Convective flow of a magnetohydrodynamic second-grade fluid past a stretching surface with Cattaneo–Christov heat and mass flux model

Nomenclature

u , v

velocity components

x , y

coordinates

B 0

strength of magnetic field

u w

stretching velocity

T , T f , T ∞

surface, fluid, and free stream temperature

C , C f , C ∞

surface, fluid, and free stream concentration

H , H f , H ∞

surface, fluid, and free stream microorganism concentration

h f

heat transfer coefficient

h g

mass transfer coefficient of the nanoparticles

h m

mass transfer coefficient of the microorganisms

λ

second-grade fluid parameter

M

magnetic factor

γ 1

thermal Biot number

γ 2

nanoparticle concentration Biot number

γ 3

microorganism concentration Biot number

Nb

Brownian motion factor

Nt

thermophoresis factor

Pr

Prandtl number

Rd

radiation factor

Sc

Schmidt number

E

activation energy

Pe

Peclet number

Le

Lewis number

δ

concentration difference factor

a

stretching constant

α e , α c

thermal and mass relaxation time factors

D B , D T , D m

Brownian, thermophoresis, and microorganism diffusion coefficients

Nu x , Sh x , Nm x

Nusselt, Sherwood, and microorganism density numbers

σ T

temperature difference factor

Abbreviations

PDEs, ODEs

partial and ordinary differential equations

SWCNTs

single-wall carbon nanotubes

MHD

magneto-hydrodynamics

EMHD

electro-magneto-hydrodynamics

HAM

homotopy analysis method

1 Introduction

During past few decades, non-Newtonian fluid has turn into a significant field of inquiry as such fluid has widespread applications in technology and engineering, for instance, biological liquid motion, lubricants’ production, and plastics fabrication. Such fluids are highly viscous and are extensively used in various fabrication phenomena. Biswal et al. [1] considered radiated non-Newtonian fluid motion on a nonlinear elongating sheet and concluded that with expansion in magnetic effects, the velocity panels have deteriorated. Iqbal et al. [2] analyzed thermal transportation for Maxwell fluid flow on a vertically elongating sheet with impacts of heat and mass flux model suggested by Cattaneo–Christov. Sharma and Shaw [3] have explored non-Newtonian magneto-hydrodynamics (MHD) liquid motion over an elongating sheet that is affected by nonlinearly radiated as well as viscously dissipated influence and have deduced that the fluid motion, temperature as well as mass diffusions have not been remained consistent in case of non-Newtonian liquid motion. Iqbal and Saleem [4] explained convective thermal transportation phenomenon of Casson flow of fluid through a vertical conduit and noted that thermal panels have augmented for expansion in Eckert number. Khalil et al. [5] have discussed the impacts of non-Newtonian fluid flow upon a stretched sheet using the effects of varying fluid characteristics upon the fluid motion and have emphasized that improvement in Deborah number has degenerated the concentration characteristics. Nandi et al. [6] have discussed computationally MHD Carreau nanofluid flow on a stretching sheet with optimization of entropy and have noted that velocity distribution has deteriorated with escalation in Weissenberg number. Asogwa et al. [7] debated on non-Newtonian electro-magneto-hydrodynamics (EMHD) fluid flow on a Riga surface using ramped energy and have proved that with growth in Schmidt and Prandtl numbers, the Sherwood and Nusselt numbers have been augmented. Farahani et al. [8] inspected the governing non-Newtonian flow of fluid with thermal transportation through a channel by employing triangular porous ribs. Rehman et al. [9] scrutinized mathematically the non-Newtonian fluid flow on an elongating sheet with dual stratified effects and have proved that velocity distribution has improved with upsurge in material factor.

