Radiative nanofluid flow over a slender stretching Riga plate under the impact of exponential heat source/sink

Abdul Hamid Ganie; Muhammad Farooq; Mohammad Khalid Nasrat; Muhammad Bilal; Taseer Muhammad; Kaouther Ghachem; Adnan

doi:10.1515/phys-2024-0020

Article Open Access

Radiative nanofluid flow over a slender stretching Riga plate under the impact of exponential heat source/sink

Abdul Hamid Ganie , Muhammad Farooq , Mohammad Khalid Nasrat , Muhammad Bilal , Taseer Muhammad , Kaouther Ghachem and Adnan

Published/Copyright: May 16, 2024

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From the journal Open Physics Volume 22 Issue 1

Abstract

Recognizing the flow behaviours across a Riga plate can reveal information about the aerodynamic efficiency of aircraft, heat propagation, vehicles, and other structures. These data are critical for optimizing design and lowering drag. Therefore, the purpose of the current analysis is to examine the energy and mass transfer across the mixed convective nanofluid flows over an extending Riga plate. The fluid flow is deliberated under the influences of viscous dissipation, exponential heat source/sink, activation energy, and thermal radiation. The Buongiorno’s concept is utilized for the thermophoretic effect and Brownian motion along with the convective conditions. The modelled are simplified into the lowest order by using similarity transformation. The obtained set of non-dimensional ordinary differential equations is then numerically solved through the parametric continuation method. For accuracy and validation of the outcomes, the results are compared to the existing studies. From the graphical analysis, it can be observed that the fluid velocity boosts with the rising values of the divider thickness parameter. The fluid temperature also improves with the effect of Biot number, Eckert number, and heat source factor. Furthermore, the effect of heat source sink factor drops the fluid temperature.

Keywords: thermal radiation; viscous dissipation; numerical solution; mixed convection; exponential heat source/sink; slender sheet

Nomenclature

T w ( x ): surface temperature
y: sheet thickness
α 1: thermal diffusivity
D B: Brownian diffusion
β T: thermal gradients factor
E _a: activation energy
Kr: chemical reaction
c p: specific heat
Nt: thermophoresis term
λ C: fixation lightness factor
β 1: dimensionless term
Q e: heat source/sink factor
Q: modified Hartman number
Ec: Eckert number
VD: viscous dissipation
C fx: shear stress
U w ( x ): uniform velocity of Riga plate
2D: two-dimensional
υ: kinematic viscosity
C w ( x ): surface concentration
K: thermal conductivity
τ: thermal capacity
j 0: current density
D T: thermophoresis diffusion
λ T: lightness parameter
β C: concentration expansion factor
K c: factor of chemical reaction
Sc: Schmidt number
Bi: Biot number
Nb: Brownian constant
Sh x: Sherwood number

1 Introduction

The study of the fluid flow over a Riga plate contributes to the development of techniques for controlling boundary layer disparity, which is critical for enhancing the efficiency of multiple engineering systems and reducing drag force. Recognizing the flow behaviours across a Riga plate can reveal information about the aerodynamic efficiency of aircraft, vehicles, and other structures. These data are critical for optimizing design and lowering drag. Researchers may enhance heat conduction and thermal proficiency by analyzing the heat exchange features of fluid flow across a Riga plate [1,2,3]. Abdul Hakeem et al. [4] numerically simulated the three-dimensional (3D) flow of nanofluids containing aluminium oxide (Al₂O₃) and magnetite (Fe₃O₄) nanoparticles through a Riga plate. It was found that the radiation parameter increases the efficiency of energy transportation. Ishtiaq et al. [5] investigated the rate-type fluid under the effect of heat radiation on passing within a Riga plate and concluded that arising Prandtl numbers decrease the temperature curve, whereas higher radiation factor increases the temperature curve. Asogwa et al. [6] assessed the mass and energy transportation phenomena caused by the parabolic fluid flow on an upward extending Riga surface. The Hartmann number has a momentous impression on fluid flow, as observed with the involvement of the Riga sheet. Adnan et al. [7] studied the thermal features of a water-based nano liquid containing oxide and metallic nanoparticles (SiO₂, Au, MoS₂) through a cylinder under combined convection. The study found that increasing the quantity of MoS₂ tiny particles from 0.1 to 0.5%, as well as the Eckert number, significantly improved the fluidity of the functional fluid. Bani-Fwaz et al. [8] evaluated the nanofluid thermal transfer characteristics within a channel generated by expanding and contracting walls. The velocity of stretching walls is measured at its maximum in the channel centre. Waqas et al. [9] and Hamad et al. [10] reported the mass and energy transference within nano liquid flow across a tilted prolonging sheet, under the upshot of magnetic fields and chemical reactions. The fluid velocity was found to increase with the change of the Tangent fluid parameter and Richardson numbers. Some further results are reported by refs. [11–14].

