Home Significance of gyrotactic microorganisms on the MHD tangent hyperbolic nanofluid flow across an elastic slender surface: Numerical analysis
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Significance of gyrotactic microorganisms on the MHD tangent hyperbolic nanofluid flow across an elastic slender surface: Numerical analysis

  • Khalid Abdulkhaliq M. Alharbi , Muhammad Bilal , Aatif Ali EMAIL logo , Sayed M. Eldin , Alhanouf Alburaikan and Hamiden Abd El-Wahed Khalifa
Published/Copyright: August 5, 2023
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Nanotechnology Reviews
From the journal Nanotechnology Reviews Volume 12 Issue 1

Abstract

In the current study, we numerically analyze the significance of motile microbes on the magnetohydrodynamic steady convective streams of tangent hyperbolic (TH) nanofluid flow across an elastic nonlinearly stretching surface of an irregular thickness. The consequences of an external magnetic field, thermal radiation, and thermal conductivity are also examined on the TH nanofluid. The governing system of equations (nonlinear set of partial differential equations) is transfigured into a system of ordinary differential equations (ODEs) by using the similarity variable conversions. Furthermore, the reduced form of nonlinear ODEs is numerically computed through the parametric continuation method (PCM) using MATLAB software. The relative evaluation is carried out to authenticate the numerical outcomes. It has been observed that the energy field accelerates with the Rayleigh number, Weissenberg number, and Brownian motion. The mass propagation ratio improves with the effect of activation energy and decreases with the influence of chemical reactions. Furthermore, the motile microbes’ profile declined with the outcome of the Peclet and Lewis numbers. The skin friction increases up to 7.3% with various magnetic values ranging from 0.5 to 1.5. However, the energy transfer rate declines to 5.92%. The thermal radiation boosts the energy propagation rate and flow velocity by up to 11.23 and 8.17%, respectively.

Keywords: bio-convection; MHD; tangent hyperbolic; nanofluidics; elastic slender sheet; PCM; activation energy

Nomenclature

g

gravitational force

T w

surface temperature

2D

two-dimensional

m

power law index

k r 2

second-order chemical reaction

E a

activation energy

β

thermal expansion factor

τ

nanoparticles’ capacities ratio

T

temperature

n

motile microbes’ density

q r

thermal radiation

M

magnetic term

Rb

Rayleigh number

Nr

buoyancy factor

Pr

Prandtl number

Lb

Lewis number

Pe

Peclet number

Ω

motile microorganisms’ density

ε ( x )

variable thickness

B 0

magnetic field

Q 0

heat source term

σ

electrical conductivity

ρ f

density

ν

kinematic viscosity

C p

specific heat

K T

thermal conductivity

C

nanoparticle concentration

Wc

Weissenberg number

Le

Lewis number

Nb

Brownian motion

λ

mixed convection

Nt

thermophoresis term

χ

wall thickness

Rd

radiation term

Kr

chemical reaction factor

1 Introduction

Energy systems, automobiles, power generation, disease detection, and microelectronic device preservation are just a few of the many industries that rely on energy transfer. However, a regular fluid is not capable of satisfying these criteria of energy transition. Therefore, the inclusion of nano-particulates into the base fluid makes them more effective. Since the turn of the century, nanofluids have come to the forefront of several scientific and technological fields, most notably in the areas of energy transfer thermodynamics, solar collectors, and hydropower rotors [1]. Yang et al. [2] examined the thermal performance of TiO2-, SiO2-, and Al2O3-based nanoliquids numerically and experimentally. The results demonstrated that nanofluids are more efficient at lowering the surface heat. Thermophysical and electrical characteristics of nanoliquids comprising carbon nano-particulates in ethene were studied by Sobczak et al. [3]. The obtained results indicate that the specific surface area strongly affects these essential features of nanofluids made with carbon nanoparticles. Mebarek-Oudina et al. [4] described the outcome of the buoyant force on the nanofluid flow within a circular region. The results of the buoyancy force on the flow of a magnetized nanoliquid in circular porous media with a Cassini oval were investigated by Jalili et al. [5]. According to the findings, the convection process weakens as the volume fraction of solid nanoparticles increases. Amar et al. [6] evaluated the influence on energy transmission. Using motor oil, Alharbi et al. [7] documented the results of a turbulent, viscous flow of silver–gold hybrid nanofluids across a squeezing cylinder. It was noticed that the performance of hybrid nanoliquids is tremendously higher than that of conventional nanofluids [8]. The rectangular curved tubes of the reactors’ super-critical water-based nanofluid containing various nanoparticles were examined by Behzadnia et al. [9]. The outcomes reveal that the spherical alumina nanoparticles are the most effective. Atif et al. [10] estimated the energy propagation through the tangent hyperbolic (TH) nanoliquid flow. Shafiq et al. [11] recognized the properties of water’s B nanoliquid flow above the Riga plate. Goud and Nandeppanavar [12] reported the effect of magnetohydrodynamic (MHD) on energy transmission across a permeable medium. Some remarkable results may be found in previous studies [13–17].

The phenomenon known as bio-convection takes place when microorganisms move in a colony or single cell. The physical implication of bio-convection is effectively distributed across numerous industrial and environmental systems, including biofuel biotechnology related to mass transportation, enzyme biosensors, and ethanol [18,19]. Waqas et al. [20] evaluated the TH flow across a Riga plate with motile microbes. The findings demonstrate that the profile of microorganisms is enhanced with a first-order slip factor, while the Lewis number for bio-convection decreases. The MHD flow of the nanoliquid over a stretchy sheet was investigated by Elattar et al. [21], along with the effects of porous materials and the rate of entropy formation. The results indicate that an increase in Peclet and Schmidt numbers degrades the microbial density features. Waqas et al. [22] assessed the flow of the nanoliquid carrying floating microbes through a strained surface. The MHD bioconvective flow over an extending sheet was observed by Waqas et al. [23]. Bio-convection effects were assessed by Shah et al. [24] on the nanoliquid flow and motile microbes over an elongating surface. The findings show that for high numbers of Prandtl fluid parameters, the velocity gradient of both fluids exhibits positive conduct. The thermal transference of viscous incompressible nanoliquid flow with floating motile microbes was studied by Nabwey et al. [25]. Some detailed explanations regarding this can be found in previous studies [26–31].

