Numerical Investigation of Magnetohydrodynamic Flow of Reiner–

Philippoff Nanofluid with Gyrotactic Microorganism Using Porous

Medium

S.K. PRASANNA LAKSHMI1, S. SREEDHAR1*, S.V.V RAMA DEVI2

1Department of Mathematics, GITAM School of Science,

GITAM Deemed University, Visakhapatnam, Andhra Pradesh, INDIA

1*Department of Mathematics, GITAM School of Science,

GITAM Deemed University, Visakhapatnam, Andhra Pradesh, INDIA

2Department of Mathematics, Raghu Engineering College (A), Visakhapatnam, Andhra Pradesh,

INDIA

Abstract: Nanoparticles facilitate the enrichment of heat transmission, which is crucial in many

industrial and technical phenomena. The suspension of nanoparticles with microbes is another

intriguing study area that is pertinent to biotechnology, health sciences, and medicinal applications. In

the dispersion of nanoparticles, the conventional non-Newtonian fluid Reiner-Philippoff flows across

a stretching sheet, which is examined in this article using numerical analysis. This study investigates

the numerical investigation of Arrhenius reaction, heat radiation, and vicious variation variations on a

Reiner-Philippoff nanofluid of MHD flow through a stretched sheet. Thus, for the current nanofluid,

nanoparticles and bio-convection are highly crucial. The set of nonlinear differential equations is

translated into Ordinary Differential Equations (ODEs) utilizing the requisite translation of

similarities. These collected simple ODE are solved using the MATLAB computational tool bvp4c

method. The graphical results for the velocity, concentration, motile microorganisms, and temperature

profile are defined using the thermophoresis parameter and the Brownian motion respectively.

Consider a tube containing gyrotactic microbes and a regular flow of nanofluid which is electrically

conducted through a porous stretched sheet surface. This nonlinear differential problem is solved by a

hybrid numerical solution method using fourth-order Runge-Kutta with shooting technique. The

optimization method also performs well in terms of predicting outcomes accurately. As a result, the

research applies the Bayesian Regularization Method (BRM) to improve the accuracy of the prediction

results. Physical constraints are plotted against temperature, velocity, concentration, and

microorganism profile trends and they are briefly described.

Keywords: Non-Newtonian Fluid, Reiner-Philippoff Nanofluid, Bioconvection, Gyrotactic

Microorganism, Thermal Radiation, MHD, Bayesian Regularization, Fourth Order Runge-Kutta.

Received: April 29, 2022. Revised: May 18, 2023. Accepted: June 19, 2023. Published: July 11, 2023.

1. Introduction

Researchers have focused a lot of emphasis on

the study of non-Newtonian fluids because of

its numerous demand in science and

engineering. Because of the complex

interactions between stress and shear rate

strain, it is challenging to develop a single

relationship that entirely accounts for the

behaviour of non-Newtonian fluids [1]. The

Reiner-Philippoff fluid exhibits shear-thinning

(Pseudo-plastic), shear-thinning (dilatant), and

Newtonian behaviour. In the same category as

Power-law, Sisko, Powell-Eyring, Prandtl-

Eyring and Sutterby are some more fluids that

fall under the same category. On a stretching

sheet, Gangadhar thought about using heat

radiation and micropolar Ferrofluid [3]. Nawaz

discovered that the index power-law of the

velocity field grows when a Sutterby fluid

impinges on an extendible surface, taking into

account the impacts of the nanoparticles

together with Ohmic effect and viscous

dissipation [4]. A slippery extensible surface

was used in Jyothi's (2020) investigation into

the nanoparticles characterization of gold on

Sisko fluid. Eyring-Powell fluid with

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nanoparticles of aluminium oxide, and found

that the thermal boundary thickness increases

when the radiation parameter is increased [5].

Non-Newtonian heat transfer has a variety of

applications, including paper production, wire

coating, heating and flavouring food. The

effects of nanoparticles on thermal conductivity

phenomena are enhanced. The well-known

substance known as a nanofluid is used in a

number of technical, commercial, and scientific

domains [6]. They have creative thermal

applications. The thermophysical

characteristics of nanofluids are followed by

certain dynamic applications in nuclear heating

and cooling devices, solar concerns, magnetic

retention, astronomy and safety, automated

operation, etc. [7].

Typically, it is assumed that the

nanoparticles are tiny metallic particles slighter

than 100 nm. The nanofluids, as opposed to the

more traditional viscous liquids, make up for

increased energy transmission. Choi was the

first researcher to pay close attention to the

thermal aspect of nanofluids [8]. There are

numerous industrial and medical uses for

nanofluids, which has drawn the attention of

researchers worldwide. Mahanthesh (2019)

studied magneto nanofluids with a rotating disc

and convective boundary conditions. The

Reiner-Philippoff models [9] are the most

important models for understanding the

characteristics of these fluids. The Reiner-

Philippoff model pertains to the

pseudoplastic/shear-thinning fluids class.

