Analysis for time-dependent flow of Carreau nanofluid over an accelerating surface with gyrotactic microorganisms: Model for extrusion systems

Abstract

Recent progress in nanotechnology and nanoscience has attended the researchers’ interest due to its inspiring industrial and technological applications. This contribution investigates the bioconvection flow of Carreau fluid over a periodically moving stretched surface with important feature of magnetic field and thermal radiation. Bioconvection in non-Newtonian nanofluid is utilized with sustaining the joint features of buoyancy and magnetic forces. In addition, thermal radiation consequences are summarized in the energy equation while further computations have been operated under “one parametric approach.” The attended problem is contested with a famous convergent technique having excellent accuracy. A comprehensive graphical approach is constituted to describe the thermo-physical features of interesting physical parameters. The reported results may use in enhancement of extrusion system, bio-fuels, and biotechnology.

Keywords

Carreau nanofluid bioconvection thermal radiation gyrotactic microorganisms

Introduction

In the current century, the worldwide development in the nanotechnology has introduced new fluids type aimed to improve thermo-physically properties. Nanofluids bring global advances in the era of fluid dynamics and are amalgamated by merging nanoparticles with conservative base liquids. The doping results enhanced thermal conductivity and improved the viscosity features.¹ The interaction of such nanoparticles constitute an attractive attention in various industrial processes, engineering science, bio-engineering, diagnoses of diseases, treatment and damage of cancer tissues, chemical reactors, automobile engine cooling, solar collectors, cooling agents, and many more. Recently, nanofluid mechanics problems are successfully presented, which configured two- and three-dimensional flows, steady and unsteady flow, and laminar flow. For instance, Sandeep et al.² explored the magnetized nanoparticles Fe₃O₄ characteristics which are influenced by magnetic force. In their analysis, two familiar non-Newtonian fluids namely Jeffrey fluid and Oldroyd B fluid were supposed, and the desired solution has been constituted via Runge–Kutta numerical procedure. Sheikholeslami and Bhatti³ determined the interesting thermo-physical features in convective flow of nanoparticles with different shape aspects. Turkyilmazoglu⁴ described the flow of nanoparticles in channel, where relevant particle flux near the surface is imagined to be zero. In this continuation, the slip mechanism perspectives were explained by employing familiar Buongiorno nanofluid model. The features of ferrofluid suspended in Williamson and Casson fluids in addition to Joule heating and thermal radiation impacts were justified by Reddy et al.,⁵ and Khan et al.⁶ used the activation energy and entropy-generation consequences in the Prandtl–Eyring nanofluid flow over a moving configuration. Abbasi and colleagues^7,8 described the energy optimization and Hall impacts in peristaltic nanofluids flows. Reddy and Chamkha⁹ claimed that an excellent enhancement in heat transportation has been noted upon interaction of Al₂O₃ and TiO₂ nanoparticles in base liquid. The phenomenon of natural convection in the transportation of Al₂O₃ nanoparticles with base liquid in presence of variable thermal conductivity configured by a vertical cone has been numerically simulated by Ghalambaz et al.¹⁰ Reddy et al.¹¹ investigated the slip flow of Cu-water and Ag-water nanoparticles over a rotating disk along with the utilization of thermal radiation and chemical reaction. Another contribution suggested by Ahmad and Khan¹² directed the flow of Sisko fluid with nanoparticles in existence of activation energy over a curved surface. The numerical results have been reported via shooting technique. In another work, Ahmad et al.¹³ settled the Joule heating effects in Sisko nanofluid. They executed the numerical solutions of this problem.