The Fourier’s law of thermal conduction is a substantial model that is applied in physics for mass as well as thermal transportation processes. There are numerous industrial and engineering applications of these processes, such as coolant of electric devices and processing of food. Fourier [10] has deliberated on heat transportation phenomenon, whereas Fick [11] has exposed the transportation of mass. But these phenomena [11] were not appropriate due to the fact that deviations in relaxation have been affected critically by diffusions of mass as well as temperature. As a consequence, Cattaneo [12] has modified these concepts that were improved further by Christov [13]. Waqas et al. [14] have debated on numerical solution for nano-liquid using the influence of heat flux and concluded that temperature flow has diminished with expansion in Cattaneo and Christov factors. Gireesha et al. [15] examined the consequences of Cattaneo–Christov model of heat/mass flux and noted that fluid heat flow properties have decayed with the augmentation of Cattaneo–Christov components. Reddy et al. [16] deliberated the Cattaneo–Christov model of heat/mass flux characteristics for nanofluid flow upon gyrating cylinder and concluded that temperature has been weakened with intensification in temperature slip factor. The influences of Cattaneo–Christov model of heat flux upon MHD Maxwell nanoparticle flow upon bi-directional stretching sheet were scrutinized in a previous study [17]. Related ideas can be perceived in previous studies [18–23].

The least quantity of energy that is needed for origination of a reaction is identified as activation energy. In the process of mass transportation, the mutual effects of chemical reactivity and activation energy are quite significant. This concept has widespread applications at industrial level, such as coolant of nuclear reactant, thermal oil recovery, chemical engineering, and geothermal reservoirs. Raza-E-Rabbi et al. [24] scrutinized MHD fluid flow using activated energy as well as nonlinear radiations and have concluded that with growth in thermophoresis and Brownian factors, the temperature is upsurge. Zeeshan et al. [25] studied numerically the consequences of MHD on flow of nano-liquid on penetrable stretched cylinder with impact of activated energy and deduced that with upsurge in activation energy parameter, the Sherwood number has grown up. Khan et al. [26] have inspected bioconvective nanofluid flow with activated energy and viscously dissipated effects and underlined that mass diffusion has dropped with thermophoretic and Brownian factors, whereas it has supported by growth in activation energy factor. Azam [27] has inspected nonlinearly thermally radiated Maxwell MHD fluid flow using activation energy as well as the influence of Cattaneo–Christov model of heat flux. Kumar et al. [28] have debated on the fabrication of entropy for slip fluid flow of nanoparticles past a gyrating disk for nonlinearly mixed convective activation energy.

The collection of microorganism is combined with a fluid for the prevention of particles accumulation. For improved bio-convection phenomenon, a better fluid mixture is needed, that is why, microorganisms are added to fluid. The phenomenon of bio-convection is a consequence of density gradient of microorganism. This process has extensive applications, for instance, oil recovery technique, oil and gas industry, and cancer treatment. Bhatti et al. [29] deliberated on spinning of microorganism in nanofluid flow through gyratory plates, placed in a permeable medium, and have concluded that growth in Peclet number has augmented the profiles of microorganism, whereas it has supported by the upsurge in bio-convective Schmidt number. Upreti et al. [30] have explored the Buongiorno model effects on flow of MHD nanoparticles on a Riga surface with influence of microorganisms and exposed that thermal flow amplified, whereas fluid velocity has declined for expansion in magnetic factor. The thermal behavior of nanofluid in a permeable conduit using the influence of microbes’ presence upon liquid motion was considered in a previous study [31]. Khan et al. [32] scrutinized bio-convective nano-liquid flow past a thin rotary needle. Faizan et al. [33] have surveyed the formation of entropy and flow of fluid on a Riga surface using gyrotactic microorganisms and proved that velocity panels upsurge with Deborah number. Madhukesh et al. [34] debated on nanoparticle flow with the influence of carbon nanotubes (CNTs) and microorganisms on a Riga surface using heat sink/source.