There aren't many study works in the literature

that discuss the Reiner-Philippoff fluid

boundary layer properties. MHD (Magneto-

hydrodynamics) is the research of the motion of

the electrically conducting fluid induced by

external magnetic forces [10]. The MHD flow

problem of a nanofluid in a boundary layer

across a porous medium across an

exponentially extending sheet was the focus of

earlier investigations. It has been demonstrated

that unstable viscous nanofluid boundary-layer

flow along a vertically extended sheet may

transfer mass and heat in the aspect of heat

generation, magnetic field, chemical reaction,

and thermal radiation [11]. They found that the

velocity, concentration profiles, and

temperature of the unstable flow varied from

those of the corresponding components in the

steady-state flow scenario. The energy

emission is known as radiation, where the

particles or waves through a physical medium,

such as space [12].

Smaller-scale physical laws that result in

larger-scale events are known as

bioconvection. During the bioconvection

process, microorganisms with a density greater

than water flow higher, resulting in a top-heavy

density stratification that commonly

destabilises [13]. Many kinds of microbes

include those that travel through oxygen

(oxytaxis), rotate and spin (gyrotaxis), or move

under the influence of gravity (gravitaxis).

Interestingly, gyrotactic microorganisms boost

the stability of the suspension when added to

nanofluid [14]. Magnetic fields and microbes

have an impact on the issue of flowing natural

boundary layer convection in porous media

around a vertical cone saturated by a nanofluid

generated by gyrotactic microorganisms. Self-

propelled microbe suspensions are the subject

of bioconvection. It is crucial to understand

how minute, heavier-than-water particles affect

the stability suspension of gyrotactic, motile

bacteria in a finite depth horizontal fluid layer.

The motile bacteria interaction and

nanoparticles cause’ nanofluid bioconvection,

which results from the magnetic field

interaction and buoyancy forces [15]. Recently,

researchers examined Stefan blow impacts and

numerous slides on bioconvection nanofluid

buoyancy-driven movement of

microorganisms. Computationally, a lot of

researchers have written about these

remarkable events. A rise in the concentrations

of motile microorganisms establishes the

efficiency of bio convection. They found that

fluid velocity raises as the unsteadiness effect

is enhanced while decreasing the viscoelastic

term [16]. Consequently, the study discusses

the dynamic contribution of Reiner Phillipoff

nanofluids' Arrhenius reaction, thermal

radiation, and viscosity variation features. To

achieve this, the hybrid numerical solution is

used to transform the nonlinear differential

equation into an ODE. After that, the

optimization approach is utilized to

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successfully forecast accurate results. The

MATLAB computational tool bvp4c technique

is used to solve these acquired simple ODE.

2. Literature Survey

Mohamed E. Nasr et al [17] suggested that the

magnetohydrodynamics (MHD), the thermal

energy, and the mass transport boundary layer

flow characteristics are numerically

investigated in non-Newtonian Reiner-

Philippoff fluid. The species reaction with

respect to the activation energy, double

diffusions of Cattaneo-Christov, surface

convective conditions, and nonlinear radiation

are all explored in terms of energy and mass

transfer. Using the right similarity variables,

the specified governing system of PDEs is

drained into a non-linear differential system.

Numerical solutions to the determined flow

equations have been found. The electrically

conductive nanofluid flow of viscosity change

is examined by O.A.Famakinwa et al [18], in

which arrhenius reaction modifications, and

thermal radiation across a convectively heated

surface. In the built-in bvp4c software package

of MATLAB, the shooting method and the 4th

order Runge-Kutta formula are used to solve

the nonlinear coupled ODE that results from the

mathematical model governing the fluid flow.

The data point statistical tool is also used to

introduce the slope of linear regression.

Taseer Muhammad et al [19] explore the

properties of the nanofluid named Jeffrey while

taking the influence of motile microorganisms

and activation energy into consideration. This

review has taken into consideration the

magnetic field influence, another significant

physical factor in the flow analysis.

Thermophoresis is relevant for mass

transportation processes in operating systems

with substantially larger temperature gradients.

