Brownian and thermal diffusivity impact due to the Maxwell nanofluid (graphene/engine oil) flow with motile microorganisms and Joule heating

Nomenclature

B 0

magnetic field strength

B 1

volume of microorganisms

( C p ) nf

specific heat of nanofluid ( J kg − 1 K − 1 )

D B

Brownian motion coefficient

D m

microorganisms diffusivity

D T

thermophoresis coefficient

Ec

Eckert number

f

base fluid

Gr

Grashof number

H ∞

ambient concentration

k

permeability of medium

k nf

thermal conductivity of nanofluid ( W m − 1 K − 1 )

Lb

bioconvection Lewis number

Le

Lewis number

M 2

magnetic parameter

Nb

Brownian motion factor

Nr

buoyancy ratio factor

Nt

thermophoresis factor

p

solid nano-particles

Pe

Peclet number

Pr

Prandtl number

Q

heat source/sink

R

radiation parameter

Rb

bioconvection Rayleigh number

Re

Reynolds number

S

nanofluid’s microorganism profile

S ∞

ambient gyrotactic microorganisms

U , V

velocity component ( m s − 1 )

U w

stretching velocity ( m s − 1 )

V T

thermophoretic velocity ( m s − 1 )

X , Y

space coordinate ( m )

β

Deborah number

β ∗

volumetric thermal expansion coefficient ( K − 1 )

γ

chemical reaction parameter

λ

porosity parameter

λ 1

Maxwell relaxation time parameters

ρ m

motile density ( kg m − 3 )

ρ nf

nanofluid’s density ( kg m − 3 )

μ nf

nanofluid’s absolute viscosity ( kg m − 1 K − 1 )

ν f

kinematic viscosity

σ nf

nanofluid’s electrical conductivity

ϕ

volume fraction parameter

ℑ

nanofluid’s temperature ( K )

ℑ ∞

ambient temperature ( K )

1 Introduction

In the past few decades, scientists have shown a profound concern in non-Newtonian liquid problems for their widespread applications in the industrial and mechanical engineering area. A liquid with non-Newtonian behavior has a vast array of applications in manufacturing such as food processing, paper production, rubber sheet production, plastic polymers, hot rolling, cosmetics, and optical fibers. That is why scientists and researchers mostly performed research addressing the non-Newtonian fluid model. Eldabe et al. [1] have presented the heat transport flow of non-Newtonian third-grade nanoliquid through the influence of thermal and mass diffusivity and initiated that the enhancement in third-grade fluid factor amplified the fluid particle speed. Bouslimi et al. [2] used silver (Ag) and copper (Cu) nanoparticles in a sodium alginate base liquid to discuss the flow of non-Newtonian Sutterby nanofluids. Further, they have discussed the applications of solar collectors in different areas of industries. Sindhu and Gireesha [3] considered the thermal radiation and buoyancy impacts on the tangent hyperbolic non-Newtonian fluids induced by a vertical surface. They have attained that the bigger estimates of the Soret parameter augmented the liquid solutal profile. Rasheed et al. [4] have discoursed the implication of the variable viscosity and Joule heating effects in a chemically reactive non-Newtonian Casson liquid flow because of the permeable stretched surface. It can be perceived that the escalation in thermal conductivity led to enhanced heat transport. Nawaz et al. [5] have exploited the finite element method technique to deliberate the psychical significance of heat–mass transference on radiative non-Newtonian Maxwell fluid flow because of the stretchy surface and they have accomplished that the thermal diffusivity enhanced the liquid thermal profile. Punith Gowda et al. [6] explicated the physical aspects of activation energy over the Marangoni-driven non-Newtonian second-grade nanofluid flow due to the permeable surface with chemical reaction. In this observation, the declined role of heat transference is detected due to the change in the second-grade liquid factor. Mallawi et al. [7] have evaluated the modeling of heat flux and thermal radiation in a thermally stratified non-Newtonian liquid due to the Riga plate along the upshot of the heat sink/source phenomena. The lower behavior of the mass transference is noticed due to the increase in the solutal relaxation factor. Hafeez et al. [8] have used Fick’s and Fourier laws to deliberate the heat as well as mass transference features of the non-Newtonian nanofluid flow over the spinning disk. Further, they have used the concept of Von-Karman similarity transformation for the simulation of their model. Mahato et al. [9] deliberated on irreversibility of Casson nanofluid in the stagnant point flow with the impression of an inclined magnetic field due to extending sheet in a permeable medium. Nayak et al. [10] studied the irreversibility analysis of non-linear 3D radiative Casson nanofluid flow with slip velocity past a permeable stretching sheet. It can be seen in this article that the entropic generation improves with the improvement in magnetic parameter.