Thermal radiations are produced whenever heat is converted to EMHD radiations during the movement of charge particles in the materials. Rehman et al. [35] discussed a relative work on thermal transportation for MHD fluid flow subjected to thermal radiations upon plane as well as cylindrical surfaces and have concluded that the thermal distribution is higher for cylindrical surface. Hussain et al. [36] have discussed that the temperature distribution has improved with upsurge in radiation. Shaw et al. [37] have explored hydro-magnetic flow of fluid using thermal radiations and arbitrary Prandtl number and emphasized that thermal distributions have amplified with progression in radiations factor for Prandtl number in the range 10 − 10 ≤ Pr ≤ 10 4 . Lone et al. [38] argued on MHD fluid flow on a surface with thermal radiation effects and presumed that thermal panels have been improved with upsurge in Eckert number as well as magnetic and radiation factors. Ibrahim et al. [39] surveyed on MHD liquid flow along a stretching plate subject to thermal radiations. Bilal et al. [40] observed the influences of thermal radiations on a stretched permeable sheet. Dogonchi et al. [41] scrutinized the water-based nanoparticle flow amid two triangular enclosures. Further related studies can be perceived in previously published works [17,42,43,44,45]. Adnan et al. [46] inspected the thermal performance stagnant point nanofluid flow on a permeable and vertically placed cylinder with impacts of nonlinear thermal radiations. Turkyilmazolu [47] inspected the fluid flow on porous slider and has discovered that when a flat slider expands, it reduces lift and drag, while the opposite occurs when it contracts. This has practical implications as expanding the slider can result in less frictional resistance during operation. Rehman et al. [48] surveyed on MHD 3D flow of nanofluid on a rotary disk by considering uniform suction with outcome as dominant behavior of magnetic factor on velocity distribution. Turkyilmazolu [49] focused on thermal transference development capability for the Cattaneo–Christov heat flux model, with the incorporation of non-Fourier effects. Muhammad et al. [50] discussed computationally partial differential equation (PDE) hydrodynamics fluid flow using finite volume method.

The homotopy analysis method (HAM) stands as a powerful and flexible analytical approach for addressing nonlinear differential equations [51,52]. At its core, HAM capitalizes on the concept of constructing a homotopy a continuous mapping from a known, solvable problem to the target nonlinear problem. The essence of HAM lies in its ability to transform a complex nonlinear problem into a more manageable one by introducing an auxiliary linear problem [53]. HAM’s versatility is exemplified by its successful application across diverse fields such as fluid dynamics, heat transfer, biology, and engineering, where nonlinearities abound. Notably, the method’s analytical nature allows researchers to derive explicit expressions, providing not only solutions but also valuable insights into the underlying physical phenomena. Its efficiency, combined with the adaptability to various nonlinear scenarios, positions HAM as a valuable tool in the researcher’s arsenal [54]. Furthermore, HAM’s educational value cannot be overstated, serving as a pedagogical instrument to elucidate the intricacies of analytical methods for nonlinear problem-solving. HAM represents a sophisticated yet accessible analytical framework that has proven instrumental in unraveling the complexities of nonlinear systems across scientific and engineering disciplines. This approach provides the solution in function series form [55].

This study explores for the first time the convective flow of a second-grade fluid on a stretching surface, incorporating the Cattaneo–Christov model for heat and mass flux with impacts of gyrotactic microorganisms, thermal radiations along with chemically reactivity, thermophoresis, Brownian motion, and activated energy. The magnetic field impact is also considered in this work. In this investigation, a mathematical model is meticulously formulated in PDE form and then converted to ordinary differential equations (ODEs) using appropriate variables. Finally, HAM becomes instrumental in solving these transformed equations. This work will answer the following research questions:

1. How does the inclusion of the Cattaneo–Christov heat and mass flux model influence the convective flow characteristics of a second-grade fluid?

2. What role does the stretching surface play in shaping the convection patterns of the second-grade fluid, considering the Cattaneo–Christov model for heat and mass flux?

3. How do the Cattaneo–Christov parameters impact the heat and mass transfer processes in the convective flow on the elongating sheet?

4. What are the impacts of emerged factors on the flow panels when the nanofluid flows over a stretching surface?

5. What are the impacts of the emerged factors on the heat and mass transfer rates and density number?

Thus, to engage with the above research questions, a mathematical model for the proposed flow problem is designed, which is specified in Section 2. The solution of the modeled equations is described in Section 3. The discussion is presented in Section 4. Finally, the outcomes of current analysis are shown in Section 5.

2 Formulation of the problem

Consider the flow of an incompressible second-grade fluid on an elongating sheet. The velocity of stretching surface, denoted by u w = a x , is taken along x-direction, while y-axis is normal to the fluid flow. u and v are the velocity components along x- and y-directions, respectively. T , T w , and T ∞ represent the temperature, surface temperature, and free-stream temperature, respectively. Similarly C , C w , and C ∞ represent the nanoparticle concentration, surface concentration, and free-stream concentration, respectively. Also, H , H w , and H ∞ represent the microorganism concentration, microorganism surface concentration, and microorganism free-stream concentration, respectively. A magnetic field of strength B 0 is applied normal to the fluid flow (along y-axis). The Cattaneo–Christov heat and mass flux model is considered to investigate the heat and mass transfer rates. Furthermore, the thermophoresis, Brownian motion, chemical reaction, and activation energy impacts are taken into consideration. Convective conditions at the surface of the sheet are considered in this analysis. By employing the aforementioned assumptions, we have the following equations [55,56,57,58] (Figure 1):

Figure 1

Geometrical view of flow.

