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



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.


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.


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.

1 Background

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 (C6H9NAO7) 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.

2 Methods

2.1 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 \(\theta_{\infty }\) and concentration \(C_{\infty }\). The flow configuration is presented in Fig. 1. The deformation in plate at time \(\tau = 0^{ + }\) is noted to be \(u\left( {0,\tau } \right) = A\tau^{2}\). The fluctuation in temperature as well as concentration is assumed to be \(\theta \left( {y,0} \right) = \theta_{\infty }\) to \(\theta \left( {0,\tau } \right) = \theta_{w}\),\(C\left( {y,0} \right) = C_{\infty }\), \(C\left( {0,\tau } \right) = C_{w}\), respectively.

Fig. 1
figure 1

Flow configuration of problem

The flow equations are [9, 16]:

$$\rho \frac{{\partial u\left( {y,\tau } \right)}}{\partial \tau } = \mu \frac{{\partial^{2} u(y,\tau )}}{{\partial y^{2} }} + \rho g\beta_{\theta } (\theta (y,\tau ) - \theta_{\infty } ) + \rho g\beta_{c} (C(y,\tau ) - C_{\infty } ) - \varphi B_{0}^{2} u(y,\tau )$$
$$\rho c_{p} \frac{\partial \theta (y,\tau )}{{\partial \tau }} = K\frac{{\partial^{2} \theta (y,\tau )}}{{\partial y^{2} }} + Q_{0} (\theta (y,\tau ) - \theta_{\infty } )$$
$$\frac{\partial C(y,\tau )}{{\partial \tau }} = D\frac{{\partial^{2} C(y,\tau )}}{{\partial y^{2} }} - K_{1} (C(y,\tau ) - C_{\infty } )$$


$$\left. \begin{gathered} u\left( {y,0} \right) = 0,\,\,\,\,\,\,\,\,\,\,\,\,\,\theta \left( {y,0} \right) = \theta_{\infty } ,{\text{ C}}\left( {y,0} \right) = C_{\infty } ,\, \hfill \\ u\left( {0,\tau } \right) = A\tau^{2} ,\,\,\,\,\,\,\,\,\,\theta \left( {0,\tau } \right) = \theta_{w} ,\,\,{\text{ C}}\left( {0,\tau } \right) = C_{w} ,\,\, \hfill \\ u\left( {\infty ,\tau } \right) = 0,\,\,\,\,\,\,\,\,\,\,\,\,\theta \left( {\infty ,\tau } \right) = \theta_{\infty } ,{\text{ C}}\left( {\infty ,\tau } \right) = C_{\infty } .\,\,\,\,\,\, \hfill \\ \end{gathered} \right\}$$

with \(\mu\) (viscosity), k (thermal conductivity), \(g\) (gravity), \(Q_{0}\) (heat generation), \(B_{0}^{{}}\) (magnetic field), \(\beta_{c}\) (concentration coefficient), \(\beta_{\theta }\) (volumetric thermal expansion), \(\theta (y,\tau )\), \(D\) (mass diffusivity), \(\rho\) (density), \(c_{p}\)(heat capacitance), and \(k_{1}\) (reaction constant).

Intervening new variables [9]:

$$u^{*} = \frac{u}{{\left( {v^{2} A} \right)^{\frac{1}{5}} }},\,{\text{ y}}^{*} = \frac{{yA^{\frac{1}{5}} }}{{\nu^{\frac{3}{5}} }},\, \, \tau^{*} = \frac{{\tau A^{\frac{2}{5}} }}{{\nu^{\frac{1}{5}} }},\,\,\, \, \,\theta^{*} (y,\tau ) = \frac{{\theta - \theta_{\infty } }}{{\theta_{w} - \theta_{\infty } }},{\text{ C}}^{*} (y,\tau ) = \frac{{C - C_{\infty } }}{{C_{w} - C_{\infty } }}$$

