Document Type : Full Lenght Research Article
Authors
Kohima Science College, Jotsoma, Nagaland 797002, India
Abstract
Keywords
Main Subjects
Semnan University 
Journal of Heat and Mass Transfer Research Journal homepage: http://jhmtr.semnan.ac.ir 

Effects of Radiation Absorption, Soret and Dufour on Unsteady MHD Mixed Convective Flow past a Vertical Permeable Plate with Slip Condition and Viscous Dissipation
Temjennaro Jamir [*] , Hemanta Konwar
Kohima Science College, Jotsoma, Nagaland 797002, India.
Paper INFO 

ABSTRACT 
Paper history: Received: 20221018 Revised: 20230128 Accepted: 20230203 
Objective: The objective of current study is to discuss the effects of the Soret and Dufour with radiation absorption, applied heat source and viscous dissipation on an unsteady MHD mixed convective flow with velocity slip condition across a semiinfinite vertical permeable plate in porous medium. Method: A similarity transformation is used to turn the governing partial differential equations with proper boundary conditions into coupled, nonlinear ordinary differential equations with variable coefficients. The inbuilt MATLAB solver bvp4c is used to generate numerical solutions. Findings: The effects on momentum, thermal and solutal boundary layers for various parametric values are graphically depicted. Skin friction, Nusselt number and Sherwood number are all tabulated and discussed in detail. An improvement in radiation absorption corresponds to enhancement of the heat transfer rate up to 59% while leading to a decline in mass transfer rate around 20%. The momentum, thermal and solutal boundary layers are all found to be boosted when the Soret effect is higher. For higher estimation of slip effect, the skin friction is found to decay around 23%. Also, as more time goes by the thermal and concentration boundary layers are enhanced. Novelty: Results obtained in this studied has also been compared and verified with available scientific literature and is found to be in good agreement, which establishes assurance in the numerical results reported in the study. 



Keywords: Radiation absorption; Porous medium; Slip flow; Soret; Dufour; Viscous Dissipation. 


