Thermo-bioconvection flow of Walter's B nanofluid over a Riga plate involving swimming motile microorganisms

M. S. Alqarni; M. S. Alqarni

doi:10.3934/math.2022886

AIMS Mathematics

2022, Volume 7, Issue 9: 16231-16248. doi: 10.3934/math.2022886

Previous Article Next Article

Research article Special Issues

Thermo-bioconvection flow of Walter's B nanofluid over a Riga plate involving swimming motile microorganisms

M. S. Alqarni ^,

Department of Mathematics, College of Science, King Khalid University, Abha 61413, Saudi Arabia

Received: 19 March 2022 Revised: 16 May 2022 Accepted: 27 May 2022 Published: 04 July 2022
MSC : 35Qxx, 76Dxx, 76Mxx

The novelty of the current paper is to study the bioconvection effects in Walter's B nanofluid flow due to stretchable surface, which leads to important properties, i.e., thermal radiation, activation energy, motile microorganisms and convective boundary constraints. The considered analysis is explained via partial differential equations (PDEs), which are first embedded into the dimensionless system of nonlinear ODEs through suitable transformations. The governing equations are solved in MATLAB using the bvp4c solver. The impact of interesting parameters on the velocity field, thermal field, concentration of species and concentration of microorganisms is exhibited in graphical and tabular forms. The velocity field increases for higher estimations of the modified Hartmann and mixed convection parameters. The thermal field decays for a higher magnitude of the Prandtl number, while it is enhanced for a larger deviation of the thermal conductivity parameter. The volumetric concentration of nanoparticles enhances the larger activation energy and thermophoresis parameters. The microorganism concentration diminishes for higher Peclet number. The current model is more useful in various fields such as tissue engineering, recombinant proteins, synthetic biology, and biofuel cell and drug delivery devices.

Keywords:

Walter's B nanofluid,
Riga plate,
activation energy,
variable thermal conduction and concentration diffusion,
thermal radiation,
swimming motile microorganisms,
bioconvection,
numerical solution

Citation: M. S. Alqarni. Thermo-bioconvection flow of Walter's B nanofluid over a Riga plate involving swimming motile microorganisms[J]. AIMS Mathematics, 2022, 7(9): 16231-16248. doi: 10.3934/math.2022886

Related Papers:

[1]	Madeeha Tahir, Ayesha Naz, Muhammad Imran, Hasan Waqas, Ali Akgül, Hussein Shanak, Rabab Jarrar, Jihad Asad . Activation energy impact on unsteady Bio-convection nanomaterial flow over porous surface. AIMS Mathematics, 2022, 7(11): 19822-19845. doi: 10.3934/math.20221086
[2]	Latifa I. Khayyat, Abdullah A. Abdullah . The onset of Marangoni bio-thermal convection in a layer of fluid containing gyrotactic microorganisms. AIMS Mathematics, 2021, 6(12): 13552-13565. doi: 10.3934/math.2021787
[3]	Abdelraheem M. Aly, Abd-Allah Hyder . Fractional-time derivative in ISPH method to simulate bioconvection flow of a rotated star in a hexagonal porous cavity. AIMS Mathematics, 2023, 8(12): 31050-31069. doi: 10.3934/math.20231589
[4]	Nadeem Abbas, Wasfi Shatanawi, Taqi A. M. Shatnawi . Innovation of prescribe conditions for radiative Casson micropolar hybrid nanofluid flow with inclined MHD over a stretching sheet/cylinder. AIMS Mathematics, 2025, 10(2): 3561-3580. doi: 10.3934/math.2025164
[5]	Fu Zhang Wang, Muhammad Sohail, Umar Nazir, Emad Mahrous Awwad, Mohamed Sharaf . Utilization of the Crank-Nicolson technique to investigate thermal enhancement in 3D convective Walter-B fluid by inserting tiny nanoparticles on a circular cylinder. AIMS Mathematics, 2024, 9(4): 9059-9090. doi: 10.3934/math.2024441
[6]	Imran Siddique, Yasir Khan, Muhammad Nadeem, Jan Awrejcewicz, Muhammad Bilal . Significance of heat transfer for second-grade fuzzy hybrid nanofluid flow over a stretching/shrinking Riga wedge. AIMS Mathematics, 2023, 8(1): 295-316. doi: 10.3934/math.2023014
[7]	Muhammad Imran Asjad, Muhammad Haris Butt, Muhammad Armaghan Sadiq, Muhammad Danish Ikram, Fahd Jarad . Unsteady Casson fluid flow over a vertical surface with fractional bioconvection. AIMS Mathematics, 2022, 7(5): 8112-8126. doi: 10.3934/math.2022451
[8]	Nadeem Abbas, Wasfi Shatanawi, Fady Hasan, Taqi A. M. Shatnawi . Numerical analysis of Darcy resistant Sutterby nanofluid flow with effect of radiation and chemical reaction over stretching cylinder: induced magnetic field. AIMS Mathematics, 2023, 8(5): 11202-11220. doi: 10.3934/math.2023567
[9]	Haifaa Alrihieli, Musaad S. Aldhabani, Ghadeer M. Surrati . Enhancing the characteristics of MHD squeezed Maxwell nanofluids via viscous dissipation impact. AIMS Mathematics, 2023, 8(8): 18948-18963. doi: 10.3934/math.2023965
[10]	Asifa, Poom Kumam, Talha Anwar, Zahir Shah, Wiboonsak Watthayu . Analysis and modeling of fractional electro-osmotic ramped flow of chemically reactive and heat absorptive/generative Walters'B fluid with ramped heat and mass transfer rates. AIMS Mathematics, 2021, 6(6): 5942-5976. doi: 10.3934/math.2021352

Abstract

Nomenclature

$u, v$	Velocity-components $\left[m.{s}^{-1}\right]$	$S{n}_{x}$	Microorganism density number
$x, y$	Space-coordinate $\left[m\right]$	${N}_{\infty }$	Ambient-microorganism
${B}_{0}$	Magnetic-field strength $\left[N.{m}^{-1}.{A}^{-1}\right]$	${T}_{\infty }$	Ambient-temperature $\left[K\right]$
${U}_{w}\left(x\right)$	Stretching velocity $\left[m.{s}^{-1}\right]$	${C}_{\infty }$	Ambient-concentration
${\left(\rho c\right)}_{p}$	Nanoparticles specific-heat $\left[J.{kg}^{-3}.{K}^{-1}\right]$	${D}_{B}$	Brownian-diffusion coefficient $\left[{m}^{2}.{s}^{-1}\right]$
${\left(\rho c\right)}_{f}$	Thermal capacity of the fluid $\left[J.{kg}^{-3}.{K}^{-1}\right]$	${D}_{m}$	Microorganism-diffusion coefficient $\left[{m}^{2}.{s}^{-1}\right]$
$T$ ${E}_{a}$ ${\sigma }^{**}$ ${\rho }_{p}$	Temperature $\left[K\right]$ Coefficient of activation energy Stefan-Boltzmann constant Density of nanoparticles	${D}_{T}$ $K$ ${k}_{1}$ ${\rho }_{m}$	Thermophoresis-diffusion coefficient $\left[{m}^{2}.{s}^{-1}\right]$ Thermal conductivity Mean-absorption coefficient Microorganism density
${J}_{0}$ ${h}_{f}$ ${M}_{0}$	Density of the current for electrodes Coefficient of heat transfer Magnetization of the permanent magnets of Riga plate	$\nu$ $c$ $a$	Kinematic-viscosity $\left[{m}^{2}.{s}^{-1}\right]$ Stretching rate Distance across the magnets electrodes of Riga plate
$Nc$	Bioconvective Rayleigh-number	$b$	Chemotaxis-constant $\left[m\right]$
$Nr$	Buoyancy-ratio number	$n$	Fitted-rate constant
$Nb$	Brownian-movement number	$\Omega$	Microorganism-difference number
${B}^{*}$	Mixed-convective parameter	$\sigma$	Electrical conductivity $\left[S.{m}^{-1}\right]$
$Nt$	Thermophoretic parameter	$\alpha$	Thermal-diffusivity $\left[{m}^{2}.{s}^{-1}\right]$
$E$	Activation-energy number	${\sigma }^{*}$	Chemical-reaction number
$Le$	Lewis parameter	${W}_{c}$	Cell-swimming-speed $\left[m.{s}^{-1}\right]$
$We$	Weissenberg parameter	$\rho$	Density $\left[kg.{m}^{-3}\right]$
$Pe$	Peclet-parameter	${N}_{w}$	Microorganism at wall
$Lb$	Bioconvective Lewis parameter	${T}_{f}$	Hot fluid temperature $\left[K\right]$
$N{u}_{x}$	Nusselt number	$N$	Microorganism
${C}_{f}$	Skin friction	$K{r}^{2}$	Chemical-reaction-constant
$S{u}_{x}$ $\beta$ $\delta$ $Bi$ $Pr$	Sherwood number Dimensionless parameter Heat generation/absorption parameter Thermal-Biot number Prandtl number	$M$ $S$ $Q$ $Rd$ ${\delta }_{0}$	Magnetic parameter Velocity ratio number Modified Hartmann number Thermal radiation parameter Temperature-difference number

1. Introduction

The description of non-Newtonian fluids has fascinated the interest of various authors due to their wide variety of practical uses in various fields. Walter's B liquid is subclass of non-Newtonian liquids with several manufacturing process applications in industry sectors such as chemical science, biosystems, biophysical techniques of thermal conduction in tissues, chemical manufacturing, and bioengineering. Walters ^[1] was a pioneer of the Walters'-B liquid relation. Nandepanavar et al. ^[2] discussed the impact of nanomaterials in Walters'-B liquid. The effect of nanoparticles in the Walters-B fluid was investigated by Nadeem et al. ^[3]. The effect of the radiative-Walters-B liquid flow and stretched wedge was investigated by Hayat et al. ^[4]. In addition, several scientists have applied the Walters-B relation, which shows the phenomenon of different polymer flows (see ^{[5,6,7,8,9,10]}).

