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Heat and mass transmission of an Oldroyd-B nanofluid flow through a stratified medium with swimming of motile gyrotactic microorganisms and nanoparticles

Abstract

This paper focuses on the research of motile microorganism rates in the bioconvective Oldroyd-B nanoliquid flow over a vertical stretching sheet with mixed convection and inclined magnetic field. Additionally, interesting characteristics of thermophoresis, Brownian motion, viscous dissipation, Joule heating, and stratification are examined. Similarity transformations are employed to reduce the mathematical model to higher-order ODE. The convergent serious solution is applied to solve the nonlinear differential system. The analysis of temperature, velocity, motile microorganisms’ density, and nanoparticle concentration are represented through graphs. Local Nusselt number, density number of motile microorganisms, and Sherwood number are examined via contour plots.

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Abbreviations

\( a \) :

Stretching rate \( \left( {{\text{s}}^{ - 1} } \right) \)

\( A_{1} \) :

Relaxation time \( \left( {-} \right) \)

\( A_{2} \) :

Retardation time \( \left( {-} \right) \)

\( b \) :

Chemotaxis constant \( \left( {\text{m}} \right) \)

\( \left\{ {\begin{array}{*{20}l} {b_{1} , b_{2} } \hfill \\ {d_{1} ,d_{2} } \hfill \\ {e_{1}, e_{2} } \hfill \\ \end{array} } \right. \) :

Dimensionless constants \( \left( {-} \right) \)

\( B_{0} \) :

Constant magnetic field \( \left( {{\text{kg}}\;{\text{s}}^{ - 2 } \;{\text{A}}^{ - 1} } \right) \)

\( \widehat{C} \) :

Concentration \( \left( {{\text{kg}}\;{\text{m}}^{ - 3} } \right) \)

\( \widehat{C}_{0} \) :

Reference concentration of nanoparticles \( \left( {{\text{kg}}\;{\text{m}}^{ - 3} } \right) \)

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

Specific heat \( \left( {{\text{J}}\;{\text{kg}}^{ - 1} \;{\text{K}}^{ - 1} } \right) \)

\( \widehat{C}_{\infty } \) :

Ambient concentration \( \left( {{\text{kg}}\;{\text{m}}^{ - 3} } \right) \)

\( \widehat{C}_{\text{w}} \) :

Surface concentration of nanoparticles \( \left( {{\text{kg}}\;{\text{m}}^{ - 3} } \right) \)

\( D_{\text{B}} \) :

Brownian diffusion coefficient \( \left( {{\text{m}}^{2} \;{\text{s}}^{ - 1} } \right) \)

\( D_{\text{T}} \) :

Thermophoretic diffusion coefficient \( \left( {{\text{m}}^{2} \;{\text{s}}^{ - 1} } \right) \)

\( D_{\text{m}} \) :

Microorganism’s diffusion coefficient \( \left( {{\text{m}}^{2} \;{\text{s}}^{ - 1} } \right) \)

\( {\text{E}}_{\text{C}} \) :

Eckert number \( \left( {-} \right) \)

\( F\left( \eta \right) \) :

Velocity similarity function \( \left( {-} \right) \)

\( G\left( { \eta } \right) \) :

Temperature similarity function \( \left( {-} \right) \)

\( k \) :

Thermal conductivity \( \left( {{\text{m}}\;{\text{kg}}\;{\text{s}}^{ - 3} \;{\text{K}}^{ - 1} } \right) \)

\( {\text{Le}} \) :

Lewis number \( \left( {-} \right) \)

\( {\text{L}}_{\text{b}} \) :

Bioconvection Lewis number \( \left( {-} \right) \)

\( M \) :

Magnetic parameter \( \left( {-} \right) \)

\( N_{\text{b}} \) :

Brownian motion parameter \( \left( {-} \right) \)

\( N_{\text{t}} \) :

Thermophoresis parameter \( \left( {-} \right) \)

\( \widehat{n}_{\text{w}} \) :

Surface concentration of microorganisms \( \left( {{\text{kg}}\;{\text{m}}^{ - 3} } \right) \)

\( \widehat{n}_{\infty } \) :

Ambient concentration of microorganisms \( \left( {{\text{kg}}\;{\text{m}}^{ - 3} } \right) \)

\( \widehat{n}_{0} \) :

Reference concentration of microorganisms \( \left( {{\text{kg}}\;{\text{m}}^{ - 3} } \right) \)