© 2022 Published by Semnan University Press. All rights reserved. 
Navier explored a boundary condition of fluid slip at a solid surface such that the solid surface velocity is proportional to the shear stress at the surface in the history of fluid flow via channels, that is , where the slip coefficient is h and the velocity along the xaxis is [1]. The flow regime is known as the slip flow regime, and the significance of its consequences in applications such as micro channel and mechanical device lubrication, where a little layer of lubricant is applied to the surfaces to prevent them from rubbing against one another, or when the surfaces are coated with a particular type of coating to lessen friction. Furthermore, in polymer melts, which frequently display tiny wall slip, make the insufficiency of the noslip criterion abundantly clear. The synchronisation of the velocities and stresses is one of the boundary requirements that must be met at the contact between a porous material and fluid layer.
Beavers and Joseph [2] analysed fluid flow at the boundary between a porous medium and fluid layer in an experimental research and established a slip boundary conditions at the porous interface.
Pal and Talukdar [3] also contributed a numerical analysis on the influence of ohmic heating and magnetic field for a mixed convective fluid flow problem taking into consideration the boundary slip effect. A numerical study of the velocity and thermal wall slip on MHD boundary layer viscous flow over a nonlinearlystretching sheet was also considered by Ramya et al. [4].
Numerous research has also focused on issues with heat and mass transfer in radiative convective flow. Because of its farreaching contributions in the realm of engineering, such as its use in MHD bearings and MHD pumps. Surface heat transfer is significantly impacted by thermal radiation. Researchers' attention has recently been drawn to thermal radiation as a mode of energy transfer, and the necessity of including radiative transfer in these processes has been highlighted by recent developments in hypersonic flights, power plants for interplanetary flight, missile reentry rocket combustion chambers, and gas cooled nuclear reactors. The radiative influence on heat transmission in a vertical porous material was investigated by Raptis [5], Hossain and Takhar [6], Manjula and Muthucumaraswamy [7]. Numerous studies on the effect of radiation on fluid flow have been conducted over the years. Because of very significant contributions to humanity, such as nuclear power plants, studies on the effect of radiation absorption have recently become a trend. Ibrahim et al. [8] conducted one of the early studies on radiation absorption on MHD mixed convective flow for a semiinfinite vertical plate. Sreedevi et al. [9] studied this effect of radiation absorption in the present of hall current for a stretching sheet. We also find the work by Aly et al. [10] who accounted unsteady convective flow with radiation effect for a vertical plate embedded in porous medium, Matta et al. [11] who stressed on the unsteady MHD flow with chemical and radiation effects for a porous plate, also fluid flow past a stretching sheet with heat source and temperature dependent fluid properties was analysed by Hemanta [12]. Khan et al. [13] considered the entropy analysis for hydromagnetic radiative flow of ternary nanofluid by an exponentially stretching surface subject to entropy generation. Some recent studies based on radiative effects are also shown in Refs. [1418].
The rate of heat generation in the boundary influences the rate of thermal distribution in flow. This is evident in a variety of manufacturing applications, including fire modelling, the fabrication of electronic chips, etc. Surface temperature and flow velocity both affect how heatwaves spread. As a result, when thermal radiation occurs, a significant difference in temperature is present somewhere in the system.
Because of this, heat generation and absorption for many systems are not constant and rely on the environment and temperature of the system. As found in studies conducted by Nemati et al. [19] who accounted periodic magnetic field effect with heat generation/absorption for a NonNewtonian fluid, Elsayed et al. [20] presents the effects of heat generation absorption on boundary layer flow of Nano fluid for a stretching cylinder, Khan et al. [21] who examined the variable heat source for unsteady stagnationpoint flow of magnetized OldroydB fluid. The impact of ohmic heating for a double diffusive Nano fluid flow for stretching cylinder is considered in the study conducted by Yasir et al. [22]. Also as found in the investigation conducted by Sheikholeslami [23] for thermal solar system, the thermal performance was found to be incremented upon elevating heat absorption.
With the increasing need for advancement, betterment and efficient technology there arises a need for advancement in research approaches and hence making every small progress an important contribution to research and advancement. Due to the temperature and mass differences in a medium there arises a mass and heat flux which are also known as the Soret and Dufour effect respectively, which are important mechanism in the transport phenomenon; it has it varied applications in vapour deposition process for optical fibre fabrication, in fluid flow fractionation, it is used to separate distinct polymers. Okuyade et al. [24] examined the effects of Soret and Dufour for an unsteady convective flow with constant suction. The Dufour effect on unsteady MHD flow past a vertical plate embedded in porous medium with ramped temperature is addressed by Sarma and Ahmed [25]. The Soret effect for a steady MHD fluid flow through a vertical porous plate with aligned magnetic field was presented by Mopuri et al. [26]. We also find in the study of Sowmiya and Kumar [27] who considered the Dufour effect for MHD mixed convective flow with bouyancy effect. Some relevant advancements about Soret and Dufour effects are also highlighted in Refs. [2832].
The literature review cited above confirms that no attempt has been made to consider the impact of Soret and Dufour under the influence of radiation of absorption, heat source, viscous dissipation, chemical and thermal radiation, with the velocity slip condition for a semiinfinite vertical permeable plate under the presence of pressure gradient. Few researchers even worked on unsteady flow problem with slip effect. Here our main objective is to scrutinize the combined effect of radiation absorption, Soret and Dufour effects with slip paramter for unsteady fluid flow. A comparison has been made with the work presented by Pal and Talukdar [3] and thus the results found in this study are thus verified accordingly. For the purpose of graphical and tabular data, the MATLAB’s builtin solver bvp4c technique will be implemented. We prioritise the following research questions in relation to the flow problem under consideration.
What impact does the radiation absorption parameter have on the thermal and solutal boundary layers, fluid momentum, for mixed convective unsteady MHD heatabsorbing fluid flow?
When slip flow is taken into account in the mixed convective flow system for the vertical permeable plate in presence of MHD and radiation absorption, how are the skin friction, heat, and mass transfer rates are affected?
What changes in the flow field do the Soret and Dufour effects have on the velocity, heat, and mass transfer?
What effects does the time parameter have on the fluid flow over a vertical permeable plate with MHD and slip flow in terms of velocity, heat transfer, and mass transfer?
We address the unsteady twodimensional mixed convective flow of an incompressible, viscous, electrically conducting, and heatabsorbing fluid through a semiinfinite vertical permeable plate submerged in a uniform porous medium with a slip boundary condition at the interface of the porous and fluid layers.
Figure 1. Geometry of the problem
In the presence of radiation and concentration buoyancy effects in the direction of the axis, a uniform transverse magnetic field of magnitude is applied. The induced magnetic field and the Hall Effect are thought to be insignificant due to the relatively low transversely applied magnetic field and magnetic Reynolds number. It is presumptively assumed that no voltage is applied, which denotes that there is no electric field. All physical variables are independent of since the motion is twodimensional and the length of the plate is sufficient. The temperature and concentration on the wall are kept constant at and , respectively, which are greater than and of the ambient fluid. Additionally, it is assumed that the diffusing species and the fluid undergo a homogenous firstorder chemical reaction with rate constant .
The governing equations that describe the physical conditions under these hypotheses, in addition to Boussinesq's approximation, are characterized in vector form as: (Ramya et al. [4], Kebede et al. [18], Sarma and Ahmed [25]).
Conservation of Mass
(1)
Conservation of Momentum
(2)
Conservation of Energy

