Bioconvective gyrotactic microorganisms in third-grade nanofluid flow over a Riga surface with stratification: An approach to entropy minimization

Karuppusamy Loganathan; Reema Jain; S. Eswaramoorthi; Mohamed Abbas; Mohammed S. Alqahtani

doi:10.1515/phys-2023-0273

Article Open Access

Bioconvective gyrotactic microorganisms in third-grade nanofluid flow over a Riga surface with stratification: An approach to entropy minimization

Karuppusamy Loganathan , Reema Jain , S. Eswaramoorthi , Mohamed Abbas and Mohammed S. Alqahtani

Published/Copyright: August 7, 2023

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From the journal Open Physics Volume 21 Issue 1

Abstract

Interest in the thermal effects of nanofluid (NF) has increased recently due to the use of nanocomposites to magnify the thermal conductivity of conventional liquids and so boost the heat transit phenomena. Based on this fundamental concept, the current study inspects the thermal advanced third-grade fluid flow with nanocomposites with an extended surface and the inclusion of stratification, non-Fourier heat flux, mass flux, and radiation. Buongiorno’s NF model is employed to observe the thermophoresis and Brownian motion properties. The gyrotactic microorganisms, which are connected to the bioconvection phenomenon that intrigues most, are also considered to be present in the nanoparticles. The governing models are composed of partial differential equations; thereafter, the relevant transformations are applied to these equations to convert the structure into an ordinary differential model. These resultant models are solved by implementing the homotopy analysis method. It is explained in detail how the pertinent parameters are affecting the motion, temperature of fluid, nanocomposite volume, dynamic microbe density, skin friction rates, local Nusselt, and local Sherwood numbers. Applications for the flow of nanoparticles carrying gyrotactic microorganisms include enzyme biosensors, microfluidic devices, microbial fuel cells, and biotechnology.

Keywords: third grade nanofluid; gyrostatic microorganisms; radiation; Riga plate; homotopy analysis method

Nomenclature

Symbol and description

A i: material constants (–)
a: stretching rate (S⁻¹)
B: magnetic field vector (T)
Be: Bejan number (–)
Br: Brinkman number (–)
b: body forces (N m⁻³)
b ₁, b ₂, d ₁, d ₂, e ₁ and e ₂: dimensionless constants (–)
b *: chemotaxis constant (m)
C: concentration ( kg m − 3 )
C p: specific heat ( J k g − 1 K − 1 )
C ∞: ambient concentration ( kg m − 3 )
C w: fluid wall concentration ( kg m − 3 )
C 0: reference concentration ( kg m − 3 )
Cf x: skin friction coefficient (–)
D B: Brownian diffusion coefficient (m² s⁻¹)
D T: thermophoretic diffusion coefficient (m² s⁻¹)
EG: entropy generation parameter (–)
F ( η ): dimensionless velocity (–)
g: gravity (m s⁻²)
H i: kinematic tensors (–)
J: electric current (A)
k: thermal conductivity (W m⁻¹ K⁻¹)
L: spatial gradient of velocity
Le: Lewis number (–)
L b: bioconvection Lewis number (–)
M 0: magnetic property of the permanent magnets (T)
N 0: reference concentration of microorganisms (kg m⁻³)
Nb: Brownian motion parameter (–)
Nt: thermophoresis parameter (–)
N u x: Nusselt number (–)
N r: buoyancy ratio parameter (–)
P e: bioconvection Peclet number (–)
Pr: Prandtl number (–)
Rb: bioconvection Rayleigh number (–)
Rd: radiation parameter (–)
Re: Reynolds number (–)
S h x: Sherwood number (–)
S gen ''': local volumetric entropy generation rate (W m⁻³ K⁻¹)
S 0 ''': characteristic entropy generation rate (W m⁻³ K⁻¹)
S 1: thermal stratification parameter (–)
S 2: mass stratification parameter (–)
S 3: motile density stratification parameter (–)
T: temperature (K)
T ∞: ambient temperature (K)
T 0: reference temperature (K)
T *: Cauchy stress tensor (–)
u w: velocity of the sheet (m s⁻¹)
u , v: velocity components (m s⁻¹)
W c: maximum cell swimming speed (m s⁻¹)
x , y: Cartesian coordinates (m)

Greeks

α 1 , α 2: material parameters (–)
β: volume expansion coefficient (–)
β 1: materiel parameter (–)
γ i: material constant (–)
Γ 1: dimensionless heat thermal relaxation parameter (–)
Γ 2: dimensionless mass thermal relaxation parameter (–)
v *: velocity field (m s⁻¹)
μ: viscosity (N s m⁻²)
ϕ ( η ): dimensionless concentration (–)
η: dimensionless parameter (–)
λ: mixed convection parameter (–)
λ T: thermal relaxation time (energy) (–)
λ c: thermal relaxation time (mass) (–)
ν: kinematic viscosity (m² s⁻¹)
Π: dimensionless temperature difference (–)
θ ( η ): dimensionless temperature (–)
τ: ratio of the effective heat capacity
ρ: density (kg m⁻¹)
ρ f: density of nanofluid (kg m⁻³)
ρ p: density of nanoparticles (kg m⁻³)
ρ m: density of microorganism’s particles (kg m⁻³)
σ: electrical conductivity (S m)
χ: dimensionless microorganism’s (–)
ψ: stream function (m s⁻¹)
ζ: dimensionless concentration difference (–)
Ω: dimensionless microorganisms’ difference (–)

