 }}^{2}/{C}_{p }{(T}_{2}{T}_{1})\) is the Eckert number, \(G = m_{p} \mu /\rho h^{2} K\), denotes the particle mass parameter, \(R = KNh^{2} /\mu\) denotes the concentration parameter, \(L_{o} = \rho h^{2} /\mu \gamma_{T}\) denotes the temperature relaxation time parameter, \({Q}_{s}=hQ/\rho c{u}_{o}\) is the heat source parameter and \({Q}_{r}=16{\sigma }^{*}{T}_{o}^{3}/3\rho c{u}_{o}\) is the radiation parameter, \({\gamma }_{c}=h{k}_{c}/{u}_{o}\) is the chemical reaction parameter, and \(M=h\mu /\rho {u}_{o}{k}_{D}\) is the porous medium parameter.
The dimensionless initial and boundary conditions are:
At \(t \le 0\):
At \(t > 0\):
At \(t > 0\):
Methods
The system of governing partial differential Eqs. (18)â€“(24) with the corresponding boundary conditions (25) are increasingly nonlinear and coupled, consequently, the precise solution cannot be found. Therefore, the mathematical solution to these equations can be obtained through the finite difference technique. The CrankNicolson implicit approach is employed and is accomplished via applying an average of the schemes of central divided difference at successive time points. The main definition and procedure of this technique were formerly presented in several research (Abdeen et al. 2013; Alkot and Abbas 2017).
Results
The physical quantities for engineering interest, the local skin friction (\(\tau\)), local Nusselt number (Nu), and local Sherwood number (Sh) are defined as (Pandya et al. 2017):

skin friction coefficient: \(\tau = {\left(\frac{\partial u}{\partial y}\right)}_{y=0}\)

Nusselt number: \(Nu= {\left(\frac{\partial T}{\partial y}\right)}_{y=0}\)

Sherwood number: \(Sh= {\left(\frac{\partial C}{\partial y}\right)}_{y=0}\)
To validate the accuracy of presented results and methodology, the current results are compared with the previously published available results in some special cases, which was reported by Pandya et al. (2017). We recorded the compared values that skin friction coefficient values, Nusselt number and Sherwood number for various values of some parameters impact, which are illustrated in Tables 1 and 2. Generally, it is clear that, from this comparison the results are in excellent agreement.
The current study was conducted to investigate the flow with mass transfer and heat of an unsteady MHD dusty fluid under thermal radiation, viscous dissipation, Joule, chemical, and ion slip influences between parallel plates.
The finite differences technique was adopted for acquiring the fluid/dust particles' temperature, concentration, and velocity as a function of some physical aspects. The graphical outcomes are demonstrated at Figs. 2, 3, 4, 5, 6, 7, 8, 9, 10, and 11. Calculations were accomplished for Câ€‰=â€‰â€‰âˆ’â€‰5, Râ€‰=â€‰0.5, Reâ€‰=â€‰1, Loâ€‰=â€‰0.7, Gâ€‰=â€‰0.8, \({P}_{r}\) = 1.
Discussion
The time variation profiles against the fluid/dust main velocities components u, u_{p}, secondary velocities components w, w_{p}, and temperatures T, T_{p}, and fluid concentration Î“, respectively, are depicted in Figs. 2, 3, 4, and 5 for Biâ€‰=â€‰3, Beâ€‰=â€‰3, Haâ€‰=â€‰1, Mâ€‰=â€‰1, Sâ€‰=â€‰1. In all these Figures, it is seen that the fluid/dust velocities, temperatures, and concentration increase with time till it reaches its steady state. It is found in Figs. 2, 3, and 4 that, the main velocities of dust and fluid attain steady state quicker relative to secondary velocities that attain it quicker than fluid temperature. It may be because of the fluid main velocity represents the source of the secondary velocity, and both are the sources of the fluid temperature. Moreover, it is seen that the fluid's temperature and velocity acquire steady state early relative to the particle phase that ascribes the fluid velocity as the source of the particle flow.
Figure 6 illustrates the particle and fluid temperature profiles for different thermal radiation values. It was observed that the temperature profiles are improved via rising the thermal radiation parameter values. However, Fig. 7 shows the fluid concentration progression with the time at the center of the channel (yâ€‰=â€‰0), for different chemical reaction parameter scores. It was recognized that the concentration is decreased by rising the chemical reaction parameter value. Figures 8, 9, 10, and 11 display the time progression of the fluid/dust temperature and main velocity profiles at yâ€‰=â€‰0, at different ion slip and Hall parameters values. Figures 8 and 9 demonstrate the influence of \(\mathrm{Be}\) and Bi parameters on the particle and fluid main velocity profiles. The particle and fluid main velocity profiles are recognized to increase with increasing the Be and Bi values. This can be owed to the decreasing of the effective conductivity in response to increasing Be or Bi values (\(\sigma\)/((1â€‰+â€‰\(\mathrm{BeBi}\))^{2}â€‰+â€‰Be^{2})). Furthermore, the Be and Bi parameters influences on the particle and fluid temperatures distribution are illustrated in Figs. 10 and 11. These Figures specify the rising value of \(Be\) or \(Bi,\) as the temperature of the dust and fluid decrease. A rise of Be or Bi reduces the Joule dissipation which is proportional to (\(\sigma\)/((1â€‰+â€‰\(\mathrm{BeBi}\))^{2}â€‰+â€‰Be^{2})).
Conclusions
On an unsteady Magnetohydrodynamic flow with mass transfer and heat of a dusty fluid was investigated. Thermal radiation, chemical reaction, ion slip, Hall current, Joule and viscous dissipation, and heat generation are considered in the current study. The mathematical solution of the controlling nonlinear partial differential equations was obtained and solved by the finite difference method.
Numerical findings are demonstrated subjected to different influential variables. The summary of the main outcomes is given below.

The thermal radiation parameter improved the thermal distribution.

Elevating the Hall parameter increases the velocity distribution of the fluid and dust main velocities.

The particle and fluid main velocities increase by increasing the ion slip parameter.

The concentration decreased with the increasing chemical reaction parameter value.

The particle/fluid temperatures are inversely influenced by differences in Hall and ion slip parameter.
Availability of data and materials
Not applicable.
Abbreviations
 MHD:

Magnetohydrodynamics
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MA team leader who proposed the present paper concept and derived the controlling equations. SB accomplished the analytical solution. OK performed the parametric study. WA performed the numerical solution and MS discussed the results and revised the manuscript. The manuscript was read and agreed by all authors. All authors read and approved the final manuscript.
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Abbas, W., Khaled, O., Beshir, S. et al. Analysis of chemical, ion slip, and thermal radiation effects on an unsteady magnetohydrodynamic dusty fluid flow with heat and mass transfer through a porous media between parallel plates. Bull Natl Res Cent 47, 49 (2023). https://doi.org/10.1186/s4226902301024x
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DOI: https://doi.org/10.1186/s4226902301024x