(3) 
Conservation of Species
(4)
We now make the standard boundarylayer approximation, based on a scale analysis, and write the governing equations(Pal and Talukdar [3], Ibrahim et al. [8])

(5) 

(6) 

(7) 

(8) 
The radiative heat flux can be represented as follows using the Roseland approximation (Raptis^{ }[5]):

(9) 
It is considered that the temperature variation within the flow is thought to be minimal enough that can be linearized by using the Taylors series to expand around and disregarding terms of higher order. Hence

(10) 
The appropriate boundary conditions for the said fluid problem are: (Pal and Talukdar [3])

(11) 
, as 
(12) 
Suction velocity is either a timedependent function or a constant, as shown by Eq. (5).

(13) 
Here, suction towards the plate is shown by the negative sign. Outside the boundary layer (Kim [33]) Eq. (6) gives:

(14) 
The governing equations and their accompanying boundary conditions must now be converted. The following nondimensional quantities are displayed in a dimensionless manner.

(15) 
Eq. (6) to Eq. (8) can be converted to the dimensionless version using the above dimensionless variables:

(16) 

(17) 

(18) 
The boundary conditions update as:

(19) 
, as 
(20) 
It is impossible to solve the partial differential Eq. (16) –Eq. (18) with boundary conditions Eq. (19) and Eq. (20) in closed form. It may, however, be solved if these equations are reduced to a set of dimensionless ordinary differential equations by modelling the (velocity), (temperature), and (concentration) as:

(21) 

(22) 

(23) 
Substitute Eq. (21)  Eq. (23) into Eq. (16) Eq. (20), equalize the harmonic and nonharmonic components, considering only the first order terms and simplifying to yield the following sets of equations for and .

(24) 

(25) 

(26) 

(27) 

(28) 

(29) 
Here the prime represents regular differentiation with respect to .
The associated boundary conditions are:

(30) 
as 
(31) 
Now using bvp4c, a builtin solver in MATLAB(Shampine et al.[34]) the highly coupled nonlinear ordinary differential Eq. (24) – Eq. (29) along with the boundary condition Eq. (30) and Eq. (31) are solved by setting:
,

(32) 
By using the equation Eq. (32) into Eq. (24) to Eq. (29) the following system of first order ordinary differential equations, Eq. (33) –Eq. (38) may be derived, along with boundary conditions equations Eq. (39) and Eq. (40):

(33) 

(34) 

(35) 

(36) 

(37) 

(38) 
Boundary Conditions:

(39) 
as 
(40) 
The most essential measures for the current condition are the skinfriction coefficient , Nusselt number and Sherwood number . (Pal and Talukdar [3])

(41) 

(42) 

