Skip to content
BY 4.0 license Open Access Published by De Gruyter Open Access March 17, 2023

Bioconvection effect in the Carreau nanofluid with Cattaneo–Christov heat flux using stagnation point flow in the entropy generation: Micromachines level study

  • Shuguang Li , Farhan Ali , A. Zaib , K. Loganathan , Sayed M. Eldin and M. Ijaz Khan EMAIL logo
From the journal Open Physics

Abstract

The addition of gyrotactic microbes in the nanoparticles is essential to embellish the thermal efficiency of many systems such as microbial fuel cells, bacteria-powered micro-mixers, micro-volumes like microfluidics devices, enzyme biosensor, and chip-shaped microdevices like bio-microsystems. This analysis investigates the second law analysis in the bioconvection flow of a Carreau nanoliquid through a convectively stretching surface. The heat transports characteristics encountered with Cattaneo–Christove heat flux and thermal radiation. The Buongiorno model is used for nanoliquid, which comprises the Brownian motion and thermophoretic. The appropriate transformation is invoked to change the system of the partial differential equation into ordinary differential equations. Afterward, these equations are classified analytically with the help of the homotopy analysis method. The influence of numerous physical variables is interpreted and elaborated via graphs. The tabular result shows the numerical consequences of different physical flow parameters. It is examined that a more significant Weissenber number We results in deprecation in the velocity field. It is appraised that the temperature profile reduces to augment the value of thermal relaxation time. Justification of the current work has existed through previous publishing results. The utilization of Carreau nanoparticles in the shear rate-dependent viscous fluid is of significant importance due to their potential to improve heat and mass transmission.

Nomenclature

We

local Weissengberg parameter

n

power index parameter

A

dimensional velocity

Nb

Brownian moment

Nt

thermophoretic parameter

Pr

Prandtl number

Rd

radiation parameter

κ

chemical reaction parameter

Sc

Schmidt number

Bi

Biot number

C f

skin friction coefficient

N u x

Nusselt number

S h x

Sherwood number

N h x

motile density

θ

dimensionless temperature

ϕ

dimensionless concentration

f'

dimensionless velocity

N

dimensionless motile density

Ω

motile microorganism density

η

similarity variable

g

gravitational acceleration ( m s 2 )

u , v

velocity component ( m s 1 )

τ

effective heat capacity ( J K 1 )

υ

kinematic viscosity ( m s 2 )

α

thermal relaxation parameter

α m

thermal diffusivity ( m s 2 )

D B

mass diffusivity ( m s 2 )

D T

thermophoresis diffusivity ( m s 2 )

( ρ d ) f

fluid heat capacity

( ρ d ) p

nanoparticles heat capacity

B o

magnetic field strength (K)

T , T

temperature of fluid (K)

C , C

concentration susceptibility (K)

T w

variable temperature (K)

C w

variable concentration (K)

ρ

density of fluid (kg m−3)

U e

free stream velocity (m s−2)

1 Introduction

The notion of nanoliquid was coined by Choi [1], who suggested suspending nanoparticles in standard fluid-like ethylene glycol, oil, and water. Nanomaterials have various consequences for heat transfer, like hybrid energy engines, microprocessors, energy engines, and temperature diminution. Later, Buongiorno [2] introduced two new features in the mathematical formulation: Brownian movement and thermophoretic force. Islam et al. [3] deliberated the MHD flow of micropolar nanofluid with a thermal mechanism inside the two. Alempour et al. [4] described an elliptic cross section for the behavior of nanoliquid with spinning wall tubes. The entropy analysis for the Darcy–Forchheimer flow of nanoliquid past the nonlinear sheet has been conducted by Rasool et al. [5]. The numerical investigation of the 3D flow of Carreau and Carreu nanoliquidpast convective boundary conditions was scrutinized by Naga Santoshi et al. [6]. The heat transport enhancement with the nanoliquid flow has been scrutinized by Wang et al. [7]. Many researchers described the nanoliquid flow with thermal conductivity and heat transfer [8,9,10,11,13]. The Hall effect and chemical reaction with hybrid nanofluid using physical properties are discussed by Mkhatshwa et al. [14]. Tlili et al. [15] explained the 3D flow of the Eyring–Powell nanolqiuid in the Darcy–Forchheimer over a permeable medium. The micropolar flow of Carreau nanoliquid with heat transport over the stretching surface was described by Atashafrooz [16]. Khan et al. [17] conducted the MHD Oldroyd-B fluid on the thermal slip flow for the axisymmetric past rotating disk. Jamshed et al. [18] observed the time-dependent flow of hybrid nanoparticles with the double diffusion theory. Eid [19] studied the magnetic effect of a hybrid nanofluid on the three-dimensional flow of the radiative Cattaneo–Christov heat flux theory. Shazad et al. [20] deliberated the Burgers nanoliquid and Cattaneo–Christov heat flux theory having motile microorganisms. Islam et al. [21] discussed the effects of Hall current on micropolar nanofluid. Shah et al. [22] conducted micropolar flow CNT with viscous dissipation.

An investigation of non-Newtonian fluid at the current scenario has paid special consideration to engineers and scientists owing to its great features in industry and technology. A few of the various non-Newtonian fluid models are Maxwell fluid, Eyring–Powell fluid, viscoelastic fluid, and many more. The complexity of these models has faced challenges due to their single constitutive relation. There is a nonlinear relationship between the shear stress and shear strain of the non-Newtonian fluid. However, in the research, the non-Newtonian fluid is considered under many physical phenomena in Refs. [2329]. One non-Newtonian fluid model is the Carreau fluid model [30,31]. It explains the shear thinning attribute at a low shear rate and the shear thickening feature at a higher shear rate. The Carreau model has the great focus of various scholars due to its implication in polymer and aqueous solutions. Numerous researchers invested their effort in analyzing the behavior of the Carreau fluid with multiple geometries in light of the relevance of such a model. Kefayati and Tang [32] studied the MHD boundary layer flow of the Carreau fluid between two circular cylinders. The heat transfer for the Carreau fluid due to vertical porous has been applied by Olajuwon [33]. The convective heat transport on the Carreau flow has been scrutinized by Hayat et al. [34]. The impact of activation energy near the stagnation point in the Carreau fluid due to nonlinear starching is examined by Khan and Hashim [35]. The MHD flow of the Carreau fluid using activation energy due to a starching sheet is described by Hsiao [36]. The 3D Carreau fluid of zero nanoparticles using the analytical method was surveyed by Khan et al. [37]. The study of the three-dimensional Carreau fluid flows with variable viscosity, and zero mass flux condition has been obtained by Irfan et al. [38]. The nonlinear thermal radiative flow of the Carreau nanoliquid past a wedge surface was reported by Jyothi et al. [39]. The entropy production for the Carreau nanoliquid with the heated convective condition is presented by Khan et al. [40]. Naz et al. [41] identified the second law analysis on the MHD Carreau nanoliquid for the impact of stratification. Rooman et al. [42] perceived features of heat transfer of the Carreau nanofluid for the Renal tubule. Li et al. [43] presented the impact of thin-film Carreau nanoparticles due to an inclined surface.

Another exciting area of research is the bioconvection nanofluid, which has a broader range of applications, including microfluidic devices, microsystems, biodiesel fuels, biosensors, biotechnology, microsystem, and polymer construction. The motile microorganism is denser than water with up-swimming in a haphazard motion, leading to the bioconvection development. A thick layer of bacteria occurs on the top surface of the lighter fluid due to the buildup of these microorganisms. The relationship between microorganisms, convection nanoparticles, and buoyancy force is tied to the sample growth. It has been shown on several occasions that the stability of nanoparticle suspension has been successfully boosted when gyrotactic microorganisms are present. The term “bioconvection” was brought to the attention of the public by Wager [44] and Platt [45]. The initial concept regarding microorganisms and oxytactic via isothermal reaction is explored by Kuznetsov [46]. Uddin et al. [47] inspected the mixed convective flow of thermal slip nanoliquid having microorganisms over a permeable medium. The flow of Sisko nanoparticles under microorganisms with mixed convection was examined by Farooq et al. [48]. The unsteady flow of nanoparticles along microorganisms was pointed out by Waqas et al. [49]. The bioconvection flow of nanoparticles due to cone is conducted by Khan et al. [50]. The significance of the Stefan flow of nanoparticles with the effect of bio-convection flow of nanofluid was studied by [43,5153]. The result of viscous dissipation with the thermal radiative bio-convection flow of Carreau nanoliquid past suction/injection is discussed by Kiari et al. [54]. Shaw et al. present the influence of oxytactic microorganisms due to the porous medium [55]. Mustafa et al. [56] investigated stagnation point flow of nanomaterial over a stretched surface.