In engineering and science, both heat and mass transfer have achieved considerable interest from researchers and engineers due to their wide applications. Idowu and Falodun [11] explained the non-Newtonian Casson liquid by considering the significance of heat–mass transport in an inclined plate and showed that the heat–mass transference rate is weakened for Casson fluid factor. Wang et al. [12] inspected the heat–mass transference phenomenon due to the flow of non-Newtonian liquid jet by considering the behavior of the power-law index and gas-to-liquid axial velocity ratio. Gutam et al. [13] established the Carreau fluid model along a vertical plate, and in their investigation heat–mass transmission and Dufour and Soret impacts are taken into account. They found that the concentration becomes lower but temperature becomes higher on the Dufour effects. Zeeshan et al. [14] worked on the heat–mass transference in non-Newtonian biofluids by the occurrence of the chemical reaction and also performed a comparative analysis between Newtonian and non-Newtonian fluids. Sulaiman et al. [15] studied some other physical features of heat–mass transference by using the motile microorganism effects in a three-dimensional Oldroyd-B nanoliquid and showed that, for fluid velocity, the retardation time and relaxation time parameters are opposite to each other.

The investigation of magnetohydrodynamics (MHD) plays a significant role in the field of mechanical engineering, petroleum engineering, drug industries, geology, and astrophysics and in important applications such as MHD generator, cosmology metallic sensor and seismology, aerodynamics, geothermal energy, and plasma experiments. As a result, the majority of researchers in the literature used MHD in their research. Eldabe et al. [16] described the influence of MHD liquid flow by using heat–mass transfer effects and a porous media and also obtained the stability analysis of their problem. Ahmad et al. [17] used the Caputo and Caputo–Fabrizio approach for the solution of MHD non-Newtonian liquid along with a chemical reaction. Kumar et al. [18] evaluated the MHD nanoliquid flow in a stretchy sheet by considering the Brownian and thermal diffusivity and initiated that the Brownian diffusivity decreases the fluid solutal profile.