Fluids can be split into two broad categories: those that follow Newton’s laws and those that do not. Non-Newtonian fluids are distinguished from Newtonian fluids by the fact that they do not respond to the Newtonian law of viscosity. There are many examples of these fluids all around us, each with its nonlinear stress–strain relationship. It has a wide variety of uses, from the ordinary to the technological. Examples of non-Newtonian fluids with high viscosity include condiments like mayonnaise and ketchup as well as paint, syrup, condensed milk, printing ink, glues, melts, mud, emulsions, soaps, and even toothpaste. In light of this essential use, scientists have investigated many features of non-Newtonian fluid using a wide range of model fluids, including the power-law fluid, second-grade fluid, and Prandtl fluid [32]. Alsallami et al. [33] evaluated the flow of a Marangoni Maxwell nanoliquid with entropy formation and Arrhenius activation energy through a turning disc. Shafiq et al. [34] numerically designated the TH nanoliquid flow across an expanding sheet. Mei et al. [35] observed the MHD nanoliquid flow in a microchannel. The thermal transmission through a non-Newtonian liquid was analyzed by Dong and Liu [36]. The results demonstrate that compared to a Newtonian fluid, non-Newtonian fluid is superior in terms of energy dispersion and thermal transit. The energy transfer characteristics of Oldroyd-B and Jeffrey fluids were explored by Sarada et al. [37] using a stretching sheet. The results show that the Oldroyd-B fluid has a faster velocity decrease and higher heat transfer than the Jeffry fluid at lower magnetic parameter values. Through parallel inclined plates, Shehzad et al. [38] explored multilayer coatings with a steady fluid flow. Shafiq et al. [39] statistically evaluated the bioconvective TH fluid flow. In the cervical canal, Ibrahim [40] elaborated on the results of magnetic strength on the thermal behavior of the Carreau nanofluid. Employing a double-diffusive Cattaneo–Christov model, Puneeth et al. [41] examined the properties of the Casson fluid with an impact of gyrotactic microorganisms. Such types of nanoliquid flows are recently reported in previous studies [42,43,44,45,46].

MHD is the study of how magnetic fields affect the motion of fluids in electrically conductive substrates. MHD is defined by a set of equations that combines Maxwell’s and the Navier–Stokes equations of electromagnetism and fluid dynamics, respectively. Some well-known applications of MHD can be found in the fields of geophysics, nuclear reactors, earthquake research, astrophysics, MHD pumps, geothermal energy extraction, blood flow measurements, engineering, sensor technology, and magnetic drug targeting [47]. Shafiq et al. [48] have conducted a numerical analysis of the fluid flow for thermal applications under the action of electromagnetohydrodynamics and the Darcy–Forchheimer medium. The effects of mass evaporation and heat radiation on the motion of a non-Newtonian nanoliquid in three dimensions over a porous sheet was addressed by Assiri et al. [49] and Shankar Goud et al. [50]. Abbas Khan et al. [51] assessed the applications of a micropolar nanofluid flow over a stretching sheet. The MHD micropolar nanoliquid was explored by Ali et al. [52] over an inclined stretched surface. The findings demonstrate that when the number of kinetic dust particles increases, the flow field’s velocity decreases, while the dust phase exhibits the opposite tendency. The hydrodynamic flow of the Casson fluid with various properties across an elongating surface was reported by Atif et al. [53]. According to the results, the velocity curve diminishes under the effect of a magnetic field. Recently, MHD fluid flow has been documented in previous studies [42,44,54–60].

The aim of this study is to numerically evaluate the significance of motile microbes on the MHD steady convective streams of the TH nanoliquid flow across an elastic stretching surface. The significance of thermal radiation, external magnetic field, and thermal conductivity is also examined on the TH nanofluid. The governing system of nonlinear partial differential equations (PDEs) is reformed to a system of ordinary differential equations (ODEs) using the similarity variable conversions. Furthermore, the reduced form of nonlinear ODEs is numerically computed by the PCM approach using MATLAB software. In the following sections, the phenomena are modeled, solved, and discussed. Finally, some novel findings are pointed out.

2 Mathematical formulations

The current analysis for the nanoliquid flow is shown in Figure 1. A 2-D steady MHD TH nanoliquid flow over an elastic surface with thickness ε ( x ) = 2 c ( x + b ) 1 − n 2 is studied. The thin sheet is vertically stretched by a gravitational force g. The magnetic field B ( x ) = B 0 ( x + b ) n − 1 2 is applied in the y-direction. The temperature T ∞ is kept constant, indicating that the nanoliquid is in a quasi-rest condition. Due to the poor electrical performance of the TH nanofluid, the magnetic Reynolds number is negligible. We expect that the addition of nano-particulates does not disturb the floating speed of motile microbes. The Hillesdon and Pedley [61] model is utilized to calculate the bioconvective transference with oxytactic bacteria. The modeled equations are formulated as follows [62–64]:

(1) ∂ u ∂ x + ∂ v ∂ y = 0 ,

(2) u ∂ u ∂ x + v ∂ v ∂ y = υ ( 1 − m ) ∂ 2 u ∂ y + 2 Γ υ m ∂ u ∂ y ∂ 2 u ∂ y 2 − σ B 2 ( x ) ρ + g β ρ f ( T − T ∞ ) ( 1 − C ∞ ) − ( ρ p − ρ f ) ( C − C ∞ ) − γ ( n − n ∞ ) ( ρ m − ρ f ) ,

(3) ( ρ C p ) u ∂ T ∂ x + v ∂ T ∂ y = K T ∂ 2 T ∂ y 2 + ∂ K T ∂ T ∂ T ∂ y 2 + τ D B ∂ C ∂ y + D T T ∞ ∂ T ∂ y 2 + ∂ qr ∂ y + Q 0 ( ρ c ) f ( T − T ∞ ) ,

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

(5) u ∂ n ∂ x + v ∂ n ∂ y − D n ∂ 2 T ∂ y 2 = d w c C w − C ∞ ∂ ∂ y n ∂ C ∂ y ,

where Q 0 is the heat source term; m is the power-law index; g is the gravitational acceleration; σ is the electrical conductivity; ρ f , ρ p , and ρ m are the fluid, nanoparticles mass, and microbes particle density; k r 2 is the second-order chemical reaction; E a is the activation energy; β is the thermal expansion volumetric factor; C p is the specific heat; τ is the capacity ratio of nanoparticles; K T is the present thermal conductivity; C, C w , and C ∞ are the nanoparticle, wall, and ambient concentrations; T , T ∞ , T w are the fluid, ambient, and wall temperatures; and n , n ∞ , n w are the motile microbes density, ambient motile microbes, and microbe concentration at the surface, respectively.

Figure 1 The TH nanoliquid flow across an elastic slender surface.
Figure 1

The TH nanoliquid flow across an elastic slender surface.

The thermal conductivity instead of thermophysical variable. K T is expressed as [63]

(6) K T = K 1 + A k T − T ∞ T w − T ∞ .

Furthermore, q r is defined as

(7) q r = − 4 ∂ e 3 β R ∂ T 4 ∂ y .

Now, by using the Taylor series,

(8) T 4 ≈ 4 T ∞ 4 T − 3 T ∞ 4 .

By using equations (7) and (8), we obtain

(9) ∂ q r ∂ y = 16 σ e T ∞ 3 3 β R ∂ 2 T ∂ y 2 ,

The boundary conditions are as follows:

(10) u = U w = U 0 ( b + x ) n , v = 0 , C = C w , T = T w , n = n w at y = ε ( x ) 2 .