Second law of thermodynamics and energy losses through boundary layer has many uses and daily life applications. Friction forces within the system cause energy losses, which involve entropy minimization. Entropy minimization is the measure of irreversible heat transportation. Such entropy minimization has produced tremendous achievement in different fields such as cooling, air separation, fuel cell, and electrochemistry. The word was coined by Bejan [57]. Khan et al. [58] reported entropy minimization on mixed convective flow with thermal radiation and slip flow. Recently, Kumar et al. [59] discussed the mathematical expression for entropy minimization on Carreau nanofluid toward a stagnation point. Most recently, the bioconvection flow of Sutterby nanofluid subject to the entropy generation was explained by Ramzan et al. [60].

In light of the literature mentioned earlier, no endeavor has been considered to study the heat transport characteristics with the bio-convection flow of the Carreau nanoliquid toward a stagnation point over a stretching sheet. So, the current investigation intends to examine the effect of bio-convection flow in a Carreau nanoliquid toward a stagnation point on a stretching sheet. In addition, the energy equation is constructed by applying an improved form of Fourier’s law known as Cattaneo–Christov heat flux. Further, a chemical reaction of the first order is also taken in the concentration. The highlighted equations have been modeled in depth by including the dimensionless parameters and adding some necessary similarity transformations. The analytical solution with the homotopy analysis method (HAM) method is used for dimensionless equations. The influence of physical parameters reveals graphical outcomes. This has potential applications in a variety of fields, including the production of materials, the defrosting of plastics, the manufacture of pharmaceutical products and polymerization fluids, the creation of chemicals, toxic and radioactive facilities, manufacturing equipment, and the food industry. It is present in a wide variety of items, including blood, nectar, scents, adhesives, petroleum diesel, tarmac, cream, and many others. This kind of fluid was found in dental paste, molten monomers, blood serum, pulps, and various other components. Applications in the fields of microbiology, bio-microsystems, and biosciences are also possible with this.

2 Description of the physical model

The Caushy stress tensor for use in the Carreau rheological theory is considered as follows:

(1) τ ̆ = pI + μ C 1 .

Here,

(2) μ = μ + ( μ 0 μ ) [ 1 + μ ( γ δ ̈ ) 2 ] n 1 2 ,

where p , I , μ 0 , μ , and n are pressure, tensor identity, the zero shear rate, infinity shear rate, and power index, respectively.

The symbol δ ̈ , which stands for the shear rate, is defined as follows:

(3) δ ̈ = 1 2 tr ( C 1 2 ) ,

where C 1 is the Rivlin–Erickson tensor:

(4) C 1 = ( grad V ) + ( grad V ) T .

2.1 Mathematical construction

In this investigation, it is considered that a steady incompressible flow of the Carreau nanofluid with motile microorganisms approaches a convectively heated sheet, the flow will split into two streams while maintaining the stagnation point at its origin. Heat transfer phenomena elaborate thermal radiation and heat source-sink, and mass transfer phenomena are examined by the chemical reaction. Further, the Buongiorno model is used to study how Brownian motion and thermophoresis affect the flow of a fluid. The x and y coordinates with velocities U w = ax and U e = bx are considered along with origin O near the stagnation point. T W , C W , and χ w are the wall of temperature, concentration, and motile density, respectively, whereas T is the ambient temperature, C is the ambient concentration, and χ is the ambient microorganisms, respectively. The geometrical interpretation is shown in Figure 1. The governing modeled equations and heat transfer under thermal radiation are given as follows [6165]:

(5) u x + v y = 0 ,

(6) u u x + v u y = ν 2 u y 2 1 + n 1 2 Γ 2 u y 2 + ν ( n 1 ) Γ 2 2 u y 2 u y 2 1 + n 3 2 Γ 2 u y 2 U e d U e d x ,

(7) u T x + v T y α f 2 T y 2 + δ Ω = τ D B C y T y + D T T T y 2 + Q 0 ( ρ C p ) f ( T T ) 1 ( ρ C p ) f q y ,

(8) u C x + v C y = D B 2 c y 2 + D T T 2 T y 2 K 0 ( C C ) ,

(9) u χ x + v χ y + b χ c ( C w C ) y χ C y = D m 2 χ y 2 .

Figure 1 
                  Physical model of the problem.
Figure 1

Physical model of the problem.

The Rosseland approximation is explained by

(10) q r = T 4 y 4 σ * 3 k * .

Here, σ * and k * represent Stefan–Boltzmann radiation and the absorption coefficient. So, the Taylor series expansion can be written as follows:

(11) T 4 = T 4 + 4 T 4 ( T T ) + 6 T 2 ( T T ) .

By ignoring the larger term ( T T ) to obtain the expression, we obtain

(12) T 4 4 T 3 T 3 T 4 .

By using Eq. (10) in Eq. (12), we obtain

(13) q r = 2 T y 2 16 σ * T 3 3 ( ρ C p ) f k * .

The aforementioned equation can be written as follows [63]:

(14) Ω = u u x T x + v v y T y + u 2 2 T x 2 + v 2 2 T y 2 + 2 u v 2 T x y + u v x T y + v u y T x ,

where u and v denote the velocity component along x- and y-axis, v is the kinematic viscosity, Γ is the material time constant, ρ f represents the density of fluid, We represents the Weissenberg fluid parameter, n is the power law index, g represents the gravity, α f = k ( ρ c p ) f represents thermal diffusivity, τ = ( ρ c p ) p ( ρ c p ) f shows that the effective heat capacity is proportional to heat capacity of the fluid, D B is the Brownian diffusion, D T is the thermophoretic coefficient, Q 0 is the heat source/sink, and δ is the fluid relaxation time.

The relevant boundary conditions are assumed to be the following form:

(15) u = u w ( x ) , v = 0 , k T y = h ( T w T ) , C = C w , χ = χ w as y = 0 , u U e , T T , C C , χ χ at y .

By using the transformation, we obtain

(16) u = ax f ( η ) , v = av f ( η ) , η = y a v , θ ( η ) = T T T w T , ϕ ( η ) = C C C w C , N ( η ) = χ χ χ w χ .

By using Eq. (14), Eqs. (2)–(4) become

(17) 1 + n 1 2 We f 2 f + 2 n 1 2 We f 2 1 + n 3 2 We f 2 + f f f 2 + A 2 = 0 = 0 ,

(18) θ 1 + 4 3 Rd + PrNt θ 2 + PrNb θ ϕ Pr α ( f f θ + f 2 θ ) Pr ε θ = 0 ,

(19) ϕ + Sc f ϕ + Nt Nb θ Sc κ ϕ = 0 ,

(20) N Pe [ ϕ ( N + Ω ) + ϕ N ] + Lb fN = 0 .

The corresponding boundary conditions are as follows:

(21) f ( 0 ) = 0 , f ( 0 ) = 1 , θ ( 0 ) = Bi ( θ ( 0 ) ) Bi , ϕ ( 0 ) = 1 , N ( 0 ) = 1 f ( ) A , θ ( ) 0 , ϕ ( ) 0 , N ( ) 0 .

The governing variables are as follows: Weissenberg number We = ( Γ a 2 ) , stretching velocity parameter A = a c , Brownian parameter, Nb = τ D B ( C w C ) v , thermophoresis parameter, Nt = τ D t ( T w T ) T v , Scimdth number, Sc = v D B , Biot number, Bi = h f k v a , thermal relaxation time, α = ( δ a ) , Prandtl number Pr = v α , Peclet number, Pe = b W c D m , and bioconvection Lewis number, Lb = v D m .