1 Introduction

Considering how crucial heat transmission is to design and industry, researchers are interested in finding out more about how to maximize it. Electrical devices, mechanical assemblies, and heat dissipators can all benefit from the thermal transmission of base liquors like H 2 O , ethylene glycol, and oil. But these base liquids have poor heat conductivity. Experts from a variety of fields are working to address this problem by introducing a special kind of nanometer-sized particle into a base fluid called “nanofluid” (NF) in an effort to increase the heat conductivity of newly cited fluids, as seen in the study by Choi [1]. Khan and Pop [2] observed how a NF flowed on a flat surface. The non-Newtonian NF flow that passes through a porous medium was examined by Barnoon and Toghraie [3]. Ghalambaz et al. [4] showed the NF flow via a heated plate, and they deduced that the fluid motion declines as the thermophoresis parameter increases. The properties of a NF’s spontaneous convection flow over a plate were studied by Aziz and Khan [5]. They discovered that the temperature profiles were relatively insensitive to changes in Brownian motion and thermophoresis parameters. Ahmad et al. [6] considered a NF flow over a thin needle. They established that the Brownian motion parameter reduces the NF volume fraction. The flow of Jeffery NF past a surface was discussed by Prasannakumara et al. [7]. They found that the maximum amount of the thermophoresis parameter enriched the thickness of the thermal boundary layer.

The bioconvection event is a fluid dynamics process that can happen in the macroscopic convective fluid flow when a group of microorganisms work together to make a density gradient in the fluid. Owing to how they move, these bacteria are sorted as chemotactic, oxytactic, or gyrotactic. Close to the upper fluid flake, these self-moving, movable bacteria assemble, creating a thicker surface that is shaky or unsterile. Bioconvection is employed in many manufacturing processes, including polymer synthesis, water treatment facilities, sustainable fuel cell skills, and microbially amended oil recovery. The two-dimensional radiative tangent hyperbolic NF flow via the Riga plate harboring gyrotactic microorganisms was demonstrated by Waqas et al. [8]. They discovered that the density of movable bacteria drops when the bio-convection Lewis number is enriched. Uddin et al. [9] showed the effects of Stefan blowing on the bioconvective NF flow through a porous medium. They notice that the density of movable bacteria rises with an upsurge in the wall suction parameter. Alshomrani et al. [10] examined the hydromagnetic flow of a cross-NF holding gyrotactic motile microorganisms. They discovered that the dynamic microorganisms become inhibited when the Peclet number rises. Muhammad et al. [11] presented the numerical method for time-independent hydromagnetic flow in Carreau NFs with bioconvection. They found that increasing the Peclet parameter resulted in a reduction in the number of local motile cells.

Researchers in the contemporary period have concentrated a large amount of their attention on the process of heat transfer as a result of the various uses that it has in the fields of industry and engineering. These applications include, but are not limited to, the production of electricity, the cooling of electronic equipment, the cooling of nuclear reactors, and the creation of energy, among many others. Fourier [12] first presented the heat transmission law. The drawback of this law is that it results in a parabolic energy equation. Cattaneo [13] revised the Fourier equation to include the relaxation time heat flow element to fix this problem. Christov [14] also utilized the Oldroyd upper convected model and modified the Cattaneo model by including thermal relaxation time. Salahuddin et al. [15] inspected the heat transport analysis of the 2D flow of cross-NF with Cattaneo–Christov theory and identified that mass relaxation parameter causes the NF concentration to decrease. The effects of hydromagnetic flow on radiative NF using Cattaneo–Christov (C–C) theory were inspected by Farooq et al. [16]. They demonstrated that increasing the thermal relaxation parameter causes the temperature of the fluid to decrease. Waqas et al. [17] studied the thermally radiative hybrid NF flow with C–C heat flux theory

So far, no research has been done on the mixed convective flow of a third-grade NF over a Riga plate with stratification despite the fact that NFs have been the subject of substantial study. To address this, the current work illustrates how the movement of a surface can induce the thermally radiative third-grade NF flow holding microorganisms.

The equations for calculating energy and the concentration of nanoparticles are framed by using a modified version of Fourier’s law.
With the purpose of finding the nonlinear equations in an analytical manner, the homotopy analysis approach [18,19,20,21,22,23,24,25] is used.
It is possible that the outcomes of the simulations will have certain applications in the areas of thermal procedures, energy structures, nuclear structures, and other related areas.