Boundary conditions are described as [59]:

In the aforementioned equations, μ is the dynamic viscosity, ρ is the density, α 1 is the material parameter, σ is the electrical conductivity, C p is the specific heat, k is the thermal conductivity, σ ⁎ is the Stefan–Boltzmann constant, k ⁎ is the mean absorption coefficient, D B is the coefficient of nanoparticles’ Brownian diffusion, D T is the thermophoretic coefficient, α 2 is the relaxation time of heat diffusion, α 3 is the relaxation time of mass diffusion, k r is the chemical reaction coefficient, E a is the activation energy coefficient, b c is the chemotaxis coefficient, W c is the maximum cell swimming speed, D m is the Brownian diffusion of microorganisms, h f is the heat transfer coefficient, h g is the nanoparticle’s mass transfer coefficient, and h m is the microorganisms’ mass transfer coefficient.

The transformable variables are depicted as:

Using the aforementioned similarity transformations, we have the following:

with related constraints at boundaries

In the aforementioned equations, λ = α 1 b μ is the second-grade fluid parameter; M = σ B 0 2 ρ b is the magnetic factor; γ 1 = h f k υ b , γ 2 = h g D B υ b , and γ 3 = h m D m υ b are the thermal, concentration, and microorganism concentration of Biot numbers, respectively; α e ( = b α 2 ) and α c ( = b α 3 ) are the thermal and mass relaxation time factors, respectively; Nt = τ D T ( T f − T ∞ ) υ T ∞ is the thermophoresis factor; Nb = τ D B ( C f − C ∞ ) T ∞ is the Brownian motion factor; Rd = 16 σ ⁎ T ∞ 3 3 k ⁎ is the thermal radiation factor; Sc = υ D B is the Schmidt number; σ T = T f − T ∞ T ∞ is the temperature difference factor; Pr = υ ( ρ C p ) k is the Prandtl number; E = E a k B T ∞ is the activation energy factor; Pe = b c W c D m is the Peclet number; Lb = υ D m is the Lewis number; and δ = H f − H ∞ H ∞ is the concentration difference factor.

The interested engineering quantities are described as:

Eqs. (14–16) are reduced as:

3 HAM solution

HAM is a semi-numerical approach used for solving nonlinear differential equations. Introduced by Prof. Shijun Liao in the early 1990s, HAM involves constructing a homotopy, a continuous deformation from a known, easily solvable problem to the original nonlinear problem. This is achieved by introducing an embedding parameter, and the solution is expressed as a series in powers of this parameter. The homotopy equation combines the original nonlinear problem with an auxiliary linear problem, and as the parameter varies, the solution deforms from the known solution of the linear problem to the desired solution of the nonlinear problem.

3.2 Convergence of HAM

For convergence of HAM has the auxiliary factors, which are described by ℏ f , ℏ θ , ℏ ϕ , and ℏ ψ . In this method, the region of convergence has been established by employing ℏ -curves corresponding to a 15th order of approximation, as illustrated in Figure 2. It has been noted from this figure that the zone of velocity convergence is − 3.1 ≤ ℏ f ≤ 0.0 , for temperature, it is − 3.5 ≤ ℏ θ ≤ 0.5 , while for concentration, it is given by the range − 3.0 ≤ ℏ Φ ≤ 0.5 . Moreover, the zone of convergence for motile panels is given as − 2.9 ≤ ℏ ψ ≤ 0.5 .

Figure 2

h-curves for f ″ ( 0 ) , θ ′ ( 0 ) , Φ ′ ( 0 ) , and ψ ′ ( 0 ) .