\( N_{\text{r}} \) :

Buoyancy ratio parameter \( \left( {-} \right) \)

\( {\text{Nu}}_{\text{x}} \) :

Nusselt number \( \left( {-} \right) \)

\( { \Pr } \) :

Prandtl number \( \left( {-} \right) \)

\( {\text{P}}_{\text{e}} \) :

Bioconvection Peclet number \( \left( {-} \right) \)

\( {\text{R}}_{\text{b}} \) :

Bioconvection Rayleigh number \( \left( {-} \right) \)

\( S_{1} \) :

Thermal stratification parameter \( \left( {-} \right) \)

\( S_{2} \) :

Mass stratification parameter \( \left( {-} \right) \)

\( S_{3} \) :

Motile density stratification parameter \( \left( {-} \right) \)

\( {\text{Sh}}_{\text{x}} \) :

Sherwood number \( \left( {-} \right) \)

\( \widehat{T} \) :

Temperature \( \left( {\text{K}} \right) \)

\( \widehat{T}_{\infty } \) :

Ambient temperature \( \left( {\text{K}} \right) \)

\( \widehat{T}_{0} \) :

Reference temperature \( \left( {\text{K}} \right) \)

\( u_{\text{w}} \) :

Velocity of the sheet \( \left( {{\text{m}}\;{\text{s}}^{ - 1} } \right) \)

\( u,\;v \) :

Velocity components \( \left( {{\text{m}}\;{\text{s}}^{ - 1} } \right) \)

\( W_{\text{c}} \) :

Maximum cell swimming speed \( \left( {{\text{m}}\;{\text{s}}^{ - 1} } \right) \)

\( x,\;y \) :

Cartesian coordinates \( \left( {\text{m}} \right) \)

\( \alpha \) :

Inclination angle of magnetic field

\( \alpha_{1} \) :

Dimensionless relaxation time parameter \( \left( {-} \right) \)

\( \beta \) :

Volume expansion coefficient \( \left( {-} \right) \)

\( \beta_{1} \) :

Dimensionless retardation time parameter \( \left( {-} \right) \)

\( \phi \left( { \eta } \right) \) :

Concentration similarity function \( \left( {-} \right) \)

\( \eta \) :

Similarity parameter \( \left( {-} \right) \)

\( \Gamma \left( { \eta } \right) \) :

Microorganisms similarity function \( \left( {-} \right) \)

\( \lambda \) :

Mixed convection parameter \( \left( {-} \right) \)

\( \nu \) :

Kinematic viscosity \( \left( {{\text{m}}^{2} \;{\text{s}}^{ - 1} } \right) \)

\( \tau \) :

Ratio of the effective heat capacity \( \left( {-} \right) \)

\( \rho \) :

Density \( \left( {{\text{kg}}\;{\text{m}}^{ - 1} } \right) \)

\( \rho_{\text{f}} \) :

Density of nanofluid \( \left( {{\text{kg}}\;{\text{m}}^{ - 3} } \right) \)

\( \rho_{\text{p}} \) :

Density of nanoparticles \( \left( {{\text{kg}}\;{\text{m}}^{ - 3} } \right) \)

\( \rho_{\text{m}} \) :

Density of microorganism’s particles \( \left( {{\text{kg}}\;{\text{m}}^{ - 3} } \right) \)

\( \sigma \) :

Electrical conductivity \( \left( {{\text{S}}^{3} \;{\text{m}}^{2} \;{\text{kg}}^{ - 1} } \right) \)

\( \psi \) :

Stream function \( \left( {{\text{m}}\;{\text{s}}^{ - 1} } \right) \)

\( \Omega \) :

Microorganisms concentration difference parameter \( \left( {-} \right) \)

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Acknowledgements

The authors are grateful to Prof. K. Loganathan, Department of Mathematics, Faculty of Engineering, Karpagam Academy of Higher Education, Coimbatore, India for his valuable support throughout this project work.

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Elanchezhian, E., Nirmalkumar, R., Balamurugan, M. et al. Heat and mass transmission of an Oldroyd-B nanofluid flow through a stratified medium with swimming of motile gyrotactic microorganisms and nanoparticles. J Therm Anal Calorim 141, 2613–2623 (2020). https://doi.org/10.1007/s10973-020-09847-w

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Keywords

  • Bioconvection
  • Gyrotactic microorganisms
  • Oldroyd-B nanofluid
  • Stratification
  • Inclined magnetic field