A nanofluid is a fluid that consists of a base fluid with nanosized particles (1–100 nm) suspended in it. Thus, Choi and Eastman ^[11] discussed the fundamental idea of nanomaterials with enhanced thermophysical aspects. Buongiorno ^[12] explained seven-slipping procedures in the movement of nanomaterials, including thermophoresis and Brownian diffusion. Mixed convective nanoliquid transport with the impact of a magnetic field was observed by Hsiao ^[13]. Rashidi et al. ^[14] numerically premeditated the convection characteristics of a nanoliquid through a nonlinear isothermal stretching sheet. Sheikholeslami and Bhatti ^[15] identified the impact of nanomaterials on the transport of nanoliquids via forced-convection through gravitational-force. Turkyilmazoglu ^[16] analyzed nanomaterial transport by vertical surfaces. Ellahi et al. ^[17] noted Jeffrey MHD nanoliquid motion between two parallel-disks. Pantokratoras et al. ^[18] discussed mixed convective fluids with nanoparticles. Rashid et al. ^[19] deliberated the behavior of activation energy in the MHD flow of Maxwell nanoliquid. Tayebi et al. ^[20] explored the magnetic hydrodynamic heat-transport of nanoliquids when wave conduction rings occurred. Hayat et al. ^[21] discussed the third grade magnetohydrodynamic nanoliquid flow with activation-energy. Muhammad et al. ^[22] explored the 3D radiative Eyring-Powell nanoliquid transport in the presence of activation-energy by a Riga plate. The transport of second-order slip nanoliquids under the effect of the Stefan blow was examined by Alamri et al. ^[23]. Khan et al. ^[24] discussed the stratification and heat generation in mixed convective Prandtl liquid flow. Anwar et al. ^[25] observed the nonlinear radiative heat transport with MHD nanoliquid spray. Gailitis ^[26] developed a Riga electromagnetic surface, which comprised clearly constructed electrode and magnet sets. Ahmad et al. ^[27] investigated the impact of nanoliquid flow through Riga plate. The features of microorganisms in nanoliquids across Riga plate were analyzed by Iqbal et al. ^[28]. Recent works have experimentally explored nanofluids in ^{[29,30,31,32,33,34,35]}.

Bioconvection resulting from the combined density gradient of the microorganism simulates macroscopic fluid convection movements. The presence of these self-contained motile microorganisms enhances the primary-density of the fluid through swimming. This important thought certainly escorts to a delicate low-density surface. There are various distinct and related properties of nanoparticles and motile microorganisms. Immunology-microsystems such as enzyme biomaterials usually include bioconvection technologies. Therefore, Kuznetsov ^[36] recommended that nano-organisms should be involved in the development of bio microsystems, where they played a significant role in the dispensation of mass transport. Li et al. ^[37] studied the bioconvective flow of second- grade nanoliquids because of Wu's slipping. Muhammad et al. ^[38] analyzed the influence of bioconvection in Carreau nanoliquid under slip formed by a wedge. Khan et al. ^[39] examined the 2-D couple stress nanoliquid transport with magnetic-field and gyrotactic motile microorganisms. The characteristics of activation energy in radiative nanoliquid flow with bioconvection features through shrinking/stretching disks were addressed by Zhang et al. ^[40]. Thermal radiative Oldroyd-B nanoliquid flow subject to motile microorganisms by rotating disk was examined by Waqas et al. ^[41]. Khan et al. ^[42] examined the bioconvective aspect between stretchable moving-disks subject to entropy generation and nanofluids. Mamatha et al. ^[43] discussed the movement of magneto-hydrodynamic liquid by a stretched surface. Ferdows et al. ^[44] observed the heat and mass transportation of viscous liquid via cylinder with motile microorganisms. Amirsom et al. ^[45] addressed the 3-D motion of bioconvection nanoliquids consisting of gyrotactic microorganisms. Kasaragadda et al. ^[46] illustrated the consequence of strong hydrophobic surfaces on the recognition of structures of nanoparticle-reinforced biomaterials. Ansari et al. ^[47] explored the significance of motile microorganisms and biomaterials on bioconvective Casson liquid flow by a nonlinear extended boundary. Recent work on bioconvection is given in investigations ^{[48,49,50,51,52,53,54,55]}.

In this paper, we generalize the analysis of ^[56] in four directions. First, we model the flow-analysis in the presence of nanoparticles. Attention is mainly given to Brownian and thermophoretic diffusion. Second, we consider the swimming gyrotactic motile microorganisms. Third, we analyzed the concentration and temperature in the presence of variable thermal-conductivity and concentration-diffusion. Fourth, we develop the numerical solution using the MATLAB bvp4c solver, which follows the Lobatto-IIIa formula. The consequences of parameters of interest versus the flow field are presented through graphs and tabular data.

2. Mathematical modeling

The two-dimensional magnetohydrodynamic flow of Walter's B nanoliquid containing gyrotactic motile microorganisms configured by a Riga plate in the occurrence of variable thermal conduction and concentration diffusion is considered. Features of heat generation/absorption and Arrhenius activation energy are also accounted for in the considered flow problems. The movement of fluid is caused by a stretchable surface. The features of thermal radiation are employed. The significance of Brownian and thermophoresis movements is considered. The physical flow configuration is depicted in Figure 1.

Figure 1. Physical description of the current problem.

DownLoad: Full-Size Img PowerPoint

Based on these assumptions, the governing expressions and boundary constraints are given as ^[1,28,55]:

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

(1)

$u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y} = {U}_{e}\frac{d{U}_{e}}{dx}+v\frac{{\partial }^{2}u}{\partial {y}^{2}}-\frac{{k}_{0}}{\rho }\left[u\frac{{\partial }^{3}u}{\partial x\partial {y}^{2}}+v\frac{{\partial }^{3}u}{\partial {y}^{3}}+\frac{\partial u}{\partial x}\frac{{\partial }^{2}u}{\partial {y}^{2}}-\frac{\partial u}{\partial y}\frac{{\partial }^{2}u}{\partial x\partial y}\right]-\frac{\sigma {B}_{0}^{2}}{\rho }\left(u-{u}_{e}\right)\\+\frac{\pi {j}_{0}{M}_{0}}{8\rho }\mathit{exp}\left(-\frac{\pi }{a}y\right)+\frac{1}{{\rho }_{f}}\left[\begin{array}{l}\left(1-{C}_{\infty }\right){\rho }_{f}{\beta }^{*}g\left(T-{T}_{\infty }\right)-\left({\rho }_{p}-{\rho }_{f}\right)g\left(C-{C}_{\infty }\right)\\ -\left(N-{N}_{\infty }\right)g{\gamma }^{*}\left({\rho }_{m}-{\rho }_{f}\right)\end{array}\right],$

(2)

$u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y} = \frac{k}{\rho {c}_{p}}\left(\frac{16{\sigma }^{**}{T}_{\infty }^{3}}{3{k}_{1}}\right)\frac{{\partial }^{2}T}{\partial {y}^{2}}+\frac{1}{{\left(\rho c\right)}_{f}}\frac{\partial }{\partial y}\left[K\left(T\right)\frac{\partial T}{\partial y}\right]\\+\frac{{Q}_{0}}{\rho {c}_{p}}\left(T-{T}_{\infty }\right)+\tau \left\{{D}_{B}\left(\frac{\partial T}{\partial y}\frac{\partial C}{\partial y}\right)+\frac{{D}_{T}}{{T}_{\infty }}{\left(\frac{\partial T}{\partial y}\right)}^{2}\right\},$

(3)

$u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial y} = \frac{\partial }{\partial y}\left[D\left(C\right)\frac{\partial C}{\partial y}\right]+{D}_{B}\frac{{\partial }^{2}C}{\partial {y}^{2}}+\frac{{D}_{T}}{{T}_{\infty }}\frac{{\partial }^{2}T}{\partial {y}^{2}}-K{r}^{2}\left(C-{C}_{\infty }\right){\left(\frac{T}{{T}_{\infty }}\right)}^{n}\mathit{exp}\left(-\frac{{E}_{a}}{kT}\right),$

(4)