The similarity transformations are [65] as follows:

(11) v = − U 0 ( x + b ) n − 1 ( n + 1 ) v ∞ 2 F ( ζ ) + n − 1 n + 1 ζ F ′ ( ζ ) , ϕ ( ζ ) = C − C ∞ C w − C ∞ , u = U w ( x + b ) n F ′ ( ζ ) , ζ = U 0 ( x + b ) n − 1 ( n + 1 ) v ∞ 2 v ∞ y , θ ( ζ ) = T − T ∞ T w − T ∞ , ψ = 2 v ∞ ( x + b ) n + 1 U 0 ( n + 1 ) F ( ζ ) , ℏ ( ζ ) = n − n ∞ n w − n ∞ .

We obtain the following:

(12) ( W e m f ″ + ( 1 − m ) ) f ‴ − 2 n n + 1 F ′ 2 + F F ″ − 2 M n + 1 F ′ + ω ( θ − Nr ϕ − Rb ℏ ) = 0 ,

(13) ( 1 + Rd ) θ ″ + Pr F θ ′ + A k θ θ ″ + A k θ ′ 2 + Nb θ ′ ϕ ′ + Nt θ ′ 2 + hs θ = 0 ,

(14) ϕ ″ + Le Pr F ϕ ′ + Nt Nb θ ″ − Sc σ ( 1 + δ θ ) n φ exp − E 1 + δ θ = 0 ,

(15) ℏ ″ + Pr Lb F ℏ ′ − Pe ( ℏ ′ ϕ ′ + Ω ϕ ″ + ϕ ″ ℏ ) = 0 .

The boundary conditions become

(16) lim ζ → χ F ( ζ ) = 1 − m 1 + m χ , lim ζ → χ F ′ ( ζ ) = θ ( χ ) , lim ζ → χ ϕ ′ ( ζ ) = lim ζ → χ ℏ ( χ ) = 1 , lim ζ → ∞ F ′ ( ζ ) = θ ( χ ) , lim ζ → ∞ ϕ ′ ( ζ ) = lim ζ → ∞ ℏ ( χ ) = 0 .

For further oversimplifications,

(17) ζ = η + χ , F ( ζ ) = F ( η + χ ) = f ( ζ ) , θ ( ζ ) = θ ( η + χ ) = Θ ( ζ ) , ϕ ( ζ ) = ϕ ( η + χ ) = φ ( ζ ) , ℏ ( ζ ) = ℏ ( η + χ ) = ƛ ( ζ ) .

As a result of equation (17), equations (12)–(15) become

(18) ( ( 1 − m ) + m Wc f ″ ) f ‴ − 2 n n + 1 f ′ 2 + f f ′ ′ − M f ′ + λ ( Θ − Nr φ − Rb ζ ) = 0 ,

(19) ( 1 + Rd ) Θ ″ + Pr f Θ ′ + A k Θ Θ ″ + A k Θ ′ 2 + Nb Θ ′ φ ′ + Nt Θ ′ 2 + hs Θ = 0 ,

(20) φ ″ + Le Pr f φ ′ + Nt Nb Θ ″ − Sc σ ( 1 + δ Θ ) n φ exp − E 1 + δ Θ = 0 ,

(21) ƛ ″ + Pr Lb f ƛ ′ − Pe ( ƛ ′ φ ′ + Ω φ ″ + φ ″ ƛ ) = 0 .

The modified boundary conditions are as follows:

(22) lim ς → χ f ( ς ) = 1 − m 1 + m χ , lim ς → χ f ′ ( ς ) = Θ ( η ) , lim ς → χ φ ′ ( ς ) = lim ς → χ ƛ ( ς ) = 1 , lim ς → ∞ f ′ ( ς ) = Θ ( χ ) , lim ς → ∞ φ ′ ( ς ) = lim ς → ∞ ƛ ( ς ) = 0 ,

where Wc is the Weissenberg number, M is the magnetic term, Le is the Lewis number, Rb is the Rayleigh number, λ is the mixed convection, Nr is the buoyancy factor, Pr is the Prandtl number, Nt is the thermophoresis term, χ is the wall thickness, Lb is the Lewis number, Pe is the Peclet number, Rd is the radiation term, Nb is the Brownian motion, Ω is the density ratio of the motile microorganisms, E is the activation energy, Kr is the chemical reaction factor:

Wc = Γ 2 ( 1 + n ) u w 3 ν 1 2 , M = σ B 0 2 u 0 ρ , Le = k D B , Rb = ( ρ m − ρ f ) ( n w − n ∞ ) β ρ ( 1 − C ∞ ) ( T w − T ∞ ) , Nb = τ D B ( C w − C ∞ ) k , λ = 2 g β ( T w − T ∞ ) ( 1 − T ∞ ) u ∞ 2 ( m + 1 ) , Nr = ( ρ p − ρ f ) ( C w − C ∞ ) β ρ ( 1 − C ∞ ) ( T w − T ∞ ) , Pr = ( ρ c p ) k , Nt = τ D T ( T w − T ∞ ) T ∞ k , Lb = k D N , χ = c u 0 ( 1 + m ) 2 ν ∞ 1 2 , Pe = d Wc D N , Rd = 16 σ e T ∞ 3 3 k β R , Ω = n ∞ ( n w − n ∞ ) , E = E a κ T ∞ , δ = T w − T ∞ T ∞ , Kr = k T 2 c .

3 Physical quantity

The physical quantities are as follows:

(23) C f x = 2 τ w ρ f U w 2 ( x ) , Nu x = ( x + b ) q w k ( T w − T ∞ ) , Sh x = ( x + b ) q m D B ( C w − C ∞ ) , Nn x = ( x + b ) q n D n ( n w − n ∞ ) ,

where τ w , q w , q m , and q n are expressed as

(24) τ w = μ ( 1 − m ) ∂ u ∂ y y = e ( x ) 2 + μ m Γ 2 ∂ u ∂ y y = e ( x ) 2 3 ,

(25) q w = − k T + 16 σ e T ∞ 3 3 β R ∂ T ∂ y y = e ( x ) 2 ,

(26) q m = − D B ∂ C ∂ y y = e ( x ) 2 ,

(27) q n = − D n ∂ n ∂ y y = e ( x ) 2 .

Using equations (24)–(27) in equation (23),

(28) Re x 1 2 C f x = − n + 1 2 ( 1 − m ) f ″ ( 0 ) − m W e 2 f ″ ( 0 ) 3 ,

(29) Re x − 1 2 Nu x = − n + 1 2 ( Λ k g ( 0 ) + ( 1 + Rd ) Θ ′ ( 0 ) ) ,

(30) Re x − 1 2 Sh x = n + 1 2 ( − φ ′ ( 0 ) ) ,

(31) Re x − 1 2 Nn x = n + 1 2 ( − ƛ ′ ( 0 ) ) .