Nowadays, a lot of scientists and researchers place a high priority on approaches and procedures that improve heat transmission in several heat exchanger operations. To meet these needs, researchers created a novel fluid class called nanofluid, which is a fluid containing nanoparticles. The primary components of the nanoparticles used in nanoliquids are metals, their oxides, carbides, or carbon nanotubes. Nanofluids have an extensive choice of uses, such as in microelectronic devices, fuel cells, medical procedures, hybrid powered machine, thermal controller, heating systems, and gas exhaust from chimneys. Numerous theoretical and experimental studies are being conducted by various scientists as a result of the significance of nanofluids. Islam et al. [19] have discussed the Maxwell nanofluid in the existence of Joule heating mechanism toward the stretching cylinder by using convective boundary conditions and observed that the nanofluid concentration becomes lower for Schmidt number Sc . Waqas et al. [20] formulated the two-dimensional viscoelastic nanofluid by considering the heat generation/absorption and radiation aspects. They examined that for radiation and heat generation parameter thermal field and Nusselt number are reduced. Kanti et al. [21] have explored the heat transport characteristics and thermophysical features of coal fly ash nanoliquid. Ramzan et al. [22] have discussed the physical characteristics of CuO and Fe 3 O 4 in an MHD Williamson nanoliquid flow because of the stretchy surface and have shown that the solutal profile is enhanced for higher activation energy. Siddique et al. [23] explicated the consequence of thermal radiation on the Casson fluid flow with molybdenum disulfide MoS 2 nanoparticle and engine oil base fluid. They have determined that the lubrication and heat rate transport of the engine oil is increased when the molybdenum disulfide MoS 2 nanoparticle is mixed up in engine oil. Waqas et al. [24] have expounded the uses of Ag and Au nanoparticles in the mathematical modeling of the nanofluid by using the stenotic artery. In this analysis, they have examined that the suction/injection factor deteriorated the nanoliquid Nusselt number. Khan et al. [25] have studied the nanofluid model over the spinning disk with several kinds of nanoparticles such as copper Cu , titanium dioxide TiO , iron oxide Fe 3 O 4 , and aluminum oxide Al 2 O 3 . They have found that fluid speed is reduced when the role of Lorentz force is applied. Mabood et al. [26] scrutinized the physical implication of heat–mass transport using Fick’s and Fourier laws in a bidirectional nanoliquid flow. In this study, Ag and Cu are the nanoparticles. They used the R-K shooting scheme to discuss the computational solution of their model. Khan et al. [27] scrutinized the problem of microorganisms and Maxwell nanoliquid over the spinning disk using Cattaneo–Christov double diffusion and Hall effects. The homotopic approach is utilized for the convergence solution of nanofluid model. Khan et al. [28] established the irreversibility analysis of bioconvection flow of Maxwell fluid with chemically reactive Arrhenius activation energy and Hall current.

In the past three decades, a medical study has examined bioconvection. Traditional upward-moving bacteria, which are somewhat thicker than water, are responsible for this occurrence. When microorganism buildup makes the top surface of fluid too dense, causing the occurrence of bioconvection, the suspension becomes unstable. Aldabesh et al. [29] analyzed the numerical study of Williamson nanoliquid including the feature of activation energy and gyrotactic microorganism. Waqas et al. [30] focused on the investigation of gyrotactic microorganisms in a Maxwell fluid in the presence of convective conditions and their outcomes show that the microorganism distribution is diminished for Lewis and Peclet numbers. Shahzad et al. [31] have presented a numerical study of the motile microorganism in a radiative magnetized Walter-B nanoliquid flow along the extended surface under the use of convective conditions and evaluated that the increment in activation energy augmented the nanoliquid concentration. Waqas et al. [32] dissected the impact of motile microorganism on the Carreau-nanoliquid flow along with the physical properties of the magnetic field and Lorentz force. They have deliberated that the Marangoni parameter has decayed the motile microorganism profile. Muhammad et al. [33] analyzed the bioconvective Jeffrey nanoparticle flow via the extended sheet with the applications of thermal-solutal diffusivity and motile microorganisms. Further, they have found the behavior of mass transmission with the use of the thermophoresis impact. Ahmad et al. [34] have constructed the hybrid nanofluid model over the extended surface which consists of MnZnFe 2 O 4 and NiZnFe 2 O 4 with the prevalence of the motile microorganism and Darcy–Forchheimer flow. Waqas et al. [35] have revealed the numerical and statistical investigation of the heat flux and thermal radiation in a pseudoplastic nanoparticle flow over the Riga sheet with the physical aspect of the motile microorganism and debated that the amplifying estimation of the Biot factor has increased the temperature. Fiazan et al. [36] have offered the idea of heat flux and motile microorganisms on the Sutterby nanofluid flow via the Riga plate with the occurrence of the entropy generation. Further, they have discussed some significant uses of their research work in different arenas of manufacturing such as nuclear reactor designs, flow meters, and thermal nuclear reactors Shah et al. [37] have used numerous kinds of nanoparticles such as titanium dioxide TiO 2 and copper Cu to study the bioconvective Prandtl hybrid nanoliquid flow due to the extending sheet and discussed that the heat transmission is lower for the higher Brownian diffusivity effects. Algehyne et al. [38] have used the permeable stretching surface to scrutinize the bioconvective Darcy–Forchheimer nanofluid flow with the manifestation of microorganism. In this work, it is noted that an increase in permeability factor declined the fluid speed. Dhlamini et al. [39] presented the mathematical model of bioconvection flow with microbial activity and activation energy. In this, it is seen that the concentration of microbes enhances with the increase in the Brownian parameter. Magagula et al. [40] demonstrated the double-dispersed bioconvective Casson nanofluid via convectively stretching sheet with the saturation of motile microorganisms. The solution of coupled equations is seized through spectral quasilinearization method. Sangeetha and De [41] anticipated the impact of motile microorganisms saturated in MHD nanoliquid with chemical reaction and activated energy past a non-Darcian permeable medium. De [42] investigated the bioconvection nanofluid because of the motile gyrotactic microorganisms with the inclusion of ohmic heating in a permeable medium.