Bioconvection is a sub-branch of thermofluid sciences and is considered as a convective transport of nanoparticles induced by the movement of motile microorganisms in the primary fluid. The appearance of bioconvection resulted from hydrodynamic instability due to upward motion of microorganism in relatively upper fluid surface. Microorganisms can impel themselves in various surrounding consequences like gravity, magnetic force, or chemical reactions. The nanoparticles fail to sustain the mechanisms of self-repulsion as they are signified by the features of thermophoresis and Brownian motion. Therefore, the utilization of motile microorganisms in nanoparticles suspension covers joint effeteness like control of microorganisms as well as improving the microscopic thermo-physical nanofluid features. The designing of ecological fuels, fuel cells, and photo bio-reactors promised the prestigious applications of this phenomenon. In the recent years, the bio-convective fluid dynamics is conquering a determining attention by imposing diverse mathematical models. Kuznetsov¹⁴ determined a shallow nanoparticles layer in utilization of oxytactic microorganisms, where bioconvection is of oscillatory nature. Uddin et al.¹⁵ carried out their attention in exploring the bioconvection of non-Newtonian fluid in addition to saturated porous media. Uddin et al.¹⁶ also contributed to the physical consequences of slip and blowing effects in the bioconvection of nanoparticles configured over a horizontal plate. They interestingly simulated the numerical solution via finite difference procedure before explaining the graphical explanation of involved parameters. Basir et al.¹⁷ reported the Peclet number effects flow of nanofluid over a stretching cylinder. Rashad and Nabwey¹⁸ implemented convective conditions in order to evaluate the mixed convection flow of nanofluid with the addition of motile microorganism. Xun et al.¹⁹ thrashed out the involvement of dependent viscosity in bioconvection of nanoparticles in a rotating system. Mosayebidorcheh et al.²⁰ operated a modified least square technique in order to examine the bioconvection utilization in nanoparticles configured in a horizontal channel. Sk et al.²¹ examined the various slip trends in nanofluid flow under the impulsion of gyrotactic microorganisms. Khan et al.²² indorsed rheology of Oldroyd B nanofluid in the presence of gyrotactic microorganisms over a stretched surface which moves with sinusoidal velocity. Recently, Waqas et al.²³ numerically inspected the bioconvection of rate-type fluid (Maxwell fluid) flow with the assistance of shooting procedure. Sivaraj et al.²⁴ investigated the interesting features of thermoelectric effects in bioconvection flow of Cuo-water nanoparticles subjected to paraboloid configuration. Some more scientific contribution based on bioconvection of nanoparticles can be seen in few previous studies.^25–28

The non-Newtonian fluid flow is fundamental to a variety of chemical, mechanical, environmental, and bio-medical engineering operations. It brings forth miscellaneous applications like ketch up, medicine, mining industries, blood, slurries, motor oils, lubricants, corn starch, and hydraulic fluids. Among such nonlinear fluid models, Carreau fluid is one which successfully exhibited the rheology of viscoelastic materials which are nonlinear in nature like polymer solutions. The inventive idea of Carreau fluid was intended by Carreau,²⁹ which was further worked out by numerous investigators.^30–34

In the current contribution, we attempt, for the first time, a bioconvection of Carreau nanofluid over an unsteady and periodically oscillatory stretched surface. Furthermore, linearized thermal radiation effects are also attributed in the current analysis, which are preceded via “one parametric approach.” The formulated equations are rendered in dimensionless forms which are objected with homotopy analysis procedure using the Mathematica software. The physical interpretations are documented for each physical parameter.

Physical problem

Here, we assumed the stretching flow of viscoelastic fluid which is characterized by Carreau nanofluid with utilization of gyrotactic microorganisms. The confined source of flow is the periodic motion of a stretched surface which moved with uniform velocity $u = bx \sin ϖ t$ , $ϖ$ being the frequency while $a$ inferred the stretching rate. In a coordinate system, $x - axis$ utilizes along the surface and $y - axis$ is presumed normally. The fluid particles are influenced with magnetic force of $B_{0}$ . The nanoparticles impulsion is confirmed by exploiting Buongiorno’s model which sustains Brownian motion and thermophoresis slip mechanisms. The conservation equations relating the momentum, energy, nanoparticles concentration, and gyrotactic microorganisms are transmuted as^22,32