The skin friction C f x , Nusselt number Nu x , Sherwood number Sh x , and motile density Nh are given by

(22) C f x = τ w p u w 2 , N u x = x q w k ( T w T ) , S h x = x q m D b ( C w C ) and W h x = x q n D m ( X w X ) ,

τ w is the surface shear stress, q w is the surface heat flux, q m is the surface mass flux, and q n is the motile density, which are given as follows:

(23) τ w = μ 0 u y + n 1 2 Γ 2 u y u x 2 + 3 v x u y 2 , y = 0 q w = k + 16 σ * T 3 3 k * T y y = 0 , q m = k C y y = 0 and q n = D m χ y y = 0 .

The dimensionless form of the aforementioned parameters is expressed as follows:

(24) C f x Re x 0 . 5 = 1 + n 1 2 We f ( 0 ) , N u x Re x 1 / 2 = 1 + 4 3 Rd θ ( 0 ) , S h x Re x 1 2 = ϕ ( 0 ) and Wn Re x 1 2 = N ( 0 ) ,

where Re x = x U w v is the local Reynolds number.

2.2 Entropy generation

The entropy minimization of Carreau nanofluid is given as follows:

(25) S g = k f T 2 1 + 16 σ T 3 3 k k T y 2 irreversibility effect of heattransfer + 1 + ( n 1 2 ) Γ 2 u y 2 u y 2 Fluid friction Irreversiblity + RD C C y 2 + RD T C y T y + RD C χ y 2 irreversibilty due to the diffision effect .

The impact of entropy production is

(26) E 0 = β 1 ( T a T b ) 2 T b 2 .

Entropy in the recreating form is expressed as follows:

(27) E 0 = E Gen E 0 .

In view of Eq. (12), Eq. (24) becomes

(28) N g = θ 2 1 + 4 3 Rd δ 1 + Y f 2 1 + n 1 2 We f 2 + Λ Y δ θ ϕ + Λ Y δ 2 ϕ 2 + Λ ω δ θ χ + Λ ω δ 2 χ 2 .

The assessment of the Bejan number Be is a ratio of entropy minimization due to the heat transfer irreversible to accumulated entropy minimization.

(29) Be = θ 2 1 + 4 3 Rd δ 1 + Λ Y δ θ ϕ + Λ Y δ 2 ϕ 2 + Λ ω δ θ χ + Λ ω δ 2 χ 2 . θ 2 1 + 4 3 Rd δ 1 + Y f 2 1 + n 1 2 Γ 1 f 2 + Λ Y δ θ ϕ + Λ Y δ 2 ϕ 2 + Λ ω δ θ χ + Λ ω δ 2 χ 2 .

2.3 Solution of the homotopy expression

The linear operators, including initial guesses, are as follows [66]:

(30) f 0 ( η ) = A η + ( 1 A ) ( 1 e η ) , θ ( η ) = Bi e η 1 + Bi , ϕ ( η ) = e η , χ ( η ) = e η , L ( f ) = f f , L ( θ ) = θ θ , L ( ϕ ) = ϕ ϕ , L ( χ ) = N N .

The aforementioned linear operators exhibit the property:

(31) L ( f ) ( B 1 + B 2 e η + B 3 e η ) = 0 , L ( θ ) ( B 4 e η + B 5 e η ) = 0 , L ( ϕ ) ( B 6 e η + B 7 e η ) = 0 , L ( χ ) ( B 8 e η + B 9 e η ) = 0 ,

where B i = 1 9 indicates coefficient constants.

2.4 Zeroth-order deformation

The zeroth-order deformation is given as follows:

(32) ( 1 c ) [ f ( η , c ) f 0 ( η ) ] = c f N f [ f ( η , c ) , θ ( η , c ) , ϕ ( η , c ) , χ ( η , c ) ] , ( 1 c ) [ θ ( η , c ) θ 0 ( η ) ] = c θ N θ [ f ( η , c ) , θ ( η , c ) , ϕ ( η , c ) ] , ( 1 c ) [ ϕ ( η , c ) ϕ 0 ( η ) ] = c ϕ N ϕ [ f ( η , c ) , θ ( η , c ) , ϕ ( η , c ) ] , ( 1 c ) [ N ( η , c ) N ( η ) ] = c χ N χ [ f ( η , c ) , θ ( η , c ) , ϕ ( η , c ) , N ( η , c ) ] . ,

(33) f ( 0 , c ) = 0 , f ( 0 , c ) = 1 , f ( , c ) = 0 , θ ( 0 , c ) = Bi ( [ θ ( 0 , c ) 1 ] ) , θ ( , c ) = 0 , ϕ ( 0 , c ) = 1 , ϕ ( , c ) = 0 , N ( 0 , c ) = 1 , N ( , c ) = 0 .

Here, c is an embedding parameter and f , θ , ϕ , and N are nonzero auxiliary variables.

N f , N θ , N ϕ , and N N denote the nonlinear operators that can be produced through Eq. (32):

(34) N f [ f ( η , c ) ] = 1 + 1 + n 1 2 We 2 f ( η , c ) η 2 2 3 f ( η , c ) η 3 + 2 n 1 2 We 2 f ( η , c ) η 2 2 1 + n 3 2 We 2 f ( η , c ) η 2 2 + f ( η , c ) 2 f ( η , c ) η 2 2 f ( η , c ) η 2 2 A 2 N θ [ θ ( η , c ) , ϕ ( η , c ) ] = 1 + 4 3 Rd 2 θ ( η , c ) η 2 + PrNb θ ( η , c ) η ϕ ( η , c ) η + PrNt θ ( η , c ) η 2 + Pr f ( η , c ) f ( η , c ) η 2 f ( η , c ) η 2 + f ( η , c ) 2 2 θ ( η , c ) η 2 + Pr ε θ ( η , c ) N ϕ [ f ( η , c ) , θ ( η , c ) , ϕ ( η , c ) ] = 2 ϕ ( η , c ) η 2 + Sc f ( η , c ) ϕ ( η , c ) η + Sc Nt Nb 2 θ ( η , c ) η 2 κ ϕ ( η , c ) N N [ f ( η , c ) , ϕ ( η , c ) , N ( η , c ) ] = 2 N ( η , c ) η 2 + Lb f ( η , c ) N ( η , c ) η + Pe N ( η , c ) η ϕ ( η , c ) η + 2 N ( η , c ) η 2 ( ϖ + N ( η , c ) )

When c = 0 and c = 1 , we obtain

(35) f ( η ; 0 ) = f 0 ( η ) , θ ( η ; 0 ) = θ 0 ( η ) , ϕ ( η ; 0 ) = ϕ 0 ( η ) , N ( η ; 0 ) = N 0 ( η ) f ( η ; 1 ) = f 0 ( η ) , θ ( η ; 1 ) = θ 0 ( η ) , ϕ ( η ; 1 ) = ϕ 1 ( η ) , N ( η ; 1 ) = N 0 ( η ) .

If the c is enhanced from 0 to 1, so f ( η , c ) , θ ( η , c ) , ϕ ( η , c ) , and N ( η , c ) fluctuate from f 0 ( η ) , θ 0 ( η ) , ϕ 0 ( η ) , N ( η ) to f ( η ) , θ ( η ) , ϕ ( η ) , and N ( η ) . By the Taylor expansion series,

(36) f ( η , c ) = f 0 ( η ) + m = 1 f m ( η ) c m f m ( η ) = 1 m ! m f ( η , c ) η m c = 0 θ ( η , c ) = θ 0 ( η ) + m = 1 θ m ( η ) c m θ m ( η ) = 1 m ! m θ ( η , c ) η m c = 0 ϕ ( η , c ) = ϕ 0 ( η ) + m = 1 ϕ m ( η ) c m f m ( η ) = 1 m ! m ϕ ( η , c ) η m c = 0 N ( η , c ) = N 0 ( η ) + m = 1 χ m ( η ) c m χ m ( η ) = 1 m ! m N ( η , c ) η m c = 0 .