Into Eqs. (1)–(4),

$$\frac{\partial u(y,\tau )}{{\partial \tau }} = \frac{{\partial^{2} u(y,\tau )}}{{\partial y^{2} }} + Gr\theta (y,\tau ) + GmC(y,\tau ) - Hau(y,\tau )$$
$$\Pr \frac{\partial \theta (y,\tau )}{{\partial \tau }} = \frac{{\partial^{2} \theta (y,\tau )}}{{\partial y^{2} }} + \delta \Pr \theta (y,\tau )$$
$$Sc\frac{\partial C(y,\tau )}{{\partial \tau }} = \frac{{\partial^{2} C(y,\tau )}}{{\partial y^{2} }} - \omega ScC(y,\tau )$$
$$\left. \begin{gathered} u\left( {y,0} \right) = 0,\,\,\,\,\,\,\,\,\,\,\,\,\,\theta \left( {y,0} \right) = 0,{\text{ C}}\left( {y,0} \right) = 0\, \hfill \\ u\left( {0,\tau } \right) = \tau^{2} ,\,\,\,\,\,\,\, \, \theta \left( {0,\tau } \right) = 1,\,\,{\text{ C}}\left( {0,\tau } \right) = 1\,\, \hfill \\ u\left( {\infty ,\tau } \right) = 0,\,\,\,\,\,\,\,\,\,\,\,\theta \left( {\infty ,\tau } \right) = 0,{\text{ C}}\left( {\infty ,\tau } \right) = 0\,\,\,\,\,\, \hfill \\ \end{gathered} \right\}$$

where \(Gr = g\beta_{\theta } \Delta \theta A^{{ - \frac{3}{5}}} \upsilon^{{ - \frac{1}{5}}}\) is (thermal Grashof number), \({\text{ Sc = }}\frac{\upsilon }{D}\) (Schmidt constant), \(Gm = g\beta_{c} \Delta CA^{{ - \frac{3}{5}}} \upsilon^{{ - \frac{1}{5}}}\) (mass Grashof constant), \({\text{Pr}} = \frac{{\mu {\text{c}}_{{\text{p}}} }}{{\text{K}}}\) (Prandtl factor), \(\delta = \frac{{Q_{0} }}{{\rho c_{p} }}A^{{ - \frac{2}{5}}} \upsilon^{\frac{1}{5}}\) (heat generation),\(\, \omega = k_{1} A^{{ - \frac{2}{5}}} \upsilon^{\frac{1}{5}}\) (reaction parameter), \(H_{a} = \frac{\varphi }{\rho }B_{0}^{2} A^{{ - \frac{2}{5}}} \upsilon^{\frac{1}{5}}\) (Hartmann constant).

2.2 Entropy generation phenomenon

The creation of entropy generation is described by the following relations:

$$E_{gen} = \frac{K}{{\theta_{\infty }^{2} }}\left( {\frac{\partial \theta }{{\partial y}}} \right)^{2} + \frac{D}{{C_{\infty }^{2} }}\left( {\frac{\partial C}{{\partial y}}} \right)^{2} + \frac{\mu }{{\theta_{\infty } }}\left( {\frac{\partial u}{{\partial y}}} \right)^{2} + \frac{{\varphi B_{0}^{2} }}{{\theta_{\infty } }}u^{2}$$

Suggesting the new variables:

\(\partial \theta /\partial y = \Delta \theta A^{\frac{1}{5}} \nu^{{ - \,\frac{3}{5}}} \partial \theta^{*} /\partial y^{*}\)\(\partial C/\partial y = \Delta CA^{\frac{1}{5}} \nu^{{ - \,\frac{3}{5}}} \partial C^{*} /\partial y^{*}\), \(\partial u/\partial y = A^{\frac{2}{5}} \nu^{{ - \,\frac{1}{5}}} \partial u^{*} /\partial y^{*}\) and added Eq. (9) leads to:

$$N_{s} = \left[ {\frac{{D\zeta^{2} }}{{k\Omega^{2} }}\left( {\frac{\partial C}{{\partial y}}} \right)^{2} + \left( {\frac{\partial \theta }{{\partial y}}} \right)^{2} + H_{a} \frac{{B_{r} }}{\Omega }u^{2} + \frac{{B_{r} }}{\Omega }\left( {\frac{\partial u}{{\partial y}}} \right)^{2} } \right]$$