$u\frac{\partial N}{\partial x}+v\frac{\partial N}{\partial y}+\frac{b{W}_{c}}{\left({C}_{f}-{C}_{\infty }\right)}\left[\frac{\partial }{\partial y}\left(N\frac{\partial C}{\partial y}\right)\right] = {D}_{m}\left(\frac{{\partial }^{2}N}{\partial {y}^{2}}\right).$

(5)

In expressions (3) and (4), the variable thermal conduction and concentration diffusion are ^[55]:

$K\left(T\right) = {k}_{\infty }\left[1+{\lambda }_{1}\frac{T-{T}_{\infty }}{{T}_{f}-{T}_{\infty }}\right] \text{ and } D\left(C\right) = {D}_{\infty }\left[1+{\lambda }_{2}\frac{C-{C}_{\infty }}{{C}_{\infty }}\right] .$

(6)

The boundary restrictions of the current flow are given by ^[28,50]:

$u\left(x, 0\right) = {U}_{w}\left(x\right) = cx, v\left(x, 0\right) = 0, -K\frac{\partial T}{\partial y} = {h}_{f}\left({T}_{f}-{T}_{\infty }\right), {D}_{B}\frac{\partial C}{\partial y}+\frac{{D}_{T}}{{T}_{\infty }}\frac{\partial T}{\partial y} = 0, N = {N}_{w}aty = 0,$

$u\to {U}_{e}\left(x\right) = cx, T\to {T}_{\infty }, C\to {C}_{\infty }, N\to {N}_{\infty }asy\to \infty .$

(7)

The suitable transformations are ^[28,50]:

$\left.\begin{array}{c}u\left(x, y\right) = cx{f}^{{'}}\left(\zeta \right), v\left(x, y\right) = -\sqrt{c\nu }f\left(\zeta \right), \\ \zeta = y\sqrt{\frac{c}{\nu }}, \theta = \frac{T-{T}_{\infty }}{{T}_{f}-{T}_{\infty }}, \phi = \frac{C-{C}_{\infty }}{{C}_{\infty }}, \chi = \frac{N-{N}_{\infty }}{{N}_{w}-{N}_{\infty }}.\end{array}\right\}$

(8)

By implementing these transformations, governing partial differential equations are transformed into dimensionless ordinary differential equations i.e., ^[1,28,50,55]:

${f}^{‴}+{S}^{2}-{\left({f}^{{'}}\right)}^{2}+f{f}^{″}-We\left[2{f}^{{'}}{f}^{‴}-f{f}^{iv}-{\left({f}^{″}\right)}^{2}\right]+Q\mathit{exp}\left(-\beta \zeta \right)-M{f}^{{'}}+MS+\\{B}^{*}\left(\theta -Nr\phi -Nc\chi \right) = 0,$

(9)

$\left(\left(1+Rd{\left(1+\left({\theta }_{f}-1\right)\theta \right)}^{3}\right){\theta }^{{'}}+{\lambda }_{1}\theta \right){\theta }^{″}+Prf{\theta }^{{'}}+\mathit{Pr}\delta \theta +Pr\left(Nb{\theta }^{{'}}{\phi }^{{'}}+Nt{\theta }^{{'}2}\right) = 0,$

(10)

$\left(1+{\lambda }_{2}\phi \right){\phi }^{″}+LePrf{\phi }^{{'}}+\frac{Nt}{Nb}{\theta }^{″}-PrLe{\sigma }^{*}{\left[1+{\delta }_{0}\theta \right]}^{n}\mathit{exp}\left[\frac{-E}{1+{\delta }_{0}\theta }\right]\phi = 0,$

(11)

$\chi {'}{'}+Lb\left(f{\chi }^{{'}}\right)-Pe\left[\phi {'}{'}\left(\chi +\Omega \right)+\chi {'}\phi {'}\right] = 0.$

(12)

The dimensionless boundary restrictions are ^[28,50]:

${f}^{{'}}\left(0\right) = 1, f\left(0\right) = 0, {\theta }^{{'}}\left(0\right) = -Bi\left(1-\theta \left(0\right)\right), \\Nb{\phi }^{{'}}\left(0\right)+Nt{\theta }^{{'}}\left(0\right) = 0, \chi \left(0\right) = 1,$

${f}^{{'}}\left(\infty \right)\to S, \theta \left(\infty \right)\to 0, \phi \left(\infty \right)\to 0, \chi \left(\infty \right)\to 0.$

(13)

The physical flow parameters in equations (9)-(13) are expressed by

$\left.\begin{array}{l}M\left( = \sqrt{\frac{\sigma {\beta }_{0}^{2}}{\rho c}}\right), Nr\left( = \frac{\left({\rho }_{p}-{\rho }_{f}\right)\left({C}_{\infty }\right)}{\left(1-{C}_{\infty }\right)\left({T}_{f}-{T}_{\infty }\right)}\right), S\left( = \frac{b}{c}\right), We\left( = \frac{{k}_{0}c}{{\mu }_{0}}\right), \beta \left( = \sqrt{\frac{{\pi }^{2}v}{{a}^{2}c}}\right), \\ Q\left( = \frac{\pi {j}_{0}{M}_{0}}{8\rho {U}_{w}^{2}}\right), \mathit{Pr}\left( = \frac{\rho {c}_{p}}{{k}_{\infty }}\right), {B}^{*}\left( = \frac{\left(1-{C}_{\infty }\right)\gamma \left({T}_{f}-{T}_{\infty }\right){\beta }^{**}}{c{U}_{w}}\right), Nc\left( = \frac{\gamma g\left({\rho }_{m}-{\rho }_{f}\right)\left({N}_{w}-{N}_{\infty }\right)}{\left(1-{C}_{\infty }\right)\left({T}_{f}-{T}_{\infty }\right){\beta }^{**}}\right), \\ \mathit{Nb}\left( = \frac{\tau {D}_{B}\left({C}_{\infty }\right)}{\nu }\right), Rd\left( = \frac{4{\sigma }^{**}{T}_{\infty }^{3}}{{k}_{1}{k}_{\infty }}\right), \mathit{Nt}\left( = \frac{\tau {D}_{T}\left({T}_{f}-{T}_{\infty }\right)}{\nu }\right), \delta \left( = \frac{{Q}_{0}}{\rho {c}_{p}}\right), \\ E\left( = \frac{{E}_{a}}{k{T}_{\infty }}\right), {\delta }_{0}\left( = \frac{{T}_{f}-{T}_{\infty }}{{T}_{\infty }}\right), {\sigma }^{*}\left( = \frac{\nu K{r}^{2}}{c}\right), Pe\left( = \frac{b{W}_{c}}{{D}_{m}}\right), Lb\left( = \frac{\nu }{{D}_{m}}\right), \Omega \left( = \frac{{N}_{\infty }}{{N}_{w}-{N}_{\infty }}\right), \\ Bi\left( = \frac{{h}_{f}}{{k}_{\infty }}\sqrt{\frac{\nu }{c}}\right).\end{array}\right\}$

(14)

The quantities of interest, e.g., skin coefficient ${C}_{f}$ , Nusselt number $N{u}_{x}$ , Sherwood number $S{u}_{x}$ and microorganism number $S{n}_{x}$ are ^[27,28,56]:

${C}_{f}\left( = \frac{{\tau }_{w}}{\rho {U}_{w}^{2}}\right), N{u}_{x}\left( = \frac{x{q}_{w}}{K\left({T}_{w}-{T}_{\infty }\right)}\right), S{u}_{x} = \frac{x{J}_{w}}{{D}_{B}\left({C}_{w}-{C}_{\infty }\right)}, S{n}_{x} = \frac{x{J}_{n}}{{D}_{m}\left({N}_{w}-{N}_{\infty }\right)},$

(15)

where

${\tau }_{w} = {\left[{\mu }_{0}\frac{\partial u}{\partial y}-{k}_{0}{\left(u\frac{{\partial }^{2}u}{\partial x\partial y}+v\frac{{\partial }^{2}u}{\partial {y}^{2}}+2\frac{\partial u}{\partial x}\frac{\partial u}{\partial y}\right)}^{3}\right]}_{y = 0},$

(16)

${q}_{w} = -K{\left(\frac{\partial T}{\partial y}\right)}_{y = 0}, {J}_{w} = -{D}_{B}{\left(\frac{\partial C}{\partial y}\right)}_{y = 0}, {J}_{n} = -{D}_{m}{\left(\frac{\partial N}{\partial y}\right)}_{y = 0}.$

(17)

Hence the dimensionless forms of the above physical quantities are:

${\left(R{e}_{x}\right)}^{\frac{1}{2}}{C}_{f} = \left(1-3We\right){f}^{″}\left(0\right),$

(18)

${\left(R{e}_{x}\right)}^{-\frac{1}{2}}N{u}_{x} = -{\theta }^{{'}\left(0\right)},$

(19)

$R{e}^{\frac{-1}{2}}S{u}_{x} = -{\phi }^{{'}}\left(0\right),$

(20)

$R{e}^{\frac{-1}{2}}S{n}_{x} = -{\chi }^{{'}}\left(0\right).$

(21)

Here ${\mathit{Re}}_{x}\left( = \frac{c{x}^{2}}{v}\right)$ is the local Reynolds number.