\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0

(1)

\begin{matrix} \frac{\partial u}{\partial t} & + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} = ν \frac{\partial^{2} u}{\partial y^{2}} \\ + ν \frac{3 (n - 1)}{2} λ_{1}^{2} \frac{\partial^{2} u}{\partial y^{2}} {(\frac{\partial u}{\partial y})}^{2} - \frac{σ_{e} B_{0}^{2}}{ρ_{f}} u \\ + \frac{1}{ρ_{f}} [\begin{matrix} (1 - C_{\infty}) ρ_{f} β^{*} g (T - T_{\infty}) - (ρ_{p} - ρ_{f}) g (C - C_{\infty}) \\ - (n - n_{\infty}) g γ^{*} (ρ_{m} - ρ_{f}) \end{matrix}] \end{matrix}

(2)

\begin{matrix} \frac{\partial T}{\partial t} + u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} \\ = (α_{f} + \frac{16 σ^{*} T_{\infty}^{3}}{3 k^{*} {(ρ c)}_{f}}) \frac{\partial^{2} T}{\partial y^{2}} + τ_{1} [D_{B} \frac{\partial C}{\partial y} \frac{\partial T}{\partial y} + \frac{D_{T}}{T_{\infty}} {(\frac{\partial T}{\partial y})}^{2}] \end{matrix}

(3)

\frac{\partial C}{\partial t} + u \frac{\partial C}{\partial x} + v \frac{\partial C}{\partial y} = D_{B} \frac{\partial^{2} C}{\partial y^{2}} + \frac{D_{T}}{T_{\infty}} \frac{\partial^{2} T}{\partial y^{2}}

(4)

\frac{\partial n}{\partial t} + u \frac{\partial n}{\partial x} + v \frac{\partial n}{\partial y} + \frac{b_{1} W_{e}}{(C_{w} - C_{\infty})} [\frac{\partial}{\partial y} (n \frac{\partial C}{\partial y})] = D_{m} (\frac{\partial^{2} n}{\partial y^{2}})

(5)

where $λ$ is the time constant, $σ_{e}$ is the electrical conductivity, $ρ_{f}$ is the fluid density, $β^{*}$ is the base fluid volumetric thermal coefficient, $g$ illustrates gravity, $ρ_{p}$ signifies the nanoparticles density, $ρ_{m}$ is the motile microorganism particles density, $T$ is the nanoparticles temperature, $α_{f}$ is the thermal diffusivity, $D_{B}$ is the diffusion constant, $D_{T}$ is the thermodiffusion constant, $C$ is the nanoparticles concentration, $b_{1}$ is the chemotaxis constant, and $W_{e}$ denotes maximum cells swimming. Now the constituted problem is further preceded by the following boundary conditions

\begin{matrix} u = u_{ω} = bx \sin ϖ t, v = 0, T = T_{w}, \\ C = C_{w}, n = n_{w} at y = 0, t > 0 \end{matrix}

(6)

u \to 0, T \to T_{\infty}, C \to C_{\infty}, n \to n_{\infty} at y \to \infty

(7)

In order to require the dimensionless form of the comprised flow problem, following apposite variable have been advised²²

u = bx f_{ξ} (ξ, τ), v = - \sqrt{ν b} f (ξ, τ), ξ = \sqrt{\frac{b}{ν}} y, τ = t ϖ

(8)

\begin{matrix} θ (ξ, τ) & = \frac{T - T_{\infty}}{T_{w} - T_{\infty}}, ϕ (ξ, τ) \\ = \frac{C - C_{\infty}}{C_{w} - C_{\infty}}, χ (ξ, τ) = \frac{n - n_{\infty}}{n_{w} - n_{\infty}} \end{matrix}

(9)