To convert PDE into ODE conveniently, the

necessary similarity transformation is used. To

estimate the numerical results of the obtained

normal system of flow, the well-known

shooting technique is used. To determine the

solution, the MATLAB built-in program's

bvp4c strategy is used to integrate the

governing dimensionless equations. In

Muhammad Jebran Khan et al [20] study, the

numerical study of a model including

bioconvection processes and a gyrotactic

motile microbe of magnetohydrodynamics is

examined. The model employs the change in

thickness effect and the thermal conductivity

feature. Nanofluid bioconvection is beneficial

for bioscience applications such as content

detection, drug delivery, micro-enzymes,

biosensors, and blood flow. For the linear

regression designed for the suggested model,

the New Iterative nuMerical (NIM) technique

is adopted and used for numerical simulation.

Najiyah SafwaKhashi et al [21] explore the

effects of MHD and dissipation on viscous in

Reiner-Philippoff fluid flow radiative heat

transfer through a nonlinearly contracting

sheet. The multivariable differential equations

partial derivatives are converted into similarity

equations of a certain form by using the proper

similarity transformations. In MATLAB

software, the resulting mathematical model is

explained using the bvp4c method. Meanwhile,

raising the suction parameter value that has

been demonstrated to enhance the efficiency of

heat transfer and the skin friction coefficient,

the Reiner-Philippoff fluid is greatly impacted

by the suction action. The first solution validity

is supported by the stability analysis that results

from the establishment of the dual solutions.

In this paper, Iskandar Waini et al [22]

investigate numerically the flow of radiative

non-Newtonian fluid across a decreasing sheet

while there is an aligned magnetic field.

Multivariable differential equations governing

partial derivatives are converted into a

particular class of similarity equations. The

bvp4c method is employed to get the numerical

results. The analysis showed that as the suction

parameter increased, the skin friction rate and

heat transfer increased as well. When the

alignment angle and magnetic parameter are

taken into account, an identical pattern shows

up. However, the performance of heat

transmission is harmed by the adjunct to

Bingham number, Reiner-Philippoff fluid, and

thermal radiation factors. An investigation of

stability confirms the correctness of the

solution when the dual solutions are formed.

An analysis of the bioconvection

phenomenon for Reiner-Philippoff nanofluid-

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induced Darcy-Forchheimer flow is presented

by Yun-XiangLi et al [23]. For the flow, the

impact of slip through higher relations is

broken down. The radiative pattern for the

thermally produced flow was examined.

Modified Cattaneo Christov expressions have

been exploited to scrutinize the evaluation of

mass and heat transfer. The dimensionless form

is created by transforming the flow equations

related to momentum, volumetric friction, and

motile microbe density. Using the computing

programme MATLAB, transformed

dimensionless non-linear equations are tracked

using the shooting approach, and the outcomes

of significant parameters are depicted via

various graphs. The accessible dynamic

property of Reiner-Philippoff nanofluid is

discussed by Sami UllahKhan et al [24] with

bioconvection phenomenon applications. The

Reiner-Philippoff nanomaterial's radiative

analysis is also designed to be performed using

the activation energy features and magnetic

force impact. To examine the flow, the higher-

order relations of slip features are included. To

suggest variations from the energy equation,

thermal radiation and its nonlinear relationship

are used. The flow equations are numerically

solved by applying a shooting strategy, which

leads to their non-dimensionless form. For the

endorsed parameters, a detailed thermal

analysis is presented. To analyse the variations

in motile density, mass and heat function,

numerical data is obtained.

The flow of Eyring-nanofluid Powell's

through a porous plate vulnerable to surface

suction and heat radiation is said to produce a

flow of two-dimensional gyrotactic

microorganisms suggested by Naseer M. Khan

et al [25]. The nonlinear approximation of

Rosseland was introduced to incorporate solar

radiation parameters into the energy equations,

while the Buongiorno model of nanofluid was

developed to include the momentum and

energy equations. The problem's numerical

solution was discovered using the MATLAB

"bvp4c" scheme. The analysis is done on how

different physical parameters affect the

distribution of velocity, temperature, and

concentration. Although suction decreases the

temperature, it speeds up the heat transmission.

A thixotropic fluid flow of a gravity-driven

system containing both gyrotactic

microorganisms and nanoparticles beside a

vertical surface was examined by Olubode

Kolade Koriko et al [26]. Special instances of

controls nanoparticles of passive and active are

studied to further explain the transport

phenomena. The governing ODE of the

gyrotactic microorganisms' momentum,

energy, concentration, and density are

parameterized and turned into a PDE system.

The OHA method is later employed to deduce

the series solutions. With regard to the

characteristics of mobile microorganisms, the

relevant crucial parameters are examined and

displayed.