3. Numerical approach

The coupled nonlinear dimensionless system (9)–(12) with boundary constraints (13) is solved using the Matlab bvp4c scheme for different variations of important physical prominent parameters ^{[57,58,59,60,61]}. The bvp4c-function is a finite-difference code that follows the three steps of Lobatto-IIIa formula. Until the process begins, problems with higher-order boundary value are converted into an initial value problem by adding several new variables. Considering

$\left.\begin{array}{l}f = q, {f}^{{'}} = {q}_{1}, {f}^{″} = {q}_{2}, {f}^{‴} = {q}_{3}, {f}^{iv} = {q}_{3}^{{'}}, \theta = {q}_{4}, {\theta }^{{'}} = {q}_{5}, \\ {\theta }^{″} = {q}_{5}^{{'}}, \phi = {q}_{6}, {\phi }^{{'}} = {q}_{7}, {\phi }^{″} = {q}_{7}^{{'}}\chi = {q}_{8}, {\chi }^{{'}} = {q}_{9}, \chi = {q}_{9}^{{'}}\end{array}\right\},$

(22)

${q}_{3}^{{'}} = \frac{\begin{array}{c}-{q}_{3}-{S}^{2}+{\left({q}_{1}\right)}^{2}-q{q}_{2}+We\left[2{q}_{1}{q}_{3}-{\left({q}_{2}\right)}^{2}\right]-Q\mathit{exp}\left(-\beta \zeta \right)+M{q}_{2}-MS\\ +{B}^{*}\left({q}_{4}-Nr{q}_{6}-Nc{q}_{8}\right)\end{array}}{Weq},$

(23)

${q}_{5}^{{'}} = \frac{-\mathit{Pr}q{q}_{5}-\mathit{Pr}\delta {q}_{4}-Pr\left(Nb{q}_{5}{q}_{7}+Nt{q}_{5}^{2}\right)}{\left(\left(1+Rd{\left(1+\left({\theta }_{f}-1\right){q}_{4}\right)}^{3}\right){q}_{5}+{\lambda }_{1}{q}_{4}\right)},$

(24)

${q}_{7}^{{'}} = \frac{-Le\mathit{Pr}q{q}_{7}-\frac{Nt}{Nb}{q}_{5}^{{'}}+PrLe{\sigma }^{*}{\left[1+{\delta }_{0}{q}_{4}\right]}^{n}\mathit{exp}\left[\frac{-E}{1+{\delta }_{0}{q}_{4}}\right]{q}_{6}}{\left(1+{\lambda }_{2}{q}_{6}\right)},$

(25)

${q}_{9}^{{'}} = -Lb\left(q{q}_{9}\right)+Pe\left[{q}_{7}^{{'}}\left({q}_{8}+\Omega \right)+{q}_{9}{q}_{7}\right],$

(26)

with

${q}_{1}\left(0\right) = 1, q\left(0\right) = 0, {q}_{5} = -Bi\left(1-{q}_{4}\left(\zeta \right)\right), \\Nb{q}_{7}\left(\zeta \right)+Nt{q}_{5}\left(\zeta \right) = 0, {q}_{9}\left(0\right) = 1, at\zeta = 0,$

${q}_{1}\left(\infty \right) = S, {q}_{4}\to 0, {q}_{6}\to 0, {q}_{8}\to 0as\zeta \to \infty .$

(27)

4. Results and discussion

Significant contributions of the prominent numbers against the velocity of the Walter's B nanoliquid, thermal, concentration and micro-organism profiles are graphically demonstrated and discussed as follows:

4.1. Velocity profile

is constructed to display the outcomes of $M$ and $Nc$ against velocity ${f}^{{'}}$ . Velocity field ${f}^{{'}}$ declines when $M$ and $Nc$ increase. The resistive forces are upgraded with the augmentation of $M$ ; hence, the velocity profile reduces. Basically, the magnetic number involves the Lorentz force. This resistive type of force is responsible for the decay in fluid velocity ^[39]. predicts the impacts of $We$ and $Nr$ against velocity profile ${f}^{{'}}$ . With increasing estimations of $We$ , the velocity ${f}^{{'}}$ diminishes. Similarly, velocity ${f}^{{'}}$ shows a decreasing behavior for $Nr$ . The modified Hartmann parameter $Q$ and mixed convective number ${B}^{*}$ affect the velocity of Walter's B nanofluid ${f}^{{'}}$ past a Riga plate, as clarified in . Both parameters have similar effects on velocity field ${f}^{{'}}$ . shows the impact of velocity ratio parameter $S$ versus velocity field ${f}^{{'}}$ , where there is boundary layer for both $S < 0$ and $S > 0$ . Velocity ${f}^{{'}}$ increases for larger velocity ratio parameter $S$ . A larger velocity ratio number increases the velocity of the fluid. When the velocity ratio is zero, the usual profile at the free stream is obtained far from the plate.

Figure 2. (a) Outcomes of

${f}^{{'}}$ against

$M\& Nc$ ; (b) Outcomes of

${f}^{{'}}$ against

$We\& Nr$ ; (c) Outcomes of

${f}^{{'}}$ against

$Q\& {B}^{*}$ ; (d) Outcomes of

${f}^{{'}}$ against

$S$ .

DownLoad: Full-Size Img PowerPoint

4.2. Thermal profile

anticipates the inspiration of the Biot number $Bi$ and thermal conductivity parameter ${\lambda }_{1}$ across the thermal field $\theta$ . Temperature $\theta$ increases for larger estimations of thermal Biot number $Bi$ . Basically, a larger Biot number corresponds to more heat provided to the working fluid, which leads to a stronger temperature field ^[30]. Thermal function $\theta$ is enhanced to increase the thermal conductivity ${\lambda }_{1}$ . depicts the contributions of $Pr$ and ${\theta }_{f}$ to the temperature of species $\theta$ . The thermal profile of species $\theta$ in Walter's B nanoliquid decreases with increasing estimations of Prandtl number $Pr$ , while it increases for a larger temperature ratio parameter ${\theta }_{f}$ .

Figure 3. (a) Outcomes of

$\theta$ against

${B}_{i}\& {\lambda }_{1}$ ; (b) Outcomes of

$\theta$ against

$\mathit{Pr}\& {\theta }_{f}$ .

DownLoad: Full-Size Img PowerPoint

4.3. Nanoparticles concentration profile

characterizes the impact of $E$ and ${\lambda }_{2}$ against the nanoparticle concentration $\phi$ . The nanoparticle concentration $\phi$ increases when $E$ increases. Furthermore, $\phi$ grows with larger concentration diffusivity ${\lambda }_{2}$ . reveals that $Nt$ and $Nb$ increase with the concentration of species $\phi$ . Here, the solutal profile of species $\phi$ decreases with increasing Brownian motion parameter $Nb$ , while it increases with increasing thermophoresis parameter $Nt$ . The natures of $Pr$ and $Le$ against the concentration field of species $\phi$ are examined in . Here, $\phi$ decreases with increasing $Pr$ and $Le$ .

Figure 4. (a) Outcomes of

$\phi$ against

$E\& {\lambda }_{2}$ ; (b) Outcomes of

$\phi$ against

$Nb\& Nt$ ; (c) Outcomes of

$\phi$ against

$Pr\& Le$ .

DownLoad: Full-Size Img PowerPoint

4.4. Microorganism profile

examines the outcome of bioconvection Lewis parameter $Lb$ and Peclet parameter $Pe$ versus the microorganism field $\chi$ . The microorganism field $\chi$ decreases with higher estimations of both parameters $Lb$ and $Pe$ .

Figure 5. Outcomes of

$\chi$ against

$Pe\& Lb$ .

DownLoad: Full-Size Img PowerPoint

4.5. Tabular data

– are captured to check the behaviors of skin friction $-{f}^{″}\left(0\right)$ , Nusselt $-{\theta }^{{'}}\left(0\right)$ , Sherwood $-{\phi }^{{'}}\left(0\right)$ and microorganism $-{\chi }^{{'}}\left(0\right)$ numbers versus various interesting parameters. displays the contribution of $-{f}^{″}\left(0\right)$ versus ${B}^{*}$ , $Q$ , $M$ , $Nr$ and $Nc$ . Here $-{f}^{″}\left(0\right)$ decays by augmenting estimations of ${B}^{*}$ , $Q$ and $M$ . compares ${f}^{″}\left(0\right)$ for varying ${B}^{*}$ with Ahmad et al. ^[27]. Here, we found agreement between the presented bvp4c solution and the shooting solution in Ahmad et al. ^[27] in the limiting case. shows the contribution of $-{\theta }^{{'}}\left(0\right)$ versus $Pr$ , $Nb$ , $Nt$ , $Le$ , $M$ , $Bi$ , ${B}^{*}$ , $Nr$ and $Nc$ . We note that $-{\theta }^{{'}}\left(0\right)$ increases as $Nb$ and $Pr$ are increased. In , $-{\phi }^{{'}}\left(0\right)$ decreases for $Le$ and increases for larger $Nt$ . shows that $-{\chi }^{{'}}\left(0\right)$ decreases for $Pe$ and $Lb$ .