The oil’s tribological qualities are enhanced by the inclusion of graphene, making it more appropriate for high-pressure, high-stress environments. Additionally, graphene has excellent heat transfer capabilities, which are crucial for making lubricants safer at greater temperatures. When all of these impacts are taken into account, engine noise and fuel consumption are reduced. The graphene-based lubricants will make engines safer for a longer amount of time due to their improved performance and stability. Graphene nanoparticles are immersed in engine oil to make a homogeneous solution. In the view of the abovementioned works, it is clear that no consideration is paid to the investigation of chemical reaction and gyrotactic microorganisms in an MHD flow of Maxwell non-Newtonian nanoliquid toward the vertical stretched plate. The radiation and internal heat absorption impacts are taken into account. Brownian and thermophoresis behaviors are also considered. The homotopic technique is used for the generation of the solution of an existing model. Heat–mass transport characteristics in the current flow problem are calculated for different flow parameters. Nanofluid’s flow outlines with respect to the numerous relevant flow parameters are deliberated with detailed physical significance.

2 Methods

2.1 Basic equations

Consider an incompressible, viscous, and electrically conducting flow of Maxwell non-Newtonian nanoliquid due to the extending surface through the porous media. The fluid motion is generated due to stretching of sheet with a velocity U ˜ w = a X ⌢ along with a positive constant a . Magnetic effects of strength B 0 have been used in a direction normal to surface. The magnetic Reynolds number is very minor in the present analysis; therefore, the induced magnetic effect is ignored. For Maxwell nanofluid, the surface variable temperature, concentration, and motile microorganisms are ℑ w , H w , and S w while their ambient values are ℑ ∞ , H ∞ , and S ∞ with ℑ w > ℑ ∞ , H w > H ∞ , and S w > S ∞ , respectively. Further, the direction of microorganisms and velocity is not affected by the nanoparticles, both nanoparticles and pure fluid have remained in the thermal stability state. The velocities of the motile microorganisms, base fluid, and nanoparticles are equivalent. The physical interpretation of the present model is debated in Scheme 1.

Scheme 1

Geometry of the model.

The leading equations in response to the aforementioned suppositions are described as follows [43].

(1) ∂ U ˜ ∂ X ⌢ + ∂ V ˜ ∂ Y ⌢ = 0 ,

(2) U ˜ ∂ U ˜ ∂ X ⌢ + V ˜ ∂ U ˜ ∂ Y ⌢ + λ 1 U ˜ 2 ∂ 2 U ˜ ∂ X ⌢ 2 + V ˜ 2 ∂ 2 U ˜ ∂ Y ⌢ 2 + 2 U ˜ V ˜ ∂ 2 U ˜ ∂ X ⌢ ∂ Y ⌢ = υ nf ∂ 2 U ˜ ∂ Y ⌢ 2 − υ nf k U ˜ − σ nf B 0 2 ρ nf U ˜ + λ 1 V ˜ ∂ U ˜ ∂ Y ⌢ + g β ∗ ( 1 − H ∞ ) ( ℑ − ℑ ∞ ) − g ( ρ p − ρ f ) ( H − H ∞ ) − g B 1 ( ρ m − ρ f ) ( S − S ∞ ) ,