Thermal radiation pertains to the mechanism of heat transfer via electromagnetic waves emitted by a fluid due to its temperature. When a fluid (like a liquid or a gas) is heated, its molecules gain energy and move more rapidly. As a result, these molecules emit electromagnetic radiation, primarily in the form of infrared radiation. This thermal radiation process is one of the ways heat is conveyed from a fluid to its ambiance or from one part of the fluid to another. It is crucial to observe that while thermal radiation can occur in a fluid, it is not exclusive to fluids and can also happen in solids and a vacuum [15]. Jamshed et al. [16] conducted an in-depth exploration of the time-dependent nanoliquid focusing on its entropy characteristics and thermal transport. Abbas et al. [17] focused on investigating heat transference and fluid motion across an inclined movable surface particularly examining the Sakiadis flow, while their analysis included factors like changing density, thermal radiation, and magnetization. Yu and Wang [18] presented the dynamics of heat and fluid flow between two spinning and stretching disks submerged in a water-based fluid containing carbon nanotubes, and their findings indicated that elevated Reynolds number (Re) values result in a diminution in the velocity near the disk surfaces considering thermal radiation as well. Sneha et al. [19] and Salawu et al. [20] scrutinized the impact of hydromagnetic flow consisting of CNTs. The study unveiled that enhancing the volume fraction of hybrid nanoparticles contributes to improved energy optimization of the system. Some prominent and recent literature regarding thermal radiation effects have been highlighted in refs. [21–26].

Exponential heat sources/sinks (EHSS) are widely used in heat transport analysis to simulate a variety of physical phenomena. EHSS can be used in thermal management, geothermal energy systems, nuclear reactor design, and environmental studies [27,28,29]. Elangovan et al. [30] debated the impact of HSS on the fluid flow across a channel with an irregular wall temperature. The results show that the electromagnetic radiation factor reduces temperature, while the higher Brinkman number causes temperature to increase. Nabwey et al. [31] and Maranna et al. [32] evaluated the impacts of EHSS on the 2D flow of Walter’s B and second-grade ternary nanofluids across a porous reducing the flat substrate. Hussain et al. [33] assessed the HSS, buoyancy impacts, and heat transport through an upward flat surface in a Darcy porous media. The results showed that the boundary effect speeds up because of the superior effect of the permeation variable. Adnan et al. [34] proposed a nanofluid model based on ZnO-SAE50, the nano-lubricant, with the additional impacts of thermal radiations, EHSS, and resistive heating and magnetic field. The results show that employing ZnO-SAE50 nano-lubricant maximizes heat transportation, whereas traditional SAE50 fails to accomplish the ideal heat exchange rate. Some valuable updated results are presented by refs. [35–39].

The objective of the current analysis is to examine the energy and mass transfer across the mixed convective nanofluid flows over an extending Riga plate. By recognizing the flow behaviours across a Riga plate can reveal information about the aerodynamic efficiency of aircraft, heat propagation, vehicles, and other structures. These data are critical for optimizing design and lowering drag. The fluid flow is studied under the influences of viscous dissipation, EHSS, activation energy, and thermal radiation. The modelled are simplified into the lowest order by using similarity transformation, which is numerically solved through the parametric continuation method (PCM). For accuracy and validation of the outcomes, the outcomes are equated to the existing studies. In the next portion, the problem is mathematically designed and solved

2 Mathematical formulation

The 2D incompressible fluid flow across a vertical heated Riga plate is considered. The permanent alternating anodes and magnetization are employed by the Riga plate towards the x-axis as demonstrated in Figure 1. The flow is induced by the stretching plate along the y-axis. The plate is supposed to be moving with uniform velocity U w ( x ) . The thickness of sheet is defined as y = A ( x + b ) ( 1 − m ) 2 , where m ≠ 1 and A is a constant that defines the sheet stretching. The surface temperature and mass are expressed as T w ( x ) = A 1 ( x + b ) r and C w ( x ) = A 2 ( x + b ) s . The 2D incompressible flow based on the aforementioned presumptions are stated as follows [40,41,42]:

(1) ∂ u ∂ x = − ∂ v ∂ y ,

(2) u ∂ u ∂ x + v ∂ u ∂ y − υ ∂ 2 u ∂ y 2 = π j 0 M 0 8 ρ exp − π a 1 x y + g ( T − T ∞ ) β T + g ( C − C ∞ ) β C ,

(3) u ∂ T ∂ x + v ∂ T ∂ y − α 1 ∂ 2 T ∂ y 2 = τ D B ∂ C ∂ y ∂ T ∂ y + D T T ∞ ∂ T ∂ y 2 + 1 ( ρ C p ) f 4 σ ⁎ 3 k ⁎ ∂ 2 ∂ z 2 ( 4 T ∞ 3 T ) + Q e ⁎ ( ρ C p ) f ( T − T ∞ ) exp − a υ f n y + μ ( ρ C p ) f ∂ u ∂ y 2 ,

(4) u ∂ C ∂ x + v ∂ C ∂ y = D B ∂ 2 C ∂ y 2 + D T T ∞ ∂ 2 T ∂ y 2 − k r 2 ( C − C 0 ) T T ∞ n exp − E a κ T .

Figure 1

Physical illustration of stretching sheet and Riga plate.

The boundary conditions (BCs) are as follows:

u = U 0 ( x + b ) m = U w ( x ) , v = 0 − K ∂ T ∂ y = h s ( x ) ( T w − T ) , D B ∂ C ∂ y = − D T T ∞ ∂ T ∂ y , at y = A ( x + b ) ( 1 − m ) 2

(5) u → U e ( x ) = U ∞ ( b + x ) m , C → C ∞ , T → T ∞ } as y → ∞ .

The variable magnetization, changing width of magnets and electrodes, and heat transfer are stated as follows:

(6) [ M 0 ( x ) = M 0 ( x + b ) 2 m − 1 , K 1 ( x ) = K 1 ( x + b ) ( m − 1 ) , a 1 ( x ) = a 1 ( x + b ) ( 1 − m ) 2 , h s ( x ) = h s ( x + b ) ( 1 − m ) 2 .

The non-dimensional variables are expressed as follows [43]:

(7) η = y U 0 ( x + b ) m − 1 υ ( m + 1 ) 2 , Θ ( η ) = T − T ∞ T w − T ∞ , ψ = U 0 ( x + b ) ( m + 1 ) υ 2 m + 1 F ( η ) , Φ ( η ) = C − C ∞ C w − C ∞ .

The velocity and stream function components are as follows:

(8) v = − ∂ ψ ∂ x = − υ ∞ 2 m + 1 U 0 ( x + b ) ( m − 1 ) ( m − 1 ) η 2 F ′ ( η ) + m + 1 2 F ( η ) , u = ∂ ψ ∂ y = U 0 ( x + b ) m F ′ ( η ) .

The insertion of Eqs. (7) and (8) in Eqs. (1)–(5) are stated as follows:

(9) F ‴ + F F ″ − 2 m ( m + 1 ) ( F ′ ) 2 + 2 ( m + 1 ) Q e − ( β 1 η ) + 2 ( m + 1 ) λ T ( Θ + λ C Φ ) = 0 ,

(10) ( 1 + Rd ) Θ ″ + Pr Θ ′ ( Nb Θ ′ + Nt Θ ′ + F ) − 2 Pr m + 1 ( ( 2 m − 1 ) F ′ Θ ) + Q e exp ( − n η ) + PrEc ( F ″ ) 2 = 0 ,

(11) Φ ″ + Nt Nb Θ ″ − 2 ( 2 m − 1 ) ( m + 1 ) Sc F ′ Φ + Sc F Φ ′ − Sc K c ( 1 + δ θ ) n Φ exp − E 1 + δ θ = 0 .

The BCs for the reduced system of ODEs are follows:

(12) F ( α ) α − ( 1 − m ) ( 1 + m ) = 0 , Θ ′ ( α ) = − Bi ( 1 − Θ ( α ) ) , F ′ ( α ) = 1 , Θ ′ ( α ) = − Nt Nb Θ ′ ( α ) , F ′ ( ∞ ) → 0 , Φ ( ∞ ) → 0 , Θ ( ∞ ) → 0 .