2 Problem development

2.1 Fluid model

The continuity and motion equations for a body force-based incompressible fluid model are as follows:

(1) div v * = 0 ,

(2) ρ d v d t = div T * + ρ b + J × B ,

where ρ is the “fluid density,” v * is the “velocity field,” b is “body forces,” J is the “electric current,” T * is the “third-grade incompressible fluids Cauchy stress tensor,” and B is the magnetic field vector.

Among the numerous constitutive assumptions that have been used to study non-Newtonian fluid behavior, one class that has gained support from both experimentalists and theorists is that of the differential type of complexity n − a class that was first considered in detail by Rivlin and Ericksen [26]. The stress constitutive assumption for an incompressible fluid characterizes this category.

(3) T * = − pI + μ H 1 + A 1 * H 2 + A 2 * H 1 2 + γ 1 H 3 + γ 2 ( H 1 , H 2 + H 2 H 1 ) + γ 3 ( tr H 1 2 ) H 1 ,

where μ , ( H 1 , H 2 , H 3 ) , and A i * , γ i are “viscosity coefficient,” “kinematics tensors,” and “material moduli’s,” respectively, as follows:

(4) H 1 = L + ( L ) T * ,

(5) H n = d d t H n − 1 + H n − 1 L + ( L ) T * H n − 1 , n = 2 , 3 ,

(6) L = ∇ · v * ,

where L represents the spatial gradient of velocity and d d t is stated as the material time derivative.

(7) d ( ) d t = ∂ ( ) ∂ t + v * . ∇ ( ) .

According to Fosdick and Rajagopal [27], the Clausius–Duhem inequality and fluids that are compatible with thermodynamics:

(8) μ ≥ 0 , A 1 * ≥ 0 , γ 1 = γ 2 = 0 , γ 3 ≥ 0 ,

(9) | A 1 * + A 2 * | ≤ 2 6 μ γ 3 ,

(10) T * = − pI + μ H 1 + A 1 * H 2 + A 2 * H 1 2 + γ 3 ( tr H 1 2 ) H 1 .

Pakdemirli [28] considered both the normal and Boussinesq boundary layer approximations.

2.2 Problem statement

Let us examine the hydrodynamic flow on third-grade viscous incompressible NF flow to the Riga plate (Figure 1a). The flow appears through the Riga plate at y = 0 , which is shown in Figure 1b. Let ( u , v ) be the velocity components with vertical and horizontal axes, appropriately. The Riga plate consists of magnets and electrodes was arranged on a plain sheet. The spin-wise adjustment on magnets improves Lorentz forces. This free (stream) velocity is given by u = ax , where “a” as the positive factor. Radiation, non-Fourier heat flux, and mass flux are taken in to the account. The gyrotactic microorganisms are considered along with nanoparticles.

(11) ∂ u ∂ x + ∂ v ∂ y = 0 ,

(12) u ∂ u ∂ x + v ∂ u ∂ y = ν ∂ 2 u ∂ y 2 + A 1 * ρ u ∂ 3 u ∂ y 2 ∂ x + v ∂ 3 u ∂ y 3 + ∂ u ∂ x ∂ 2 u ∂ y 2 + 3 ∂ u ∂ y ∂ 2 u ∂ x ∂ y + 2 A 2 * ρ ∂ u ∂ y ∂ 2 u ∂ x ∂ y + 6 γ 3 ρ ∂ u ∂ y 2 ∂ 2 u ∂ y 2 + 1 ρ f ⌊ ( 1 − C ∞ ) ρ f β g ( T − T ∞ ) − ( ρ p − ρ f ) g ( C − C ∞ ) − ( N − N ∞ ) g γ * ( ρ m − ρ f ) ⌋ + π J M 0 8 ρ exp − π a 1 y ,

(13) u ∂ T ∂ x + v ∂ T ∂ y + λ T u 2 ∂ 2 T ∂ x 2 + v 2 ∂ 2 T ∂ y 2 + u ∂ u ∂ x ∂ T ∂ x + v ∂ u ∂ y ∂ T ∂ x + 2 uv ∂ 2 T ∂ x ∂ y + u ∂ v ∂ x ∂ T ∂ y + v ∂ v ∂ y ∂ T ∂ y = k ρ c p ∂ 2 T ∂ y 2 + 1 ρ c p 16 σ * 3 k * ∂ ∂ y T 3 ∂ T ∂ y + τ D B ∂ C ∂ y ∂ T ∂ y + D T T ∞ ∂ T ∂ y 2 ,

(14) u ∂ C ∂ x + v ∂ C ∂ y = D B ∂ 2 C ∂ y 2 + D T T ∞ ∂ 2 T ∂ y 2 + λ C u 2 ∂ 2 C ∂ x 2 + v 2 ∂ 2 C ∂ y 2 + u ∂ u ∂ x ∂ C ∂ x + v ∂ u ∂ y ∂ C ∂ x + 2 uv ∂ 2 C ∂ x ∂ y + u ∂ v ∂ x ∂ C ∂ y + v ∂ v ∂ y ∂ C ∂ y ,

(15) u ∂ N ∂ x + v ∂ N ∂ y − D m ∂ 2 N ∂ y 2 = − b * W c ( C W − C ∞ ) ∂ ∂ y N ∂ C ∂ y .