4 Solution procedure

Here equations (18)–(21) are further cracked through the PCM approach, which is as follows [66–68]:

Step 1: Simplifying equations (18)–(21) to first order:

(32) ξ 1 ( η ) = f ( η ) , ξ 2 ( η ) = f ′ ( η ) , ξ 3 ( η ) = f ″ ( η ) , ξ 4 ( η ) = Θ ( η ) , ξ 5 ( η ) = Θ ′ ( η ) , ξ 6 ( η ) = φ ( η ) , ξ 7 ( η ) = φ ′ ( η ) , ξ 8 ( η ) = ƛ ( η ) , ξ 9 ( η ) = ƛ ′ ( η ) . .

By substituting equation (32) into equations (18)–(22), we obtain

(33) ( ( 1 − m ) + m W e ξ 3 ( η ) ) ξ ′ 3 ( η ) − 2 n n + 1 ( ξ 2 ( η ) ) 2 + ξ 1 ( η ) ξ 3 ( η ) − M ξ 2 ( η ) + λ ( ξ 4 ( η ) − Nr ξ 6 ( η ) − Rb ξ 8 ( η ) ) = 0 ,

(34) ( 1 + Rd + Λ k ξ 4 ( η ) ) ξ ′ 5 ( η ) + Pr ξ 1 ( η ) ξ 5 ( η ) + Λ k ( ξ 5 ( η ) ) 2 + Nb ξ 5 ( η ) ξ 7 ( η ) + Nt ( ξ 5 ( η ) ) 2 + hs ξ 4 ( η ) = 0 ,

(35) ξ ′ 7 ( η ) + Le Pr ξ 1 ( η ) ξ 7 ( η ) + Nt Nb ξ ′ 5 ( η ) − Sc σ ( 1 + δ ξ 4 ( η ) ) n ξ 7 ( η ) exp − E 1 + δ ξ 4 ( η ) = 0 ,

(36) ξ ′ 9 ( η ) + Pr Lb ξ 1 ( η ) ξ 9 ( η ) − Pe ( ξ 9 ( η ) ξ 7 ( η ) + Ω ξ ′ 7 ( η ) + ξ ′ 7 ( η ) ξ 8 ( η ) ) = 0 .

Along the modified boundary constraints,

(37) lim η → χ ξ 1 ( η ) = 1 − m 1 + m χ , lim η → χ ξ 2 ( η ) = ξ 4 ( η ) , lim η → χ ξ 7 ( η ) = lim η → χ ξ 8 ( η ) = 1 , lim η → ∞ ξ 2 ( η ) = ξ 4 ( η ) , lim η → ∞ ξ 7 ( η ) = lim η → ∞ ξ 8 ( η ) = 0 .

Step 2: Introducing parameter p in equations (33)–(36):

(38) ( ( 1 − m ) + m W e ξ 3 ( η ) ) ξ ′ 3 ( η ) − 2 n n + 1 ( ξ 2 ( η ) ) 2 + ξ 1 ( η ) ( ( ξ 3 ( η ) − 1 ) p + 1 ) − M ξ 2 ( η ) + λ [ ξ 4 ( η ) − Nr ξ 6 ( η ) − Rb ξ 8 ( η ) ] = 0 ,

(39) ( 1 + Rd + Λ k ξ 4 ( η ) ) ξ ′ 5 ( η ) + Pr ξ 1 ( η ) ( ( ξ 5 ( η ) − 1 ) p + 1 ) + Λ k ( ξ 5 ( η ) ) 2 Nb ξ 5 ( η ) ξ 7 ( η ) + Nt ( ξ 5 ( η ) ) 2 hs ξ 4 ( η ) = 0 ,

(40) ξ ′ 7 ( η ) + Le Pr ξ 1 ( η ) ( ( ξ 7 ( η ) − 1 ) p + 1 ) + Nt Nb ξ ′ 5 ( η ) − Sc σ ( 1 + δ ξ 4 ( η ) ) n ξ 7 ( η ) exp − E 1 + δ ξ 4 ( η ) = 0 ,

(41) ξ ′ 9 ( η ) + ( Pr Lb ξ 1 ( η ) − Pe ξ 7 ( η ) ) ( ( ξ 9 ( η ) − 1 ) p + 1 ) − Pe [ Ω ξ ′ 7 ( η ) + ξ ′ 7 ( η ) ξ 8 ( η ) ] = 0 .

5 Results and discussion

This section expresses the physical mechanism and reasons behind increasing and decreasing trends in figures and tables.

5.1 Velocity interpretation

Figures 2–6 highlight the nature of the velocity field f ′ ( η ) versus the power-law index m, mixed convection λ , magnetic term M, and Rb and Weissenberg numbers Wc, respectively. It has been detected that the velocity curve decreases with the increase of the power-law index, while it enhances with the influence of mixed convection as presented in Figures 2 and 3. Physically, the increasing effect of λ magnifies the buoyancy force and gravitational acceleration, which causes the promotion of the velocity field. Figure 4 shows that the velocity curve decays with the intensifying effect of the magnetic strength. Physically, the resistive force between the stretching slender sheet and fluid, which is generated due to the magnetic field, weakens the flow stream. Figures 5 and 6 show that the influence of Rb and Wc also decreases the velocity curve.

Figure 2 Significance of the power-law index m on the velocity curve f ′ ( η ) . f^{\prime} (\eta ). When λ = 0.5 , \lambda =0.5, M = 1.0 , M=1.0, Rb = 0.1 , \text{Rb}=0.1, Wc = 0.2 , \text{Wc}=0.2, Hs = 2.0 , \text{Hs}=2.0, Nt = 1.0 , \text{Nt}=1.0, Nb = 1.0 , \text{Nb}=1.0, Ea = 0.1 , \text{Ea}=0.1, Kr = 0.2 , \text{Kr}=0.2, Ea = 0.1 , \text{Ea}=0.1, Ω = 0.2 , \Omega =0.2, and Pe = 0.5 . \text{Pe}=0.5.
Figure 2

Significance of the power-law index m on the velocity curve f ′ ( η ) . When λ = 0.5 , M = 1.0 , Rb = 0.1 , Wc = 0.2 , Hs = 2.0 , Nt = 1.0 , Nb = 1.0 , Ea = 0.1 , Kr = 0.2 , Ea = 0.1 , Ω = 0.2 , and Pe = 0.5 .

Figure 3 Significance of λ \lambda on the velocity curve f ′ ( η ) . f^{\prime} (\eta ). When m = 0.1 , m=0.1, M = 1.0 , M=1.0, Rb = 0.1 , \text{Rb}=0.1, Wc = 0.2 , \text{Wc}=0.2, Hs = 2.0 , \text{Hs}=2.0, Nt = 1.0 , \text{Nt}=1.0, Nb = 1.0 , \text{Nb}=1.0, Ea = 0.1 , \text{Ea}=0.1, Kr = 0.2 , \text{Kr}=0.2, Ea = 0.1 , \text{Ea}=0.1, Ω = 0.2 , \Omega =0.2, and Pe = 0.5 . \text{Pe}=0.5.
Figure 3

Significance of λ on the velocity curve f ′ ( η ) . When m = 0.1 , M = 1.0 , Rb = 0.1 , Wc = 0.2 , Hs = 2.0 , Nt = 1.0 , Nb = 1.0 , Ea = 0.1 , Kr = 0.2 , Ea = 0.1 , Ω = 0.2 , and Pe = 0.5 .