(43) 
We plot velocity, temperature, and concentration profiles to determine the physical significance of the problem and display the tabular statistics for for distinct values of parameters involved. For the sake of this study, we will use the following default values for computation:
The present calculated numerical values for the skinfriction coefficient, Nusselt number, and Sherwood number that are shown in Table 1, Table 2 and Table 3 and are in excellent agreement with the computed results of Pal and Talukdar [3] which validates the results and data discussed below.
Figure 23 depict the actions of magnetic parameter and permeability parameter on velocity within the boundary layer. Because of the Lorentz force that acts as the resistive force due to magnetic field reduces the thickness of the momentum boundary layer. We find that upon incrementing , the resistive force of the porous medium decreases that aids the fluid to move faster hence incrementing fluid velocity. The result is applicable in reservoir engineering for enhanced oil recovery. Table 4 shows a decay about 8% in inside , whereas undergoes a growth upto 0.4% within the range .
Figures 45 show the impact of Prandtl number taken as 0.71 (air), 1.00 (electrolytic solution), 7.00 (water), 11.40 (water at 4^{0}C) on velocity and temperature respectively. As grows, the thickness of the thermal boundary layer decreases, just as a higher value of makes the fluid more viscous and decreases the thickness of the momentum boundary layer; slowing the fluid velocity. Thus, as a result, average temperature and velocity are lower within the boundary layer. This is because fluids with a greater Prandtl number have higher viscosity, which reduces velocity. In addition, fluids with smaller values of are highly conductive. Table 4 shows that for values of within the range , undergoes negative growth about 7%, while is drastically reduced about 140% whereas we see a positive growth upto 50% in .
Figure 67 show the impact of Eckert number on velocity and temperature. In fluid flow is a quantity that establishes the kinetic energy of a flow's relative relevance in a heat transfer scenario. The expression is simplified to a balance of convection and conduction when the terms in the energy expressions represent, making it possible to disregard the effects of pressure fluctuations, dissipation, and body forces on energy balance. As a result, a rise in shows that dissipative heat or additional kinetic energy is being accumulated in the fluid particles as a result of frictional heating, which improves the fluid's total velocity and temperature. It is also evident from Table 4 that and is varied by a growth upto 0.1% and 2.2% respectively while is effected by a decay around 1% within the range .
Consequently, upon varying values of radiation absorption , both velocity and temperature appear to be increasing, which is due to the buoyancy force generated by the absorption of radiation parameter that subsequently increases the flow rate and temperature profile as seen in Figures 89. Taking in account Table 4 the influence of on Nu is noted to differ by a positive growth upto 8% and 59% respectively while to be undergoing a decay around 20% within .
The temperature response is plotted against in Figure 10, and it is revealed that the thermal boundary layer is thickened upon improving this is due to the fact that an increase in denotes a general increase in the concentration gradient over temperature gradient. The temperature field therefore rises as the concentration gradient increases. Consequently, we observe from Table 4 that is improved around 11%, while reduces around 4% for . The impact of both and , as seen in Figure 11, is to increase the fluid flow velocity throughout the boundary layer. However, a distinct velocity overshoot exists near the plate, and thereby the profile falls to attain its boundary condition. We also observe that is improved upto 1.5% and 5.6% for and respectively.
The effects of Soret on temperature and concentration are seen in Figures 1213. As seen in figure the fluid flow temperature and species concentration are found to rise as the Soret effect improves, which is due to mass transfer from lower to greater solute concentration driven by the temperature gradient hence higher Soret effect indicate stronger fluid molecule diffusivity. From Table 4 it is interesting to note that in and both get augmented about 5% and 45% respectively with ranging within .
Figures 1415 demonstrate velocity and concentration profiles for varied Schmidt number values of 0.22 (hydrogen), 0.30 (helium), 0.60 (water vapour), and 0.78 (ammonia). The velocity profile lowers because of the heavier diffusing species. It is also worth noting that with lower value the peak value (at ) is relatively more far from the plate than the highest value (at ) for larger , and the concentration distribution was established to fall faster as increased, resulting in the velocity and concentration boundary layers being lowered at the same time. Physically stated, a decrease in molecular diffusion results from an increase in . As a result, the concentration of the species is higher for low values and decrease for high values. Furthermore, as presented in Table 4, as the values of vary from it can be stated that and values undergo decay around 11%, 3% and 181% respectively.
As we increase the values of chemical reaction , we see in Figures 1618 a rise in the velocity and concentration profile, which finally leads to a simultaneous depletion in the velocity and the concentration boundary layer thickness. On the graph, upon increasing the value of the chemical reaction has a noticeable impact on the concentration boundary layer. The figure makes it abundantly obvious that the concentration of species that is slightly larger than one at the initial point that gradually falls to satisfy the boundary condition at
. Additionally, it has been shown that incrementing the value of results in a drop in the species concentration in the boundary layer. In addition, we notice a fall in the temperature profile, it is clear because when the increases, the thermal boundary layer loses energy, causing the fluid temperature to drop. Furthermore, from Table 4, as the values of vary from it can be concluded that and values are decremented about 7.1%, 7% and 64% respectively.
The effect of slip parameter on the momentum boundary layer is seen in Figure19. Rising causes the velocity gradient near the surface to increase, before decrementing to its free stream value. Table 4 also shows that within the range , the experiences a decay around 23%.
It is also worth noting that when time increases, each of these factors have opposing effects on velocity, temperature and concentration as seen in Figures 20 22. The velocity gradually approaches its greatest value near the plate, and then the profile falls till it reaches the bare minimum at . All times are relevant to this circumstance. It is also noticed that the peak value grows with time. In the flow area, temperature rises with passing time. This occurs as a result of the fluid capacity to store heat energy as time goes on. The solutal boundary layer species concentration rises with time, increasing the layer thickness in the process.