Substituting equations (8) and (9) in equations (2)–(4), we have

\begin{matrix} f_{ξ ξ ξ y} & - S f_{ξ τ} - f_{y}^{2} + f f_{ξ ξ} + \frac{3}{2} (n - 1) Γ f_{ξ ξ} f_{ξ ξ}^{2} \\ - Ha f_{ξ} + λ (θ - N_{r} ϕ - R_{b} χ) = 0 \end{matrix}

(10)

(\frac{1 + Rd}{\Pr}) θ_{ξ ξ} - [S ϕ_{τ} + f ϕ_{ξ} + Nb θ_{ξ} ϕ_{ξ} + Nt {(θ_{ξ})}^{2}] = 0

(11)

ϕ_{yy} - S (Sc) ϕ_{τ} + Scf ϕ_{ξ} + \frac{Nt}{Nb} θ_{ξ ξ} = 0

(12)

χ_{ξ ξ} - SLb χ_{τ} + Lb χ_{ξ} - Pe [ϕ_{ξ ξ} (χ + σ) + χ_{ξ} ϕ_{ξ}] = 0

(13)

Similarly, the boundary conditions in non-dimensional forms are

\begin{matrix} f_{ξ} (0, τ) = \sin τ, f (0, τ) = 0, θ (0, τ) = 1, \\ ϕ (0, τ) = 1, χ (0, τ) = 1 \end{matrix}

(14)

f_{ξ} (\infty, τ) \to 0, θ (\infty, τ) \to 0, ϕ (\infty, τ) \to 0, χ (\infty, τ) \to 0

(15)

The dimensionless parameters material constant $Γ$ , mixed convection constant $λ$ , Hartmann number $Ha$ , buoyancy ratio constant $N_{r}$ , oscillating frequency to stretching rate ratio $S$ , bioconvection Rayleigh number $R_{b}$ , thermal radiation $Rd$ , Brownian motion parameter $Nb$ , thermophoresis parameter $Nb$ , Prandtl number $\Pr$ , radiation parameter $Rd$ , Schmidt number $Sc$ , Peclet number $Pe$ , and biconvection Lewis number $Lb$ are mathematically related as

\begin{matrix} Γ = {λ_{1}}^{2} b^{2} x^{2} / ν, S = ϖ / b, Ha = σ^{*} B_{0}^{2} / ρ_{f} b, λ = β (T_{w} - T_{\infty}) (1 - C_{\infty}) / b^{2} x \\ \Pr = ν / α_{f}, Nt = τ_{1} D_{T} (T_{w} - T_{\infty}) / T_{\infty} ν, Sc = ν / D_{B}, Rd = 16 σ^{*} T_{\infty}^{3} / 3 k k^{*}, Nb = τ_{1} D_{B} (C_{f} - C_{\infty}) / ν \\ N_{r} = (ρ_{p} - ρ_{f}) (C_{w} - C_{\infty}) / β^{*} ρ_{f} (1 - C_{\infty}) T_{\infty} β, R_{b} = γ^{*} (n_{w} - n_{\infty}) (ρ_{m} - ρ_{f}) / β^{*} ρ_{f} (1 - C_{\infty}) (T_{w} - T_{\infty}) \\ Pe = b W_{e} / D_{m}, Lb = ν / D_{m} \end{matrix}}

It is remarked that parameters material constant $Γ$ and mixed convection constant $λ$ are locally valid. With the use of one parametric approach in the case of linearized thermal radiation, we modify equation (11) as follows

θ_{ξ ξ} - \Pr_{eff} [Nb θ_{ξ} ϕ_{ξ} + Nt {(θ_{ξ})}^{2} + S ϕ_{τ} + f ϕ_{ξ}] = 0

(16)

where $\Pr_{eff} = (1 + Rd) / \Pr$ stands for effective Prandtl number.