Table 1. Comparative outcomes of

$-{f}^{″}\left(0\right)$ for variations of

${B}^{*}$ ,

$Q$ ,

$M$ ,

$Nr$ and

$Nc$ .

${B}^{*}$	$Q$	$M$	$Nr$	$Nc$	$-{f}^{″}\left(0\right)$
0.1 0.8 1.6	2.0	0.5	0.1	0.2	0.6271 0.5301 0.4249
0.2	1.0 1.6 2.2	0.5	0.2	0.2	0.6243 0.6176 0.6059
0.2	2.0	0.1 0.6 1.2	0.2	0.2	0.4912 0.6384 0.7657
0.2	2.0	0.5	0.1 1.0 2.0	0.2	0.6123 0.6174 0.6243
0.2	2.0	0.5	0.1	0.1 1.0 2.0	0.6098 0.6360 0.6697

| Show Table

DownLoad: CSV

Table 2. Outcomes of

${f}^{″}\left(0\right)$ for variations of

${B}^{*}$ when

$Q$ = 0.5,

$Pr$ = 5,

$\beta$ = 0.5,

$Bi$ = 50,

$Nt$ = 0.5,

$Nb$ = 0.5,

$Nr$ = 0.1,

$LePr$ = 0.5,

$S$ = 1 and

$We$ =

$Nc$ =

$M$ = 0.

${B}^{*}$	Ahmad et al. ^[27]	Present results
0 1 2 3 4	1.5394682 1.9023442 2.2416224 2.5631502 2.8705968	1.5394732 1.9023488 2.2416273 2.5631452 2.8706019

| Show Table

DownLoad: CSV

Table 3. Outcomes of

$-{\theta }^{{'}}\left(0\right)$ for variations of

$Pr$ ,

$Nb$ ,

$Nt$ ,

$Le$ ,

$M$ ,

$Bi$ ,

${B}^{*}$ ,

$Nr$ and

$Nc$ .

$Pr$	$Nb$	$Nt$	$Le$	$M$	$Bi$	${B}^{*}$	$Nr$	$Nc$	$-{\theta }^{{'}}\left(0\right)$
2 3 4	0.2	0.3	2.0	0.5	2.0	0.2	0.2	0.2	0.7709 0.8729 0.9450
1.2	0.1 0.5 1.0	0.3	2.0	0.5	2.0	0.2	0.2	0.2	0.6454 0.6456 0.6457
1.2	0.2	0.1 0.5 1.0	2.0	0.5	2.0	0.2	0.2	0.2	0.6518 0.6394 0.6240
1.2	0.2	0.3	1.0 1.8 2.6	0.5	2.0	0.2	0.2	0.2	0.6481 0.6460 0.6445
1.2	0.2	0.3	2.0	0.1 0.6 1.2	2.0	0.2	0.2	0.2	0.6566 0.6444 0.6336
1.2	0.2	0.3	2.0	0.5	0.3 1.0 1.8	0.2	0.2	0.2	0.2299 0.4898 0.6245
1.2	0.2	0.3	2.0	0.5	2.0	0.1 0.8 1.6	0.2	0.2	0.6456 0.6525 0.6594
1.2	0.2	0.3	2.0	0.5	2.0	0.2	0.1 1.0 2.0	0.2	0.6468 0.6460 0.6452
1.2	0.2	0.3	2.0	0.5	2.0	0.2	0.2	0.1 1.0 2.0	0.6468 0.6453 0.6434

| Show Table

DownLoad: CSV

Table 4. Outcomes of

$-{\phi }^{{'}}\left(0\right)$ for variations of

$Pr$ ,

$Nb$ ,

$Nt$ ,

$Le$ ,

$M$ ,

${B}^{*}$ ,

$Bi$ ,

$Nr$ and

$Nc$ .

$Pr$	$Nb$	$Nt$	$Le$	$M$	${B}^{*}$	$Bi$	$Nr$	$Nc$	$-{\phi }^{{'}}\left(0\right)$
2 3 4	0.2	0.3	2.0	0.5	2.0	0.5	0.2	0.2	1.1564 1.3094 1.4175
1.2	0.1 0.5 1.0	0.3	2.0	0.5	2.0	0.5	0.2	0.2	1.9363 0.3874 0.1937
1.2	0.2	0.1 0.5 1.0	2.0	0.5	2.0	0.5	0.2	0.2	0.3259 1.5984 3.1199
1.2	0.2	0.3	1.0 1.8 2.6	0.5	2.0	0.5	0.2	0.2	0.9721 0.9690 0.9668
1.2	0.2	0.3	2.0	0.1 0.6 1.2	2.0	0.5	0.2	0.2	0.9849 0.9585 0.9503
1.2	0.2	0.3	2.0	0.5	0.1 0.8 1.6	0.5	0.2	0.2	0.9684 0.9788 0.9892
1.2	0.2	0.3	2.0	0.5	2.0	0.1 0.5 1.0	0.2	0.2	0.3434 0.7346 0.9367
1.2	0.2	0.3	2.0	0.5	2.0	0.5	0.1 0.8 1.6	0.2	0.9701 0.9691 0.9679
1.2	0.2	0.3	2.0	0.5	2.0	0.5	0.2	0.1 1.0 2.0	0.9703 0.9679 0.9652

| Show Table

DownLoad: CSV

Table 5. Outcomes of

$-{\chi }^{{'}}\left(0\right)$ for variations of

$Pe$ ,

$Lb$ ,

$M$ ,

$Nr$ and

$Nc$ .

$Pe$	$Lb$	$M$	$Nr$	$Nc$	$-{\chi }^{{'}}\left(0\right)$
0.1 1.0 2.0	2.0	0.5	0.1	0.2	1.0873 1.7010 2.4250
0.6	1.0 1.6 2.2	0.5	0.2	0.2	1.1303 1.3814 1.1546
0.6	2.0	0.1 0.6 1.2	0.2	0.2	1.4608 1.4302 1.4029
0.6	2.0	0.5	0.1 1.0 2.0	0.2	1.4358 1.4340 1.4320
0.6	2.0	0.5	0.1	0.1 1.0 2.0	1.4360 1.4318 1.4270

| Show Table

DownLoad: CSV

5. Conclusions

The current work discusses the bioconvection nonlinear flow of Walter's B nanoliquid over a Riga plate with variable thermal conduction and concentration diffusion features. The aspects of thermal radiation and activation energy are considered. First, the coupled partial differential equations are embedded into the dimensionless system of nonlinear ODEs through suitable transformations ^{[62,63,64,65]}. The achieved dimensionless system of ODEs is solved by using the MATLAB built-in bvp4c solver, which follows the Lobatto-IIIa formula. The main results of the considered problem are summarized below.

● Velocity has an enlarging effect for higher estimations of mixed convection and modified Hartmann numbers.

● The temperature of Walter's B nanoliquid increases via larger estimations of Biot number and thermal conductivity parameter.

● The nanoparticle concentration is improved for larger concentration diffusion and activation energy parameters.

● Microorganism fields have similar features for both Peclet and microorganism Lewis parameters.

● The current bioconvective nonlinear Walter's B nanoliquid model has a significant role in the power generation, medical sciences, energy manufacturing, metallurgical industry, thermal recovery of oil, heat storage devices, etc. ^{[66,67,68,69,70]}.

Acknowledgment

The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University, Abha, Saudi Arabia for funding this work through Large Groups Project under grant number RGP.2/206/43.

Conflict of interest

The author declares no conflict of interest.