3. Research Problem Definition and

Motivation

Due to its numerous industrial and

technological applications, the study of heat

transfer and boundary layer flow over a

stretching surface had significant success over

the decades. Recently, several industrial

processes have benefited greatly from the

research of Nanofluids and convective heat

transfer. The nanofluid's thermal conductivity

has been the subject of several theoretical and

experimental studies by many scientists. Over

the past ten years, nanofluids have gained

importance due to their wide range of

applications, particularly in the problem of

boundary layer flow. Nanofluids are useful for

improving the thermo-physical characteristics

of the governing fluid, such as thermal

diffusivity, convection, and conductivity. The

word "bioconvection" refers to the additional

mobility of swimming microorganisms caused

by the macroscopic movement of the fluid on

account of the spatial variation of density over

a region. A bio-convective stream is created

when the self-driven motile microorganisms

improve the base fluid in a certain direction.

Smaller density microorganisms float, which is

the basis of the bioconvection process. The

non-uniform instability structure is typically

too responsible for the microorganisms

swimming in the top region.

Due to biological germs swimming on the

higher surface, the upper section becomes

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unstable which enlightens the increased

stratification density. The interplay between

nanoparticles and bioconvection aspects is

crucial for microfluidic applications since

swimming microbes cannot travel in

nanoparticles' uniform motion. Additionally,

the interplay of buoyancy forces,

nanomaterials, and microorganisms result in

bio convection of nanoparticles, which is

related to pattern formation and stratification

density. The nanoparticle suspension stability

is frequently observed to be greatly increased

when gyrotactic bacteria are present. Numerous

important physical processes are regularly

modelled using nonlinear differential equations

in the relevant field of science and technology.

These equations are frequently difficult or

impossible to solve analytically. However,

analytical approximation techniques for

obtaining reasonably correct results have

grown significantly in importance recently.

This study then describes how bioconvection

can be used in Reiner-Philippoff nanofluid flow

while also dealing with thermal radiation and

viscosity variation. The thermal applications of

the Reiner-Philippoff nanofluid are addressed

via nonlinear radiation relations. With

intriguing applications, the physical

perspective of parameters is graphically

operated out. Additionally, the numerical

calculations are exposed to show how mass,

motile density, and heat rate change in relation

to flow parameters.

4. Research Proposed Methodology

The objective of the ongoing research is to

create a thin Reiner-Philippoff nanofluid MHD

flow over a stretching sheet using thermal

radiation and heat transfer. The physically

demonstrated geometry yields boundary-layer

equations. Thermophoresis effects and

Brownian motion are also encircled with

different physical parameters. With the aid of

new variables, a similar solution is attained,

which causes a complex model to be reduced to

a straightforward coupled ODE. The simplified

system is solved using an analytical approach.

For converting the nonlinear differential

equation into regular equations, Runge Kutta's

fourth-order and shooting method are

suggested. The outcomes are plotted, tabulated,

and systematically discussed using a number of

different physical characteristics. The

optimization approach is also used to increase

the precision of the numerical solution.

Figure 1: Proposed Work Flow Diagram

The flow diagram for the suggested task is

shown in Figure 1. This article examines the

numerical investigation of changes in

Arrhenius reaction, thermal radiation, and

viscous variation on a Reiner-Philippoff

nanofluid of MHD flow using the stretching

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sheet. Nanoparticles and bio-convection are

therefore quite important for the current

nanofluid. The required translation of

similarities is used to convert the set of

nonlinear PDE into ODE for this purpose, this

also mentioned the parameter analysis of

Brownian and thermophoresis. The numerical

solution method of the Runge-Kutta method of

fourth order with shooting technique is used to

solve this nonlinear differential problem. The

article introduced the BR optimization

Algorithm to increase the suggested technique's

accuracy.

The MATLAB computational tool bvp4c

technique is used to solve these acquired simple

ODEs. By using the thermophoresis parameter

and the Brownian motion parameter,

respectively, the graphical results are defined

for the velocity, motile microorganisms,

concentration and temperature profile. The

study considers the thermal radiation effect on

a continuous Reiner-Philippoff nanofluid flow

that is two-dimensional and transports

nanoparticles and gyrotactic microorganisms

through a vertical porous medium of the

stretched sheet. Both the horizontal y-axis and

the vertical x-axis are subjected to a fluctuating

magnetic field. Assume that the fluid is being

subjected to a strong magnetic field at a

particular rate, and the surface is stretched in

the x-direction. The Buongiorno model allows

for the observation of the Brownian and

thermophoresis distinctiveness.

4.1 Mathematical Modelling

With velocity components  that are rejected

along with the sheet, the surface is stretched in

the guidance of flow, whereas  is taken in the

normal flow direction. The magnetic force

often affects the flow direction. In the flow

domain, the nanofluid's temperature (),

concentration (), and motile density () are all

regarded as being uniform. Utilizing specific

relations for the analysis of thermally radiative

flow modifies the energy equation.