\(N_{s} = \frac{{E_{gen} \nu^{\frac{6}{5}} \theta^{2}_{\infty } }}{{kA^{2/5} (\Delta \theta )^{2} }},\, \, Br = \frac{{\mu A^{\frac{2}{5}} \nu^{\frac{4}{5}} }}{k\Delta \theta },\,\, \, \Omega = \frac{\Delta \theta }{{\theta_{\infty } }} = \frac{{\theta_{w} - \theta_{\infty } }}{{\theta_{\infty } }}, \, \zeta = \frac{\Delta C}{{C_{\infty } }} = \frac{{C_{w} - C_{\infty } }}{{C_{\infty } }}\).

2.3 Bejan numbers

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

$$BN = \frac{{\frac{K}{{\theta_{\infty }^{2} }}\left( {\frac{\partial \theta }{{\partial y}}} \right)^{2} }}{{\frac{K}{{\theta_{\infty }^{2} }}\left( {\frac{\partial \theta }{{\partial y}}} \right)^{2} + \frac{D}{{C_{\infty }^{2} }}\left( {\frac{\partial C}{{\partial y}}} \right)^{2} + \frac{\mu }{{\theta_{\infty } }}\left( {\frac{\partial u}{{\partial y}}} \right)^{2} + \frac{{\varphi B_{0}^{2} }}{{\theta_{\infty } }}u^{2} }}$$


$$BN = \frac{{\left( {\frac{\partial \theta }{{\partial \eta }}} \right)^{2} }}{{\left( {\frac{\partial \theta }{{\partial y}}} \right)^{2} + \frac{{D\zeta^{2} }}{{k\Omega^{2} }}\left( {\frac{\partial C}{{\partial y}}} \right)^{2} + \frac{{B_{r} }}{\Omega }\left( {\frac{\partial u}{{\partial y}}} \right)^{2} + H_{a} \frac{{B_{r} }}{\Omega }u^{2} }}$$

2.4 Implementation of Laplace transform scheme

The assessment via Laplace transform is essentially evaluated for Eqs. (5)–(8). Utilizing the Laplace operator as:

$$q\mathop u\limits^{ - } (y,q) = \frac{{\partial^{2} \mathop u\limits^{ - } (\eta ,q)}}{{\partial y^{2} }} + Gr\mathop \theta \limits^{ - } (y,q) + Gm\mathop C\limits^{ - } (y,q) - H_{a} \overline{u} (y,q)$$
$$\overline{u} \left( {0,q} \right) = \frac{2}{{q^{3} }},\,\,\,\,\,\,\overline{u} \left( {\infty ,q} \right) = 0$$
$$\Pr q\overline{\theta } \left( {y,q} \right) = \frac{{\partial^{2} \overline{\theta } (y,q)}}{{\partial y^{2} }} + \delta \Pr \overline{\theta } \left( {y,q} \right)$$
$$\overline{\theta } \left( {0,q} \right) = \frac{1}{q},\,\,\,\,\,\,\overline{\theta } \left( {\infty ,q} \right) = 0$$
$$Sc \, q\overline{C} \left( {y,q} \right) = \frac{{\partial^{2} \overline{C} (y,q)}}{{\partial y^{2} }} - Sc \, \omega \overline{C} \left( {y,q} \right)$$
$$\overline{C} \left( {0,q} \right) = \frac{1}{q},\,\,\,\,\,\,\overline{C} \left( {\infty ,q} \right) = 0$$

The resulted solution is:

$$\overline{\theta } \left( {y,q} \right) = \frac{{e^{{ - y\sqrt {(q - \delta )\Pr } }} }}{q}$$
$$\theta \left( {y,\tau } \right) = \frac{1}{2}\left[ {e^{{ - y\sqrt { - \delta \Pr } }} erfc\left( {\frac{{y\sqrt {\Pr } }}{2\sqrt \tau } - \sqrt { - \delta \Pr t} } \right) + e^{{y\sqrt { - \delta \Pr } }} erfc\left( {\frac{{y\sqrt {\Pr } }}{2\sqrt \tau } + \sqrt { - \delta \Pr t} } \right)} \right]$$
$$\overline{C} \left( {y,q} \right) = \frac{{e^{{ - y\sqrt {(q + \omega )Sc} }} }}{q}$$

which leads to:

$$C\left( {y,\tau } \right) = \frac{1}{2}\left[ {e^{{ - y\sqrt {\omega Sc} }} erfc\left( {\frac{{y\sqrt {Sc} }}{2\sqrt \tau } - \sqrt {\omega Sct} } \right) + e^{{y\sqrt {\omega Sc} }} erfc\left( {\frac{{y\sqrt {Sc} }}{2\sqrt \tau } + \sqrt {\omega Sct} } \right)} \right]$$