Figure 1

(a) Physical model of Riga plate and (b) geometry of the third-grade fluid flow.

With the boundary points

(16) u = u w = ax , v = 0 , T = T w = T 0 + b 1 x , C = C W = C 0 + d 1 x , N = N w = N 0 + e 1 x at y = 0 , u → 0 , T = T ∞ = T 0 + b 2 x , C = C ∞ = C 0 + d 2 x , N = N ∞ = N 0 + e 2 x as y → ∞ .

Modifications are stated as described. The stream functions are stated as follows:

ψ = ( a ν ) 1 2 F ( η ) , u = ∂ ψ ∂ y , v = − ∂ ψ ∂ x ,

and we obtain the following:

η = y a ν 1 2 , u = u w F ′ ( η ) , v = − ( u w ) 1 2 F ( η ) ,

(17) θ = T − T ∞ T w − T 0 , ϕ ( η ) = C − C ∞ C w − C 0 , χ ( η ) = N − N ∞ N w − N 0 .

The governing nonlinear equations are as follows:

(18) F ″ + F F ″ − F ′ 2 + α 1 ( 2 F ′ F ′ ″ − F F iv ) + ( 3 α 1 + 2 α 2 ) F ″ 2 + 6 β 1 Re F ′ ″ F ″ 2 + Ha e − δ η + λ ( θ − Nr ϕ − Rb χ ) = 0 ,

(19) 1 + 4 3 Rd θ ″ + Pr F θ ′ − Pr F ′ θ − Pr S 1 F ′ − Pr Γ 1 [ F ′ 2 θ + S 1 F ′ 2 − F F ′ θ ′ − F F ″ θ − S 1 F F ″ + F 2 θ ″ ] + PrNb θ ′ ϕ ′ + PrNt θ ′ 2 = 0 ,

(20) ϕ ″ + Le F ϕ ′ − Le F ′ ϕ − Le S 2 F ′ − Le Γ 2 [ F ′ 2 ϕ + S 2 F ′ 2 − F F ′ ϕ ′ − F F '' ϕ − S 2 F F '' + F 2 ϕ ″ ] + Nt Nb θ ″ = 0 ,

(21) χ ″ + L b [ F χ ′ − F ′ χ ] − L b S 3 F ′ − P e [ ϕ ″ ( χ + Ω ) + ϕ ′ χ ′ ] = 0 .

The limitations are described as follows:

(22) F ( 0 ) = 0 , F ′ ( 0 ) = 1 , θ ( 0 ) = 1 − S 1 , ϕ ( 0 ) = 1 − S 2 , χ ( 0 ) = 1 − S 3 , F ′ ( ∞ ) = 0 , θ ( ∞ ) = 0 , ϕ ( ∞ ) = 0 , χ ( ∞ ) = 0 .

The nondimensional variables are described in Table 1.

Table 1

The nondimensional variables

Nondimensional form
α 1 , α 2 , β 1	a A 1 * ν , a A 2 * ν , a γ 3 ν
Re	u w x ν
Pr	ρ C p / k
Rd	( 4 σ * T ∞ 3 ) / ( k k * )
Γ 1	λ T a
Γ 2	λ C a
λ	β γ ( 1 ‒ C ∞ ) ( T w ‒ T 0 ) a u w
Nr	( ρ p ‒ ρ f ) β ρ f ( C w ‒ C 0 ) ( T ˆ w ‒ T ˆ 0 )
Rb	γ ( N w ‒ N 0 ) ( ρ m ‒ ρ f ) β ρ f ( 1 ‒ C ∞ ) ( T w ‒ T 0 )
Nt	τ D T ( T w ‒ T 0 ) υ
Nb	τ D B ( C w ‒ C 0 ) ν
Le	ν D B
Lb	ν D m
Pe	b W c D m
Ω	N ∞ N w ‒ N 0
S 1	b 2 b 1
S 2	d 2 d 1
S 3	e 2 e 1

Utilization of physical assets is presented as follows:

(23) C f Re − 0 . 5 = F '' ( 0 ) + α 1 F ′ ( 0 ) F ''' ( 0 ) + β 1 Re [ F '' ( 0 ) ] 3 ,

(24) Nu x Re − 0 . 5 = − 1 + 4 3 Rd θ ′ ( 0 ) ,

(25) S h x Re − 0 . 5 = − ϕ ′ ( 0 ) ,

(26) Nn x Re − 0 . 5 = − χ ′ ( 0 ) .