Figure 2: Geometrical Configurations

The geometrical arrangement of a two-

dimensional nanofluid flow above the

stretching sheet is exposed in Figure 2.

Between these two values of viscosity , the

non-Newtonian character in Reiner-Philippoff

fluid is presented. The Reiner-Philippoff fluid

model's deformation rate and shear stress

relationships are defined as

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























s

y

u









1

0

(1)

with shear stress



and deformation rate

y

u



. The boundary layer assumptions form the foundation of

the equation system for the current task.

0







y

u

x

u

(2)

 

   

 





































fmfp

ff

c

fNNCC

TTC

g

B

yy

u

v

x

u













1

11

2

0

(3)

 











































































2

3

2

3

16

y

T

D

y

T

y

C

D

c

y

T

ycy

T

y

T

v

x

T

uT

B

f

p

f







`

(4)

 























































Tk

E

T

CCKr

y

T

D

y

T

y

C

D

y

C

v

x

C

ua

n

T

B

1

2

exp

(5)

 

2

ˆ

y

N

D

y

C

N

yCC

wb

y

N

v

x

N

um

w











































(6)

The nonlinear thermal relations for the

radiative phenomenon are incorporated into the

energy equation using the Roseland

approximations. Additionally, the activation

energy is represented by the final term in

Equation (6), which is obtained by applying the

Arrhenius relations.

The boundary conditions are

 

































0,,,

0,, 2

2

3

1

yatNN

y

T

DT

y

C

DTTh

y

T

k

vat

y

u

B

y

u

Auuaxxuu

wBff

slslw

(7)

  NNCCTTu ,,,0

as

y

(8)

Physical quantities such as fluid density

f



, kinematic viscosity

v

, Stefan Boltzmann

constant





, nanoparticles heat capacity

 

p

c



, applied Boltzmann constant





,

thermal diffusivity

m



, swimming cells speed

w

ˆ

, electrical conductivity

c



, volume

suspension coefficient , diffusion constant

B

D

, chemotaxis constant

b

ˆ

, mean absorption

coefficient



k

, microorganisms diffusion

constant

m

D

, magnetic field strength

0

B

,

gravity is represented as

g

, heat capacity

 

f

p

c



, motile microorganisms is denoted as

m



and nanofluid density as

p



. For the

current flow model, the following non-

dimensional set of expressions are anticipated:

   



























NN

C

CC

TT

vgaavx

x

y

v

a

w

ww

w





,

,,, 3

3

2

31

(9)

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22







g

Ag

fg

(10)

The equation of momentum is articulated as follows

 

0

3

2

3

12

















RbNrGrfMfffg

(11)

      

 

 

 

0Pr

111411

3

4

1

2

23





































t

www

NNbf

RdRd

(12)

 

0

1

exp1PrPr 



































E

Le

N

fLe n

b

t

(13)

According to the Concentration Equation,

 

 

0

1

















PeLbf

            

     

       





































0,0,0,0

10,000

010,0010,00 21







f

N

Biffff

b

t

(14)

Brownian constant

   

 

v

c

CC

DcN

f

w

B

p

b

















, bioconvection Rayleigh number

 













TC

NN

Rb

f

wfm

1





, Philippoff fluid parameter







0

A

, microorganism difference

parameter





NN

N

w

1



, buoyancy ratio parameter

 











TC

CC

Nr

f

fPw

1





, Bingham number

va

s

3







, Prandtl number

f

v



Pr

, Peclet number

m

D

wb

Pe ˆ

ˆ



, thermophoresis constant

   

 

vTc

TTDc

N

f

wT

p

t









, activation energy

Tk

E

Ea

1



, chemical reaction parameter

a

kr2





,

temperature difference parameter





T

TTw



, bioconvection Lewis number

m

DvLe 

are some of

the other parameters that are used.

The dimensionless version of the decreased Sherwood number, Nusselt number, and motile density

number is given as follows:

        

   































0Re

0011

3

4

1Re

5.0

35.0





x

wx

Nn

RdNu

(15)

󰇛󰇜󰇛󰇜

4.2 Numerical Solution of Governing

Equation

The shooting approach and the fourth-order

Runge-Kutta integration scheme are used to

solve the two-point boundary value problem

created by the nonlinear coupled differential

Equation (7) and the boundary conditions

(Equation 8), which is then transformed into an

initial value problem. The governing equations

are initially converted into ODEs by using an

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appropriate similarity transformation. After

that, this system of ODEs is numerically solved

using the Runge-Kutta-Fehlberg technique of

fourth order. The unbounded domain





,0

has

been changed for numerical solutions by

 

max

,0



where

max



is a real number selected

hence that there are no appreciable fluctuations

in the solution for

max





. The fact that

7

max 



guarantees the anticipated level of

convergence for all the numerical results

described in this paper is remarkable. The

momentum equations (11) and (12) will be

solved collectively using the shooting method,

with the temperature and concentration

equations solution using

f

as a known

function.