Equation (13) leads to:

$$\begin{gathered} \mathop u\limits^{ - } (y,q) = \frac{2}{{q^{{^{3} }} }}e^{{ - y\sqrt {H_{a} + q} }} + \frac{1}{{a_{0} }}\frac{1}{q - A}e^{{ - y\sqrt {H_{a} + q} }} - \frac{1}{{a_{0} }}\frac{1}{q}e^{{ - y\sqrt {H_{a} + q} }} - \hfill \\ \, \frac{1}{{b_{0} }}\frac{1}{q + B}e^{{ - y\sqrt {H_{a} + q} }} + \frac{1}{{b_{0} }}\frac{1}{q}e^{{ - y\sqrt {H_{a} + q} }} - \frac{1}{{a_{0} }}\frac{1}{q - A}e^{{ - y\sqrt {\Pr } \sqrt {q - \delta } }} + \hfill \\ \, \frac{1}{{a_{0} }}\frac{1}{q}e^{{ - y\sqrt {\Pr } \sqrt {q - \delta } }} + \frac{1}{{b_{0} }}\frac{1}{q + B}e^{{ - y\sqrt {Sc} \sqrt {q + \omega } }} - \frac{1}{{b_{0} }}\frac{1}{q}e^{{ - y\sqrt {Sc} \sqrt {q + \omega } }} \hfill \\ \, \hfill \\ \end{gathered}$$


\(a_{0} = \frac{{\delta {\text{ Pr + H}}_{a} }}{Gr}, \, a_{1} = \frac{{\text{ Pr - 1}}}{Gr}{, }b_{0} = \frac{{\omega {\text{ Sc - H}}_{a} }}{Gm}{\text{, b}}_{1} = \frac{{\text{ Sc - 1}}}{Gm}{\text{, A = }}\frac{{a_{0} }}{{a_{1} }}{\text{, B = }}\frac{{b_{0} }}{{b_{1} }}\).