4 Results and discussions

This study investigated the second-grade fluid flow encompassing gyrotactic microbes on a stretched sheet. The fluid flow is affected by thermal radiation, chemical reactivity, and activation energy factors. The Cattaneo–Christov thermal and mass flux model is also incorporated in the considered flow problem. During the non-dimensionalization process of the modeled equations, some dimensionless factors are found, and the foremost aim of this analysis is to study the influences of these factors on the flow profiles. The default values of the embedded factors are chosen as λ = 0.6 , M = 1.3 , γ 1 = 0.5 , γ 2 = 0.5 , γ 3 = 0.5 , α e = 0.7 , α c = 0.6 , Nt = 0.1 , Rd = 0.3 , Nb = 0.1 , Pr = 0.72 , Sc = 0.5 , σ T = 1.1 , E = 1.0 , Pe = 1.5 , Le = 1.1 , and δ = 1.1 . The impact of these parameters on various panels of flow will be deliberated in forthcoming paragraph.

4.1 Velocity characteristics

The effects of several emerging factors upon the velocity profiles of fluid are discussed graphically in Figures 3(a) and (b). In Figure 3(a), it is detected that with improvement in magnetic factor M , the fluid motion declines. It is because of the fact that greater M generates the Lorentz force, which retards the fluid motion. The contrasting force that is produced due to Lorentz force increases the friction force at the sheet’s surface and causes a reduction in the fluid velocity. Therefore, the greater M reduces the velocity characteristic, as depicted in Figure 3(a). In Figure 3(b), the influence of second-grade fluid parameter λ on the velocity panels is displayed. In this figure, it is observed that fluid motion upsurges with the growth in λ . The second-grade fluid parameter characterizes the material’s rheological properties, impacting its response to shear stress. As λ grows, the fluid exhibits a more pronounced shear-thinning behavior, leading to enhanced velocity gradients. Consequently, the overall velocity distribution across the fluid is heightened. Hence, expansion in λ causes intensification in fluid motion, as shown in Figure 3(b).

Figure 3

(a) Impact of magnetic factor M on velocity distribution f ′ ( η ) , and (b) impact of second-grade fluid factor λ on velocity distribution f ′ ( η ) .

4.2 Temperature characteristics

The impression of numerous flow parameters upon temperature characteristic is shown in Figure 4(a)–(e). The effect of Rd on energy curve is shown in Figure 4(a). It is detected in Figure 4(a) that with the growth in Rd , there is an augmentation in thermal panels. Actually, as the thermal radiation factor rises, the emission of electromagnetic waves also increases, signifying a greater release of thermal energy. This heightened radiation contributes to an elevated thermal distribution across the system, resulting in increased temperatures. The interpretation underscores the direct relationship between the thermal radiation factor and the thermal profile, emphasizing the influential role of radiative heat transfer in shaping temperature distributions within the system. In Figure 4(b), the impact of Pr on temperature characteristic is described. It is apparent from this figure that thermal flow declines for expansion in Pr . This can be explained physically as Pr that gives the ratio of momentum diffusivity to thermal diffusivity, which influences the efficiency of heat transfer in the fluid. A higher Prandtl number suggests the reduced thermal diffusivity relative to momentum diffusivity, slowing down thermal distribution. This phenomenon implies that heat transfer becomes less effective, leading to a delayed spread of thermal energy within the fluid as portrayed in Figure 4(b). The impact of thermophoresis factor Nt upon temperature characteristic is illustrated in Figure 4(c). Since in case of thermophoresis factor we have mathematically as Nt = τ D T ( T f − T ∞ ) / υ T ∞ , which can be interpreted as for grater values of Nt , there is higher gradient in temperature between the thermal diffusion at surface of sheet to free stream. Hence, due to the growth in Nt , more heat diffuses, which supports the strength of thermal layer at boundary. Therefore, with higher values of Nt , temperature characteristic upsurges as portrayed in Figure 4(c). The impact of Nb upon fluid temperature is portrayed in Figure 4(d). It is professed in this figure that with the expansion in Nb , the temperature profile augments. Actually, for higher values of Nb , there are more collisions among the nanoparticles due to which more heat diffuses. Because during the growth in Brownian motion, the internal energy of a fluid particles is converted to heat energy that is responsible for maximum diffusion of temperature. Hence, with the higher values of Nb , the temperature characteristic upsurges is portrayed in Figure 4(d). The influence of α e upon temperature profile is described in Figure 4(e). With the higher values of α e , the augmenting behavior in temperature characteristic is perceived. Actually, with the growth in α e , more time is required for heat to be diffused that gradually weakens the thermal layer at the boundary of flow system. Hence, with the growth in α e , the temperature characteristic reduces, as depicted in Figure 4(e).