2.5 mth order deformation

Considering Eq. (36) for homotopy at the mth order

(37) L f [ f m ( η ; c ) X m f m 1 ( η ) ] = f R f , m ( η ) L θ [ θ ( η ; c ) X m θ m 1 ( η ) ] = θ R θ , m ( η ) L ϕ [ ϕ m ( η ; c ) X m ϕ m 1 ( η ) ] = ϕ R ϕ , m ( η ) L N [ N m ( η ; c ) X m N m 1 ( η ) ] = N R N , m ( η ) ,

with the mth order boundary conditions:

(38) f m ( 0 ) = 0 , f m ( 0 ) = 0 , f m ( ) = 0 , θ m ( 0 ) Bi θ m ( 0 ) = 0 , θ m ( ) = 0 , ϕ m ( 0 ) = 0 , ϕ m ( ) = 0 , N m ( 0 ) = 0 , N m ( ) = 0 ,

(39) R f , m ( η ) = 1 + n 1 2 Γ 1 f '' 2 f ''' + 2 n 1 2 Γ 1 f 2 1 + n 3 2 Γ 1 f '' 2 + A 2 . R f , θ ( η ) = θ m 1 + Pr k = 0 m 1 f m 1 k θ k NbNt θ m 1 k ϕ k + Nt θ m 1 k θ k ε θ m 1 k Pr α k = 0 m 1 ( f m 1 k f k θ k + f m 1 k f k θ m 1 ) R f , ϕ ( η ) = ϕ m 1 + Sc k = 0 m 1 f m 1 k ϕ k + Nt Nb θ m k κ ϕ m 1 R f , N ( η ) = N m 1 + Lb k = 0 m 1 ( f m 1 k N k ) Pe k = 0 m 1 ( N m 1 k ϕ k + ϕ m 1 '' ( ϖ + f N 1 k ) ) ,

(40) X m = 0 , m 1 1 , m > 1 .

The solutions can be explained in general

(41) f m f m * = B 1 + B 2 e η + B 3 e η θ m θ m * = B 4 e η + B 5 e η ϕ m ϕ m * = B 6 e η + B 7 e η N m N m * = B 8 e η + B 9 e η ,

where f m * , θ m * , ϕ m * , and N m * have the particular solution.

2.6 Homotopic solution of convergence

The study of auxiliary variables f , θ , ϕ , and N depends on adjusting and controlling convergence. The appropriate magnitude of these variables, sketch the graph of the –curve at different order of approximation for the auxiliary parameters are displayed via Figure 1(a–d). Figure 2(a–d) shows that the appropriate values of f , θ , ϕ , and N are −2.0< f < −0.1, −1.4 < θ < 0.1, −1.0 < ϕ < 0.1, −1.0 < N <−0.1. Table 1 evaluates the series solution of convergence of the concerning when f = −0.60, θ = ϕ = −0.55, N = −0.7 (Figure 2).

Table 1

Convergence solution of HAM for A = 0.1 , We = 1 , n = 1.0 , N b = N t = 0.4 ,   S c = P r = 1.0 , ε = 0.4 , P e = 0.4 , L b = 1.0 , Ω = 0.1 .

Order of m f ( 0 ) θ ( 0 ) ϕ ( 0 ) N ( 0 )
1 0.940500 0.167218 1.229710 0.960000
5 0.968455 0.168943 1.434720 1.065450
10 0.969384 0.168784 1.44170 1.109280
15 0.969387 0.168831 1.441760 1.112850
20 0.969387 0.168831 1.441760 1.112850
25 0.969387 0.168831 1.441760 1.112850
30 0.969387 0.168831 1.441750 1.112850
35 0.969387 0.168831 1.441750 1.112850
40 0.969387 0.168831 1.441750 1.112850
Figure 2 
                  (a) 
                        
                           
                           
                              
                                 
                                    ℏ
                                 
                                 
                                    f
                                 
                              
                           
                           {\hslash }_{f}
                        
                      – sketch over 
                        
                           
                           
                              
                                 
                                    f
                                 
                                 
                                    ″
                                 
                              
                              (
                              0
                              )
                           
                           {f}^{^{\prime\prime} }\left(0)
                        
                     ; (b) 
                        
                           
                           
                              
                                 
                                    ℏ
                                 
                                 
                                    θ
                                 
                              
                           
                           {\hslash }_{\theta }
                        
                      – sketch over 
                        
                           
                           
                              θ
                              ′
                              (
                              0
                              )
                           
                           \theta \left^{\prime} \left(0)
                        
                     ; (c) 
                        
                           
                           
                              
                                 
                                    ℏ
                                 
                                 
                                    ϕ
                                 
                              
                           
                           {\hslash }_{\phi }
                        
                      – sketch over 
                        
                           
                           
                              ϕ
                              ′
                              (
                              0
                              )
                           
                           \phi \left^{\prime} \left(0)
                        
                     ; (d) 
                        
                           
                           
                              
                                 
                                    ℏ
                                 
                                 
                                    N
                                 
                              
                           
                           {\hslash }_{N}
                        
                      – sketch over 
                        
                           
                           
                              N'
                              (
                              0
                              )
                           
                           {N\text{'}}\left(0)
                        
                     .
Figure 2

(a) f – sketch over f ( 0 ) ; (b) θ – sketch over θ ( 0 ) ; (c) ϕ – sketch over ϕ ( 0 ) ; (d) N – sketch over N' ( 0 ) .

3 Results and discussion

The significance of numerous physical variables is elucidated in Figures 39. These physical parameters are sketched for velocity, temperature, concentration, and microorganism profiles for numerous magnitudes of variables. To identify the current estimation, a comparison of the current study with numeric outcomes is revealed in Table 2.

Figure 3 
               (a and b) Influence of 
                     
                        
                        
                           We
                        
                        {\rm{We}}
                     
                   and 
                     
                        
                        
                           A
                        
                        A
                     
                   on 
                     
                        
                        
                           
                              
                                 f
                              
                              
                                 ′
                              
                           
                           (
                           η
                           )
                        
                        {f}^{^{\prime} }\left(\eta )
                     
                  .
Figure 3

(a and b) Influence of We and A on f ( η ) .

Figure 4 
               (a–f) Influence of 
                     
                        
                        
                           A
                           ,
                           Rd
                           ,
                           Nt
                           ,
                           Nb
                           ,
                           α
                        
                        A,{\rm{Rd}},{\rm{Nt}},{\rm{Nb}},\alpha 
                     
                  , and 
                     
                        
                        
                           Bi
                        
                        {\rm{Bi}}
                     
                   on 
                     
                        
                        
                           θ
                           
                              
                                 (
                                 
                                    η
                                 
                                 )
                              
                           
                        
                        \theta (\eta )
                     
                  .
Figure 4

(a–f) Influence of A , Rd , Nt , Nb , α , and Bi on θ ( η ) .

Figure 5 
               (a–d) Influence of 
                     
                        
                        
                           Nb
                           ,
                           Nt
                           ,
                           κ
                        
                        {\rm{Nb}},{\rm{Nt}},\kappa 
                     
                  , and 
                     
                        
                        
                           Sc
                        
                        {\rm{Sc}}
                     
                   on 
                     
                        
                        
                           ϕ
                           
                              
                                 (
                                 
                                    η
                                 
                                 )
                              
                           
                        
                        \phi (\eta )
                     
                  .
Figure 5

(a–d) Influence of Nb , Nt , κ , and Sc on ϕ ( η ) .

Figure 6 
               (a–c) 
                     
                        
                        
                           Pe
                           ,
                           Lb
                           ,
                        
                        {\rm{Pe}},{\rm{Lb}},
                     
                   and 
                     
                        
                        
                           ϖ
                        
                        \varpi 
                     
                   on 
                     
                        
                        
                           N
                           
                              
                                 (
                                 
                                    η
                                 
                                 )
                              
                           
                        
                        N(\eta )
                     
                  .
Figure 6

(a–c) Pe , Lb , and ϖ on N ( η ) .

