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.
Similar content being viewed by others
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 \( \left( {-} \right) \)
References
Reddy GJ, Kumar M, Anwar Beg O. Effect of temperature dependent viscosity on entropy generation in transient viscoelastic polymeric fluid flow from an isothermal vertical plate. Phys A. 2018;510:426–45.
Dash GC, Ojha KL. Viscoelastic hydromagnetic flow between two porous parallel plates in the presence of sinusoidal pressure gradient. Alex Eng J. 2018;57:3463–71.
Hayat T, Kiyani MZ, Ahmad I, Khan MI, Alsaedi A. Stagnation point flow of viscoelastic nanomaterial over a stretched surface. Results Phys. 2018;9:518–26.
Bhatnagar RK, Gupta G, Rajagopal KR. Flow of an Oldroyd-B fluid due to a stretching sheet in the presence of a free stream velocity. Int J Non-Linear Mech. 1995;30:391–405.
Sajid M, Abbas Z, Javed T, Ali N. Boundary layer flow of an Oldroyd-B fluid in the region of a stagnation point over a stretching sheet. Can J Phys. 2010;88:635–40.
Shehzad SA, Alsaedi A, Hayat T, Alhuthali MS. Three-dimensional flow of an Oldroyd-B fluid with variable thermal conductivity and heat generation/absorption. PloS One. 2013;8:e78240.
Abbasbandy S, Hayat T, Alsaedi A, Rashidi MM. Numerical and analytical solutions for Falkner–Skan flow of MHD Oldroyd-B fluid. Int J Numer Methods Heat Fluid Flow. 2014;24:390–401.
Xu H, Cui J. Mixed convection flow in a channel with slip in a porous medium saturated with a nanofluid containing both nanoparticles and microorganisms. Int J Heat Mass Transf. 2018;125:1043–53.
Hayat T, Aziz A, Muhammad T, Alsaedi A. An optimal analysis for Darcy–Forchheimer 3D flow of nanofluid with convective condition and homogeneous–heterogeneous reactions. Phys Lett A. 2018;382:2846–55.
Hayat T, Aziz A, Muhammad T, Alsaedi A. An optimal analysis for Darcy–Forchheimer 3D flow of Carreau nanofluid with convectively heated surface. Results Phys. 2018;9:598–608.
Khan M, Irfan M, Khan WA. Impact of nonlinear thermal radiation and gyrotactic microorganisms on the Magneto-Burgers nanofluid. Int J Mech Sci. 2017;130:375–82.
Alsaedi A, Khan MI, Farooq M, Gull N, Hayat T. Magnetohydrodynamic (MHD) stratified bioconvective flow of nanofluid due to gyrotactic microorganisms. Adv Powder Technol. 2017;28:288–98.
Abdelmalek Z, Khan SU, Waqas H, et al. A mathematical model for bioconvection flow of Williamson nanofluid over a stretching cylinder featuring variable thermal conductivity, activation energy and second-order slip. J Therm Anal Calorim. 2020. https://doi.org/10.1007/s10973-020-09450-z.
Tham L, Nazar R, Pop I. Mixed convection flow over a solid sphere embedded in a porous medium filled by a nanofluid containing gyrotactic microorganisms. Int J Heat Mass Transf. 2013;62:647–60.
Aziz A, Khan WA, Pop I. Free convection boundary layer flow past a horizontal flat plate embedded in porous medium filled by nanofluid containing gyrotactic microorganisms. Int J Therm Sci. 2012;56:48–57.
Xu H, Pop I. Fully developed mixed convection flow in a horizontal channel filled by a nanofluid containing both nanoparticles and gyrotactic microorganisms. Eur J Mech B Fluids. 2014;46:37–45.
Abdelmalek Z, Khan SU, Awais M, et al. Analysis of generalized micropolar nanofluid with swimming of microorganisms over an accelerated surface with activation energy. J Therm Anal Calorim. 2020. https://doi.org/10.1007/s10973-020-09474-5.
Kuznetsov AV. The onset of nanofluid bioconvection in a suspension containing both nanoparticles and gyrotactic microorganisms. Int Commun Heat Mass Transf. 2010;37:1421–5.
Kuznetsov AV. Nanofluid biothermal convection: simultaneous effects of gyrotactic and oxytactic micro-organisms. Fluid Dyn Res. 2011;43:055505.
Muhammad T, Alamri SZ, Waqas H, et al. Bioconvection flow of magnetized Carreau nanofluid under the influence of slip over a wedge with motile microorganisms. J Therm Anal Calorim. 2020. https://doi.org/10.1007/s10973-020-09580-4.
Khan WA, Uddin MJ, Ismail AIM. Free convection of non-Newtonian nanofluids in porous media with gyrotactic microorganisms. Transp Porous Med. 2013;97:241–52.
Tausif MS, Das K, Kundu PK. Multiple slip effects on bioconvection of nanofluid flow containing gyrotactic microorganisms and nanoparticles. J Mol Liq. 2016;220:518–26.
Bhatti MM, Michaelides EE. Study of Arrhenius activation energy on the thermo-bioconvection nanofluid flow over a Riga plate. J Therm Anal Calorim. 2020. https://doi.org/10.1007/s10973-020-09492-3.
Siddiqa S, Sulaiman M, Hossain MA, Islam S, Gorla RSR. Gyrotactic bioconvection flow of a nanofluid past a vertical wavy surface. Int J Therm Sci. 2016;108:244–50.
Mutuku WN, Makinde OD. Hydromagnetic bioconvection of nanofluid over a permeable vertical plate due to gyrotactic microorganisms. Comput Fluid. 2014;95:88–97.