Figure 4

(a) Impact of thermal radiation factor Rd on temperature distribution θ ( η ) , (b) impact of Pr on energy profile θ ( η ) , (c) impact of thermophoresis factor Nt on temperature distribution θ ( η ) , (d) impact of Brownian motion factor Nb on temperature distribution θ ( η ) , and (e) impact of thermal relaxation time factor α e on temperature distribution θ ( η ) .

4.3 Concentration characteristics

The impacts of numerous factors on Φ ( η ) are described in Figure 5(a)–(f). The impact of concentration relaxation time factor α c upon concentration profile is shown in Figure 5(a). A declining behavior is perceived in this figure. Similar to thermal relaxation time factor, more time is required for concentration to be diffused. Hence, with the upsurge in α c , the concentration characteristic reduces as portrayed in Figure 5(a). The influence of activation energy factor E upon concentration profile is shown in Figure 5(b). It is observed in this figure that with the growth in E , there is an augmentation in concentration profile. Actually, the enhancement of E results in the decay of Arrhenius-modified function that eventually endorsed the generation of chemical reaction. In this physical process, the concentration of fluid particles upsurges as portrayed in Figure 5(b). The effect of chemical reaction factor ω upon concentration profile is portrayed in Figure 5(c). Actually, greater activation energy requirement is necessary for molecules to overcome the energy barriers and traverse the concentration gradients. In practical terms, this manifests as a more pronounced resistance to the flow of substances, impacting various processes such as diffusion and chemical reactions. The augmented concentration profile reflects the increased energy investment needed for particles to overcome obstacles, indicating a more energetically demanding environment that influences the rate and efficiency of molecular transport and reactions within the system. Moreover, main reason for this change is that the number of solute molecules is experiencing chemically reactive effects and gets augmented with the growth in ω . In this process, the concentration characteristic diminishes. Figure 5(d) demonstrates the influence of Nb on concentration characteristic. In Figure 5(d), it is perceived that mass profile diminishes with the growing values of Nb . Actually with the upsurge in Nb , the zigzag motion among the fluid particles augments due to which more collisions among the nanoparticles occur. In this process, little mass disseminates and deteriorates the width of concentration boundary layer and ultimately reduced the concentration distribution. The impact of thermophoresis factor Nt upon concentration characteristic is shown in Figure 5(e). Here, it is perceived that growth in Nt augments the concentration distribution. Actually, for upsurge in Nt , more mass diffuses, which strengthens the concentration boundary layer. Hence, with the upsurge in Nt , there is an augmentation in concentration distribution as depicted in Figure 5(e). The influence of Sc upon concentration distribution is portrayed in Figure 5(f). From this figure, it has observed that with the growing values of Sc , there is a retardation in concentration characteristic. Actually, higher values of Sc are related to the weaker solute diffusions, and the concentration characteristic declines for growth in Sc . Thus, the thickness of solute boundary is larger for smaller values of Sc , and vice versa. Finally, it should be noted that the movement of molecules inside fluids is known as convection. It is one of the primary techniques for transferring heat and mass, which uses diffusion and the random Brownian motion of various liquid components. Convection in this sense refers to all advective and diffusive transport combined. It is only interpreted as an advective phenomenon, though. Convective heat transfer is the process through which heat is transferred while fluids are moving in bulk. Heat that is being transmitted and dispersed is highlighted.

Figure 5

(a) Impact of α c on concentration distribution Φ ( η ) , (b) impact of activation energy factor E on concentration distribution Φ ( η ) , (c) impact of chemical reaction factor ω on concentration distribution Φ ( η ) , (d) impact of Brownian motion factor Nb on concentration distribution Φ ( η ) , (e) impact of thermophoresis factor Nt on concentration distribution Φ ( η ) , and (f) impact of Schmidt number Sc on concentration distribution Φ ( η ) .

