Entropy generation analysis for magnetohydrodynamic flow of chemically reactive fluid due to an accelerated plate

Background The mixed convection flow of viscous fluid due to an oscillating plate is inspected. The external heating effects and chemical reaction assessment are predicted. Moreover, the flow applications of the entropy generation phenomenon are claimed. Results The dimensionless system is expressed in partial differential forms, which are analytically treated with the Laplace scheme. The physical aspects of the flow model are graphically observed. The optimized phenomenon is focused on flow parameters. The results for the Bejan number are also presented. The dynamic of heat transfer and entropy generation phenomenon is observed with applications of Bejan number. Conclusions It is claimed that an enhancement of entropy generation phenomenon is noticed due to heat and mass Grashof coefficients. The Bejan number declined due to mass Grashof number. Furthermore, the velocity profile boosted due to Grashof constant.

The mixed convection phenomenon is associated with the fluid flow subject to interaction of forced convection and natural convection.The forced convection is endorsed by external force, while buoyancy forces are subject to the natural convective flow.The role of mixed convection is commonly noted in the different fluid systems as well as industrial processes.A novel role of this phenomenon is also attributed to engineering problems.Basak et al. [1] examined the evaluation in heating phenomenon based on cavity flow with mixed convection effects.Khanafer and Aithal [2] executed analysis for the cylinder flow via significance of mixed convection contribution.Huang et al. [3] explored the mixed convection analysis with convergent channel.Maughan et al. [4] attributed the experimental approach for suggesting the expression for mixed convection in inclined channel.Ahmed et al. [5] suggested the double-driven flow of metallic particles under the mixed convection features.A numerical treatment toward the mixed convention approach was proceeded by Rehman et al. [6].Hayat et al. [7] intended the optimized aspects of mixed convection problem with Soret features.Sharma et al. [8] elaborated the chemical reaction species for mixed convection flow in inclined surface.Chen et al. [9] suggested the inclined surface flow subject to mixed convection analysis.Ige et al. [10] investigated the transient flow with blood liquid additional forced by mixed convection consequences.
The transport for heat fluctuation phenomenon is important in dynamical systems, engineering mechanisms and thermal devices.The fluctuated impact of thermal generation is preserved for diverse change in heat transfer.The heat transfer subject to various flow configurations has been performed in recent years.For instance, Saeed et al. [11] developed a fractional model for Oldroyd-B fluid with heat transfer impact.The analysis was subject to the ramped thermal constraints.Rehman et al. [12] performed heat transfer analysis for second-grade fluid via fractional approach.Saeed et al. [13] observed thermal flow due to chemically reactive Casson fluid with extended boundary constraints.The fractional investigation for heat transfer problem due to Oldroyd-B fluid in view of ramped thermal constraints was analyzed by Saeed et al. [11].It is commonly observed that for thermal phenomenon, the optimized assessment becomes prime objective to control the loss of heat transfer.Novel role regarding the entropy production is proceeded in the thermal era.For maintaining the fundamental and desirable thermal achievements in electronic systems, sustainability of heat cannot be neglected.The theory of thermodynamics helps in observing the thermal impact for designing various engineering devices and extrusion framework.The first thermodynamics approach conveys the zero energy loss against the transmission of energy.However, this concept is totally unable to predict the fundamental of energy production.The loss and control of thermal energy can be effectively controlled via second thermodynamics approach.The dynamic of entropy production is significantly attributed in the nuclear progress, thermal devices, chemical engineering, extrusion processes, etc. Bejan [14] taken the initiative role for entropy generation by providing the fundamental concept.Li et al. [15] determined the entropy outcomes for porous medium flow with interaction of nanoparticles.Abdelhameed [16] provided the analyzation of entropy generation with sodium alginate material.Shah et al. [17] described the swirling flow under the optimized analysis due to boosted heat transfer.Wang et al. [18] predicted the autocatalysis mechanism based on entropy generation of viscous liquid.Alsallami et al. [19] discussed the rotatory transport with entropy profile via disk flow.Turkyilmazoglu [20] attributed the slip effects with entropy outcomes in channel surface.Batool et al. [21] determined the bioconvective analyzation under optimized impact.Liu et al. [22] identified the grooved flow with contributing the entropy production analysis.The square cavity flow with distribution of entropy judgment was provided by Iftikhar et al. [23].Sheikholeslami [24] investigated the Lorentz force aspects for entropy generation analysis in porous media.In another work, Sheikholeslami [25] addressed the optimized mechanisms in nanofluid by using the aluminum particles.Sheikholeslami [26][27][28] presented applications of entropy generation with nanofluid.Abdelhameed [16] addressed the entropy generation of MHD flow of sodium alginate (C 6 H 9 NAO 7 ) fluid in thermal engineering.
In the above-mentioned literature survey, it is observed that different investigations are inspected for heat and mass transfer phenomenon.The objective of current work is to discuss the mixed convection flow of viscous fluid with assessment of entropy generation phenomenon.The flow observations are supported with the impact of chemically reactive species and external heating source.The Bejan number association with variation of flow parameters is predicted.The solution procedure via Laplace transform is performed.The motivations for performing the current work are to control the loss of heat transfer which is essential in various heat and fluid problems.The simulated results convey applications in thermal systems, solar collectors, heat and fluid problems, chemical engineering, electronic equipment, heat control systems, etc.