References

[1]	K. Walters, Non-newtonian effects in some elastic-viscous liquids whose behaviour at small rates of shear is characterized by a general linear equation of state, Quart. J. Mech. Appl. Math., 15 (1962), 63-76. https://doi.org/10.1093/qjmam/15.1.63 doi: 10.1093/qjmam/15.1.63
[2]	M. M. Nandeppanavar, M. S. Abel, J. Tawade, Heat transfer in a Walter's liquid B fluid over an impermeable stretching sheet with non-uniform heat source/sink and elastic deformation, Commun. Nonlinear Sci. Numer. Simul., 15 (2010), 1791-1802. https://doi.org/10.1016/j.cnsns.2009.07.009 doi: 10.1016/j.cnsns.2009.07.009
[3]	S. Nadeem, R. Mehmood, S. S. Motsa, Numerical investigation on MHD oblique flow of a Walter's B type nano fluid over a convective surface, Int. J. Therm. Sci., 92 (2015), 162-172. https://doi.org/10.1016/j.ijthermalsci.2015.01.034
[4]	T. Hayat, S. Qayyum, M. Imtiaz, A. Alsaedi, Radiative Falkner-Skan flow of Walter-B fluid with prescribed surface heat flux, J. Theor. Appl. Mech., 55 (2017), 117-127. https://doi.org/10.15632/jtam-pl.55.1.117 doi: 10.15632/jtam-pl.55.1.117
[5]	A. Majeed, T. Javed, S. Shami, Numerical analysis of Walters-B fluid flow and heat transfer over a stretching cylinder, Can. J. Phys., 94 (2016), 522-530. https://doi.org/10.1139/cjp-2015-0511 doi: 10.1139/cjp-2015-0511
[6]	T. Hayat, S. Asad, M. Mustafa, H. H. Alsulami, Heat transfer analysis in the flow of Walters' B fluid with a convective boundary condition, Chinese Phys. B, 23 (2014), 084701. https://doi.org/10.1088/1674-1056/23/8/084701 doi: 10.1088/1674-1056/23/8/084701
[7]	M. Ijaz, M. Yousaf, A. M. El Shafey, Arrhenius activation energy and Joule heating for Walter-B fluid with Cattaneo-Christov double-diffusion model, J. Therm. Anal. Calorim., 143 (2021), 3687-3698. https://doi.org/10.1007/s10973-020-09270-1
[8]	K. Loganathan, N. Nithyadevi, P. Boopathi, K. Mohana, Inquiry of inclined magnetic field effects on Walter-B nanofluid flow with heat generation/absorption, IOP Conf. Ser.: Mater. Sci. Eng., 872 (2020), 012097. https://doi.org/10.1088/1757-899X/872/1/012097 doi: 10.1088/1757-899X/872/1/012097
[9]	M. Mueller, O. A. Igbokwe, B. Walter, C. L. Pederson, S. Riechelmann, D. K. Richter, et al., Testing the preservation potential of early diagenetic dolomites as geochemical archives, Sedimentology, 67 (2020), 849-881. https://doi.org/10.1111/sed.12664
[10]	B. Meier, A. Schmidt, N. Glaser, A. Meining, B. Walter, A. Wannhoff, et al., Endoscopic full-thickness resection of gastric subepithelial tumors with the gFTRD-system: A prospective pilot study (RESET trial), Surg. Endosc., 34 (2020), 853-860. https://doi.org/10.1007/s00464-019-06839-2
[11]	S. U. S. Choi, J. A. Eastman, Enhancing thermal conductivity of fluids with nanoparticles, ASME Pub Fed., 231 (1995), 99-106.
[12]	J. Buongiorno, Convective transport in nanofuids, J. Heat Transfer, 128 (2006), 240-250. https://doi.org/10.1115/1.2150834 doi: 10.1115/1.2150834
[13]	K. L. Hsiao, Stagnation electrical MHD nanofuid mixed convection with slip boundary on a stretching sheet, Appl. Therm. Eng., 98 (2016), 850-861. https://doi.org/10.1016/j.applthermaleng.2015.12.138 doi: 10.1016/j.applthermaleng.2015.12.138
[14]	M. M. Rashidi, N. Freidoonimehr, A. Hosseini, O. A. Beg, T. K. Hung, Homotopy simulation of nanofuid dynamics from a non-linearly stretching isothermal permeable sheet with transpiration, Meccanica, 49 (2014), 469-482. https://doi.org/10.1007/s11012-013-9805-9 doi: 10.1007/s11012-013-9805-9
[15]	M. Sheikholeslami, M. M. Bhatti, Forced convection of nanofluid in presence of constant magnetic field considering shape effects of nanoparticles, Int. J. Heat Mass Tran., 111 (2017), 1039-1049. https://doi.org/10.1016/j.ijheatmasstransfer.2017.04.070 doi: 10.1016/j.ijheatmasstransfer.2017.04.070
[16]	M. Turkyilmazoglu, Condensation of laminar film over curved vertical walls using single and two-phase nanofluid models, Eur. J. Mech.-B/Fluids, 65 (2017), 184-191. https://doi.org/10.1016/j.euromechflu.2017.04.007 doi: 10.1016/j.euromechflu.2017.04.007
[17]	R. Ellahi, Recent developments of nanofluids, MDPI-Multidisciplinary Digital Publishing Institute, 2018. https://doi.org/10.3390/books978-3-03842-834-3
[18]	A. Pantokratoras, Discussion: "Computational analysis for mixed convective flows of viscous fluids with nanoparticles"(Farooq, U., Lu, DC, Ahmed, S., and Ramzan, M., 2019, ASME J. Therm. Sci. Eng. Appl., 11(2), p. 021013), J. Thermal. Sci. Eng. Appl., 11 (2019), 055503. https://doi.org/10.1115/1.4043092
[19]	M. Rashid, A. Alsaedi, T. Hayat, B. Ahmed, Magnetohydrodynamic flow of Maxwell nanofluid with binary chemical reaction and Arrhenius activation energy, Appl. Nanosci., 10 (2020), 2951-2963. https://doi.org/10.1007/s13204-019-01143-w doi: 10.1007/s13204-019-01143-w
[20]	T. Tayebi, A. J. Chamkha, Magnetohydrodynamic natural convection heat transfer of hybrid nanofluid in a square enclosure in the presence of a wavy circular conductive cylinder, J. Therm. Sci. Eng. Appl., 12 (2020), 031009. https://doi.org/10.1115/1.4044857 doi: 10.1115/1.4044857
[21]	T. Hayat, R. Riaz, A. Aziz, A. Alsaedi, Influence of Arrhenius activation energy in MHD flow of third grade nanofluid over a nonlinear stretching surface with convective heat and mass conditions, Physica A, 549 (2020), 124006. https://doi.org/10.1016/j.physa.2019.124006 doi: 10.1016/j.physa.2019.124006
[22]	T. Muhammad, H. Waqas, S. A. Khan, R. Ellahi, S. M. Sait, Significance of nonlinear thermal radiation in 3D Eyring-Powell nanofluid flow with Arrhenius activation energy, J. Therm. Anal. Calorim., 143 (2021), 929-944. https://doi.org/10.1007/s10973-020-09459-4 doi: 10.1007/s10973-020-09459-4
[23]	S. Z. Alamri, R. Ellahi, N. Shehzad, A. Zeeshan, Convective radiative plane Poiseuille flow of nanofluid through porous medium with slip: An application of Stefan blowing, J. Mol. Liq., 273 (2019), 292-304. https://doi.org/10.1016/j.molliq.2018.10.038 doi: 10.1016/j.molliq.2018.10.038
[24]	I. Khan, A. Hussain, M. Y. Malik, S. Mukhtar, On magnetohydrodynamics Prandtl fluid flow in the presence of stratification and heat generation, Physica A, 540 (2020), 123008. https://doi.org/10.1016/j.physa.2019.123008 doi: 10.1016/j.physa.2019.123008
[25]	S. E. Awan, M. A. Z. Raja, A. Mehmood, S. A. Niazi, S. Siddiqa, Numerical treatments to analyze the nonlinear radiative heat transfer in MHD nanofluid flow with solar energy, Arab. J. Sci. Eng., 45 (2020), 4975-4994. https://doi.org/10.1007/s13369-020-04593-5 doi: 10.1007/s13369-020-04593-5
[26]	A. Gailitis, O. Lielausis, On a possibility to reduce the hydrodynamic resistance of a plate in an electrolyte, Appl. Magnetohydrodynamics Rep. Inst. Riga, 13 (1961), 143-146.
[27]	R. Ahmad, M. Mustafa, M. Turkyilmazoglu, Buoyancy effects on nanofluid flow past a convectively heated vertical Riga-plate: A numerical study, Int. J. Heat Mass Tran., 111 (2017), 827-835. https://doi.org/10.1016/j.ijheatmasstransfer.2017.04.046 doi: 10.1016/j.ijheatmasstransfer.2017.04.046
[28]	Z. Iqbal, Z. Mehmood, E. Azhar, E. N. Maraj, Numerical investigation of nanofluid transport of gyrotactic microorganisms submerged in water towards Riga plate, J. Mol. Liq., 234 (2017), 296-308. https://doi.org/10.1016/j.molliq.2017.03.074 doi: 10.1016/j.molliq.2017.03.074
[29]	R. Ellahi, M. Hassan, A. Zeeshan, Aggregation effects on water base Al₂O₃-nanofluid over permeable wedge in mixed convection, Asia‐Pac. J. Chem. Eng., 11 (2016), 179-186. https://doi.org/10.1002/apj.1954 doi: 10.1002/apj.1954
[30]	R. M. Kasmani, S. Sivasankaran, M. Bhuvaneswari, Z. Siri, Effect of chemical reaction on convective heat transfer of boundary layer flow in nanofluid over a wedge with heat generation/absorption and suction, J. Appl. Fluid Mech., 9 (2015), 379-388. https://doi.org/10.18869/acadpub.jafm.68.224.24151 doi: 10.18869/acadpub.jafm.68.224.24151
[31]	M. Khan, M. Azam, A. S. Alshomrani, Effects of melting and heat generation/absorption on unsteady Falkner-Skan flow of Carreau nanofluid over a wedge, Int. J. Heat Mass Tran., 110 (2017), 437-446. https://doi.org/10.1016/j.ijheatmasstransfer.2017.03.037 doi: 10.1016/j.ijheatmasstransfer.2017.03.037
[32]	A. K. Pandey, M. Kumar, Chemical reaction and thermal radiation effects on a boundary layer flow of nanofluid over a wedge with viscous and Ohmic dissipation, St. Petersburg Polytechnical Univ. J.: Phys. Math., 3 (2017), 322-332. https://doi.org/10.1016/j.spjpm.2017.10.008 doi: 10.1016/j.spjpm.2017.10.008
[33]	M. Khan, M. Azam, A. S. Alshomrani, Unsteady slip flow of Carreau nanofluid over a wedge with nonlinear radiation and new mass flux condition, Res. Phys., 7 (2017), 2261-2270. https://doi.org/10.1016/j.rinp.2017.06.038 doi: 10.1016/j.rinp.2017.06.038
[34]	A. Chamkha, S. Abbasbandy, A. M. Rashad, Non-Darcy natural convection flow for non-Newtonian nanofluid over cone saturated in porous medium with uniform heat and volume fraction fluxes, Int. J. Numer. Method. Heat Fluid Flow, 25 (2015), 422-437. https://doi.org/10.1108/HFF-02-2014-0027 doi: 10.1108/HFF-02-2014-0027
[35]	M. Macha, N. Kishan, Boundary layer flow of viscoelastic nanofluid over a wedge in the presence of buoyancy force effects, Comput. Therm. Sci.: An Int. J., 9 (2017), 257-267. https://doi.org/10.1615/ComputThermalScien.2017016742 doi: 10.1615/ComputThermalScien.2017016742
[36]	A. V. Kuznetsov, Thermo bioconvection in a suspension of oxytactic bacteria, Int. Commun. Heat Mass Tran., 32 (2005), 991-999. https://doi.org/10.1016/j.icheatmasstransfer.2004.11.005 doi: 10.1016/j.icheatmasstransfer.2004.11.005