The material quantities for heat transfer, mass transfer of nanoparticles, and mass transfer of gyrotactic microorganisms which are subsequent to local Nusselt number, local Sherwood number, and motile density number are articulated with following mathematical forms

\begin{matrix} N u_{x} = \frac{x q_{s}}{k (T_{w} - T_{\infty})}, S h_{x} = \frac{x j_{s}}{D_{B} (C_{w} - C_{\infty})}, \\ N n_{x} = \frac{x j_{n}}{D_{n} (n_{w} - n_{\infty})} \\ q_{s} = - k {(\frac{\partial T}{\partial y})}_{y = 0}, j_{s} = - D_{B} {(\frac{\partial C}{\partial y})}_{y = 0}, \\ j_{n} = - D_{m} {(\frac{\partial N}{\partial y})}_{y = 0} \end{matrix}

(17)

\begin{matrix} {Re}_{x}^{- 1 / 2} {Nu}_{x}^{*} & = - θ_{ξ} (0, τ), {Re}_{x}^{- 1 / 2} S h_{x} \\ = - ϕ_{ξ} (0, τ), {Re}_{x}^{- 1 / 2} N n_{x} = - χ_{ξ} (0, τ) \end{matrix}

(18)

where ${Nu}_{x}^{*},$ $S h_{x}$ , and $N n_{x}$ indicate the effective local Nusselt number, local Sherwood number, and motile density number, respectively.

Solution methodology

This section intends to scrutinize the adopted procedure for solution of set of partial differential equations (PDEs) (10)–(13) with boundary constraints (14) and (15). In order to achieve the analytical solution, we have employed the homotopy analysis technique which was primarily established in 1995 by Liao³⁵ and later on, many problems are treated via this approach.^36–41 The recommend initial guesses are

\begin{matrix} f_{0} (ξ, τ) = \sin τ (1 - e^{- ξ}), θ_{0} (ξ) = e^{- ξ}, \\ ϕ_{0} (ξ) = e^{- ξ}, χ_{0} (ξ) = e^{- ξ} \end{matrix}

(19)

In order to precede the simulations, the auxiliary linear operators are assessed as

£_{f} = \frac{\partial^{3}}{\partial ξ^{3}} - \frac{\partial}{\partial ξ}, £_{θ} = \frac{\partial^{2}}{\partial ξ^{2}} - 1, £_{ϕ} = \frac{\partial^{2}}{\partial ξ^{2}} - 1, £_{χ} = \frac{\partial^{2}}{\partial ξ^{2}} - 1

(20)

satisfying

£_{f} [a_{1} + a_{2} e^{ξ} + a_{3} e^{- ξ}] = 0

(21)

£_{θ} [a_{4} e^{ξ} + a_{5} e^{- ξ}] = 0

(22)

£_{ϕ} [a_{6} e^{ξ} + a_{7} e^{- ξ}] = 0

(23)

£_{χ} [a_{8} e^{ξ} + a_{9} e^{- ξ}] = 0

(24)

where $a_{i} (i = 1, 2, \dots, 9)$ notify the arbitrary constants.

Convergence analysis and physical interpretation

The convergence of homotopy solutions can be efficiently regulated with the proper enrollment of auxiliary parameters $h_{f}$ , $h_{θ}$ , $h_{ϕ}$ , and $h_{χ} .$ Therefore, for optimum values of these parameters, $h - curves$ are plotted attributed to some specified parameters values. The permissible convergence region can be taken out from $- 2.1 \leq h_{f} \leq 0.0$ , $- 1.9 \leq h_{θ} \leq 0.0$ , $- 2 \leq h_{ϕ} \leq 0.0$ , and $- 2 \leq h_{χ} \leq - 0.2$ (Figure 1).

Figure 1.

$h - curves$ for velocity, temperature, concentration, and microorganism distributions with $Γ = 0.2$ , $λ = 0.2$ , $N_{r} = 0.1$ , $R_{b} = 0.1$ , $\Pr_{eff} = 0.7$ , $N_{t} = 0.1$ , $N_{b} = 0.2$ , $Sc = 0.2$ , $Pe = 0.1$ , $Lb = 0.2$ , and $τ = π / 2$ .