Figure 7 
               (a–c) Influence of 
                     
                        
                        
                           We
                           ,
                           Br
                           ,
                        
                        {\rm{We}},{\rm{Br}},
                     
                   and 
                     
                        
                        
                           Rd
                        
                        {\rm{Rd}}
                     
                   on 
                     
                        
                        
                           Ng
                           .
                        
                        {\rm{Ng}}.
Figure 7

(a–c) Influence of We , Br , and Rd on Ng .

Figure 8 
               (a–c) Influence of 
                     
                        
                        
                           We
                           ,
                           Br
                        
                        {\rm{We}},{\rm{Br}}
                     
                  , and 
                     
                        
                        
                           Rd
                        
                        {\rm{Rd}}
                     
                   on Be.
Figure 8

(a–c) Influence of We , Br , and Rd on Be.

Figure 9 
               (a–d) Fluctuation of (a) 
                     
                        
                        
                           We
                        
                        {\rm{We}}
                     
                   against 
                     
                        
                        
                           A
                        
                        A
                     
                  , (b) 
                     
                        
                        
                           Rd 
                           against 
                           α
                        
                        {\rm{Rd}}{\rm{against}}\alpha 
                     
                  , (c) 
                     
                        
                        
                           Sc
                        
                        {\rm{Sc}}
                     
                   against 
                     
                        
                        
                           κ
                        
                        \kappa 
                     
                  , (d) 
                     
                        
                        
                           Lb
                        
                        {\rm{Lb}}
                     
                   against 
                     
                        
                        
                           Pe
                        
                        {\rm{Pe}}
                     
                   on 
                     
                        
                        
                           
                              
                                 Re
                              
                              
                                 x
                              
                              
                                 1
                                 /
                                 2
                              
                           
                           Cf
                        
                        {{\rm{Re}}}_{x}^{1/2}{\rm{Cf}}
                     
                  , 
                     
                        
                        
                           
                              
                                 Re
                              
                              
                                 x
                              
                              
                                 −
                                 1
                                 /
                                 2
                              
                           
                           Nu
                        
                        {{\rm{Re}}}_{x}^{-1/2}{\rm{Nu}}
                     
                  , 
                     
                        
                        
                           
                              
                                 Re
                              
                              
                                 x
                              
                              
                                 −
                                 1
                                 /
                                 2
                              
                           
                           Sh
                        
                        {{\rm{Re}}}_{x}^{-1/2}{\rm{Sh}}
                     
                  , and 
                     
                        
                        
                           
                              
                                 Re
                              
                              
                                 x
                              
                              
                                 −
                                 1
                                 /
                                 2
                              
                           
                           Nh
                        
                        {{\rm{Re}}}_{x}^{-1/2}{\rm{Nh}}
                     
                  .
Figure 9

(a–d) Fluctuation of (a) We against A , (b) Rd against α , (c) Sc against κ , (d) Lb against Pe on Re x 1 / 2 Cf , Re x 1 / 2 Nu , Re x 1 / 2 Sh , and Re x 1 / 2 Nh .

Scheme 1 
                  Solution of the homotopy expression.
Scheme 1

Solution of the homotopy expression.

Table 2

Results comparison with Mustafa et al.’s paper [56]

A Mustafa et al. [56] Present
0.01 0.99823 0.998065380
0.10 0.96954 0.969387504
0.20 0.91813 0.918107131
0.50 0.66735 0.667263673
2.00 2.01767 2.017502801
3.00 4.72964 4.729282355

Figure 3(a–b) shows the impact of the Weissenberg number We and stretching ratio parameter A for the velocity field f ( η ) . Intensifying magnitude of Weissenberg number We results in depreciation in the velocity field of the Carreau nanoliquid. Physically, the larger values of We show that the velocity and momentum boundary layer thickness are decaying in contrast to the viscous fluid. The magnitude of A on f ( η ) is shown in Figure 3(b). Stretching velocity is greater than free stream velocity when the velocity and boundary layer thickness reduces with the increased value of A . Velocity and boundary layer thickness increase when the stretching velocity is depreciated than free stream velocity as the larger value of A . If A = 0, no boundary is observed.

Figure 4(a–f) sketches the curve for temperature θ ( η ) as it is attributed by A , Rd , Bi , α , Nt , and Nb . The impact of A is shown in Figure 4(a). The enhancing magnitude of A related to the more incredible free stream velocity causes the temperature profile to become thinner. The impact of thermal radiation is shown in Figure 4(b). The uplifting of Rd uplifts the thermal layer. It is the fact that the escalated value of Rd obtains inside heat, which is the reason for the acceleration of Rd. The estimation of Bi on θ ( η ) is shown in Figure 4(c). Figure 4(d) manifests that rising values of α on temperature profile θ ( η ) . Physically, it can be seen that the impact of thermal relaxation time builds the temperature profile which displays the strengthened heat transfer. An augmentation in the magnitude of Bi causes the enhancement of the thermal layer and boundary layer thickness. The Biot number is the proportion of conductive resistance inside the surface over the outside conductive resistance outside the surface. Figure 4(e) views that the temperature field enhances with enhancing the Brownian motion Nb . There is an incline in the thermal layer thickness with an increased value of Nb . So, with the rising magnitude of Nb , the strength of their hectic movement leads to enhancing the kinetic energy of the nanoparticles and therefore improves the temperature field. The thermophoretic curve for the thermal layer is sketched in Figure 4(f). The larger values of the Nt in the thermal layer increase the temperature and hence hot particles move away from the hot to the cold section.

The conceding values of Nb , Nt , κ , and Sc on the concentration curve ϕ ( η ) are plotted in Figure 5(a–d). Figure 5(a) shows the impact of Brownian movement variables Nb on the concentration curve. The behavior of the concentration curve is noted for numerous magnitudes of the Brownian motion variables. It enhances the value of Nb , which causes the striking of the fluid particles to intensify as it reduces the concentration profile. Concentration distribution against the thermophoretic variables Nt is shown in Figure 5(b). It has been noted that nanoparticles concentration inflates with the increase in thermophoretic variable Nt . Increasing the κ and Sc decrease ϕ ( η ) , as shown in Figure 5(c–d). The reason behind this is that the greater the magnitude of the κ decay the mass transfer rate due to this concentration field reduces with the larger magnitude of Sc proportion of thermal diffusivity to mass diffusivity. Figure 6(a–c) views the reducing role of microorganism density profiles N ( η ) with the increase in Pe , ϖ , and Lb . Figure 6(a) portrays the impact of Pe on microorganism’s density profile N ( η ) . Peclect number is the ratio of the rate of advection of a physical quantity by the flow to the rate of diffusion of the same quantity driven by an appropriate gradient. Diffusion is the phenomenon that accelerates from a more significant concentration to a lower concentration. It is noticed that a decrease in the diffusion of microorganism density leads to the mounting value of Pe . The depreciation in the motile density is shown in Figure 6(b) with an increasing estimation of ϖ . Figure 6(c) elucidates the consequences of the Lb on the motile microorganisms N ( η ) . An augmentation in Lb causes a depreciation in the diffusivity of microorganisms due to the reduction of the motile microorganism profile. Figure 7(a–c) shows the role of We , Br , and Rd on entropy production of Ng . Figure 7(a) shows the reduction of Ng for larger values of We . Physically, an uplift in the values of We , which is due to the quicker shear of fluid with the surface, is due to the decrease in entropy generation. Figure 7(b) exhibits the attribute of thermal radiative Rd . Physically thermal radiation Rd has larger heat to enhance the entropy minimization Ng . Figure 7(c) shows that Ng is increase with the increase in the magnitude of Br . Moreover, the Brinkmann number produces the proportion of liberated heating through viscous heat to heating transport through the conduction of molecules. So, extra heating is created in the system with the increase in the magnitude of Brikman number Br, thus consequently increasing Ng .