Implementing the Laplace operator as,

$$u(y,\tau ) = u_{1} (y,\tau ) + y_{2} (y,\tau ) + u_{3} (y,\tau ) + u_{4} (y,\tau )$$
$$\begin{gathered} u_{1} (y,\tau ) = 2\left[ {e^{{ - y\sqrt {H_{a} } }} erfc\left( {\frac{y}{2\sqrt \tau } - \sqrt {H_{a} \tau } } \right)\frac{1}{4}\left( {\tau^{2} - \frac{y}{{4\sqrt {H_{a} } }}} \right) - e^{{y\sqrt {H_{a} } }} erfc\left( {\frac{y}{2\sqrt \tau } + \sqrt {H_{a} \tau } } \right)\frac{1}{4}\left( {\tau^{2} + \frac{y}{{4\sqrt {H_{a} } }}} \right)} \right] + \hfill \\ \, \frac{1}{{2a_{0} }}e^{A\tau } \left[ {e^{{ - y\sqrt {H_{a} + A} }} erfc\left( {\frac{y}{2\sqrt \tau } - \sqrt {\left( {H_{a} + A} \right)\tau } } \right) + e^{{y\sqrt {H_{a} + A} }} erfc\left( {\frac{y}{2\sqrt \tau } + \sqrt {\left( {H_{a} + A} \right)\tau } } \right)} \right] \hfill \\ \end{gathered}$$
$$\begin{gathered} u_{2} (y,\tau ) = \frac{1}{2}\left( {\frac{1}{{b_{0} }} - \frac{1}{{a_{0} }}} \right)\left[ {e^{{ - y\sqrt {H_{a} } }} erfc\left( {\frac{y}{2\sqrt \tau } - \sqrt {H_{a} \tau } } \right) + e^{{y\sqrt {H_{a} } }} erfc\left( {\frac{y}{2\sqrt \tau } + \sqrt {H_{a} \tau } } \right)} \right] - \hfill \\ \, \frac{1}{{2b_{0} }}e^{ - B\tau } \left[ {e^{{ - y\sqrt {H_{a} - B} }} erfc\left( {\frac{y}{2\sqrt \tau } - \sqrt {\left( {H_{a} - B} \right)\tau } } \right) + e^{{y\sqrt {H_{a} - B} }} erfc\left( {\frac{y}{2\sqrt \tau } + \sqrt {\left( {H_{a} - B} \right)\tau } } \right)} \right] \hfill \\ \end{gathered}$$
$$\begin{gathered} u_{3} (y,\tau ) = - \frac{1}{{2a_{0} }}e^{A\tau } \left( {e^{{ - y\sqrt {\Pr (A - \delta )} }} erfc \, \left( {\frac{{y\sqrt {\Pr } }}{2\sqrt \tau } - \sqrt {(A - \delta )\tau } } \right) + e^{{y\sqrt {\Pr (A - \delta )} }} erfc \, \left( {\frac{{y\sqrt {\Pr } }}{2\sqrt \tau } + \sqrt {(A - \delta )\tau } } \right)} \right) + \hfill \\ \, \frac{1}{{2a_{0} }}\left( {e^{{ - y\sqrt { - \delta \Pr } }} erfc \, \left( {\frac{{y\sqrt {\Pr } }}{2\sqrt \tau } - \sqrt { - \delta \tau } } \right) + e^{{y\sqrt { - \delta \Pr } }} erfc \, \left( {\frac{{y\sqrt {\Pr } }}{2\sqrt \tau } + \sqrt { - \delta \tau } } \right)} \right) \hfill \\ \end{gathered}$$
$$\begin{gathered} u_{4} (y,\tau ) = \frac{1}{{2b_{0} }}e^{ - B\tau } \left( {e^{{ - y\sqrt {Sc(\omega - B)} }} erfc \, \left( {\frac{{y\sqrt {Sc} }}{2\sqrt \tau } - \sqrt {(\omega - B)\tau } } \right) + e^{{y\sqrt {Sc(\omega - B)} }} erfc \, \left( {\frac{{y\sqrt {Sc} }}{2\sqrt \tau } + \sqrt {(\omega - B)\tau } } \right)} \right) - \hfill \\ \, \frac{1}{{2b_{0} }}\left( {e^{{ - y\sqrt {\omega Sc} }} erfc \, \left( {\frac{{y\sqrt {Sc} }}{2\sqrt \tau } - \sqrt {\omega \tau } } \right) + e^{{y\sqrt {\omega Sc} }} erfc \, \left( {\frac{{y\sqrt {Sc} }}{2\sqrt \tau } + \sqrt {\omega \tau } } \right)} \right) \hfill \\ \end{gathered}$$

Defining the wall drag force:

$$c_{f} = \left. {\frac{\partial v(y,\tau )}{{\partial y}}} \right|_{y = 0}$$

Now defining the Nusselt quantity as:

$$Nu = \left. {\frac{\partial \theta (y,\tau )}{{\partial y}}} \right|_{y = 0}$$

3 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 compromises the observations for Bejan number \(BN\) under the impact of mass Grashof number \({\text{Gm}}\). The decreasing observations are listed for Bejan number when peak changes have been assigned to \({\text{Gm}}\). 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 \({\text{Gm}}\) shows that viscous forces play dominant role in controlling the optimized phenomenon. Figure 2b shows the entropy generation \(N_{s}\) against increasing \({\text{Gm}}{.}\) A boosted impact in \(N_{s}\) for \({\text{Gm}}\) 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 \({\text{Gm}}{.}\). An increment in velocity is noticed when \(Gm\) is enhanced. Figure 3a–c reports the onset of thermal Grashof number \({\text{Gr}}\) to report the change in entropy generation \(N_{s}\), velocity profile and Bejan number \(BN\). The reducing change in \(BN\) with large \({\text{Gr}}\) 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 increasing impact of \(Sc\) on Bejan number while less observations for entropy generation field is resulted. The control of velocity is predicted for Schmidt number. Such features are noted due to low mass diffusivity. Figure 6 reports the variation of skin friction due to distinct values of \(Gr.\) It has been noticed that skin friction reduces due to \(Gr.\) Figure 7 explains the assessment for Nusselt number 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. 2
figure 2