[37]	Y. R. Li, H. Waqas, M. Imran, U. Farooq, F. Mallawi, I. Tlili, A numerical exploration of modified second-grade nanofluid with motile microorganisms, thermal radiation, and Wu's slip, Symmetry, 12 (2020), 393. https://doi.org/10.3390/sym12030393 doi: 10.3390/sym12030393
[38]	T. Muhammad, S. Z. Alamri, H. Waqas, D. Habib, R. Ellahi, Bioconvection flow of magnetized Carreau nanofluid under the influence of slip over a wedge with motile microorganisms, J. Therm. Anal. Calorim., 143 (2021), 945-957. https://doi.org/10.1007/s10973-020-09580-4 doi: 10.1007/s10973-020-09580-4
[39]	S. U. Khan, H. Waqas, M. M. Bhatti, M. Imran, Bioconvection in the rheology of magnetized couple stress nanofluid featuring activation energy and Wu's slip, J. Non-Equil. Thermody., 45 (2020), 81-95. https://doi.org/10.1515/jnet-2019-0049 doi: 10.1515/jnet-2019-0049
[40]	T. P. Zhang, S. U. Khan, M. Imran, I. Tlili, H. Waqas, N. Ali, Activation energy and thermal radiation aspects in bioconvection flow of rate type nanoparticles configured by a stretching/shrinking disk, J. Energy Resour. Technol., 142 (2020), 112102. https://doi.org/10.1115/1.4047249 doi: 10.1115/1.4047249
[41]	H. Waqas, M. Imran, T. Muhammad, S. M. Sait, R. Ellahi, Numerical investigation on bioconvection flow of Oldroyd-B nanofluid with nonlinear thermal radiation and motile microorganisms over rotating disk, J. Therm. Analy. Calorim., 145 (2021), 523-539. https://doi.org/10.1007/s10973-020-09728-2 doi: 10.1007/s10973-020-09728-2
[42]	N. S. Khan, Q. Shah, A. Bhaumik, P. Kumam, P. Thounthong, I. Amiri, Entropy generation in bioconvection nanofluid flow between two stretchable rotating disks, Sci. Rep., 10 (2020), 4448. https://doi.org/10.1038/s41598-020-61172-2 doi: 10.1038/s41598-020-61172-2
[43]	S. U. Mamatha, K. R. Babu, P. D. Prasad, C. S. K. Raju, S. V. K. Varma, Mass transfer analysis of two-phase flow in a suspension of microorganisms, Arch. Thermodyn., 41 (2020), 175-192. https://doi.org/10.24425/ather.2020.132954 doi: 10.24425/ather.2020.132954
[44]	M. Ferdows, M. G. Reddy, F. Alzahrani, S. Y. Sun, Heat and mass transfer in a viscous nanofluid containing a gyrotactic micro-organism over a stretching cylinder, Symmetry, 11 (2019), 1131. https://doi.org/10.3390/sym11091131 doi: 10.3390/sym11091131
[45]	N. A. Amirsom, M. J. Uddin, M. F. M. Basir, A. Ismail, O. A. Beg, A. Kadir, Three-dimensional bioconvection nanofluid flow from a bi-axial stretching sheet with anisotropic slip, Sains Malays., 48 (2019), 1137-1149. http://doi.org/10.17576/jsm-2019-4805-23 doi: 10.17576/jsm-2019-4805-23
[46]	S. Kasaragadda, I. M. Alarifi, M. Rahimi-Gorji, R. Asmatulu, Investigating the effects of surface superhydrophobicity on moisture ingression of nanofiber-reinforced bio-composite structures, Microsyst. Technol., 26 (2020), 447-459. https://doi.org/10.1007/s00542-019-04507-y doi: 10.1007/s00542-019-04507-y
[47]	M. S. Ansari, O. Otegbeye, M. Trivedi, S. P. Goqo, Magnetohydrodynamic bio-convective Casson nanofluid flow: A numerical simulation by paired quasilinearisation, J. Appl. Comput. Mech., 7 (2021), 2024-2039. https://doi.org/10.22055/JACM.2020.31205.1839 doi: 10.22055/JACM.2020.31205.1839
[48]	M. S. Alqarni, S. Yasmin, H. Waqas, S. A. Khan, Recent progress in melting heat phenomenon for bioconvection transport of nanofluid through a lubricated surface with swimming microorganisms, Sci. Rep., 12 (2022), 8447. https://doi.org/10.1038/s41598-022-12230-4 doi: 10.1038/s41598-022-12230-4
[49]	S. M. H. Zadeh, S. A. M. Mehryan, M. A. Sheremet, M. Izadi, M. Ghodrat, Numerical study of mixed bio-convection associated with a micropolar fluid, Therm. Sci. Eng. Prog., 18 (2020), 100539. https://doi.org/10.1016/j.tsep.2020.100539 doi: 10.1016/j.tsep.2020.100539
[50]	M. M. Bhatti, E. E. Michaelides, Study of Arrhenius activation energy on the thermo-bioconvection nanofluid flow over a Riga plate, J. Therm. Anal. Calorim., 143 (2021), 2029-2038. https://doi.org/10.1007/s10973-020-09492-3 doi: 10.1007/s10973-020-09492-3
[51]	H. Waqas, S. U. Khan, M. Imran, M. M. Bhatti, Thermally developed Falkner-Skan bioconvection flow of a magnetized nanofluid in the presence of a motile gyrotactic microorganism: Buongiorno's nanofluid model, Phys. Scr., 94 (2019), 115304. https://doi.org/10.1088/1402-4896/ab2ddc doi: 10.1088/1402-4896/ab2ddc
[52]	A. M. Alwatban, S. U. Khan, H. Waqas, I. Tlili, Interaction of Wu's slip features in bioconvection of Eyring Powell nanoparticles with activation energy, Processes, 7 (2019), 859. https://doi.org/10.3390/pr7110859 doi: 10.3390/pr7110859
[53]	Y. Wang, H. Waqas, M. Tahir, M. Imran, C. Y. Jung, Effective Prandtl aspects on bio-convective thermally developed magnetized tangent hyperbolic nanoliquid with Gyrotactic microorganisms and second order velocity slip, IEEE Access, 7 (2019), 130008-130023. https://doi.org/10.1109/ACCESS.2019.2940203 doi: 10.1109/ACCESS.2019.2940203
[54]	M. Z. Ullah, T. S. Jang, An efficient numerical scheme for analyzing bioconvection in von-Kármán flow of third-grade nanofluid with motile microorganisms, Alex. Eng. J., 59 (2020), 2739-2752. https://doi.org/10.1016/j.aej.2020.05.017 doi: 10.1016/j.aej.2020.05.017
[55]	A. S. Alshomrani, M. Z. Ullah, D. Baleanu, Importance of multiple slips on bioconvection flow of cross nanofluid past a wedge with gyrotactic motile microorganisms, Case Stud. Therm. Eng., 22 (2020), 100798. https://doi.org/10.1016/j.csite.2020.100798 doi: 10.1016/j.csite.2020.100798
[56]	A. Shafiq, Z. Hammouch, A. Turab, Impact of radiation in a stagnation point flow of Walters' B fluid towards a Riga plate, Therm. Sci. Eng. Prog., 6 (2018), 27-33. https://doi.org/10.1016/j.tsep.2017.11.005 doi: 10.1016/j.tsep.2017.11.005
[57]	M. M. Peiravi, J. Alinejad, D. Ganji, S. Maddah, Numerical study of fins arrangement and nanofluids effects on three-dimensional natural convection in the cubical enclosure, Chall. Nano Micro Scale Sci. Technol., 7 (2019), 97-112. https://doi.org/10.22111/tpnms.2019.4845 doi: 10.22111/tpnms.2019.4845
[58]	M. M. Peiravi, J. Alinejad, Hybrid conduction, convection and radiation heat transfer simulation in a channel with rectangular cylinder, J. Therm. Anal. Calorim., 140 (2020), 2733-2747. https://doi.org/10.1007/s10973-019-09010-0
[59]	J. Alinejad, M. M. Peiravi, Numerical analysis of secondary droplets characteristics due to drop impacting on 3D cylinders considering dynamic contact angle, Meccanica, 55 (2020), 1975-2002. https://doi.org/10.1007/s11012-020-01240-z doi: 10.1007/s11012-020-01240-z
[60]	M. M. Peiravi, J. Alinejad, D. D. Ganji, S. Maddah, 3D optimization of baffle arrangement in a multi-phase nanofluid natural convection based on numerical simulation, Int. J. Numer. Method. Heat Fluid Flow, 30 (2020), 2583-2605. https://doi.org/10.1108/HFF-01-2019-0012 doi: 10.1108/HFF-01-2019-0012
[61]	M. M. Peiravi, J. Alinejad, Nano particles distribution characteristics in multi-phase heat transfer between 3D cubical enclosures mounted obstacles, Alex. Eng. J., 60 (2021), 5025-5038. https://doi.org/10.1016/j.aej.2021.04.013 doi: 10.1016/j.aej.2021.04.013
[62]	J. K. Madhukesh, A. Alhadhrami, R. N. Kumar, R. J. P. Gowda, B. C. Prasannakumara, R. S. V. Kumar, Physical insights into the heat and mass transfer in Casson hybrid nanofluid flow induced by a Riga plate with thermophoretic particle deposition, P. I. Mech. Eng. Part E: J. Proc. Mech. Eng., 2021, https://doi.org/10.1177/09544089211039305 doi: 10.1177/09544089211039305
[63]	J. K. Madhukesh, R. S. V. Kumar, R. J. P. Gowda, B. C. Prasannakumara, S. A. Shehzad, Thermophoretic particle deposition and heat generation analysis of Newtonian nanofluid flow through magnetized Riga plate, Heat Transf., 51 (2022), 3082-3098. https://doi.org/10.1002/htj.22438 doi: 10.1002/htj.22438
[64]	R. J. P. Gowda, R. N. Kumar, A. M. Jyothi, B. C. Prasannakumara, I. E. Sarris, Impact of binary chemical reaction and activation energy on heat and mass transfer of Marangoni driven boundary layer flow of a non-Newtonian nanofluid, Processes, 9 (2021), 702. https://doi.org/10.3390/pr9040702 doi: 10.3390/pr9040702
[65]	A. M. Jyothi, R. N. Kumar, R. J. P. Gowda, Y. Veeranna, B. C. Prasannakumara, Impact of activation energy and gyrotactic microorganisms on flow of Casson hybrid nanofluid over a rotating moving disk, Heat Transf., 50 (2021), 5380-5399. https://doi.org/10.1002/htj.22129 doi: 10.1002/htj.22129
[66]	B. Shaker, M. Gholinia, M. Pourfallah, D. D. Ganji, CFD analysis of Al₂O₃-syltherm oil Nanofluid on parabolic trough solar collector with a new flange-shaped turbulator model, Theor. Appl. Mech. Lett., 12 (2022), 100323. https://doi.org/10.1016/j.taml.2022.100323 doi: 10.1016/j.taml.2022.100323
[67]	F. H. Sani, M. Pourfallah, M. Gholinia, The effect of MoS₂-Ag/H₂O hybrid nanofluid on improving the performance of a solar collector by placing wavy strips in the absorber tube, Case Stud. Therm. Eng., 30 (2022), 101760. https://doi.org/10.1016/j.csite.2022.101760 doi: 10.1016/j.csite.2022.101760
[68]	A. H. Ghobadi, M. G. Hassankolaei, A numerical approach for MHD Al₂O₃-TiO₂/H₂O hybrid nanofluids over a stretching cylinder under the impact of shape factor, Heat Transf., 48 (2019), 4262-4282. https://doi.org/10.1002/htj.21591 doi: 10.1002/htj.21591
[69]	S. Shahlaei, M. G. Hassankolaei, MHD boundary layer of GO-H₂O nanoliquid flow upon stretching plate with considering nonlinear thermal ray and Joule heating effect, Heat Transf., 48 (2019), 4152-4173. https://doi.org/10.1002/htj.21586 doi: 10.1002/htj.21586
[70]	A. H. Ghobadi, M. G. Hassankolaei, Numerical treatment of magneto Carreau nanofluid over a stretching sheet considering Joule heating impact and nonlinear thermal ray, Heat Transf., 48 (2019), 4133-4151. https://doi.org/10.1002/htj.21585 doi: 10.1002/htj.21585