Figure 8(a–c) shows the effect of We , Br , and Rd on Be . From Figure 8(a), it is noted that Bejan number Be augments with the augmenting values of We . Physically, the rate of the Bejan number intensifies when the Weissenberg number increases. In Figure 8(b), Br indicates the ratio of heat transmission for conduction to heat creation through viscous heating, and greater values of Br generate more heat in the system, which leads to an increase in the temperature and hence the whole system is disordered. The effect of Rd on Be is shown in Figure 8(c). It is manifest from the figure that Be gets increased for larger magnitudes of radiation parameter Rd . Moreover, it is interestingly noted that the Bejan number Be is accelerated. Figure 9(a–d) exhibits variation of A , We , Rd , α , Sc , κ , Lb , and Pe on C f . From Figure 9(a), it is pointed out that the larger values of A and We maximizes C f . Figure 9(b) shows the influence of the Nusselt number on α against Rd . The larger values of Rd and α reduce the Nusselt number. Figure 9(c) shows the impact of κ and Sc on the Sherwood number. The figure shows that the rate of mass transport is enhanced by increasing the value of κ and Sc. The effect of Pe and Lb on Nh is shown in Figure 9(d). There is an increase in Nh for the given values of Pe and Ω.

Figures 10 (a and b) and 11 (a and b) highlight the physical impact of pertinent flow variable in the form of 3D plot on the skin friction coefficient, Nusselt number, Sherwood number, and motility number. Further, Figure 12(a–b) signifies the salient aspects of We = 1.5 and Rd = 0.5 in the form of streamlines. Figure 12(a) shows that the decrease in the values of We increases streamlines along the x-axis. In contrast, in Figure 12(b), the isotherm line stretches far from the x-axis due to the augmented value of Rd .

Figure 10 
               3D plot of (a) 
                     
                        
                        
                           
                              
                                 Re
                              
                              
                                 x
                              
                              
                                 1
                                 /
                                 2
                              
                           
                           Cf
                           ,
                        
                        {{\rm{Re}}}_{x}^{1/2}{\rm{Cf}},
                     
                   (b) 
                     
                        
                        
                           
                              
                                 Re
                              
                              
                                 x
                              
                              
                                 −
                                 1
                                 /
                                 2
                              
                           
                            Nu
                        
                        {{\rm{Re}}}_{x}^{-1/2}{\rm{Nu}}
                     
                   with variation of 
                     
                        
                        
                           A
                           ,
                           We
                        
                        A,{\rm{We}}
                     
                  , 
                     
                        
                        
                           Rd
                           ,
                           and 
                           α
                        
                        {\rm{Rd}},{\rm{and}}\alpha 
                     
                  .
Figure 10

3D plot of (a) Re x 1 / 2 Cf , (b) Re x 1 / 2 Nu with variation of A , We , Rd , and α .

Figure 11 
               3D plot 
                     
                        
                        
                           
                              
                                 Re
                              
                              
                                 x
                              
                              
                                 −
                                 1
                                 /
                                 2
                              
                           
                           Sh
                        
                        {\mathrm{Re}}_{x}^{-1/2}\text{Sh}
                     
                   and 
                     
                        
                        
                           
                              
                                 Re
                              
                              
                                 x
                              
                              
                                 −
                                 1
                                 /
                                 2
                              
                           
                           Nh
                        
                        {\mathrm{Re}}_{x}^{-1/2}\text{Nh}
                     
                   with variation of 
                     
                        
                        
                           Sc
                           ,
                           κ
                        
                        {\rm{Sc}},\kappa 
                     
                  , 
                     
                        
                        
                           Pe
                           ,
                           and
                           Lb
                        
                        {\rm{Pe}},{\rm{and}}{\rm{Lb}}
                     
                  .
Figure 11

3D plot Re x 1 / 2 Sh and Re x 1 / 2 Nh with variation of Sc , κ , Pe , and Lb .

Figure 12 
               Stream line and isotherm line when We = 1.5 and Rd = 0.5.
Figure 12

Stream line and isotherm line when We = 1.5 and Rd = 0.5.

Table 3 plots the numerical estimation C fx Re x 1 / 2 for different A values. A tending fluctuation has been obtained, C f x Re x 1 / 2 . From Table 4, the numeric simulations are performed, Nu Re x 1 / 2 . It is shown that the Nu Re x 1 / 2 reduced with the increase in α and Rd . Table 5 shows the numeric estimation of numerous magnitudes on Sh Re x 1 / 2 . Also the table shows that the Sherwood number increases for Sc and κ . Finally, fluctuation in motile density Nh Re x 1 / 2 has been shown in Table 6. Noted from the table that the motile density enhances Pe and Lb .

Table 3

Correlation of A and We on skin friction

A We n Re x 1 / 2 C fx
0.6 1.0 0.55646
1.2 0.8 0.53043
1.0 0.50400
1.2 0.48146
1.4 0.6 1.0 1.21689
0.8 1.14862
1.0 1.08600
1.2 1.03522
0.6 1.98133
1.6 0.8 2.0 1.85456
1.0 1.74600
1.2 1.60080
0.6 2.84978
1.8 0.8 2.0 2.64825
1.0 2.48400
1.2 2.35644
Table 4

Correlation of Rd and ε on Nusselt number

Rd α Nb Nt Re x 1 / 2 Nu x
0.4 0.1 0.1 0.02367
0.2 0.6 0.02263
0.8 0.02191
1.0 0.02061
0.4 0.6 0.4 0.03751
0.8 0.5 0.03645
1.0 0.6 0.03515
1.2 0.7 0.03437
0.6 0.3 0.05375
0.6 0.8 0.5 0.05130
1.0 0.7 0.05020
1.2 0.9 0.04909
0.6 1.0 0.06522
0.8 0.8 0.06413
1.0 0.06302
1.2 0.06188
Table 5

Correlation of Rd and κ on Sherwood number

Sc κ Nb Nt Re x 1 / 2 Sh x
0.6 0.1 0.3 1.47531
1.2 0.9 1.59114
1.2 1.69592
1.5 0.3 1.79245
1.4 0.6 0.5 1.57363
0.9 0.7 1.69549
1.2 0.9 1. 8095
1.5 0.5 1.90799
0.6 0.7 1.65479
1.6 0.9 0.9 1.78146
1.2 1.0 1.89646
1.5 2.00257
0.6 1.72259
1.8 0.9 1.85136
1.2 1.97186
1.5 2.01845
Table 6

Correlation of Pe and Lb on Sherwood number

Pe Lb ϖ Re x 1 / 2 Nh x
0.8 0.9 2.37845
1.1 0.9 2.47663
1.0 2.57481
1.1 2.67299
1.2 0.8 1.0 2.56700
0.9 1.1 2.66483
1.0 1.2 2.77616
1.1 2.88389
0.8 2.74472
1.3 0.9 2.86210
1.0 1.3 2.97949
1.1 3.09688
0.8 2.93050
1.4 0.9 1.4 3.05765
1.0 3.18479
1.1 3.31194

4 Conclusion

Here theoretical attempt is taken on the flow of Carreau nanofluid over a stretchable surface with microorganisms and second law of thermodynamics. The interesting significance of thermal radiative and chemical reactions is also studied in temperature and concentration profiles. The HAM has been used to obtain the series solution for the laminar flow toward the stagnation point. Some of the key remarks are given as follows:

  • The velocity profile and thermal boundary layer thickness reduce with the increase in the Carreau parameter.

  • The f ( 0 ) , θ ( 0 ) , ϕ ( 0 ) , and N ( 0 ) convergence graph for different orders of approximation is obtained.

  • Both Biot number Bi and radiation parameter Rd lead to the increasing behavior in temperature profile.

  • Concentration profile enhanced with the increase in Sc and κ .

  • Microorganism profile is reduced with the increase in Lb and Pe.

  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.