a–c: Assessment of Bejan number against \({\text{Gm}}\), b assessment of entropy phenomenon against \({\text{Gm}}\), and c assessment of velocity against \({\text{Gm}}{.}\)

Fig. 3
figure 3

a–c: Assessment of Bejan number for \({\text{Gr}}\), b assessment of entropy generation for \(Gr\), c assessment of velocity profile for \({\text{Gr}}{.}\)

Fig. 4
figure 4

a–c: a Assessment of Bejan number for \(Ha\), b assessment of entropy generation for \(Ha\), c assessment of velocity profile \(Ha\)

Fig. 5
figure 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
figure 6

Analysis for skin friction with variation of \({\text{Gr}}.\)

Fig. 7
figure 7

Analysis for Nusselt number with variation of \(\delta .\)

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. Table 2 analyzes the significance of Grashof number \(Gr\) on velocity, 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 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 optimized 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.

Table 1 FDM simulations for \(Ha\)
Table 2 FDM simulations for v Gr
Table 3 FDM simulations for \(Gm\)
Table 4 FDM simulations for \(\Omega\)

4 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.

Availability of data and materials

All data are available in the manuscript.


\(u\) :

Velocity of the fluid \({\text{(ms}}^{ - 1} )\)

\(\theta\) :

Temperature of the fluid \({\text{(K}})\)

C :

Concentration of the fluid

\(g\) :

Acceleration due to gravity \(({\text{ms}}^{ - 2} )\)

\(c_{p}\) :

Specific heat at a constant pressure \(({\text{j}}\;{\text{kg}}^{ - 1} \;{\text{K}}^{ - 1} )\)

\(Gr\) :

Thermal Grashof number \(k\)

k :

Thermal conductivity of the fluid \(({\text{W}}\;{\text{m}}^{ - 2} \;{\text{K}}^{ - 1} )\)


The mass Grashof number


The Schmidt number

\(\omega\) :

The chemical reaction parameter

\(Nu\) :

Nusselt number

\(\Pr\) :

Prandtl number

\(\theta_{\infty }\) :

Fluid temperature far away from the plate \(({\text{K}})\)

\(q\) :

Laplace transforms parameter

K :

Porosity parameter

D :

The mass diffusivity

\(k_{1}\) :

The chemical reaction parameter

\(Q_{0}\) :

The heat generation term

\(\delta\) :

The heat generation parameter

A :

Random constant \({\text{(ms}}^{{ - {2}}} )\)

\(\nu\) :

Kinematic viscosity of the fluid \(({\text{m}}^{2} {\text{s}}^{{1}} )\)

\(\mu\) :

Dynamic viscosity \(({\text{kg}}\;{\text{m}}^{{1}} {\text{s}}^{ - 1} )\)

\(\rho\) :

Fluid density \(({\text{kg}}\;{\text{ms}}^{{3}} )\)

\(\beta_{\theta }\) :

The volumetric coefficient of thermal expansion \(({\text{K}}^{{1}} )\)

\(\beta\) :

The Casson fluid parameter

\(\beta_{c}\) :

The coefficient of concentration

\(Br\) :

The Brinkman number

\(\Omega\) :

The dimensionless temperature function


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The author extends the appreciation to the Deanship of Postgraduate Studies and Scientific Research at Majmaah University for funding this research work through the project number R-2024-1053.


This work was supported by the Majmaah University (Grant No. R-2024-1053).

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Abdelhameed, T.N. Entropy generation analysis for magnetohydrodynamic flow of chemically reactive fluid due to an accelerated plate. Beni-Suef Univ J Basic Appl Sci 13, 37 (2024).

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