The non-dimensional parameters obtained from Eqs. (9)–(11) are defined in the following table.

Parameters	Symbols	Expression
Thermal Grashof number	λ T	λ T = ± g β A 1 U 0 2
Mass Grashof number	λ C	λ C = ± g β C A 2 U 0 2
Thermophoresis factor	Nt	Nt = τ D T ( T w − T ∞ ) T ∞ υ
Schmidt number	Sc	Sc = υ D B
Heat source/sink parameter	Q e	Q e = Q e ⁎ l a ( ρ C p ) f
Dimensionless constant	β 1	β 1 = π a 1 2 ( m + 1 ) υ U 0
Eckert number	Ec	Ec = U w 2 c p ( T w − T ∞ )
Chemical reaction factor	K c	K c = K 1 U 0
Thermal radiation factor	Rd	Rd = 16 σ ⁎ ⁎ T ∞ 3 ( T w − T ∞ ) 3 k ⁎ k
Brownian motion	Nb	Nb = τ D B ( C w − C ∞ ) υ
Divider thickness term	α	α = A U 0 ( m + 1 ) 2 υ
Modified Hartman number	Q	Q = π j 0 M 0 8 ρ ∞ U 0 2
Biot number	Bi	Bi = h s k 2 ( m + 1 ) υ U 0

Further, we are introducing Θ ( η ) = θ ( η − α ) = θ ( ξ ) , F ( η ) = f ( η − α ) = f ( ξ ) , where η = α illustrate the flat surface. We obtain:

(13) f ‴ + f f ″ − 2 m ( f ′ ) 2 ( m + 1 ) + 2 ( m + 1 ) Q e − β 1 ( ξ + a ) + 2 ( m + 1 ) ( λ T θ + λ C φ ) = 0 ,

(14) ( 1 + Rd ) θ ″ + Pr θ ′ ( Nb φ ′ + Nt θ ′ + f ) + Q e exp ( − n η ) + PrEc ( f ″ ) 2 − 2 Pr ( m + 1 ) ( ( 2 m − 1 ) f ′ θ ) = 0 ,

(15) φ ″ + Nt Nb θ ″ − 2 ( 2 m − 1 ) ( m + 1 ) Sc f ′ φ + Sc f φ ′ − Sc K c ( 1 + δ θ ) n Φ exp − E 1 + δ θ = 0 .

The final BCs are given as follows:

(16) f ( 0 ) α − ( 1 − m ) ( 1 + m ) = 0 , Nb φ ′ ( 0 ) = − Nt θ ′ ( 0 ) , f ′ ( 0 ) = 1 , θ ( 0 ) 1 − θ ( 0 ) = − Bi , f ′ ( ∞ ) → 0 , φ ( ∞ ) → 0 , θ ( ∞ ) → 0 .

The shear stress C fx , Nusselt number Nu x , and Sherwood number Sh x can be defined as follows:

(17) C fx = 1 U w 2 τ w , Sh x = ( x + b ) ( C w − C ∞ ) j w , Nu x = ( x + b ) ( T w − T ∞ ) q w ,

where τ w = υ ∂ u ∂ y , j w = ∂ C ∂ y and q w = ∂ T ∂ y at y = A ( x + b ) ( 1 − m ) 2 .

The dimensionless form of Eq. (17) is obtained as follows:

Re x 1 / 2 C fx = m + 1 2 1 / 2 f ″ ( 0 ) , Re x − 1 / 2 Sh x = − m + 1 2 1 / 2 φ ′ ( 0 ) , Re x − 1 / 2 Nu x = − m + 1 2 1 / 2 θ ′ ( 0 ) .

3 Numerical solution

The PCM is a numerical method, which can be applied to almost all sorts of complex problems, such as problems of nonlinear mechanics, thermo-fluids, Newtonian and Non-Newtonian fluids models, and inverse problems [44,45,46]. The detailed steps of PCM are defined as follows:

Step 1: In this step, the system of ODEs is reduced to its lowest order.

(18) ϖ 1 ( η ) = f ( η ) , ϖ 3 ( η ) = f ″ ( η ) , ϖ 5 ( η ) = θ ′ ( η ) , ϖ 7 ( η ) = φ ′ ( η ) , ϖ 2 ( η ) = f ′ ( η ) , ϖ 4 ( η ) = θ ( η ) , ϖ 6 ( η ) = φ ( η ) .