References

[1] Choi SUS, editor. Enhancing thermal conductivity of fluids with nanoparticles. ASME Publications-Fed. Vol. 231; 1995. p. 99–106.Search in Google Scholar

[2] Buongiorn J. Convective transport in nanofluids J. Heat Transf. 2006;28:240–50.10.1115/1.2150834Search in Google Scholar

[3] Islam S, Khan A, Deebani W, Bonyah E, Alreshidi NA, Shah Z. Influences of Hall current and radiation on MHD micropolar non-Newtonian hybrid nanofluid flow between two surfaces. AIP Adv. 2020;10(5):Article 055015.10.1063/1.5145298Search in Google Scholar

[4] Alempour SM, Abbasian Arani AA, Najafizadeh MM. Numerical investigation of nanofluid flow characteristics and heat transfer inside a twisted tube with elliptic cross section. J Therm Anal Calorim. 2020;140:1237–57.10.1007/s10973-020-09337-zSearch in Google Scholar

[5] Rasool G, Shafiq A, Khan I, Baleanu D, Nisar KS, Shahzadi G. Entropy generation and consequences of MHD in Darcy-Forchheimer nanofluid flow bounded by non-linearly stretching surface. Symmetry. 2021;143:2485–97.10.3390/sym12040652Search in Google Scholar

[6] Naga Santoshi P, Ramana Reddy GV, Padma P. Numerical scrutinization of three dimensionalCarreau-Carreau nano fluid flow. J Appl Comput Mech. 2020;6(3):531–42.Search in Google Scholar

[7] Wang Y, Deng K, Wu JM, Su G, Qiu S. A mechanism of heat transfer enhancement or deterioration of nanofluid flow boiling heat transfer. Int J Heat Mass Transf. 2020;158:Artic 119985.10.1016/j.ijheatmasstransfer.2020.119985Search in Google Scholar

[8] Nadeem S, Khan MN, Abbas N. Transportation of slip effects on nanomaterial micropolar fluid flow over exponentially stretching. Alex Eng J. 2020;59:3443–50.10.1016/j.aej.2020.05.024Search in Google Scholar

[9] Waqas H, Imran M, Khan SU, Shehzad SA, Meraj MA. Slip flow of Maxwell viscoelasticity-based micropolar nanoparticles with porous medium: a numerical study. Appl Math Mech. 2019;40(9):1255–68.10.1007/s10483-019-2518-9Search in Google Scholar

[10] Nadeem S, Abbas N, Malik MY. Heat transport in CNTs based nanomaterial flow of non-Newtonian fluid having electro magnetize plate. Alex Eng J. 2020;59:3431–42.10.1016/j.aej.2020.05.022Search in Google Scholar

[11] Hayat T, Yaqoob R, Qayyum S, Alsaedi A. Entropy generation optimization in nanofluid flow by variable thicked sheet. Phys A: Stat Mech Appl. 2020;124022.10.1016/j.physa.2019.124022Search in Google Scholar

[12] Alic F. Entransy dissipation analysis and new irreversibility dimension ratio of nanofluid flow through adaptive heating elements. Energies. 2020;13(1):114.10.3390/en13010114Search in Google Scholar

[13] Amjad M, Zehra I, Nadeem S, Abbas N. Thermal analysis of Carreau micropolar nanofluid flow over a permeable curved stretching surface under the stagnation region. J Therm Anal Calorim. 2020;1–13.10.1007/s10973-020-10127-wSearch in Google Scholar

[14] Mkhatshwa MP, Motsa SS, Ayano MS, Sibanda P. MHD mixed convective nanofluid flow about a vertical slender cylinder using overlapping multi-domain spectral collocation approach. Case Stud Therm Eng. 2020;18:Article 100598.10.1016/j.csite.2020.100598Search in Google Scholar

[15] Tlili I, Shahmir N, Ramzan M, Kadry S, Kim JY, Nam Y, et al. A novel model to analyze Darcy Forchheimer nanofluid flow in a permeable medium with Entropy generation analysis. J Taibah Univ Sci. 2020;14(1):916–30.10.1080/16583655.2020.1790171Search in Google Scholar

[16] Atashafrooz M. Influence of radiative heat transfer on the thermal characteristics of nanofluid flow over an inclined step in the presence of an axial magnetic field. J Therm Anal Calorim. 2020;139(5):3345–60.10.1007/s10973-019-08672-0Search in Google Scholar

[17] Khan I, Ullah KS, Chu YM, Nisar KS, Al-Khaled K. Oldroyd-B nanofluid-flow between stretching disks with thermal slip and multiple flow features. Therm Sci. 2020;24(Suppl. 1):83–94.10.2298/TSCI20S1083KSearch in Google Scholar

[18] Jamshed W, Safdar R, Ibrahim RW, Nisar KS, Eid MR, Alam MM. Shape-factor and radiative flux impacts on unsteady graphene–copper hybrid nanofluid with entropy optimisation: cattaneo–christov heat flux theory. SpringerLink. Accessed January 1, 2023. https://link.springer.com/article/. 10.1007/s12043-022-02403-1.Search in Google Scholar

[19] Eid MR. 3-D flow of magnetic rotating hybridizing nanoliquid in parabolic trough solar collector: Implementing Cattaneo-Christov heat flux theory and centripetal and coriolis forces. Mathematics. January 2022;10(15):2605. 10.3390/math10152605.Search in Google Scholar

[20] Shahzad F, Jamshed W, Sajid T, Shamshuddin MD, Safdar R, Salawu SO, et al. Electromagnetic control and dynamics of generalized burgers’ nanoliquid flow containing motile microorganisms with Cattaneo–Christov relations: Galerkin finite element mechanism. Appl Sci. January 2022;12(17):8636. 10.3390/app12178636.Search in Google Scholar

[21] Islam S, Khan A, Deebani W, Bonyah E, Alreshidi NA, Shah Z. Influences of hall current and radiation on MHD micropolar non-newtonian hybrid nanofluid flow between two surfaces.”. AIP Adv. May 2020;10(5):055015. 10.1063/1.5145298.Search in Google Scholar

[22] Shah Z, Rooman M, Jan MA, Vrinceanu N, Deebani W, Shutaywi M, et al. Radiative Darcy-Forchheimer Micropler Bödewadt flow of CNTs with viscous dissipation effect. J Pet Sci Eng. October 1, 2022;217:110857. 10.1016/j.petrol.2022.110857.Search in Google Scholar

[23] Rajagopal KR, Na TY, Gupta AS. Flow of viscoelastic fluid over a stretching sheet. Rheo Acta. 1984;23:213–5.10.1007/BF01332078Search in Google Scholar

[24] Hayat T, Abbas Z, Pop I. Mixed convection in the stagnation point flow adjacent to a vertical surface in a viscoelastic fluid. Int J Heat Mass Transf. 2008;51:3200–6.10.1016/j.ijheatmasstransfer.2007.05.032Search in Google Scholar

[25] Ishak A, Nazar R, Pop I. Heat transfer over a stretching surface with variable surface heat flux in micropolar fluids. Phys Lett. 2008;372:559–61.10.1016/j.physleta.2007.08.003Search in Google Scholar

[26] Prasad KV, Datti PS, Vajravelu K. Hydromagnetic flow and heat transfer of a non-Newtonian power law fluid over a vertical stretching sheet. Int J Heat Mass Transf. 2010;53:879–88.10.1016/j.ijheatmasstransfer.2009.11.036Search in Google Scholar

[27] Makinde D, Aziz A. Mixed convection from a convectively heated vertical plate to a fluid with internal heat generation. J Heat Transf. 2011;133:122501.10.1115/1.4004432Search in Google Scholar

[28] Vajravelu K, Prasad KV, Sujath A. Convection heat transfer in a Maxwell fluid at a non-isothermal surface. Cent Eur J Phys. 2011;9:807–15.10.2478/s11534-010-0080-2Search in Google Scholar

[29] Hsiao K-L. MHD mixed convection for viscoelastic fluid past a porous wedge. Int J Non-Linear Mech. 2011;46:1–8.10.1016/j.ijnonlinmec.2010.06.005Search in Google Scholar

[30] Carreau PJ. Rheological equations from molecular network theories. Trans Soc Rheol. 1972;16:99–127.10.1122/1.549276Search in Google Scholar

[31] Carreau PJ. An analysis of the viscous behavior of polymer solutions. Can J Chem Eng. 1979;57:135–40.10.1002/cjce.5450570202Search in Google Scholar