Before proceeding with the graphical significance of the involved parameters, first we validate our results with already presented numerical data presented by Abbas et al.⁴² and Zheng et al.⁴³ For this purpose, Table 1 is prepared which shows that present results have a convincible agreement with the numerical computations of Abbas et al.⁴² and Zheng et al.⁴³

Table 1.

Comparison of $f ″ (0, τ)$ with Abbas et al.⁴² and Zheng et al.⁴³ when $S = 1$ , $Ha = 12$ , $Γ = 0$ , $λ = 0$ , $N_{r} = 0$ , and $R_{b} = 0$ .

$τ$	Zheng et al.⁴³	Abbas et al.⁴²	Present results
$τ = 1.5 π$	11.678656	11.678656	11.678656
$τ = 5.5 π$	11.678706	11.678707	11.678706
$τ = 9.5 π$	11.678656	11.678656	11.678656

Now, in order to perform extensive physical visualization of involved parameters for non-dimensional velocity distribution $f_{y}$ , nanoparticles temperature distribution $θ$ , nanoparticles volume fraction $ϕ$ , and motile density distribution $χ$ , detailed graphical contributions are conducted. For this purpose, each parameters has assigned some theoretical varying values which list other parameters kept constant values like $Γ = 0.2$ , $λ = 0.2$ , $N_{r} = 0.2$ , $R_{b} = 0.1$ , $\Pr_{eff} = 0.7$ , $N_{t} = 0.2$ , $N_{b} = 0.2$ , $Sc = 0.2$ , $Pe = 0.2$ , $Lb = 0.2$ , and $τ = π / 2$ .

Figure 2(a)–(c) determined the relative effeteness of velocity $f_{y}$ against variation of fluid parameter $Γ$ , buoyancy ration constant $N_{r}$ , and bioconvection Rayleigh number $R_{b}$ . From Figure 2(a), an oscillatory distribution of velocity has been examined with decreasing trend as $Γ$ varies. The physical justification of such trend may be attributed, as non-Newtonian parameter involves the viscosity effects due to which distribution of velocity decreases. Figure 2(b) reports the impact of buoyancy ration constant $N_{r}$ on $f_{y}$ . Here, the velocity oscillates periodically without any phase shift with variation of $N_{r}$ . Physically, variation in $N_{r}$ involves buoyancy forces which resist the movement of fluid particles in the whole flow domain. From Figure 2(c), the effeteness of $R_{b}$ which is physically related to the buoyancy force because of bioconvection slows down the velocity distribution efficiently.

Figure 2.

Velocity variation with time for (a) $Γ$ , (b) $N_{r}$ , and (c) $R_{b}$ .

The temperature distribution $θ$ is depicted graphically in Figure 3(a)–(f) for iterative values of fluid parameter $Γ$ , buoyancy ration constant $N_{r}$ , bioconvection Rayleigh number $R_{b}$ , effective Prandtl number $P e_{eff}$ , thermophoresis constant $N_{t}$ , and Brownian movement constant $N_{b}$ . In order to point out the physical iteration of $Γ$ on $θ$ , Figure 3(a) is organized. The nanoparticles temperature boosts up with $Γ$ . From Figure 3(b), an accelerating distribution of nanoparticles temperature is observed as $N_{r}$ , which is attributed to some iterative values. Such physical perspective is due to the response of involved buoyancy forces which enhanced the nanoparticles temperature. Similar results have been reported in Figure 3(c) for $R_{b} .$ Figure 3(d) evaluates the graphical features of effective Prandtl number $\Pr_{eff}$ . Mathematically, $\Pr_{eff}$ configured the dual features of Prandtl number and radiation parameter. A well-reputed justification about Prandtl number is it occupies reverse relation with thermal diffusivity; therefore, some increasing variation in Prandtl number reveals a weaker nanoparticles distribution. Moreover, effective Prandtl number is directly constituted to the Prandtl number, and hence, a demising nanoparticles distribution is obtained. It is remarked that in case of linearized thermal radiation, the radiation influences may not alter the temperature distribution if it follows “one parametric approach.” The influence of another important parameter thermophoresis constant $N_{t}$ which is probably the most important slip mechanisms parameter in Buongiorno’s nanofluid model, on nanoparticles temperature $θ$ is executed in Figure 3(e). In thermophoresis phenomenon, the fluid particles are migrated in the cooler region because of temperature difference. Therefore, with increment of $N_{t}$ , the nanoparticles temperature boost up. The interesting observation here may conclude that the appearance of thermophoresis phenomenon can enhance the heat transfer object which has a prime role in various thermal extrusion systems. In order to find out how temperature is altered with variation of Brownian movement constant $N_{b},$ Figure 3(f) is prepared. Since Brownian motion is related to the nanoparticles random movement, which is here notified by dimensionless parameter $N_{b}$ , an increasing temperature distribution is seen out for leading values of $N_{b}$ .