3 Modeling of entropy generation

Entropy is essentially a disorder of the system and environment. Thus, entropy has been designated as the measure of irreversibility. Therefore, heat cannot be completely converted to work. We calculate the entropy generation specifically. In a dimensional system, the expression for entropy generation for the third-grade fluid model is given as follows:

(27) S gen ''' = k T ∞ 2 ∂ T ∂ x 2 + ∂ T ∂ y 2 + 16 σ * T ∞ 3 3 k k * ∂ T ∂ y 2 + μ T ∞ 2 ∂ u ∂ x 2 + ∂ v ∂ y 2 + ∂ u ∂ y + ∂ v ∂ x 2 + RD C ∞ ∂ C ∂ x 2 + ∂ C ∂ y 2 + RD T ∞ ∂ T ∂ x ∂ C ∂ x + ∂ T ∂ y ∂ C ∂ y + RD N ∞ ∂ N ∂ y 2 + RD T ∞ ∂ T ∂ x ∂ N ∂ x + ∂ T ∂ y ∂ N ∂ y .

By using boundary-layer approximations, Eq. (27) is modified.

(28) S gen ''' = k T ∞ 2 ∂ T ∂ y 2 + 16 σ * T ∞ 3 3 k k * ∂ T ∂ y 2 + μ T ∞ ∂ u ∂ y 2 + RD C ∞ ∂ C ∂ y 2 + RD T ∞ ∂ T ∂ y ∂ C ∂ y + RD N ∞ ∂ N ∂ y 2 + RD T ∞ ∂ T ∂ y ∂ N ∂ y .

E NT = k T ∞ 2 ∂ T ∂ y 2 + 16 σ * T ∞ 3 3 k k * ∂ T ∂ y 2 = “ Entropy contribution due to heat transmission . ”

E NF = μ T ∞ ∂ u ∂ y 2 = “ Entropy contribution due to fluid friction . ”

E NC = RD C ∞ ∂ C ∂ y 2 + RD T ∞ ∂ T ∂ y ∂ C ∂ y = “ Entropy contribution due to mass transmission . ”

E NM = RD N ∞ ∂ N ∂ y 2 + ∂ T ∂ y ∂ N ∂ y = “ Entropy contribution due to microorganisam s . ”

S gen ''' = N T + N f + N C + N m .

In addition, the usual rate of entropy formation is given by:

(29) S 0 ''' = k T ∞ 2 ( ∆ T ) 2 l 2 .

The dimensionless entropy production quantity can be determined as follows:

(30) EG = S gen ''' S 0 ''' .

The total entropy production quantity thus has the matching nondimensional form:

(31) EG = Re 1 + 4 3 Rd θ ′ 2 + Re Br Π F ″ 2 + Re ζ Π 2 λ ϕ ′ 2 + Re ζ Π λ ϕ ′ θ ′ + Re λ ζ Ω 2 χ ′ + Re λ ζ Ω χ ′ θ ′ .

The Bejan number (BE) is expressed as follows:

(32) Be = E NT + E NC + E NM EG .

4 Homotopy solutions

By using the homotopy analysis method (HAM) technique, the governing equations are analytically solved. To fix the initial approximation in this regard, Figure 2 shows the flowchart of the HAM procedure.

F 0 ( η ) = [ 1 − e − η ] ,

θ 0 ( η ) = [ ( 1 − S 1 ) e − η ] ,

ϕ 0 ( η ) = [ ( 1 − S 2 ) e − η ] ,

χ 0 ( η ) = [ ( 1 − S 3 ) e − η ] .

Figure 2

Flowchart of HAM.

The linear operators are as follows:

L ˆ F = d ˆ 3 F d η 3 − d F d η ,

L ˆ θ = d ˆ 2 θ d η 2 − θ ,

L ˆ ϕ = d ˆ 2 ϕ d η 2 − ϕ ,

L ˆ χ = d ˆ 2 χ d η 2 − χ ,

with the properties

L ˆ F [ Δ 1 + Δ 2 e ɳ + Δ 3 e − ɳ ] = 0 ,

L ˆ θ [ Δ 4 e ɳ + Δ 5 e − ɳ ] = 0 ,

L ˆ ϕ [ Δ 6 e ɳ + Δ 7 e − ɳ ] = 0 ,

L ˆ χ [ Δ 8 e ɳ + Δ 9 e − ɳ ] = 0 .

where Δ i [ i = 1 − 9 ] are the licentious invariable.