By placing Eq. (18) in Eqs. (13)–(15), we obtain:

(19) ϖ 3 ′ + ϖ 1 ϖ 3 − 2 m ( m + 1 ) ( ϖ 2 ) 2 + 2 ( m + 1 ) Q e − β 1 ( ξ + a ) + 2 ( m + 1 ) λ T ( ϖ 4 + λ C ϖ 6 ) = 0 ,

(20) ( 1 + Rd ) ϖ 5 ′ + Pr ϖ 5 ( Nb ϖ 7 + Nt ϖ 5 + ϖ 1 ) + Q e exp ( − n η ) + Pr Ec ( ϖ 3 ) 2 − 2 Pr ( m + 1 ) ( ( 2 m − 1 ) ϖ 2 ϖ 4 ) = 0 ,

(21) ϖ 7 ′ + Nt Nb ϖ 5 ′ − 2 ( 2 m − 1 ) ( m + 1 ) Sc ϖ 2 ϖ 6 + Sc ϖ 1 ϖ 7 − Sc K c ( 1 + δ ϖ 4 ) n ϖ 6 exp − E 1 + δ ϖ 4 = 0 .

The BCs for the first-order ODEs are as follows:

(22) ϖ 1 ( 0 ) α − ( 1 − m ) ( 1 + m ) = 0 , Nb ϖ 7 ( 0 ) = − Nt ϖ 5 ( 0 ) , ϖ 2 ( 0 ) = 1 , ϖ 4 ( 0 ) 1 − ϖ 4 ( 0 ) = − Bi , ϖ 2 ( ∞ ) → 0 , ϖ 6 ( ∞ ) → 0 , ϖ 4 ( ∞ ) → 0 .

Step 2: Presenting parameter p in Eqs. (20)–(22):

(23) ϖ ′ 3 + ϖ 1 ( ϖ 3 − 1 ) p − 2 m ( m + 1 ) ( ϖ 2 ) 2 + 2 ( m + 1 ) Q e − β 1 ( ξ + a ) + 2 ( m + 1 ) λ T ( ϖ 4 + λ C ϖ 6 ) = 0 ,

(24) ( 1 + Rd ) ϖ ′ 5 + Pr ( ϖ 5 − 1 ) p ( Nb ϖ 7 + Nt ϖ 5 + ϖ 1 ) + Q e exp ( − n η ) + Pr Ec ( ϖ 3 ) 2 − 2 Pr ( m + 1 ) ( ( 2 m − 1 ) ϖ 2 ϖ 4 ) = 0 ,

(25) ϖ ′ 7 + Nt Nb ϖ ′ 5 − 2 ( 2 m − 1 ) ( m + 1 ) Sc ϖ 2 ϖ 6 + Sc ϖ 1 ( ϖ 7 − 1 ) p − Sc K c ( 1 + δ ϖ 4 ) n ϖ 6 exp − E 1 + δ ϖ 4 = 0 .

Step 3: By applying the numerical implicit scheme.

(26) U i + 1 − U i Δ η = A U i + 1 , W i + 1 − W i Δ η = A W i + 1 , or ( I − Δ η A ) W i + 1 = W i .

Finally, we obtain the iterative form as follows:

(27) U i + 1 = ( I − Δ η A ) − 1 U i , W i + 1 = ( I − Δ η A ) − 1 ( W i + Δ η R ) .

Further, the discretized form is numerically solved through Matlab software.

Table 1 reveals the evaluation of the present work with the existing studies. It has been perceived from Table 1 that the PCM outcomes are reliable and accurate.

Table 1

Validation of the present results with published studies

A	m	Vaidya et al. [43]	Fang et al. [47]	Prasad et al. [48]	Present results
0.2	−0.3	0.0689	0.0723	0.0812	0.08124245
	−0.2	0.3011	0.5000	0.5000	0.5000021
	1.0	1.0511	1.0614	1.0524	1.0524468
	3.0	1.1003	1.0905	1.0905	1.0905843
	4.0	1.1166	1.1176	1.1166	1.1166686
0.4	−0.4	1.1645	1.1647	1.1645	1.1645123
	−0.2	1.0000	1.0000	1.0000	1.0000986
	1.0	1.0134	1.0214	1.0214	1.0214543
	2.0	1.0358	1.0359	1.0358	1.0358086
	4.0	1.0486	1.0466	1.0466	1.0466432

4 Results and discussion

The energy and mass transfer across the mixed convective nanofluid flows over an extending Riga plate is reported in the current analysis. The fluid flow is studied under the influences of viscous dissipation, EHSS, activation energy, and thermal radiation. The Buongiorno’s concept is utilized for the thermophoretic effect and Brownian motion. The results obtained through PCM are presented through figures.