Figure 4 Significance of the magnetic term M on the velocity curve f ′ ( η ) . f^{\prime} (\eta ). When λ = 0.5 , \lambda =0.5, Rb = 0.1 , \text{Rb}=0.1, Wc = 0.2 , \text{Wc}=0.2, Hs = 2.0 , \text{Hs}=2.0, Nt = 1.0 , \text{Nt}=1.0, Nb = 1.0 , \text{Nb}=1.0, Ea = 0.1 , \text{Ea}=0.1, Kr = 0.2 , \text{Kr}=0.2, Ea = 0.1 , \text{Ea}=0.1, Ω = 0.2 , \Omega =0.2, and Pe = 0.5 . \text{Pe}=0.5.
Figure 4

Significance of the magnetic term M on the velocity curve f ′ ( η ) . When λ = 0.5 , Rb = 0.1 , Wc = 0.2 , Hs = 2.0 , Nt = 1.0 , Nb = 1.0 , Ea = 0.1 , Kr = 0.2 , Ea = 0.1 , Ω = 0.2 , and Pe = 0.5 .

Figure 5 Significance of the bio-convection Rayleigh number Rb on the velocity curve f ′ ( η ) . f^{\prime} (\eta ). When λ = 0.5 , \lambda =0.5, M = 1.0 , M=1.0, Wc = 0.2 , \text{Wc}=0.2, Hs = 2.0 , \text{Hs}=2.0, Nt = 1.0 , \text{Nt}=1.0, Nb = 1.0 , \text{Nb}=1.0, Ea = 0.1 , \text{Ea}=0.1, Kr = 0.2 , \text{Kr}=0.2, Ea = 0.1 , \text{Ea}=0.1, Ω = 0.2 , \Omega =0.2, and Pe = 0.5 . \text{Pe}=0.5.
Figure 5

Significance of the bio-convection Rayleigh number Rb on the velocity curve f ′ ( η ) . When λ = 0.5 , M = 1.0 , Wc = 0.2 , Hs = 2.0 , Nt = 1.0 , Nb = 1.0 , Ea = 0.1 , Kr = 0.2 , Ea = 0.1 , Ω = 0.2 , and Pe = 0.5 .

Figure 6 Significance of Wc on the velocity curve f ′ ( η ) . f^{\prime} (\eta ). When λ = 0.5 , \lambda =0.5, M = 1.0 , M=1.0, Rb = 0.1 , \text{Rb}=0.1, Hs = 2.0 , \text{Hs}=2.0, Nt = 1.0 , \text{Nt}=1.0, Nb = 1.0 , \text{Nb}=1.0, Ea = 0.1 , \text{Ea}=0.1, Kr = 0.2 , \text{Kr}=0.2, Ea = 0.1 , \text{Ea}=0.1, Ω = 0.2 , \Omega =0.2, and Pe = 0.5 . \text{Pe}=0.5.
Figure 6

Significance of Wc on the velocity curve f ′ ( η ) . When λ = 0.5 , M = 1.0 , Rb = 0.1 , Hs = 2.0 , Nt = 1.0 , Nb = 1.0 , Ea = 0.1 , Kr = 0.2 , Ea = 0.1 , Ω = 0.2 , and Pe = 0.5 .

5.2 Energy interpretation

Figures 7–14 illustrate the effect of physical constraints such as m, λ , the heat source Hs, magnetic term M, Nb, Nt, Rb, and Weissenberg number Wc on the energy profile Θ ( η ) . Figures 7 and 8 show that the energy curve develops with the variation of the power-law index and mixed convection. Physically, the increasing effect of λ increases the buoyancy force and gravitational acceleration. Figures 9 and 10 reveal that the energy field increases with the flourishing trend of the heat source and magnetic force. Physically, the effect of the parameter H s provides some additional heat to the fluid flow, which causes an advancement of the energy curve. Similarly, due to the magnetic effect, the Lorentz force is generated, which also provides additional heat to the fluid flow due to a resistive force between the sheet and fluid. Therefore, the energy field increases under the influence of H s and M. Figures 11 and 12 show that the energy field also accelerates under the effect of Nb and Nt. Physically, the Nt outcome refers to the transfer of nano-particulates to a lower temperature from a higher one, consequently increasing the concentration curve. Figures 13 and 14 highlight that an energy field is developed due to Rb and Weissenberg number Wc.

Figure 7 Significance of the power-law index m on the energy profile Θ ( η ) . \Theta (\eta ). When λ = 0.5 , \lambda =0.5, M = 1.0 , M=1.0, Rb = 0.1 , \text{Rb}=0.1, Wc = 0.2 , \text{Wc}=0.2, Hs = 2.0 , \text{Hs}=2.0, Nt = 1.0 , \text{Nt}=1.0, Nb = 1.0 , \text{Nb}=1.0, Ea = 0.1 , \text{Ea}=0.1, Kr = 0.2 , \text{Kr}=0.2, Ea = 0.1 , \text{Ea}=0.1, Ω = 0.2 , \Omega =0.2, and Pe = 0.5 . \text{Pe}=0.5.
Figure 7

Significance of the power-law index m on the energy profile Θ ( η ) . When λ = 0.5 , M = 1.0 , Rb = 0.1 , Wc = 0.2 , Hs = 2.0 , Nt = 1.0 , Nb = 1.0 , Ea = 0.1 , Kr = 0.2 , Ea = 0.1 , Ω = 0.2 , and Pe = 0.5 .

Figure 8 Significance of mixed convection λ \lambda on energy profile Θ ( η ) . \Theta (\eta ). When M = 1.0 , M=1.0, Rb = 0.1 , \text{Rb}=0.1, Wc = 0.2 , \text{Wc}=0.2, Hs = 2.0 , \text{Hs}=2.0, Nt = 1.0 , \text{Nt}=1.0, Nb = 1.0 , \text{Nb}=1.0, Ea = 0.1 , \text{Ea}=0.1, Kr = 0.2 , \text{Kr}=0.2, Ea = 0.1 , \text{Ea}=0.1, Ω = 0.2 , \Omega =0.2, and Pe = 0.5 . \text{Pe}=0.5.
Figure 8

Significance of mixed convection λ on energy profile Θ ( η ) . When M = 1.0 , Rb = 0.1 , Wc = 0.2 , Hs = 2.0 , Nt = 1.0 , Nb = 1.0 , Ea = 0.1 , Kr = 0.2 , Ea = 0.1 , Ω = 0.2 , and Pe = 0.5 .