Figure 2. Velocity for varying Magnetic parameter 

Figure 3. Velocity for varying Permeability parameter K 

Figure 4. Velocity for varying Prandtl number Pr 

Figure 5. Temperature for varying Prandtl number Pr 

Figure 6. Velocity for varying Eckert number Ec 

Figure 7. Temperature for varying Eckert number Ec 

Figure 8. Velocity for varying absorption 

Figure 9. Temperature for varying absorption 

Figure 10. Temperature for varying Dufour Du 

Figure 11. Velocity for varying Soret Sr 

Figure 12. Temperature for varying Soret number Sr 

Figure 13. Concentration for varying Soret number Sr 

Figure 14. Velocity for varying Schmidt number Sc 

Figure 15. Concentration for varying Schmidt number Sc 

Figure 16. Velocity for varying Chemical reaction Kr 

Figure 17. Temperature for varying Chemical reaction Kr 

Figure 18. Concentration for varying Chemical reaction Kr 

Figure 19. Velocity for varying slip parameter 

Figure 20. Velocity for varying time t 

Figure 21. Temperature for varying time t 

Figure 22. Concentration for varying time t 
Table 1. Effects of on for reference values

Pal and Talukdar [3] 
Present 








0.0 
6.4047 
1.3822 
0.9087 
6.4047 
1.3822 
0.9088 
0.1 
5.3082 
1.3822 
0.9087 
5.3082 
1.3822 
0.9088 
0.3 
3.9540 
1.3822 
0.9087 
3.9540 
1.3822 
0.9088 
0.5 
3.1503 
1.3822 
0.9087 
3.1503 
1.3822 
0.9088 
Table 2. Effects of on for reference values

Pal and Talukdar [3] 
Present 








1.0 
4.1838 
0.9254 
0.9087 
4.1838 
0.9255 
0.9088 
2.0 
3.9540 
1.3822 
0.9087 
3.9540 
1.3822 
0.9088 
3.0 
3.8015 
1.7436 
0.9087 
3.8015 
1.7437 
0.9088 
4.0 
3.6901 
2.0488 
0.9087 
3.6901 
2.0489 
0.9088 
Table 3. Effects of on for reference values

Pal and Talukdar [3] 
Present 








0.00 
4.0441 
1.3400 
0.8098 
4.0439 
1.3401 
0.8099 
0.50 
3.7512 
1.4825 
1.1864 
3.7513 
1.4825 
1.1864 
0.75 
3.6744 
1.5226 
1.3178 
3.6744 
1.5227 
1.3178 
1.00 
3.6149 
1.5546 
1.4325 
3.6149 
1.5546 
1.4326 
Table 4: Effects of on
for reference values

