Figures 2–6 display the behaviour of velocity curve f ′ ( η ) versus the divider thickness parameter a, power index of velocity m, Hartmann number Q, thermal Grashof number λ T , and mass Grashof number λ C , respectively. Figure 2 illustrates that fluid velocity boosts with the mounting values of divider thickness factor a. Physically, the sheet distending velocity increases and fluid viscosity lessens with the enhancing outcome of a, as a result, such phenomena are observed. Figures 3 and 4 specify that fluid velocity declines with the variation of m, although evolving with the impact of Q. The velocity field is the reducing function of Q, however, in the current incident; the flow stream improves due to magnetic impact. Figures 5 and 6 entitle the significances of λ C and λ T on the fluid velocity, respectively. In fact, the positive variation of λ T assistant to the flow, whereas the negative variation of λ T disclose the conflicting behaviour. Moreover, the gravitational force turns into more active and stretching rate of surface drops, which results in such scenarios as revealed in Figures 5 and 6.

Figure 2

Velocity profile against divider thickness factor a.

Figure 3

Velocity profile against power index m.

Figure 4

Velocity profile against modified Hartmann number Q.

$Figure 5 Velocity profile against λ T {\lambda }_{\text{T}} .$

Figure 5

Velocity profile against λ T .

$Figure 6 Velocity profile against λ C {\lambda }_{\text{C}} .$

Figure 6

Velocity profile against λ C .

Figures 7–11 depict the nature of Rd, Bi, Eckert number Ec, Nt, and Q _e, respectively, on the θ ( η ) . Physically, the thermal radiation variable enhances fluid temperature by transferring thermal energy via electromagnetic radiation generated by the fluid. When the thermal radiation variable rises, additional heat passes through via radiation, increasing the fluid temperature. This factor impacts the rate at which heat passes within the fluid and its surrounding environment, which in turn affects the fluid’s overall temperature (Figure 7). The viscosity of the fluid dispels during flow into the kinetic energy, which increases the temperature of the fluid, and is known as viscous dissipation as displayed in Figure 8. Figure 9 reveals the effect of Bi on the thermal profile. It can be seen that the fluid temperature augments with the effect of Bi. Figure 10 demonstrates the effect of the thermophoresis term Nt on the heat profile θ ( η ) . The implication of Nt intensifies the θ ( η ) as shown in Figure 10. The thermophoresis factor elevates fluid temperature by causing atoms in a fluid to migrate to higher temperatures areas. The motion of elements can cause a rise in the temperature inside the fluid as they develop in hotter areas. The thermophoresis term influences the temperature transportation inside the fluid. Figure 11 displays the impact of Q _e on the energy field. The impact of Q _e on increasing fluid temperature is caused by the introduction of heat energy into the system, which raises the fluid temperature. When heat is introduced to a fluid via a heat source, the temperature of the fluid rises as the energy is consumed. On the other hand, the heat sink variable reduces fluid temperature through eliminating heat energy from the system. Heat sinks act as a process to remove excess heat from the fluid, causing the temperature to drop. As a result, heat source factor increase the fluid temperature by adding heat energy, whereas heat sink parameters decrease fluid temperature through expelling heat energy from the system. Figure 12 clarifies the importance of Kr and Sc on the concentration field φ ( η ) . The impacts of both terms lessen the mass profile. The dissemination ratio is concentrated with the effect of Sc, which falls the mass transport as exposed in Figure 12. Likewise, the outcome of Kc declines the mass conduction rate φ ( η ) . Figure 13 regulates the effect of Nb and E on the φ ( η ) . Physically, the smallest volume of energy requisite for a chemical reaction is named as activation energy. So, the effect of E speed up the mass distribution rate as revealed in Figure 13. Similarly, the effect of Nb also increases the mass diffusion rate φ ( η ) .

