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Bioconvection in Eyring–Powell fluid with composite features of variable viscosity and motile microorganism density

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Abstract

The yield stress in non-Newtonian fluids is intriguing. Their behaviours and properties transform under different conditions, such as pressure, temperature, concentration, and motile density. Consider a fascinating biological system where temperature differences ignite microorganism activity and production rates, explaining the need for refrigeration and similar processes. As such, the current model breaks free from constant assumptions about fluid properties and mathematically predicts changes in thermal conductivity and mass diffusivity, while viscosity and motile variation are modelled as a composite function of microorganism density and fluid temperature. The bio-convection phenomenon arises when microorganisms self-propel the Eyring–Powell fluid past a three-dimensional Riga plate, driven by stretching velocity. It is an intriguing interplay. To gain deeper insights into the flow model parameters, the weighted residual method (Galerkin approach) is employed to solve the model systems while the findings are presented through tables and graphs. Improving the temperature- and microorganism-dependent variable viscosity significantly decreases the fluid velocities and motile density movement but enhances the temperature and fluid concentration. Conversely, the response of temperature- and microorganism-dependent variable motile density variation energizes the flow momentum and decreases the fluid concentration considerably. For the variable viscosity parameter, \(\xi _1 \in [0,0.5]\), the skin drag force and local motile number increase by 22.6% and 5.98%, respectively. Additionally, 100% increment in variable motile density number downsized the skin friction by 289.36% while the local motile density appreciates by 18.90%. In general, the results obtained here are found to be applicable in biophysics, environmental science, and engineering systems.

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Abbreviations

p :

Pressure

I :

The identity tensor

\(\mu\) :

Dynamic viscosity

\(\beta _{\mathrm{m}}\) and \(\beta _{\mathrm{n}}\) :

Eyring–Powell fluid parameters

\({\textbf {J}}_1\) :

Rivlin–Ericksen tensor

\(j_0\) :

Current density

\(M_0\) :

Surface magnetic property

\(r_0\) :

Diameter of the magnets

\(\mu\) :

Variable fluid viscosity

\(\rho _{\mathrm{f}}\) :

Fluid density

\(\kappa (T)\) :

Variable temperature

\(\mu\) :

Variable fluid viscosity

\(D_{\mathrm{o}}\) :

Mass diffusivity

C :

Fluid concentration

C :

Fluid concentration

T :

Fluid temperature

\(\rho\) :

Fluid density

\(C_{\mathrm{p}}\) :

Specific heat capacity

\(\nu\) :

Kinematic viscosity

w, v, and u :

Components along

z, y, and x :

Axis

Pe:

Peclet number

\(\beta\), \(\Delta _{v}\), and \(\Delta _{u}\) :

Powell–Eyring fluid parameters

N :

Motile concentration

\(U_0\) and \(V_0\) :

Streatching velocities

Le:

Lewis number

\(\hbox {Ec}_x\) and \(\hbox {Ec}_y\) :

Local Eckert number

\(\lambda\) :

Bioconvection constant

Gn:

Gyrotactic Grashof number

Gc:

Mass Grashof number

Gr:

Thermal Grashof number

Pr:

Prandtl number

Sc:

Schmidt number

\(\epsilon _2\) :

Stretching ratio

\(\epsilon _1\) :

Material constant

\(\xi _1\) :

Variable viscosity

\(\xi _2\) :

Variable thermal conductivity

\(\xi _3\) :

Variable mass concentration

\(\xi _4\) :

Variable motile density

J :

Modified Hartman number

\(\hbox {Re}_x\) :

Reynolds number

\(f_1\) and \(f_2\) :

Dimensionless velocities

\(\Theta\) :

Dimensionless temperature

\(\gamma\) :

Dimensionless concentration

\(\chi\) :

Dimensionless motile concentration

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Acknowledgements

The authors appreciate reviewers for their valuable suggestions and comments, which have helped improve the clarity of the manuscript.

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Authors and Affiliations

  1. Department of Mathematics, Computational Mechanics and Mathematical Research Group, University of Ilorin, Ilorin, 240003, Kwara, Nigeria

    Mojeed T. Akolade

  2. Department of General Studies, School of Management Sciences, Federal Polytechnic Ayede, Ayede, Oyo, Nigeria

    John O. Olabode

  3. Department of Mathematics and Applied Mathematics, Computational Mechanics and Mathematical Research Group, Nelson Mandela University, Port-Elizabeth, South Africa

    Yusuf O. Tijani

  4. Department of Mathematics, Government College University, Faisalabad, 38000, Pakistan

    Tayyaba Akhtar

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Contributions

MTA, YOT, and JOO helped in conceptualization; MTA and YOT helped in methodology; YOT and TA worked in formal analysis and investigation; JOO, TA, and MAT contributed to writing—original draft preparation; and JOO, TA, MTA, YOT, and JOO contributed to writing—review and editing. All authors read and approve the manuscript.

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Correspondence to Mojeed T. Akolade.

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Akolade, M.T., Olabode, J.O., Tijani, Y.O. et al. Bioconvection in Eyring–Powell fluid with composite features of variable viscosity and motile microorganism density. Eur. Phys. J. Plus 138, 736 (2023). https://doi.org/10.1140/epjp/s13360-023-04361-y

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