The zeroth-order distortion:

( 1 − p ) L F [ F ( η ; p ) − F ( η ) ] = p h F N F [ F ( η ; p ) ] ,

( 1 − p ) L θ [ θ ( η ; p ) − θ 0 ( η ) ] = p h θ N θ [ θ ( η ; p ) , F ( η ; p ) , ϕ ( η ; p ) ] ,

( 1 − p ) L ϕ [ ϕ ( η ; p ) − ϕ 0 ( η ) ] = p h ϕ N ϕ [ ϕ ( η ; p ) , θ ( η ; p ) , F ( η ; p ) ] ,

( 1 − p ) L χ [ χ ( η ; p ) − χ 0 ( η ) ] = p h χ N χ [ χ ( η ; p ) , ϕ ( η ; p ) , θ ( η ; p ) , F ( η ; p ) ] ,

where p ϵ [ 0,1 ] and h F , h θ , h ϕ , and h χ are the nonzero auxiliary constants and N F , N θ , N ϕ , and N χ are the nonlinear operators given by

N F = ∂ 3 F ( η ; p ) ∂ η 3 − ∂ F ( η ; p ) ∂ η 2 + F ( η ; p ) ∂ 2 F ( η ; p ) ∂ η 2 + α 1 2 ∂ F ( η ; p ) ∂ η ∂ 3 F ( η ; p ) ∂ η 3 − F ( η ; p ) ∂ 4 f ( η ; p ) ∂ η 4 + ( 3 α 1 + 2 α 2 ) ∂ F ( η ; p ) ∂ η 2 + 6 β 1 Re ∂ 3 F ( η ; p ) ∂ η 3 ∂ F ( η ; p ) ∂ η 2 + Ha e − δ η + λ ( θ ( η ; p ) − Nr ϕ ( η ; p ) − Rb χ ( η ; p ) )

N θ = 1 + 4 3 Rd ∂ 2 θ ( η ; p ) ∂ η 2 + Pr F ( η ; p ) ∂ θ ( η ; p ) ∂ η − Pr θ ( η ; p ) ∂ F ( η ; p ) ∂ η − Pr S 1 ∂ F ( η ; p ) ∂ η − Pr Γ 1 ∂ F ( η ; p ) ∂ η 2 θ ( η ; p ) + S 1 ∂ F ( η ; p ) ∂ η 2 − F ( η ; p ) ∂ F ( η ; p ) ∂ η ∂ θ ( η ; p ) ∂ η − F ( η ; p ) ∂ 2 F ( η ; p ) ∂ η 2 θ ( η ; p ) − S 1 F ( η ; p ) ∂ 2 F ( η ; p ) ∂ η 2 + PrNb ∂ θ ( η ; p ) ∂ η ∂ ϕ ( η ; p ) ∂ η + PrNt ∂ θ ( η ; p ) ∂ η 2 ,

N ϕ = ∂ 2 ϕ ( η ;p ) ∂ η 2 + Le F ( η ;p ) ∂ ϕ ( η ;p ) ∂ η − Le ϕ ( η ;p ) ∂ F ( η ;p ) ∂ η − Le S 2 ∂ F ( η ;p ) ∂ η − Le Γ 2 ∂ F ( η ;p ) ∂ η 2 ϕ ( η ;p ) + S 2 ∂ F ( η ;p ) ∂ η 2 − F ( η ;p ) ∂ F ( η ;p ) ∂ η ∂ ϕ ( η ;p ) ∂ η − F ( η ;p ) ∂ 2 F ( η ;p ) ∂ η 2 ϕ ( η ;p ) − S 2 F ( η ;p ) ∂ 2 F ( η ;p ) ∂ η 2 + Nt Nb ∂ 2 θ ( η ;p ) ∂ η 2 ,

N χ = ∂ 2 χ ( η ;p ) ∂ η 2 + Lb F ( η ;p ) ∂ χ ( η ;p ) ∂ η − Lb χ ( η ;p ) ∂ F ( η ;p ) ∂ η − Lb S 3 ∂ F ( η ;p ) ∂ η − Pe ∂ 2 ϕ ( η ;p ) ∂ η 2 ( χ ( η ;p ) + Ω ) + ∂ ϕ ( η ;p ) ∂ η ∂ χ ( η ;p ) ∂ η ,

F ( 0 ;p ) = 0 , F ′ ( ∞ ;p ) = 0 , θ ′ ( 0 ;p ) = 0 , θ ( ∞ ;p ) = 0 , ϕ ′ ( 0 ;p ) = 0 , ϕ ′ ( ∞ ;p ) = 0 ,

χ ′ ( 0 ;p ) = 0 , χ ′ ( ∞ ;p ) = 0 .

The mth-order distortion relations are as follows:

L F [ F m ( η ) − ϱ m F m − 1 ( η ) ] = h F R F , m ( η ) ,

L θ [ θ m ( η ) − ϱ m θ m − 1 ( η ) ] = h θ R θ , m ( η ) ,

L ϕ [ ϕ m ( η ) − ϱ m ϕ m − 1 ( η ) ] = h ϕ R ϕ , m ( η ) ,

L χ [ χ m ( η ) − ϱ m χ m − 1 ( η ) ] = h χ R χ , m ( η ) ,

where

ϱ m = 0 , m ≤ 1 1 , m > 1 .