4 
2 
2 
5 
0.71 
2 
1 
2 
0.01 
0.2 
0.60 
0.5 
1 
0.2 
3.5876 
1.1492 
0.9396 
5 













3.9266 
1.1470 
0.9402 
6 













4.2658 
1.1446 
0.9410 

3 












3.9499 
1.1468 
0.9403 

4 












4.3123 
1.1441 
0.9411 


3 











3.4028 
1.1511 
0.9390 


4 











3.2975 
1.1523 
0.9387 



6 










3.5961 
1.1492 
0.9396 



7 










3.6022 
1.1491 
0.9396 




1 









3.5158 
1.3737 
0.8760 




3 









3.3234 
2.7564 
0.4704 





3 








3.7331 
0.8057 
1.0347 





4 








3.8726 
0.4742 
1.1264 






2 







3.4746 
1.4491 
0.8557 






3 







3.3902 
1.7060 
0.7830 







3 






3.4746 
1.4491 
0.8557 







4 






3.3902 
1.7060 
0.7830 








0.02 





3.5891 
1.1367 
0.9433 








0.03 





3.5906 
1.1241 
0.9470 









0.4 




3.6139 
1.0892 
0.9651 









0.6 




3.6407 
1.0246 
0.9740 










1 



3.4272 
1.2000 
1.2911 










2.62 



3.2959 
1.2194 
1.7998 











1 


3.6922 
1.1160 
0.7195 











1.5 


3.7875 
1.0870 
0.5106 












2 

3.4277 
1.2010 
1.2773 












3 

3.3332 
1.2262 
1.5412 













0.3 
3.1279 
1.1526 
0.9386 













0.4 
2.7726 
1.1548 
0.9380 
Conclusion
The following interpretations of the flow field may be formed for varying values of parameters associated in the flow field:
It is intended that the physics of flow across the permeable vertical plate with velocity slip may be used as the foundation for various engineering and scientific applications using our current model. Additionally, it can serve as inspiration for next experiments, which are now lacking.
In subsequent studies, we will use the proper velocity slip notion to further generalize comparisons to various body forms.
Nomenclature

Mean absorption coefficient 

Real positive constant 

Magnetic field flux 

Magnetic field strength 

Species concentration 

A scaled concentration 

Skin friction 

Ambient fluid's species concentration 

Wall concentration 

Specific heat at constant pressure 

Concentration susceptibility 

Molecular diffusivity coefficient 

Dufour number 

Eckert number 

Solutal Grashof number 

Thermal Grashof number 

Acceleration due to gravity 

Dimensional slip parameter 

Nondimensional slip parameter 

Current density 

thermal conductivity 

Permeability of porous medium 

Nondimensional permeability 

Dimensional chemical reaction 

Nondimensional chemical reaction 

Thermal diffusion ratio 

Magnetic parameter 

Nusselt number 

Fluid pressure 

Prandtl number 

Nondimensional heat source parameter 

Dimensional heat absorption coefficient 

Dimensional radiation of absorption 

Radiative heat flux 

Nondimensional radiation of absorption 

Radiation parameter 

Schmidt number 

Sherwood number 

Soret number 

Dimensional fluid temperature 

A scaled temperature 

Ambient fluid temperature 

Mean fluid temperature 

Dimensional time 

Nondimensional time 

A scaled velocity 

Dimensional velocities direction 

Characteristic velocity 

Non dimensional free stream 

Dimensional free stream velocity 

Dimensional velocities direction 

Scale of suction velocity 

Flow velocity vector 

Dimensional distances along the plate 

Dimensional distances perpendicular to the plate 
Greek symbols 


Concentration expansion coefficients 

Thermal expansion coefficients 

Fluid thermal diffusivity 

Fluid density 

Electrical conductivity of fluid 

Kinematic viscosity 

Dynamic viscosity 

A scaled coordinate 

StefanBoltzmann constant 

A small positive constant 

Frequency of oscillation 

A scaled frequency 

Differential operator 
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[*]Corresponding Author: Temjennaro Jamir
Email: temjennaro@kscj.ac.in