The momentum equations (11) and (12) are

transformed into the first-order ODEs,

f

is

represented as

fy 

,

1

and

3

y

is denoted by

gy ,

2

, and the missing initial condition is stated as

 































syyyyy

y

yy

y

yyy

0,

3

2

3

1

10,

00

321

2

23

2

3

22

33

2

121



(16)

The Runge Kutta fourth-order (RK4)

algorithm has been used to numerically manage

the aforementioned system (16). Additionally,

Newton's technique is used to update the

missing initial conditions until the criteria are

satisfied.

󰇝󰇛󰇜󰇞 (17)

In this case, the symbol  denotes the

positive number with the value  and

.

The temperature equation (13) is

transformed into a system, which uses

differential equations of the first order (16) with

known functions  to represent  by  and 

is represented as . With the initial conditions,

it is feasible to achieve the system of ODEs in

(18).

󰆒󰇛󰇜

󰆒󰇛󰇜󰇛󰇛󰇜󰇜

󰇛 

󰇜󰇛󰇜

󰇛󰇛󰇜󰇛 

󰇜󰇛󰇛󰇜󰇜󰇜

󰇛󰇜  (18)

To produce  and , equation (18) is treated

in the same manner as equation (16).

By designating  as  and  as , and

using  as a known function, the concentration

equation (14) is converted into the first-order

ODEs. The system of equations that results is

󰆒󰇛󰇜󰆒󰇛 

󰇜

󰇛󰇜󰇦 (19)

Furthermore, the optimization process

maintains the precision of numerical solutions.

The following is a description of the solution

process that was used based on ANN’s back

propagated with BR.

4. 3 Bayesian Regularization Optimization

Algorithm

It is decided that the solution requires a

description of the neural network-based

approach, including the layer structure, hidden

neurons, topology of the networks, and

arbitrary selection of an input and target data

set for training, testing, and validation samples.

The suggested approach based on ANN back

propagated with BR, or ANN-BR is executed

using the MATLAB software's "nftool," which

takes advantage of the neural networks

environment. The solution process includes a

description of a significant dataset and an

execution process for the suggested ANN-BR.

Figure 3 displays the solution methodology's

overall workflow.

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Figure 3: BR Network Mathematical Representation

The data point’s total number for the ANN-

BR model is 1001, which was discovered

between 0 and 1 by utilising the Runge-Kutta

method in Mathematica's "NDSolve" function

with 0.001 as the step size. To get the best

convergence, these data are randomly dispersed

for , , , and  into sets for training, testing,

and validation. The reference datasets for the

temperature profile

 



,

 





and velocity

profile

 



f

,

 



f

respectively. Additionally,

for 61 inputs, the concentration profile is

denoted as

 



and

 





, along with the

motile density, profile is represented by

 



and

 





are generated. As 85% of the data in

this problem are used for training, 10% are used

for testing, and 5% are used to validate the

suggested ANN-BR using a neural network.

Additionally, the neurons amount is mutable;

20 neurons gives the computational findings of

high accuracy.

5. Analysis of Results and Discussion

This section provides results of a systematic

analysis obtained from numerical computations

on Equations (11–14) using the MATLAB

software's bvp4c module. The analysis includes

the effects of several physical parameters that

come into play in the suggested model, with the

calculations' results shown in graphical and

tabular form. Using the default settings of the

dimensionless parameter, effects of viscosity

variation analysis, Arrhenius reaction, and

thermal radiation flow under study are

displayed as tables and figures. The effects of

several selected factors have been assessed and

shown for velocity profiles

 



f

, temperature

profiles

 



, nanoparticle concentration

profiles

 



, and motile microorganism

density profiles is denoted as

 



w

. The

governing parameters have the theoretical

values of

5.0 RbNrM

,

1

KNN nttb 



and

1.0

2K

,

0.1 PeSh

,

0.2Pr  Le

. The number of

local Nusselt is demonstrated in Table 1 to be a

decreasing function of

NbPr,

and

Nt

for the

nanoparticles.