After utilizing the mth-order HAM, we obtain

f j ( η ) = f j * ( η ) + Δ 1 + Δ 2 e ɳ + Δ 3 e − ɳ ,

θ j ( η ) = θ j * ( η ) + Δ 4 e ɳ + Δ 5 e − ɳ ,

ϕ j ( η ) = ϕ j * ( η ) + Δ 6 e ɳ + Δ 7 e − ɳ ,

χ j ( η ) = χ j * ( η ) + Δ 8 e ɳ + Δ 9 e − ɳ ,

where F j * ( η ) , θ j * ( η ) , ϕ j * ( η ) , and χ j * ( η ) are the particular solutions.

5 Convergence analysis

Figure 3 represents the convergence amounts are of h F , h θ , h ϕ , and h χ , and it designed in the convergence range level is “ − 0.4 ≤ h F ≤ − 0.1 , − 0.5 ≤ h θ , h ϕ ≤ − 0.1 , − 0.5 ≤ h ϕ ≤ 0.1 and − 0.5 ≤ h χ ≤ − 0.2 .” The convergence range of the recent result is h F = 0.35 and h θ = h ϕ = h χ = − 0 . 30. Table 2 shows the order of F'' ( 0 ) , θ ′ ( 0 ) , ϕ ′ ( 0 ) , and χ ′ ( 0 ) and found that 15 th order is sufficient for all calculations.

$Figure 3 The h h -curves for ( h F , h θ , h ϕ , & h χ ) ({h}_{F},{h}_{\theta },{h}_{\phi }\left,{\rm{\& }}{h}_{{\rm{\chi }}}) .$

Figure 3

The h -curves for ( h F , h θ , h ϕ , & h χ ) .

Table 2

Validation of C f Re − 0.5 for the limiting conditions Ha = λ = N r = Rb = 0 .

α 1	α 2	β 1	Re	Imtiaz et al. [18]	Current
0	0.1	0.1	0.1	0.04605	0.04605
0.1	0	0.1	0.1	1.0668	1.0668
0.2	0.1	0	0	1.1747	1.1747
0.1	0.2	0.1	0.1	1.1201	1.1201
0.1	0.1	0.2	0.2	1.0668	1.0668
		0.1		1.0183	1.0183
				1.0629	1.0629
				1.0668	1.0668
				1.0703	1.0703
				1.0629	1.0629
				1.0668	1.0668
				1.0706	1.0706

6 Results and discussion

The goal of this section is to investigate the impacts of different developing parameters on fluid velocity ( F ′ ), fluid temperature ( θ ), concentration ( ϕ ), microorganism profile ( χ ), skin friction coefficient ( C f ), Nusselt number ( Nu x ), tSherwood number ( S h x ), local entropy production ( EG ) , and Bejan number (Be). Table 2 shows the code validation against the study by Imtiaz et al. [18]. This validation results in the excellent agreement with the previous literature.

6.1 Velocity profile ( F ′ )

The output that takes into account changes in material parameter ( α 1 ) on F ′ is shown in Figure 4a. In more specific terms, viscosity has an inverse relationship with material parameter. For greater values of α 1 , the fluid velocity increases due to the lower viscosity. Figure 4b displays the discrepancies of F ′ for developed values of Nr . The increasing values of Nr lead to the deduction in fluid velocity. Behaviour of Pe on F ′ is shown in Figure 4c. The increasing values of Pe augments the swimming speed of the microorganism cells, which enhance the F ′ . Figure 4d exhibits the influence of stratification parameter ( S 1 ) on F ′ . The convective potential between the heated surface and away from the surface is decreased as the stratification parameter ( S 1 ) increases, and hence, F ′ declines as a result.

$Figure 4 Velocity profile for several values of (a) α 1 \alpha 1 , (b) Nr {\rm{Nr}} , (c) Pe {\rm{Pe}} , and (d) S 1 {S}_{1} .$

Figure 4

Velocity profile for several values of (a) α 1 , (b) Nr , (c) Pe , and (d) S 1 .

6.2 Temperature profile ( θ )

Figure 5a explores the difference in θ due to the change in Ha . It is noted that Ha and θ had an opposing behavior. In Figure 5b shows the effect of the Rayleigh number ( Rb ) on θ . It is demonstrated that the temperature and the thickness of the thermal layer are rising functions of Rb . It is obvious that an increase in Rb enhances buoyancy forces since it produces an increase in bioconvection, which elevates θ . The effect of the mixed convection parameter ( λ ) and the thermal stratification parameter ( S 1 ) on the θ is shown in Figure 5c and d. It is detected that the thermal boundary layer reduces with increasing buoyancy force λ , which upsurges the amount of heat transmission and subsequently lowers fluid temperature. In addition, it is noted that for small S 1 , the thermal boundary layer grows. In fact, the temperature difference between the heated surface and the nearby atmosphere diminishes as S 1 rises. Thus, a drop in θ is observed.

$Figure 5 Temperature profile for several values of (a) Ha {\rm{Ha}} , (b) Rb {\rm{Rb}} , (c) λ , \lambda , and (d) S 1 {S}_{1} .$

Figure 5

Temperature profile for several values of (a) Ha , (b) Rb , (c) λ , and (d) S 1 .