This article has been cited by:

1.	Fahad Alsharari, Mohamed M. Mousa, New application of MOL-PACT for simulating buoyancy convection of a copper-water nanofluid in a square enclosure containing an insulated obstacle, 2022, 7, 2473-6988, 20292, 10.3934/math.20221111
2.	Ebrahem A. Algehyne, Showkat Ahmad Lone, Zehba Raizah, Sayed M. Eldin, Anwar Saeed, Ahmed M. Galal, Chemically reactive hybrid nanofluid flow past a Riga plate with nonlinear thermal radiation and a variable heat source/sink, 2023, 10, 2296-8016, 10.3389/fmats.2023.1132468
3.	Kaouther Ghachem, Bilal Ahmad, Skeena Noor, Tasawar Abbas, Sami Ullah Khan, Sanaa Anjum, Norah Alwadai, Lioua Kolsi, Numerical simulations for radiated bioconvection flow of nanoparticles with viscous dissipation and exponential heat source, 2023, 100, 00194522, 100828, 10.1016/j.jics.2022.100828
4.	B. Arun, M. Deivanayaki, Selvakumar Kuppusamy Vaithilingam, B. R. Ramesh Bapu, Bioconvection Flow in the Presence of Casson Nanoparticles on a Stretching/Shrinking Vertical Sheet with Chemical Reaction, 2023, 2023, 2090-9071, 1, 10.1155/2023/6199200
5.	Muhammad Irfan, Taseer Muhammad, Numerical simulation of bio‐convection radiative heat transport flow of MHD Carreau nanofluid, 2024, 104, 0044-2267, 10.1002/zamm.202300813
6.	Nidhish Kumar Mishra, Ghulfam Sarfraz, Mutasem Z. Bani-Fwaz, Sayed M. Eldin, Dynamics of Corcione nanoliquid on a convectively radiated surface using Al2O3 nanoparticles, 2023, 148, 1388-6150, 11303, 10.1007/s10973-023-12448-y
7.	K. Vijayalakshmi, Ajmeera Chandulal, Hadil Alhazmi, A.F. Aljohani, Ilyas Khan, Effect of chemical reaction and activation energy on Riga plate embedded in a permeable medium over a Maxwell fluid flow, 2024, 59, 2214157X, 104457, 10.1016/j.csite.2024.104457
8.	M. Faizan Ahmed, A. Zaib, Farhan Ali, Umair Khan, Syed Sohaib Zafar, Thermal radiation of Walter-B magneto bioconvection nanofluid due to the stretching surface under convective condition and heat source/sink: A semi-analytical technique for the stagnation point, 2025, 18, 16878507, 101291, 10.1016/j.jrras.2025.101291
9.	Muhammad Sohail, Umar Nazir, Ibrahim Mahariq, Yasser Elmasry, Implementation of FEM and Taguchi analysis on blood flow for Casson fluid inclusion of di- and tri-Hamilton Crosser nanofluid through the cylinder with a rough surface, 2025, 29, 1385-2000, 10.1007/s11043-025-09766-z

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)

通讯作者: 陈斌, bchen63@163.com

1.
沈阳化工大学材料科学与工程学院沈阳 110142

AIMS Mathematics

1.8 3.4

Metrics

Article views(2116) PDF downloads(131) Cited by(9)

Preview PDF

Download XML

Export Citation

Article outline

Show full outline

Figures and Tables

Figures(5) / Tables(5)

AIMS Mathematics

Thermo-bioconvection flow of Walter's B nanofluid over a Riga plate involving swimming motile microorganisms

Related Papers:

Abstract

Nomenclature

1. Introduction

2. Mathematical modeling

3. Numerical approach

4. Results and discussion

4.1. Velocity profile

4.2. Thermal profile

4.3. Nanoparticles concentration profile

4.4. Microorganism profile

4.5. Tabular data

5. Conclusions

Acknowledgment

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Catalog

AIMS Mathematics

Thermo-bioconvection flow of Walter's B nanofluid over a Riga plate involving swimming motile microorganisms

Related Papers:

Abstract

Nomenclature

1. Introduction

2. Mathematical modeling

3. Numerical approach

4. Results and discussion

4.1. Velocity profile

4.2. Thermal profile

4.3. Nanoparticles concentration profile

4.4. Microorganism profile

4.5. Tabular data

5. Conclusions

Acknowledgment

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Related pages

Tools

Export File

Citation

Format

Content

Catalog