Figure 9 Significance of heat source Hs on the energy profile Θ ( η ) . \Theta (\eta ). When λ = 0.5 , \lambda =0.5, M = 1.0 , M=1.0, Rb = 0.1 , \text{Rb}=0.1, Wc = 0.2 , \text{Wc}=0.2, Nt = 1.0 , \text{Nt}=1.0, Nb = 1.0 , \text{Nb}=1.0, Ea = 0.1 , \text{Ea}=0.1, Kr = 0.2 , \text{Kr}=0.2, Ea = 0.1 , \text{Ea}=0.1, Ω = 0.2 , \Omega =0.2, and Pe = 0.5 . \text{Pe}=0.5.
Figure 9

Significance of heat source Hs on the energy profile Θ ( η ) . When λ = 0.5 , M = 1.0 , Rb = 0.1 , Wc = 0.2 , Nt = 1.0 , Nb = 1.0 , Ea = 0.1 , Kr = 0.2 , Ea = 0.1 , Ω = 0.2 , and Pe = 0.5 .

Figure 10 Significance of the magnetic term M on the energy profile Θ ( η ) . \Theta (\eta ). When λ = 0.5 , \lambda =0.5, Rb = 0.1 , \text{Rb}=0.1, Wc = 0.2 , \text{Wc}=0.2, Hs = 2.0 , \text{Hs}=2.0, Nt = 1.0 , \text{Nt}=1.0, Nb = 1.0 , \text{Nb}=1.0, Ea = 0.1 , \text{Ea}=0.1, Kr = 0.2 , \text{Kr}=0.2, Ea = 0.1 , \text{Ea}=0.1, Ω = 0.2 , \Omega =0.2, and Pe = 0.5 . \text{Pe}=0.5.
Figure 10

Significance of the magnetic term M on the energy profile Θ ( η ) . When λ = 0.5 , Rb = 0.1 , Wc = 0.2 , Hs = 2.0 , Nt = 1.0 , Nb = 1.0 , Ea = 0.1 , Kr = 0.2 , Ea = 0.1 , Ω = 0.2 , and Pe = 0.5 .

Figure 11 Significance of Nb on the energy profile Θ ( η ) . \Theta (\eta ). When λ = 0.5 , \lambda =0.5, M = 1.0 , M=1.0, Rb = 0.1 , \text{Rb}=0.1, Wc = 0.2 , \text{Wc}=0.2, Hs = 2.0 , \text{Hs}=2.0, Nt = 1.0 , \text{Nt}=1.0, Ea = 0.1 , \text{Ea}=0.1, Kr = 0.2 , \text{Kr}=0.2, Ea = 0.1 , \text{Ea}=0.1, Ω = 0.2 , \Omega =0.2, and Pe = 0.5 . \text{Pe}=0.5.
Figure 11

Significance of Nb on the energy profile Θ ( η ) . When λ = 0.5 , M = 1.0 , Rb = 0.1 , Wc = 0.2 , Hs = 2.0 , Nt = 1.0 , Ea = 0.1 , Kr = 0.2 , Ea = 0.1 , Ω = 0.2 , and Pe = 0.5 .

Figure 12 Significance of Nt on the energy profile Θ ( η ) . \Theta (\eta ). When λ = 0.5 , \lambda =0.5, M = 1.0 , M=1.0, Rb = 0.1 , \text{Rb}=0.1, Wc = 0.2 , \text{Wc}=0.2, Hs = 2.0 , \text{Hs}=2.0, Nb = 1.0 , \text{Nb}=1.0, Ea = 0.1 , \text{Ea}=0.1, Kr = 0.2 , \text{Kr}=0.2, Ea = 0.1 , \text{Ea}=0.1, Ω = 0.2 , \Omega =0.2, and Pe = 0.5 . \text{Pe}=0.5.
Figure 12

Significance of Nt on the energy profile Θ ( η ) . When λ = 0.5 , M = 1.0 , Rb = 0.1 , Wc = 0.2 , Hs = 2.0 , Nb = 1.0 , Ea = 0.1 , Kr = 0.2 , Ea = 0.1 , Ω = 0.2 , and Pe = 0.5 .

Figure 13 Significance of the bio-convection Rayleigh number Rb on the energy profile Θ ( η ) . \Theta (\eta ). When λ = 0.5 , \lambda =0.5, M = 1.0 , M=1.0, Wc = 0.2 , \text{Wc}=0.2, Hs = 2.0 , \text{Hs}=2.0, Nt = 1.0 , \text{Nt}=1.0, Nb = 1.0 , \text{Nb}=1.0, Ea = 0.1 , \text{Ea}=0.1, Kr = 0.2 , \text{Kr}=0.2, Ea = 0.1 , \text{Ea}=0.1, Ω = 0.2 , \Omega =0.2, and Pe = 0.5 . \text{Pe}=0.5.
Figure 13

Significance of the bio-convection Rayleigh number Rb on the energy profile Θ ( η ) . When λ = 0.5 , M = 1.0 , Wc = 0.2 , Hs = 2.0 , Nt = 1.0 , Nb = 1.0 , Ea = 0.1 , Kr = 0.2 , Ea = 0.1 , Ω = 0.2 , and Pe = 0.5 .

Figure 14 Significance of Wc on the energy profile Θ ( η ) . \Theta (\eta ). When λ = 0.5 , \lambda =0.5, M = 1.0 , M=1.0, Rb = 0.1 , \text{Rb}=0.1, Hs = 2.0 , \text{Hs}=2.0, Nt = 1.0 , \text{Nt}=1.0, Nb = 1.0 , \text{Nb}=1.0, Ea = 0.1 , \text{Ea}=0.1, Kr = 0.2 , \text{Kr}=0.2, Ea = 0.1 , \text{Ea}=0.1, Ω = 0.2 , \Omega =0.2, and Pe = 0.5 . \text{Pe}=0.5.
Figure 14

Significance of Wc on the energy profile Θ ( η ) . When λ = 0.5 , M = 1.0 , Rb = 0.1 , Hs = 2.0 , Nt = 1.0 , Nb = 1.0 , Ea = 0.1 , Kr = 0.2 , Ea = 0.1 , Ω = 0.2 , and Pe = 0.5 .

5.3 Mass and motile microbes’ profile

Figures 15–18 illustrate the characteristics of the mass curve φ ( η ) versus the variation of the activation energy E a, Kr, thermophoresis factor Nt, and Brownian motion Nb. Figures 15 and 16 show that the mass propagation ratio increases with the effect of the activation energy, while it decreases with the influence of chemical reactions. Physically, the activation energy effect stimulates the fluid molecules and accelerates their transmission rate, resulting in the increase of the mass profile φ ( η ) . On the one hand, chemical reactions always have an inverse relation with the activation energy; therefore, the chemical reaction diminishes the mass curve φ ( η ) , as shown in Figure 16. Figures 17 and 18 show that the mass curve augments with the impact of Nt, while it reduces by the action of Nb. Physically, the Nt outcome refers to the transfer of nano-particulates to a lower temperature from a higher one, resulting in expanding the concentration curve.