Table 1:

x

Sh

,

x

Nu

and

x

Nn

for Various

M

and

0

P

M

0

P

x

Nu

x

Sh

x

Nn

2

0

2.5231

-0.2700

2.8226

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M

0

P

x

Nu

x

Sh

x

Nn

3

1

3.4493

-0.1697

3.5544

4

10

4.2423

-0.0506

4.2734

5

100

5.0333

0.0800

4.9863

The current results also indicate that higher

convergence of the bioconvection parameter

can be attained if the porosity of the medium

rises. Due to the huge size of holes (0 <

0

P

<100), which prevent them from ingesting

microorganisms, there is no discernible

variation in the concentration

x

Nn

of motile

bacteria (Table 1). Because the porosity

parameter is not included in the concentration

and heat equation, it has little impact on the

rates of mass and heat transport. Additionally,

only a significant shear stress shift is observed,

due to the inclusion of

0

P

in the momentum

equation. The thermal radiation presence has no

influence on bioconvection across porous

surfaces, and the action of heat radiation results

in a growth in the motile microorganisms'

density

x

Nn

. The obtained results depict that

the

x

Nn

density of local motile microbes is only

stable at high porosity.

Table 2: Numerical Analysis of

 

0







Flow Parameter

Pr



Nr

Nc

Rd

Nb

Nt

 

0







0.1

0.7

1.1

0.4

0.2

0.6

0.4

0.63547

0.67878

0.71657

0.2

0.3

0.7

1.1

0.59786

0.53653

0.49775

1.0

0.3

0.5

0.8

0.58855

0.57435

0.55534

0.2

0.3

0.5

0.9

0.55764

0.52764

0.50134

0.3

0.9

1.5

0.60744

0.63765

0.64748

2.0

1.0

0.1

0.5

0.9

0.68741

0.71745

0.7326

0.1

0.7

1.3

0.62953

0.59664

0.53743

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Due to this

Pr

,

Nb

and

Nt

a rising shift in

heat transmission is observed, while

Nr

and

Rd

attain lower numerical data. Table 2's

numerical data illustrated the variation in the

 

0







assertion, the transfer rate of mass slows

with increasing values of

Nb

,

Nr

and

Nc

.

5.1 Brownian Motion and Effect of

Thermophoresis Parameter Analysis

The fluid's velocity is decreased as a

consequence of the overall effect. It has been

demonstrated that lowering the boundary layer

solute results in a Brownian motion parameter

increase. As a result, nanoparticles in the

inactive fluid began to move away from the

surfaces as the boundary layer warmed up. The

effect of thermophoresis and the Brownian

motion on the velocity, concentration profiles,

motile microorganisms, and temperature is

depicted in Figures (4–8). A fluid's fast-moving

molecules collide with suspended particles,

causing Brownian motion, which is the

random, "indecisive," movement of the

particles.

Figure 4: Velocity Profile for Brownian and Thermophoresis

Figure 4 illustrates how parameter Brownian

b

N

and thermophoresis

t

N

affect Reiner

Philippoff fluid velocity function

 



f

. Due to

the interaction of a growing opposing force, the

Lorentz force, the flow speed curve decreases

as M values rise. The estimated parameters of

brownian motion with the thermophoresis

effect on the velocity profiles which is

dimensionless for

1Pr 

,

1Le

,

5.0 NM

. It is seen that while the velocity profile

 



f

is reduced for the parameter of buoyancy ratio

r

N

, it is boosted for the bioconvection

parameter



.

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Figure 5: Temperature Profile for Brownian and Thermophoresis

The temperature distribution under the

t

N

and

b

N

is shown in Figure 5. Generally

speaking, a particle collision results from the

particles' erratic velocity. Effects of these

parameters on the dimensional fewer

temperature profiles for

1Pr 

,

1Le

,

5.0 NM

. The temperature is lowered by

means of the Prandtl number. The

dimensionless quantity of the Prandtl number

correlates with a fluid's viscosity and thermal

conductivity. Both the temperature and the

width of the boundary layer tend to decrease

due to this event. Larger values lead to a rise in

temperature, which in turn causes an increase

in surface temperature. Similar to this, the

temperature profile

 



rises as the



values in

the stretching situation increase.

Figure 6: Concentration Profile for Thermophoresis and Brownian motion

The parameter of chemical reaction

importance with

tb NN ,

on the fluids'

concentration

Kr

, was shown in Figure 6. The

improvement in these parameters of the

chemical reaction is responsible for a drop in

the concentration profile. Meanwhile, viscous

fluids (

1M

) experience a greater drop.

Figure 6 clearly shows that, in the absence of

Kr

, the concentration distribution grows as

increases in the value of

t

N

. According to

Table 3, increasing the viscous variation

parameter boosts skin friction coefficients by

0.2786 and heat transmission by 0.02703, but

decreases mass transfer by -0.0569 and motile

microbial density by -0.0321.