[32] Kefayati GHR, Tang H. MHD thermosolutal natural convection and entropy generation of Carreau fluid in a heated enclosure with two inner circular cold cylinders, using LBM. Int J Heat Mass Transf. 2018;126:508–30.10.1016/j.ijheatmasstransfer.2018.06.026Search in Google Scholar

[33] Olajuwon BI. Convective heat and mass transfer in a hydromagnetic Carreau fluid past a vertical porous plated in presence of thermal radiation and thermal diffusion. Therm Sci. 2011;15:241–52.10.2298/TSCI101026060OSearch in Google Scholar

[34] Hayat T, Sadia A, Mustafa M, Alsaedi A. Boundary layer flow of Carreau fluid over a convectively heated stretching sheet. App Math Comp. 2014;246:12–22.10.1016/j.amc.2014.07.083Search in Google Scholar

[35] Khan M, Hashim. Boundary layer flow and heat transfer to Carreau fluid over a nonlinear stretching sheet. AIP Adv. 2015;5:1–14.10.1063/1.4932627Search in Google Scholar

[36] Hsiao KL. To promote radiation electrical MHD activation energy thermal extrusion manufacturing system efficiency by using Carreau-nanofluid with parameters control method. Energy. 2017;130:486–99.10.1016/j.energy.2017.05.004Search in Google Scholar

[37] Khan M, Irfan M, Khan WA. Numerical assessment of solar energy aspects on 3D magneto-Carreau Nanofluid: a revised proposed relation. Int J Hydrog Energy. 2017;42:22054–65.10.1016/j.ijhydene.2017.07.116Search in Google Scholar

[38] Irfan M, Khan M, Khan WA. Numerical analysis of unsteady 3D flow of Carreau Nanofluid with variable thermal conductivity and heat source/sink. Results Phys. 2017;7:3315–24.10.1016/j.rinp.2017.08.029Search in Google Scholar

[39] Jyothi K, Sudarsana Reddy P, Suryanarayana Reddy M. Carreau nanofluid heat and mass transfer flow through wedge with slip conditions and nonlinear thermal radiation. J Braz Soc Mech Sci Eng. 2019;41:415.10.1007/s40430-019-1904-7Search in Google Scholar

[40] Khan MI, Kumar A, Hayat T, Waqas M, Singh R. Entropy generation in flow of Carreau nanofluid. J Mol Liq, J Mol Liquids. 2019;278:677–87.10.1016/j.molliq.2018.12.109Search in Google Scholar

[41] Naz R, Tariq S, Sohail M, Shah Z. Investigation of entropy generation in stratified MHD Carreau nanofluid with gyrotactic microorganisms under Von Neumann similarity transformations. Eur Phys J Plus. February 1, 2020;135(2):178. 10.1140/epjp/s13360-019-00069-0.Search in Google Scholar

[42] Rooman M, Shah Z, Bonyah E, Jan MA, Deebani W. Mathematical modeling of Carreau fluid flow and heat transfer characteristics in the renal tubule. J Math. May 10, 2022;2022:e2517933. 10.1155/2022/2517933.Search in Google Scholar

[43] Li S, Khan MI, Alzahrani F, Eldin SM. Heat and mass transport analysis in radiative time dependent flow in the presence of Ohmic heating and chemical reaction, viscous dissipation: An entropy modeling. Case Stud Therm Eng. 2023;42:102722.10.1016/j.csite.2023.102722Search in Google Scholar

[44] Wager H. On the effect of gravity upon the movements and aggregation of euglena viridis, Ehrb., and other micro-organisms. Philos Trans R Soc B. 1911;201:333–90.10.1098/rstb.1911.0007Search in Google Scholar

[45] Platt JR. Bioconvection pattern in cultures of free-swimming organism. Science (New York, N.Y.). 1961;133:1766–7.10.1126/science.133.3466.1766Search in Google Scholar PubMed

[46] Kuznetsov AV. Te onset of nanofuid bioconvection in a suspension containing both nanoparticles and gyrotactic microorganisms. Int Commun Heat Mass Transf. 2010;37(10):1421–5.10.1016/j.icheatmasstransfer.2010.08.015Search in Google Scholar

[47] Uddin MJ, Alginahi Y, Bég OA, Kabir MN. Numerical solutions for gyrotactic bioconvection in nanofuid-saturated porous media with Stefan blowing and multiple slip efects. Comput Math Appl. 2016;72(10):2562–81.10.1016/j.camwa.2016.09.018Search in Google Scholar

[48] Farooq S, Hayat T, Alsaedi A, Ahmad B. Numerically framing the features of second order velocity slip in mixed convective flow of Sisko nanomaterial considering gyrotactic microorganisms. Int J Heat Mass Transf. 2017;112:521–32.10.1016/j.ijheatmasstransfer.2017.05.005Search in Google Scholar

[49] Waqas H, Khan SU, Imran M, Bhatti MM. Thermally developed Falkner-Skan bioconvection flow of a magnetized nanofluid in the presence of motile gyrotactic microorganism: Buongiorno’s nanofluid model. Phys Scr. 2019;94(11):115304.10.1088/1402-4896/ab2ddcSearch in Google Scholar

[50] Khan WA, Rashad AM, Abdou MMM, Tlili I. Natural bioconvection flow of a nanofluid containing gyrotactic microorganisms about a truncated cone. Eur J Mech B Fluids. 2019;75:133–42.10.1016/j.euromechflu.2019.01.002Search in Google Scholar

[51] Jafarimoghaddam A, Turkyilmazoglu M, Pop I. Threshold for the generalized Non-Fourier heat flux model: Universal closed form analytic solution. Int Commun Heat Mass Transf. April 1, 2021;123:105204. 10.1016/j.icheatmasstransfer.2021.105204.Search in Google Scholar

[52] Turkyilmazoglu M. Heat transfer enhancement feature of the Non-Fourier Cattaneo–Christov heat flux model. J Heat Transf. July 21, 2021;143(9):094501. 10.1115/1.4051671.Search in Google Scholar

[53] Amirsom NA, Uddin MJ, Ismail AI. MHD boundary layer bionanoconvective non-Newtonian flow past a needle with Stefan blowing. Heat Transf Asian Res. 2019;48(2):727–43.10.1002/htj.21403Search in Google Scholar

[54] Kairi RR, Shaw S, Roy S, Raut S. Thermo-solutalmarangoni impact on bioconvection in suspension of gyrotactic microorganisms over an inclined stretching sheet. J Heat Transf. 2020;143(3):031201.10.1115/1.4048946Search in Google Scholar

[55] Shaw S, Kameswaran PK, Narayana M, Sibanda P. Bioconvection in a non-Darcy porous medium saturated with a nanofluid and oxytactic microorganisms. Int J Biomath. 2014;7(1):1450005.10.1142/S1793524514500053Search in Google Scholar

[56] Mustafa M, Hayat T, Pop I, Asghar S, Obaidat S. Stagnation-point flow of a nanofluid towards a stretching sheet. Int J Heat Mass Transf. 2011;54:5588–94.10.1016/j.ijheatmasstransfer.2011.07.021Search in Google Scholar

[57] Bejan A. A study of entropy generation in fundamental convective heat transfer. J Heat Trans. 1979;101:718–25.10.1115/1.3451063Search in Google Scholar

[58] Khan MI, Hayat T, Khan MI, Waqas M, Alsaedi A. Numerical simulation of hydromagnetic mixed convective radiative slip flow with variable fluid properties: A mathematical model for entropy generation. J Phys Chem Solid. 2019;125:153–64.10.1016/j.jpcs.2018.10.015Search in Google Scholar

[59] Kumar A, Tripathi R, Singh R. Entropy generation and regression analysis on stagnation point flow of Carreau nanofluid with Arrhenius activation energy. J Braz Soc Mech Sci Eng. 2019;41:306.10.1007/s40430-019-1803-ySearch in Google Scholar

[60] Ramzan M, Javed M, Rehman S, Ahmed D, Saeed A, Kumam P. Computational assessment of microrotation and buoyancy effects on the stagnation point flow of Carreau-Yasuda hybrid nanofluid with chemical reaction past a convectively heated Riga plate. ACS Omega. 2022 Aug 17;7(34):30297–312.