Figure 3.

Temperature distribution for (a) $Γ$ , (b) $N_{r}$ , (c) $R_{b}$ , (d) $P e_{eff}$ , (e) $N_{t}$ , and (f) $N_{b}$ .

Figure 4(a)–(e) addresses the consequences of dimensionless Schmidt number $Sc$ , buoyancy ratio constant $N_{r}$ , bioconvection Rayleigh number $R_{b}$ , Brownian motion constant $N_{b}$ , and thermophoresis parameter $N_{t}$ on nanoparticles concentration $ϕ$ . From Figure 4(a), the nanoparticles concentration is deliberating as $Sc$ varies from $0.5, 1.0, 2.5$ and $3.5$ . Since the definition of $Sc$ acquired that it has opposite relation with mass diffusivity, it controls the nanoparticles concentration. The results configured in Figure 4(b) show that the nanoparticles concentration uplifted for the leading values of $N_{r} .$ The variation of $R_{b}$ objected similar trend and is nominated in Figure 4(c). In order to visualize the importance of Brownian motion constant $N_{b}$ on $ϕ$ , Figure 4(d) has been deliberated. The nanoparticles concentration revolution is smaller with Brownian motion constant involvement. However, in case of thermophoresis parameter, apposite increasing nanoparticles concentration distribution is declared.

Figure 4.

Concentration distribution for: (a) $Sc$ , (b) $N_{r}$ , (c) $R_{b}$ , (d) $N_{b}$ , and (e) $N_{t}$ .

In order to specify the enrollment of various parameters like Peclet number $Pe$ , biconvection Lewis number $Lb$ , Brownian constant $N_{b}$ , and thermophoresis parameter $N_{t}$ on motile density distribution $χ$ , Figure 5(a)–(d) is sketched. From Figure 5(a), the diffusion of motile organisms $χ$ slows down with increasing Peclet number $Pe$ . Physically Peclet number is related with diffusivity of microorganisms. Larger values of $Pe$ results lower microorganisms diffusivity, which causes a decrement in $χ$ . Figure 5(b) reports that again, the profile of $χ$ seems to be depressed for peak values of $Lb$ . In fact, leading values of bioconvection Lewis number $Lb$ also causes lower motile microorganism distribution $χ$ . Figure 5(c) and (d) relates the output of Brownian constant $N_{b}$ and thermophoresis parameter $N_{t}$ on $χ$ . Here, in contrast to Figure 5(a) and (b), the motile density distribution improved.

Figure 5.

Motile density distribution for (a) $Pe$ , (b) $Lb$ , (c) $N_{b}$ , and (d) $N_{t}$ .