Makinde OD, Animasaun IL. Bioconvection in MHD nanofluid flow with nonlinear thermal radiation and quartic autocatalysis chemical reaction past an upper surface of a paraboloid of revolution. Int J Therm Sci. 2016;109:159–71.
Raees A, Raees-ul-Haq M, Xu H, Sun Q. Three-dimensional stagnation flow of a nanofluid containing both nanoparticles and microorganisms on a moving surface with anisotropic slip. Appl Math Model. 2016;40:4136–50.
Akbar NS, Khan ZH. Magnetic field analysis in a suspension of gyrotactic microorganisms and nanoparticles over a stretching surface. J Magn Magn Mater. 2016;410:72–80.
Wang N, Maleki A, Alhuyi Nazari M, Tlili I, Safdari Shadloo M. Thermal conductivity modeling of nanofluids contain MgO particles by employing different approaches. Symmetry. 2020;12:206.
Ahmadi MH, Ahmadi MA, Maleki A, Pourfayaz F, Bidi M, Açıkkalp E. Exergetic sustainability evaluation and multi-objective optimization of performance of an irreversible nanoscale Stirling refrigeration cycle operating with Maxwell–Boltzmann gas. Renew Sustain Energy Rev. 2017;78:80–92.
Hayat T, Qayyum S, Khan MI, Alsaedi A. Current progresses about probable error and statistical declaration for radiative two-phase flow using AgH2O and CuH2O nanomaterials. Int J Hydrog Energy. 2017. https://doi.org/10.1016/j.ijhydene.2017.09.124.
Ramezanizadeh M, Nazari MA, Ahmadi MH, Lorenzini G, Pop I. A review on the applications of intelligence methods in predicting thermal conductivity of nanofluids. J Therm Anal Calorim. 2019. https://doi.org/10.1007/s10973-019-08154-3.
Hayat T, Khan MI, Waqas M, Alsaedi A, Farooq M. Numerical simulation for melting heat transfer and radiation effects in stagnation point flow of carbon–water nanofluid. Comput Methods Appl Mech Eng. 2016. https://doi.org/10.1016/j.cma.2016.11.033.
Maleki A, Elahi M, Assad ME, Nazari MA, Shadloo MS, Nabipour N. Thermal conductivity modeling of nanofluids with ZnO particles by using approaches based on artificial neural network and MARS. J Therm Anal Calorim. 2020. https://doi.org/10.1007/s10973-020-09373-9.
Irandoost Shahrestani M, Maleki A, Safdari Shadloo M, Tlili I. Numerical investigation of forced convective heat transfer and performance evaluation criterion of Al2O3/water nanofluid flow inside an axisymmetric microchannel. Symmetry. 2020;12:120.
Ramezanizadeh M, Ahmadi MA, Ahmadi MH, Nazari MA. Rigorous smart model for predicting dynamic viscosity of Al2O3/water nanofluid. J Therm Anal Calorim. 2018. https://doi.org/10.1007/s10973-018-7916-1.
Khan SU, Rauf A, Shehzad SA, Abbas Z, Javed T. Study of bioconvection flow in Oldroyd-B nanofluid with motile organisms and effective Prandtl approach. Phys A. 2019;527:121179.
Komeilibirjandi A, Raffiee AH, Maleki A, Nazari MA, Shadloo MS. Thermal conductivity prediction of nanofluids containing CuO nanoparticles by using correlation and artificial neural network. J Therm Anal Calorim. 2019. https://doi.org/10.1007/s10973-019-08838-w.
Liao S, Tan Y. A general approach to obtain series solutions of nonlinear differential equations. Stud Appl Math. 2007;119(4):297–354.
Loganathan K, Rajan S. An entropy approach of Williamson nanofluid flow with Joule heating and zero nanoparticle mass flux. J Therm Anal Calorim. 2020. https://doi.org/10.1007/s10973-020-09414-3.
Freidoonimehr N, Rahimi AB. Brownian motion effect on heat transfer of a three-dimensional nanofluid flow over a stretched sheet with velocity slip. J Therm Anal Calorim. 2019. https://doi.org/10.1007/s10973-018-7060-y.
Loganathan K, Mohana K, Mohanraj M, Sakthivel P, Rajan S. Impact of 3rd-grade nanofluid flow across a convective surface in the presence of inclined Lorentz force: an approach to entropy optimization. J Therm Anal Calorim. 2020. https://doi.org/10.1007/s10973-020-09751-3.
Loganathan K, Sivasankaran S, Bhuvaneshwari M, Rajan S. Second-order slip, cross-diffusion and chemical reaction effects on magneto-convection of Oldroyd-B liquid using Cattaneo–Christov heat flux with convective heating. J Therm Anal Calorim. 2019;136:401–9.
Sadeghy K, Hajibeygi H, Taghavi SM. Stagnation-point flow of upper-convected Maxwell fluids. Int J Non-Linear Mech. 2006;41:1242.
Mukhopadhyay S. Heat transfer analysis of the unsteady flow of a Maxwell fluid over a stretching surface in the presence of a heat source/sink. Chin Phys Lett. 2012;29:054703.
Abbasi FM, Mustafa M, Shehzad SA, Alhuthali MS, Hayat T. Analytical study of Cattaneo–Christov heat flux model for a boundary layer flow of Oldroyd-B fluid. Chin Phys B. 2016;25:014701.
Fang T, Zhang J, Yao S. Slip MHD viscous flow over a stretching sheet an exact solution. Commun Nonlinear Sci Numer Simul. 2009;14:3731–7.
Makinde OD, Aziz A. Boundary layer flow of nanofluid past a stretching sheet with a convective boundary condition. Int J Therm Sci. 2011;50:1326–32.
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10973-020-09847-w