(3) U ˜ ∂ ℑ ∂ X ⌢ + V ˜ ∂ ℑ ∂ Y ⌢ = K ( ρ C p ) nf ∂ 2 ℑ ∂ Y ⌢ 2 − 1 ( ρ C p ) nf ∂ q r ∂ Y ⌢ + Q 0 ( ρ C p ) nf ( ℑ − ℑ ∞ ) + μ ( ρ C p ) nf ∂ U ˜ ∂ Y ⌢ 2 + σ nf B 0 2 ( ρ C p ) nf U ˜ 2 + τ D B ∂ H ∂ Y ⌢ ∂ ℑ ∂ Y ⌢ + D T ℑ ∞ ∂ 2 ℑ ∂ Y ⌢ 2 2 ,

(4) U ˜ ∂ H ∂ X ⌢ + V ˜ ∂ H ∂ Y ⌢ = D B ∂ 2 H ∂ Y ⌢ 2 + D T ℑ ∞ ∂ 2 ℑ ∂ Y ⌢ 2 − ∂ ( V T ) H ∂ Y ⌢ − k r ( H − H ∞ ) ,

(5) U ˜ ∂ S ∂ X ⌢ + V ˜ ∂ S ∂ Y ⌢ + b W C H − H ∞ ∂ ∂ Y ⌢ S ∂ H ∂ Y ⌢ + ∂ ∂ X ⌢ S ∂ H ∂ X ⌢ = D m ∂ 2 S ∂ X ⌢ 2 + 2 ∂ 2 S ∂ X ⌢ ∂ Y ⌢ + ∂ 2 S ∂ Y ⌢ 2 .

The corresponding conditions are as follows [44]:

(6) U ˜ = U ˜ w = a X ⌢ , − k ∂ ℑ ∂ Y ⌢ = h f ( ℑ w − ℑ ) , V ˜ = 0 , D B ∂ H ∂ Y ⌢ + D T ℑ ∞ ∂ ℑ ∂ Y ⌢ = 0 , S = S w , at Y ⌢ = 0 ,

(7) U ˜ → 0 , ℑ → ℑ ∞ , H → H ∞ , S → S ∞ , when Y ⌢ → ∞ .

Here, along the direction of X- and Y-axes, the velocity components are U ˜ and V ˜ ; the Maxwell relaxation time parameter is λ 1 ; μ stands for dynamic viscosity; ν denotes kinematic viscosity; ρ is fluid density; B 0 scrutinizes the magnetic field; β ∗ is the coefficient of volumetric volume expansion; the densities of base fluid, nanoparticles, and motile are denoted by ρ f , ρ p and ρ m , respectively; k demonstrates the permeability of medium; the electrical conductivity is denoted by σ ; the gravity acceleration is g ; the radiative heat flux is q r ; the chemical reaction is k r ; the nanofluid’s temperature is represented by ℑ ; the nanofluid’s concentration is denoted by H ; motile gyrotactic microorganisms is S ; and, ℑ ∞ , H ∞ , and S ∞ , respectively, stand for ambient temperature, concentration, and gyrotactic microorganisms. C p demonstrates the specific heat, k stands for thermal conductivity, the internal heat absorption is Q 0 , and the thermophoretic velocity is V T . The Brownian and thermophoretic diffusivity coefficients are D B and D T , respectively. The maximum amount of swimming cells is W C , b is known as the chemotaxis constant, the microorganism’s diffusivity is D m , and B 1 specifies the average volume for a gyrotactic microorganism.

3 Homotopic approach

To evaluate the analytical solution of the existing flow model, the homotopic technique is used. This method required the initial guess and linear operator and discoursed as follows:

with

where s H ( H = 1 − 9 ) are the inconsequential constants.