Figure 15 Significance of the activation energy Ea on the mass profile φ ( η ) . \varphi (\eta ). When λ = 0.5 , \lambda =0.5, M = 1.0 , M=1.0, Rb = 0.1 , \text{Rb}=0.1, Wc = 0.2 , \text{Wc}=0.2, Hs = 2.0 , \text{Hs}=2.0, Nt = 1.0 , \text{Nt}=1.0, Nb = 1.0 , \text{Nb}=1.0, Kr = 0.2 , \text{Kr}=0.2, Ea = 0.1 , \text{Ea}=0.1, Ω = 0.2 , \Omega =0.2, and Pe = 0.5 . \text{Pe}=0.5.
Figure 15

Significance of the activation energy Ea on the mass profile φ ( η ) . When λ = 0.5 , M = 1.0 , Rb = 0.1 , Wc = 0.2 , Hs = 2.0 , Nt = 1.0 , Nb = 1.0 , Kr = 0.2 , Ea = 0.1 , Ω = 0.2 , and Pe = 0.5 .

Figure 16 Significance of the chemical reaction Kr on the mass profile φ ( η ) . \varphi (\eta ). When λ = 0.5 , \lambda =0.5, M = 1.0 , M=1.0, Rb = 0.1 , \text{Rb}=0.1, Wc = 0.2 , \text{Wc}=0.2, Hs = 2.0 , \text{Hs}=2.0, Nt = 1.0 , \text{Nt}=1.0, Nb = 1.0 , \text{Nb}=1.0, Ea = 0.1 , \text{Ea}=0.1, Ea = 0.1 , \text{Ea}=0.1, Ω = 0.2 , \Omega =0.2, and Pe = 0.5 . \text{Pe}=0.5.
Figure 16

Significance of the chemical reaction Kr on the mass profile φ ( η ) . When λ = 0.5 , M = 1.0 , Rb = 0.1 , Wc = 0.2 , Hs = 2.0 , Nt = 1.0 , Nb = 1.0 , Ea = 0.1 , Ea = 0.1 , Ω = 0.2 , and Pe = 0.5 .

Figure 17 Significance of Nt on the mass profile φ ( η ) . \varphi (\eta ). When λ = 0.5 , \lambda =0.5, M = 1.0 , M=1.0, Rb = 0.1 , \text{Rb}=0.1, Wc = 0.2 , \text{Wc}=0.2, Hs = 2.0 , \text{Hs}=2.0, Nb = 1.0 , \text{Nb}=1.0, Ea = 0.1 , \text{Ea}=0.1, Kr = 0.2 , \text{Kr}=0.2, Ea = 0.1 , \text{Ea}=0.1, Ω = 0.2 , \Omega =0.2, and Pe = 0.5 . \text{Pe}=0.5.
Figure 17

Significance of Nt on the mass profile φ ( η ) . When λ = 0.5 , M = 1.0 , Rb = 0.1 , Wc = 0.2 , Hs = 2.0 , Nb = 1.0 , Ea = 0.1 , Kr = 0.2 , Ea = 0.1 , Ω = 0.2 , and Pe = 0.5 .

Figure 18 Significance of Nb on the mass profile φ ( η ) . \varphi (\eta ). When λ = 0.5 , \lambda =0.5, M = 1.0 , M=1.0, Rb = 0.1 , \text{Rb}=0.1, Wc = 0.2 , \text{Wc}=0.2, Hs = 2.0 , \text{Hs}=2.0, Nt = 1.0 , \text{Nt}=1.0, Ea = 0.1 , \text{Ea}=0.1, Kr = 0.2 , \text{Kr}=0.2, Ea = 0.1 , \text{Ea}=0.1, Ω = 0.2 , \Omega =0.2, and Pe = 0.5 . \text{Pe}=0.5.
Figure 18

Significance of Nb on the mass profile φ ( η ) . When λ = 0.5 , M = 1.0 , Rb = 0.1 , Wc = 0.2 , Hs = 2.0 , Nt = 1.0 , Ea = 0.1 , Kr = 0.2 , Ea = 0.1 , Ω = 0.2 , and Pe = 0.5 .

Figures 19–21 show the nature of gyrotactic microbes versus the Lewis number Lb, the density ratio of the motile microorganisms Ω , and the Peclet number Pe, respectively. Figures 19 and 20 show that the microorganism’s curve decline with the impact of the Lewis number and parameter Ω . Similarly, the variation in the Peclet number also declines the microorganism’s curve because physically the diffusive transport rate is less than the advection transport rate. Figure 22 expresses the skin friction and Nusselt and Sherwood numbers. It can be observed that the effect of thermal radiation enhances the energy transference rate, while the influence of mixed convection and the Peclet number diminishes the drag force and mass diffusion rate.

Figure 19 Significance of Lewis number Lb on the motile microbe’s profile λ ( η ) . \lambda (\eta ). When λ = 0.5 , \lambda =0.5, M = 1.0 , M=1.0, Rb = 0.1 , \text{Rb}=0.1, Wc = 0.2 , \text{Wc}=0.2, Hs = 2.0 , \text{Hs}=2.0, Nt = 1.0 , \text{Nt}=1.0, Nb = 1.0 , \text{Nb}=1.0, Ea = 0.1 , \text{Ea}=0.1, Kr = 0.2 , \text{Kr}=0.2, Ea = 0.1 , \text{Ea}=0.1, Ω = 0.2 , \Omega =0.2, and Pe = 0.5 . \text{Pe}=0.5.
Figure 19

Significance of Lewis number Lb on the motile microbe’s profile λ ( η ) . When λ = 0.5 , M = 1.0 , Rb = 0.1 , Wc = 0.2 , Hs = 2.0 , Nt = 1.0 , Nb = 1.0 , Ea = 0.1 , Kr = 0.2 , Ea = 0.1 , Ω = 0.2 , and Pe = 0.5 .

Figure 20 Significance of Ω \Omega on the motile microbe’s profile λ ( η ) . \lambda (\eta ). When λ = 0.5 , \lambda =0.5, M = 1.0 , M=1.0, Rb = 0.1 , \text{Rb}=0.1, Wc = 0.2 , \text{Wc}=0.2, Hs = 2.0 , \text{Hs}=2.0, Nt = 1.0 , \text{Nt}=1.0, Nb = 1.0 , \text{Nb}=1.0, Ea = 0.1 , \text{Ea}=0.1, Kr = 0.2 , \text{Kr}=0.2, Ea = 0.1 , \text{Ea}=0.1, and Pe = 0.5 . \text{Pe}=0.5.
Figure 20

Significance of Ω on the motile microbe’s profile λ ( η ) . When λ = 0.5 , M = 1.0 , Rb = 0.1 , Wc = 0.2 , Hs = 2.0 , Nt = 1.0 , Nb = 1.0 , Ea = 0.1 , Kr = 0.2 , Ea = 0.1 , and Pe = 0.5 .