6.3 Concentration profile ( ϕ ) and microorganism profile ( χ )

The importance of mass relaxation time ( Γ 2 ) on ϕ is presented in Figure 6a. A reducing tendency in ϕ is found for higher values of Γ 2 . Figure 6b reveals the role of mass stratification parameter ( S 2 ) on ϕ . This analysis shows that as S 2 rises, and the volumetric fraction of surface compared to reference nanoparticles decreases. Consequently, ϕ drops. The change in the bioconvection Peclet number ( Pe ) on the motile density profile ( χ ) is shown in Figure 7a. For increasing Pe , the χ drops. Physically, increment in Pe diminishes in the diffusivity of microbes, which in turn reduces the fluid’s motile density. The effect of the bioconvection Lewis number ( Lb ) on χ is shown in Figure 7b. It is noted that the diffusivity of microorganisms declines, and the χ decays for higher values of Lb .

$Figure 6 Concentration profile for several values of (a) Γ 2 {\Gamma }_{2} and (b) S 2 {S}_{2} .$

Figure 6

Concentration profile for several values of (a) Γ 2 and (b) S 2 .

$Figure 7 Microorganism diagram for several values of (a) Pe {\rm{Pe}} and (b) Lb . {\rm{Lb}}.$

Figure 7

Microorganism diagram for several values of (a) Pe and (b) Lb .

6.4 Entropy generation ( EG ) and Bejan number (Be)

Figure 8a illustrates the impact of thermal relaxation time ( Γ 1 ) on the entropy generation profile ( EG ) . It is evident that the Γ 1 initially falls and then rises after η = 0.4 . Further, an increment in EG is also obtained for the higher Lb in Figure 8b. Performance of thermal stratification parameter ( S 1 ) and mass relaxation time ( Γ 2 ) against Be is shown in Figure 9a and b, which shows that Be decreases as S 1 increases. For improved values of Γ 2 , the Be declines at the initial stage and then increases after η = 0.8 .

$Figure 8 Entropy production profile for several values of (a) Γ 1 {\Gamma }_{1} and (b) Lb {\rm{Lb}} .$

Figure 8

Entropy production profile for several values of (a) Γ 1 and (b) Lb .

$Figure 9 Bejan number diagram for several values of (a) S 1 {S}_{1} and (b) Γ 2 . {\Gamma }_{2}.$

Figure 9

Bejan number diagram for several values of (a) S 1 and (b) Γ 2 .

6.5 Physical entities

Figure 10a shows the combined impact of modified Hartmann number ( Ha ) and λ on Nu x with other parameters held constant. The Nu x is reduced when both Ha and λ . The graphical analysis of the Sherwood number ( Sh x ) versus the variations in Le and S 2 is shown in Figure 10b, while other parameters are kept constant. The increase in Le and S 2 increases Sh x . The graphical comparison of the microbe density number Nn x against variations in Rb and with other parameters being held constant is shown in Figure 10c. The increase in Rb and leads to an improvement in Nn x .

$Figure 10 (a) “Nusselt number” for Ha and λ \lambda , (b) “Sherwood number” for S 2 {{\rm{S}}}_{2} and Le {\rm{Le}} , and (c) “microorganism density number” for Ω and Rb \Omega {\rm{and}}{\rm{Rb}} .$

Figure 10

(a) “Nusselt number” for Ha and λ , (b) “Sherwood number” for S 2 and Le , and (c) “microorganism density number” for Ω and Rb .

7 Key outcomes

This study used computational analysis to examine the bioconvective flow of third-grade NF across a Riga plate. Other factors that affected the study are radiation, stratification, non-Fourier heat, and mass flux. The investigation reveals these intriguing findings.

The occurrence of material parameter boosts up the velocity profile.
The Rayleigh number produces the most fluctuation in the temperature profile. While thermal stratification shows the opposite impact against the temperature profile.
It also underlines that by raising the Lewis number and mass stratification parameter, the pace of mass transportation can be modified.
According to the study, the existence of thermal relaxation time parameter and bioconvection Lewis number is more useful to improve the entropy rate.
With increased thermal stratification and mass relaxation parameters, the Be falls.

Acknowledgments

The authors express their appreciation to the Research Centre for Advanced Materials Science (RCAMS) at King Khalid University, Saudi Arabia, for funding this work under the grant number RCAMS/KKU/025-23.

Funding information: This work was funded under the grant number RCAMS/KKU/025-23 by the Research Centre for Advanced Materials Science (RCAMS) at King Khalid University, Saudi Arabia
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: The data used to support the findings of this study are made available by the corresponding author upon request.

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Received: 2023-06-04

Revised: 2023-07-08

Accepted: 2023-07-10

Published Online: 2023-08-07

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

Articles in the same Issue

https://doi.org/10.1515/phys-2023-0273

Keywords for this article

third grade nanofluid; gyrostatic microorganisms; radiation; Riga plate; homotopy analysis method

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