Figure 7

Thermal profile against heat radiation Rd.

Figure 8

Thermal profile against Eckert number Ec.

Figure 9

Thermal profile against Biot number Bi.

Figure 10

Thermal profile against thermophoresis Nt.

Figure 11

Thermal profile against heat source/sink Q _e.

Figure 12

Mass curve against chemical reaction Kc and Schmidth number Sc.

Figure 13

Mass curve against Brownian motion Nb and Activation energy E.

Table 2 discloses the numerical outcomes for shear stress, Nusselt number, and Sherwood number. It can be remarked that the intensifying values of Kr and Pr augment the rate of energy and velocity transmission. The Ec has the similar behaviour as Kr and Pr, though drops the mass conveyance rate.

Table 2

Numerical outcomes for Sherwood number, skin friction, and Nusselt

Pr	Kc	Nt	Nb	Ec	λ C	λ T	Q	m	f ″ ( η )	θ ′ ( η )	φ ′ ( η )
1.0	0.3	0.5	0.5	0.1	0.2	0.1	0.1	0.4	0.028864	0.425904	0.622742
								0,8	0.788763	0.336135	0.562264
								1.2	1.078124	0.324863	0.523164
							0.1	0.4	0.311535	0.141774	0.336641
							0.4		0.164165	0.173934	0.260790
							0.7		0.025644	0.196313	0.285187
							0.1		1.009531	0.316645	0.216600
						0.1			0.749540	0.325513	0.524064
						0.2			0.687721	0.331676	0.431635
					0.2	0.3			0.792933	0.322143	0.322124
					0.4				1.001235	0.323735	0.324013
					0.6				1.010136	0.326832	0.229975
				0.1	0.2				0.621246	0.031698	0.037974
				0.3					0.604724	0.033932	0.032035
				0.5					0.589085	0.031012	0.019921
			0.5	0.1					1.001336	0.323746	0.414074
			1.0						0.795237	0.324243	0.141287
			1.5						0.790535	0.326112	0.056989
		0.5	0.5						0.794236	0.324236	0.141212
		1.0							1.006163	0.321051	0.483309
		1.5							1.006037	0.321809	1.025121
	0.3	0.5							0.685524	0.322121	0.522273
	0.5								0.888634	0.320843	0.520836
	0.7								0.785025	0.320154	0.521085
1.0	0.3								0.783963	0.282465	0.588897
2.0									0.788782	0.221309	0.621282
3.0									0.792801	0.309810	0.546144

5 Conclusions

The fluid flow across a Riga plate has several applications in the field of aerodynamic efficiency of aircraft, automobiles, and heat propagation. Therefore, the energy and mass transfer across the mixed convective nanofluid flows over an extending Riga plate are reported in the current analysis. The fluid flow is studied under the influences of viscous dissipation, EHSS, activation energy, and thermal radiation. The modelled are simplified into lowest order by using similarity transformation, which are numerically solved through the PCM. From the graphical analysis, the following findings are deducted:

The fluid velocity boosts with the mounting values of divider thickness parameter a.
The fluid velocity declines with the variation of m, although evolving with the impact of Q.
The positive variation of λ T enhances the flow, whereas the negative variation of λ T discloses the conflicting behaviour.
The thermal radiation variable enhances fluid temperature by transferring thermal energy via electromagnetic radiation generated by the fluid.
The effect of heat source sink parameter drops the energy field.
The impacts of Kr and Sc lessen the mass profile.
The effect of Nb and activation energy enhances the mass profile.
The fluid temperature also improves with the effect of Biot number, Eckert number, and heat source factor.

The current study can be expanded in the future to include more complexity and variables. Some possible directions for future studies may involve:

Including more realistic BCs.
Scrutinizing different sorts of nano-particulates.
Observing the influence of other external factor.
Reconnoitering different heat source/sink profiles.

Acknowledgments

The authors thank the KKU research unit for the financial and administrative support under grant number 593 for year 44.

Funding information: The KKU research unit for the financial and administrative support under grant number 593 for year 44.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: All data generated or analysed during this study are included in this published article.

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Received: 2024-01-10

Revised: 2024-03-12

Accepted: 2024-03-16

Published Online: 2024-05-16

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/phys-2024-0020

Keywords for this article

thermal radiation; viscous dissipation; numerical solution; mixed convection; exponential heat source/sink; slender sheet

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