Figure 21 Significance of Peclet number Pe on the motile microbe’s profile λ ( η ) . \lambda (\eta ). When λ = 0.5 , \lambda =0.5, M = 1.0 , M=1.0, Rb = 0.1 , \text{Rb}=0.1, Wc = 0.2 , \text{Wc}=0.2, Hs = 2.0 , \text{Hs}=2.0, Nt = 1.0 , \text{Nt}=1.0, Nb = 1.0 , \text{Nb}=1.0, Ea = 0.1 , \text{Ea}=0.1, Kr = 0.2 , \text{Kr}=0.2, Ea = 0.1 , \text{Ea}=0.1, and Ω = 0.2 . \Omega =0.2.
Figure 21

Significance of Peclet number Pe on the motile microbe’s profile λ ( η ) . When λ = 0.5 , M = 1.0 , Rb = 0.1 , Wc = 0.2 , Hs = 2.0 , Nt = 1.0 , Nb = 1.0 , Ea = 0.1 , Kr = 0.2 , Ea = 0.1 , and Ω = 0.2 .

Figure 22 (a) Skin friction (b) Nusselt number and (c) Sherwood number.
Figure 22

(a) Skin friction (b) Nusselt number and (c) Sherwood number.

Table 1 presents the comparative correlation between the present outcomes and the published work for accuracy purposes using skin friction Re x 1 2 C f x . It can be seen that both results are in the best settlement, which ensures that the current results are accurate and reliable. Table 2 reveals the statistical outcomes for Re x 1 2 C f x and Re x − 1 2 Nu x . It can be observed that the influence of the magnetic effect accelerates the skin friction while the Nusselt number decreases. Furthermore, the variation in the thermal radiation parameter Rd increases the Nusselt number and skin friction.

Table 1

Numerical comparison with published work for skin friction Re x 1 2 C f x while ignoring all other parameters

Wakif et al. [59] Fang et al. [65] Current results
n χ = 0.5 χ = 0.25 χ = 0.5 χ = 0.25 χ = 0.5 χ = 0.25
10 1.143320620 1.060324666 1.1433 1.0603 1.143329 1.060330
9.0 1.140392519 1.058915794 1.1404 1.0589 1.140397 1.058925
7.0 1.132285178 1.055044823 1.1323 1.0550 1.132299 1.055048
5.0 1.118590381 1.048611306 1.1186 1.0486 1.118582 1.048608
3.0 1.090492254 1.035868282 1.0905 1.0359 1.090510 1.035864
2.0 1.061402505 1.023407744 1.0614 1.0234 1.061410 1.023410
Table 2

Numerical results for the skin friction Re x 1 2 C f x and Nusselt number Re x ‒ 1 2 N u x .

m We M λ Nr Rb Rd ‒ f ″ ( 0 ) ‒ Θ ′ ( 0 )
0.2 0.1 1.0 1.2352 0.5896
0.4 1.2594 0.5727
0.6 1.3442 0.5507
0.1 1.2352 0.5896
0.2 1.2587 0.5889
0.3 1.2816 0.5881
0.5 1.2352 0.5896
1.0 1.3905 0.5635
1.5 1.5353 0.5401
0.3 1.3081 0.5767
0.5 1.2718 0.5836
0.7 1.2352 0.5896
0.1 1.2352 0.5896
0.3 1.2392 0.5888
0.5 1.2432 0.5883
1.2 1.2275 0.5907
1.4 1.2313 0.5903
1.6 1.2353 0.5896
1.0 1.3353 0.5871
2.0 1.3441 0.5910
3.0 1.4103 0.6231

6 Conclusion

We have numerically evaluated the significance of motile microbes on the MHD steady convective streams of the TH nanofluid flow across an elastically nonlinear stretching surface of an irregular thickness. The significance of an external magnetic field, thermal radiation, and thermal conductivity is also examined on the TH nanofluid. The governing system of nonlinear PDEs is reformed to a system of ODEs by using the similarity variable conversions. Furthermore, the reduced form of nonlinear ODEs is numerically computed through the PCM approach using MATLAB software. From the above studies, the following conclusions are drawn:

  • The skin friction increases by up to 7.3% with various magnetic values ranging from 0.5 to 1.5. However, the energy transfer rate declines by up to 5.92%.

  • The velocity curve reduces under the effect of the power-law index and magnetic strength, while it enhances under the influence of mixed convection.

  • The thermal radiation increases the energy propagation rate and flow velocity by up to 11.23 and 8.17%, respectively.

  • The influence of Rb and Weissenberg number Wc also reduces the velocity curve.

  • The energy profile develops with the variations of the power-law index, heat source, magnetic force, and mixed convection.

  • The energy field also increases under the effect of Rb, Weissenberg number Wc, Nb, and Nt.

  • The mass propagation ratio increases with the effect of activation energy, while it declines under the influence of chemical reactions.

  • The motile microbes’ profile declines under the effect of Lewis number and parameter Ω . Similarly, the variation of the Peclet number also decreases the microorganism’s curve because physically the diffusive transport rate is less than the advection transport rate.

  1. Funding information: The authors state no funding involved.

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: The authors state no conflict of interest.

  4. Data availability statement: All data generated or analysed during this study are included in this published article.

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Received: 2023-03-31
Revised: 2023-05-19
Accepted: 2023-07-15
Published Online: 2023-08-05

© 2023 the author(s), published by De Gruyter

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

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  139. A review of the performance of fibre-reinforced composite laminates with carbon nanotubes
  140. Research on AFM tip-related nanofabrication of two-dimensional materials
  141. Advances in phase change building materials: An overview
  142. Development of graphene and graphene quantum dots toward biomedical engineering applications: A review
  143. Nanoremediation approaches for the mitigation of heavy metal contamination in vegetables: An overview
  144. Photodynamic therapy empowered by nanotechnology for oral and dental science: Progress and perspectives
  145. Biosynthesis of metal nanoparticles: Bioreduction and biomineralization
  146. Current diagnostic and therapeutic approaches for severe acute respiratory syndrome coronavirus-2 (SARS-COV-2) and the role of nanomaterial-based theragnosis in combating the pandemic
  147. Application of two-dimensional black phosphorus material in wound healing
  148. Special Issue on Advanced Nanomaterials and Composites for Energy Conversion and Storage - Part I
  149. Helical fluorinated carbon nanotubes/iron(iii) fluoride hybrid with multilevel transportation channels and rich active sites for lithium/fluorinated carbon primary battery
  150. The progress of cathode materials in aqueous zinc-ion batteries
  151. Special Issue on Advanced Nanomaterials for Carbon Capture, Environment and Utilization for Energy Sustainability - Part I
  152. Effect of polypropylene fiber and nano-silica on the compressive strength and frost resistance of recycled brick aggregate concrete
  153. Mechanochemical design of nanomaterials for catalytic applications with a benign-by-design focus
https://doi.org/10.1515/ntrev-2023-0106
Keywords for this article
bio-convection; MHD; tangent hyperbolic; nanofluidics; elastic slender sheet; PCM; activation energy
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