Statement of the problem
Let us assume the mixed convection fluid flow due to the oscillating plate is investigated.The flow pattern is unsteady for moving plates.A static behavior of fluid with adjusted plate is noted with surface uniform temperature θ ∞ and concentration C ∞ .The flow configura- tion is presented in Fig. 1.The deformation in plate at time τ = 0 + is noted to be u(0, τ ) = Aτ 2 .The fluctuation in temperature as well as concentration is assumed to be The flow equations are [9,16]: (1)

Entropy generation phenomenon
The creation of entropy generation is described by the following relations: Suggesting the new variables: ∂θ/∂y = �θA where

Bejan numbers
The Bejan constant is important to present the prediction of entropy production and thermal saturation profile.The Bejan constant is expressed as: and

Implementation of Laplace transform scheme
The assessment via Laplace transform is essentially evaluated for Eqs. ( 5)- (8).Utilizing the Laplace operator as: (9) The resulted solution is: which leads to: Equation ( 13) leads to: as.

Results
The optimized thermal phenomenon for mixed convection problem has been discussed.The modeled problem is supported with the perturbation technique.In the current section, graphical outcomes are presented in view of different parameters.Figure 2a  Figure 2b shows the entropy generation N s against increasing Gm.A boosted impact in N s for Gm have been results.On this end, it is concluded that the mixed convection phenomenon is important for enhancing the optimized outcomes.Figure 2c deduces the results for changing velocity for Gm. .An increment in velocity is noticed when Gm is enhanced.Figure 3a-c    Table 1 shows the attribution of Bejan number, wall shear and entropy generation in view of Hartmann number and Schmidt number.A leading role of both parameters beyond the variation of Bejan number has been observed.However, less observations are claimed for entropy generation and Bejan number.due to variation of Gr have been inspected.However, the velocity field and entropy generation increase against the larger variation of Gr .Moreover, a decrement in wall shear is also noticed due to Gr .A decrement in opti- mized phenomenon and velocity in view of mass and thermal Grashof numbers have been disclosed which are represented in Tables 3 and 4. From Table 4, Bejan number decreases for temperature difference constant.

Conclusion
The heat and mass transfer phenomenon for mixed convection flow with entropy generation impact via flat plate is discussed under the influence of external heat source and chemical reaction.The solution technique is adopted with Laplace technique.The association of parameters with Bejan number, velocity and entropy generation is discussed, summarizing main observations as follows: • By enhancing both heat and mass Grashof numbers, a control of entropy generation phenomenon is noticed.• The contribution of magnetic force boosted the entropy generation.• Upon increasing mass Grashof number, the Bejan number declined • The enhancing lower change in entropy assessment and Bejan profile is noted due to Schmidt number, respectively.• The velocity field increased for mass Grashof constant as well as thermal Grashof number.• The decreasing numerical values of Bejan profile and entropy generation are being observed for Hartmann constant and Schmidt number.• These results can be further updated by utilizing the bioconvection applications and artificial neural network.Moreover, the geometry of the proposed equations will be investigated in addition to numerical investigations will involve different types of fractional differential operators will be studied.The volumetric coefficient of thermal expansion (K 1 ) β The Casson fluid parameter β c The coefficient of concentration Br The Brinkman number The dimensionless temperature function

1 5 ν − 3 5 1 5 ν − 3 5 2 5 ν − 1 5
∂θ * /∂y * ∂C/∂y = �CA ∂C * /∂y * , ∂u/∂y = A ∂u * /∂y * and added Eq. (9) leads to: compromises the observations for Bejan number BN under the impact of mass Grashof number Gm .The decreas- ing observations are listed for Bejan number when peak changes have been assigned to Gm .Physically, Grashof number presents a special attention in the mass and heat transfer phenomenon.It associated the relation between buoyancy force and viscous force in the thermal transport phenomenon.The declining change in BN due to Gm shows that viscous forces play domi- nant role in controlling the optimized phenomenon.

Fig. 3 a
Fig. 3 a-c: Assessment of Bejan number for Gr , b assessment of entropy generation for Gr , c assessment of velocity profile for Gr.
reports the onset of thermal Grashof number Gr to report the change in entropy generation N s , velocity profile and Bejan number BN .The reducing change in BN with large Gr is resulted in Fig. 3a. Figure 3b discloses the assistance of Gr on entropy generation N s .With larger change assigning to Gr , the entropy phenomenon boosted.The same results discussed for velocity are noticed in Fig. 3c. Figure 4a pronounces the insight of Bejan number via Hartmann constant Ha .An enlarge change is noticed under the larger assigning values of Ha .The enhanced trend in Bejan number is claimed in Fig. 4b.Such outcomes are associated with the Lorentz force.Some results are noted for entropy generation in view of Ha .However, a contracted behavior is observed for velocity in Fig. 4c. Figure 5a-c announces the change in Bejan number with larger variation of Schmidt number Sc on BN , N s and velocity.The

Fig. 4 a
Fig. 4 a-c: a Assessment of Bejan number for Ha , b assessment of entropy generation for Ha , c assessment of velocity profile Ha

Figure 7
explains the assessment for Nusselt with different values of Schmidt number Sc.Low Nusselt number profile is predicted due to Sc.These outcomes are associated with the low mass diffusivity.

Fig. 5 a
Fig. 5 a-c: a Assessment of Bejan number for Sc , b assessment of entropy phenomenon for Sc c assessment of velocity profile for Sc.

Fig. 6
Fig. 6 Analysis for skin friction with variation of Gr.

Fig. 7
Fig. 7 Analysis for Nusselt number with variation of δ.

Table 2 ana
lyzes the significance of Grashof number Gr on veloc- ity, Bejan number, Nusselt number and wall shear force.The analysis is performed for fixed numerical values of Ha and time t .The lower observations for Bejan number

Table 1
FDM simulations for Ha

Table 2
FDM simulations for v Gr

Table 3
FDM simulations for Gm

Table 4
FDM simulations for