4.4 Microorganism characteristics

The effects of various factors on microorganism distribution ψ ( η ) are shown in Figure 6(a)–(c). The impact of bioconvection Lewis number Lb upon ψ ( η ) is depicted in Figure 6(a). Here, it is perceived that higher values of Lb diminish the microorganism panels. This can be explained, since Lb is in inverse relation to D n (microorganisms diffusions) because of which for higher values of Lb there a retardation in microorganisms diffusion that deteriorates the strength of microbes layer at boundary. Therefore, with upsurge in Lb , there is a decline in ψ ( η ) as depicted in Figure 6(a). The impression of Peclet number Pe on ψ ( η ) is portrayed in Figure 6(b). Again, a reducing behavior in ψ ( η ) is noted for higher values of Pe . This can be physically explained as W c (speed of swimming cell) and Pe are inversely related to D n (microorganism diffusions) and both are in direct relation. Moreover, the advection and diffusion rates are directly related to Pe . So, growth in Pe declines ψ ( η ) as presented in Figure 6(b). The effect of concentration difference factor δ upon ψ ( η ) is illustrated in Figure 6(c). In this figure, it is noted that with the growth in δ , ψ ( η ) experiences a decline in their characteristic features due to intensified microbes’ gradients. In fluid with substantial concentration differences, microorganisms face greater challenges in nutrient acquisition and energy utilization. The escalating gradient demands more efficient metabolic adaptations, potentially leading to a decline in growth rates and overall microbial vitality. This response reflects the organisms’ struggle to maintain optimal physiological functions amid increasingly disparate concentrations, highlighting the complex relationship between fluid flow and microbial behavior in their quest for survival and proliferation. Actually, for upsurge in δ , the boundary-layer thickness of the microorganism distribution retards that ultimately reduces as depicted in Figure 6(c).

Figure 6

(a) Impact of bioconvection Lewis number Lb on microorganism concentration distribution ψ ( η ) , (b) impact of bioconvection Peclet number Pe on microorganism concentration distribution ψ ( η ) , and (c) impact of concentration difference factor δ on microorganism concentration distribution ψ ( η ) .

5 Conclusion

This work investigated the second-grade fluid flow over a stretching surface using convective constraints at boundary. The flow is affected by the magnetic effect in normal direction to the fluid motion. The fluid flow is influenced by thermal radiation and activation energy. The Cattaneo–Christov model of thermal and mass flux is also incorporated in this analysis. After detailed insight of the problem, it is observed that:

1. With rise in magnetic factor, the fluid motion has weakened. Also, an upsurge in second-grade fluid parameter results in a retardation in viscous forces, due to which fluid particles dislocate quite easily and quite frequently that causes growth in velocity of fluid.

2. For escalation in radiation factor, more heat diffuses, due to which maximum thermal transmission takes place from a zone of higher thermal concentration to a region of low thermal concentration. The higher values of Prandtl number have a declining impact upon thermal characteristics.

3. With the growth in thermophoresis factor, more heat diffuses that supports the strength of thermal boundary layer and boosts the temperature distribution.

4. With upsurge in Brownian motion, there are more collisions among the nanoparticles because more heat diffuses because during the growth in Brownian motion, the internal energy of fluid particles is converted into heat energy, which is responsible for maximum diffusion of temperature. With the growth in thermal relaxation time factor, more time is required for heat diffusion that gradually weakens the thermal boundary layer and declines the temperature profile.

5. With the upsurge in thermal relaxation time factor, more time is required for concentration to be diffused that declines the concentration profile. For greater effect activation energy and thermophoresis factors, there is an augmentation in concentration profiles. Upsurge in chemical reaction and Brownian motion factors are responsible for retardation in concentration characteristics. With the growth in Schmidt number, there is retardation in concentration characteristic.

6. The microorganisms’ characteristics show a reducing behavior with growth in bioconvective Lewis, Peclet numbers, and concentration difference factor.

7. From the comparative analysis, a fine agreement exists between the existing and published results.

6 Future recommendations

In the future, this work can be extended for the three-dimensional second-grade fluid flows over a stretching surface. The authors can consider the slip conditions, convective condition, convective along with zero-mass flux conditions, etc. Additionally, some external forces such as magnetic field, porous medium, mixed convection, Joule heating, viscous dissipation, thermal-dependent heat source, and space-dependent heat source are also considered.