The numerical illustration of local Nusselt number against various parameter like $Γ$ , $λ$ , $N_{r}$ , $R_{b}$ , $N_{t}$ , $N_{b}$ , and $\Pr_{eff}$ has been determined in Table 2. Maximum values of local Nusselt number are attained with variation of $\Pr_{eff}$ , while all the remaining parameters behave oppositely. The variation of local Sherwood number is enlisted in Table 3 which shows that variation in this dimensionless quantity becomes relatively slower for $N_{t}$ , $R_{b}$ , and $Γ$ . The rate of mass transportation has been improved with variation of $N_{b}$ and $Sc$ . The variation in local motile density number is illustrated in Table 4. This Table demonstrates that the local motile density number achieved the peak values for larger Pe and Lb.

Table 2.

Numerical evaluation of local Nusselt number at $τ = π / 2$ .

$Γ$	$λ$	$N_{r}$	$R_{b}$	$N_{t}$	$N_{b}$	$\Pr_{eff}$	$- θ_{y} (0, τ)$
0.2 0.4 0.6	0.1	0.2	0.2	0.4	0.3	0.3	0.65485 0.642320 0.63220
	0.20.50.7						0.556980.603250.64821
		0.10.30.5					0.586320.565210.53321
			0.20.60.8				0.589420.557410.53200
				0.10.20.4			0.587460.576540.54623
					0.10.20.4		0.567480.559680.53387
						0.20.71.0	0.684750.765010.80124

Table 3.

Numerical evaluation of local Sherwood number at $τ = π / 2$ .

$Γ$	$λ$	$N_{r}$	$R_{b}$	$N_{t}$	$N_{b}$	$Sc$	$- ϕ_{y} (0, τ)$
0.20.40.6	0.1	0.2	0.2		0.3	0.3	0.664850.6323200.60220
	0.20.50.7						0.546980.583250.62821
		0.10.30.5					0.546320.505210.43321
			0.20.60.8				0.629300.538430.50562
				0.10.30.4			0.687000.659870.62520
					0.10.30.4		0.589870.644620.67509
						0.20.71.0	0.685500.765670.80287

Table 4.

Numerical evaluation of local motile density number at $τ = π / 2$ .

$Γ$	$λ$	$N_{r}$	$R_{b}$	$σ$	$Lb$	$P e$	$- χ_{y} (0, τ)$
0.20.40.6	0.1	0.2	0.2		0.3	0.3	0.506430.4327650.40445
	0.20.50.7						0.447860.487230.52790
		0.10.30.5					0.447540.407900.43034
			0.20.60.8				0.525640.436540.40234
				0.10.50.7			0.487390.426490.59864
					0.10.30.5		0.439300.494460.56239
						0.20.71.0	0.488540.567890.60741

Conclusion

We have analyzed the flow of non-Newtonian Carreau nanofluid comprising the effects of microorganisms over an oscillating and periodically moving configuration. The desired flow problem is constituted in terms of PDEs which are solved analytically. A variety of interesting observations were obtained for variation of dimensionless parameters like material constant, buoyancy ratio constant, bioconvection Rayleigh number, thermal radiation constant, Brownian motion parameter, thermophoresis parameter, effective Prandtl number, Schmidt number, Peclet number, and biconvection Lewis number.

The velocity distribution can be alerted with variation of material constant, buoyancy ratio constant, and bioconvection Rayleigh number.

The nanoparticles temperature can be enhanced with bioconvection Rayleigh number, Brownian motion parameter, and thermophoresis parameter.

The non-Newtonian parameter involvement is more useful to improve the nanoparticles temperature and concentration profile.

The variation in motile density can be enhanced by involving Brownian motion parameter and thermophoresis parameter.

The physical quantities like local Nusselt number, local Sherwood number and motile density number decreases with variation of non-Newtonian parameter.

Finally, the proposed scientific results can provide attractive advances in the solar systems, energy consumptions, and improvement in thermal extrusion systems. The bioconvection based on interaction of nanoparticles can also be utilized in transpiration of heat for various industrial and engineering processes.

Footnotes

Handling Editor: James Baldwin

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Sabir Ali Shehzad

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