Table 3: Variation in

     

0,0,0













f

and

 

0







with

M

when

7

max 



and

1Kr

M

 

0f

 

0







 

0







 

0







1

0.6254

0.0025

1.2345

1.1423

2

0.7893

−0.0489

1.6485

1.3645

3

1.1256

−0.0421

1.5875

1.2678

5.2 Motile Microorganism Profile

The bioconvection is produced by the unstable

density stratification. According to recent

research, the flow's viscous drag and

gravitational torques influence how the

microorganisms move (gyrotaxis). The

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gyrotactic microbe with the nanoparticles was

used in the study to determine the bio

convective fluid density. The motile microbe

density is shown in the following figures,

together with thermophoresis and the Brownian

motion parameter.

Figure 7: Motile Microorganism Density Parameter Analysis of Brownian motion

For numerous estimates of the

bioconvection Lewis and buoyancy number,

Figure 7 shows the distribution of the motile

microorganisms. The concentration profiles are

typically depressed by the expanding values of

Le

and

Br

. An increase in the bioconvection

Peclet number and motile microorganism’s

parameter



values is accompanied by a

declaration in the distribution of motile

microbes

b

N

.

Figure 8: Motile Microorganism Density of Thermophoresis Parameter Analysis

Figure 8 depicts the thermophoresis

parameter analysis to show the motile

microorganism density. Inversely proportional

to

Dn

(microorganism diffusivity) and directly

proportional to one another are the Peclet

number and cell swimming speed. As a result,

increasing Peclet number values decrease the

density profile of motile microbes and increase

the flux of wall motile bacteria. The motile

microorganisms' density increased due to their

diverse variety in

Pe

and

t

N

.

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Table 4: Comparison of Various Emerging Parameters Values on Nusselt Number

Pr

Rb

w



x

Nu

Newtonian

Fluid

 

1



Dilant Fluid

 

1



Reiner

Philippoff

Fluid

 

1



0.5

1.170370

0.849563

1.271570

1.0

0.985130

0.718474

1.271570

1.5

0.727497

0.411036

1.117839

2.0

1.0

1.167123

0.605998

0.986096

1.5

1.337873

0.813748

1.648060

2.0

1.467557

0.909926

1.322524

0.5

0.865201

1.001301

1.506275

1.2

1.167123

0.635582

1.465753

1.5

1.368815

0.925486

1.635462

The obtained values of the local

x

Nu

for

innumerable values of flow issues in the current

parameters are revealed in Table 4. For Reiner

Philippoff fluid, Newtonian fluid, and dilatant

fluid acknowledged that scale back is

necessary. The value of

x

Nu

increases in the

pseudoplastic, Newtonian and dilatant fluid is

also suggested for rising

Rb

,

Pr

and

w



values.

In addition, compared to Newtonian fluid and

dilatant fluid,

x

Nu

is efficiently improved in

Reiner Philippoff fluid.

6. Research Conclusion

Nanofluids have been widely used in energy

technologies and have already demonstrated

significant promise in the thermal amplification

of numerous manufacturing industries. The

present approach aims to characterise the

features of the Reiner Philippoff nanofluid

MHD flow while considering the impact of

thermal radiation, viscosity change, and the

Arrhenius reaction over a porous stretched

sheet. Brownian motion along with the effect of

thermophoresis features show the extraordinary

properties of nanofluid. Thermophoresis is

relevant for mass transportation processes in

operating systems with substantially larger

temperature gradients. To convert PDE into

ODE conveniently, the necessary similarity

transformation is exploited. It is expected that

the process makes use of MATLAB. It is

possible to estimate the effects of velocity,

concentration, motile microorganism density,

and temperature profiles on a number of

physical parameters, including the Brownian

motion, magnetic and Prandtl number, the

thermophoresis parameter, the Lewis number,

and the Peclet number.

 The controlling dimensionless

equations are integrated using the MATLAB

built-in program’s bvp4c technique to attain the

solution.

 The shooting method of Runge Kutta's

fourth order is used to do numerical

computations for the group of equations

expressed in coupled and nonlinear forms.

 While Bingham number and

permeability of porous space show a decreasing

change in velocity, R-P fluid parameter

increases fluid velocity.

 The motile microorganisms' density

increased due to their varied variety of

Pe

and

t

N

 In the absence of

Kr

, the concentration

distribution is increased by the rising value of

t

N

.

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DOI:10.37394/232025.2023.5.8

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 The Lewis and Prandtl number, along

with the concentration relaxation parameter

reveal fading results, although the nanofluids

concentration is increased by the

thermophoresis constants and activation

energy.

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DOI:10.37394/232025.2023.5.8

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E-ISSN: 2692-5079

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Policy)

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problem to the final findings and solution.

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Scientific Article or Scientific Article Itself

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that are relevant to the content of this article.

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DOI:10.37394/232025.2023.5.8

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E-ISSN: 2692-5079

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