The onset of bioconvection contains both nanoparticles and gyrotactic microorganisms confined within a Hele-Shaw cell is investigated by incorporating the effects of Brownian diffusion and thermophoresis by using the zero flux nanoparticle boundary conditions. The linear analysis is based on the normal mode technique and the resulting equations are solved numerically by the higher order Galerkin weighted Residual method. The critical Hele-Shaw Rayleigh number is presented in terms of bioconvection parameters, nanofluid parameters, and Hele-Shaw parameters. It is found that the highly promoted disturbance in the presence of gyrotactic microorganisms enhances heat transfer in nanofluids. Gyrotactic microorganisms enhance the bioconvection and this effect is larger if both the concentration and average speed of microorganisms have larger values. Wavenumber is the function of Hele-Shaw Rayleigh number, bioconvection Péclet number and Gyrotaxis number. A comparison is also made between the different bioconvection Péclet number and bioconvection Hele-Shaw number. The present study may found applications in bio-convection nanotechnological devices.
nanofluid, Hele-Shaw cell, thermophoresis, Brownian motion, bioconvection, gyrotactic microorganism
Aniss S., Souhar M., Brancher J. P. (1995). Asymptotic study and weakly nonlinear analysis at the onset of Rayleigh–Bénard convection in Hele-Shaw cell. Physics of Fluids, Vol. 7, No. 5, pp. 926-934. http://doi.org/10.1063/1.868568
Baehr H. D., Stephan K. (2006). Heat and mass transfer. Springer, Vol. 32, pp.147-164. http://doi.org/10.1007/978-3-662-03659-4
Bees M. A., Hill N. A. (1997). Wavelengths of bioconvection patterns. The Journal of Experimental Biology, Vol. 200, No. 10, pp. 1515-1526. http://www.ncbi.nlm.nih.gov/pubmed/9319416
Buongiorno J. (2006). Convective transport in nanofluids. Journal of Heat Transfer, Vol. 128, pp. 240-250. http://doi.org/10.1115/1.2150834
Chandrasekhar S. (1961). Hydrodynamic and Hydrodynamic stability. OUP.
Childress S., Levandowsky M., Spiegel E. A. (1975). Pattern formation in a suspension of swimming micro-organisms: Equations and stability theory. Journal of Fluid Mechanics, Vol. 69, No. 3, pp. 591-613. https://doi.org/10.1017/S0022112075001577
Chol S. U. S. (1995). Enhancing thermal conductivity of fluids with nanoparticles. ASME-Publications-Fed, Vol. 231, pp. 99-106.
Das S. K., Putra N., Roetzel W. (2003). Pool boiling characteristics of nano-fluids. International Journal of Heat and Mass Transfer, Vol. 46, No. 5, pp. 851-862.
Ebrahimi S., Sabbaghzadeh J., Lajevardi M., Hadi I. (2010). Cooling performance of a microchannel heat sink with nanofluids containing cylindrical nanoparticles (carbon nanotubes). Heat and Mass Transfer/Waerme- Und Stoffuebertragung, Vol. 46, No. 5, pp. 549-553. https://doi.org/10.1007/s00231-010-0599-1
Fan X., Chen H., Ding Y., Plucinski P. K., Lapkin A. A. (2008). Potential of ‘nanofluids’ to further intensify microreactors. Green Chemistry, Vol. 10, No. 6, pp. 670-677. http://doi.org/10.1039/B717943J
Ghorai S., Hill N. A. (1999). Development and stability of gyrotactic plumes in bioconvection. Journal of Fluid Mechanics, Vol. 400, pp. 1–31. http://doi.org/10.1017/s0022112099006473
Guo J., Kaloni P. N. (1995). Double‐diffusive convection in a porous medium, nonlinear stability, and the brinkman effect. Studies in Applied Mathematics, Vol. 94, No. 3, pp. 341-358. http://doi.org/10.1002/sapm1995943341
Hartline B. K., Lister C. R. B. (1977). Thermal convection in a Hele-Shaw cell. Journal of Fluid Mechanics, Vol. 79, No. 2, pp. 379-389. http://doi.org/10.1017/S0022112077000202
Hele-Shaw H. S. (1898). The flow of water. Nature, Vol. 58, No. 1489, pp. 33-36. http://doi.org/10.1038/058034a0
Huh D., Matthews B. D., Mammoto A., Montoya-Zavala M., Hsin H. Y., Ingber D. E. (2010). Reconstituting organ-level lung functions on a chip. Science, Vol. 328, No. 5986, pp. 1662-1668. https://doi.org/10.1126/science.1188302
Kuznetsov A. V., Avramenko A. A. (2002). A 2D analysis of stability of bioconvection in a fluid saturated porous medium - Estimation of the critical permeability value. International Communications in Heat and Mass Transfer, Vol. 29, No. 2, pp. 175-184. https://doi.org/10.1016/S0735-1933(02)00308-1
Kuznetsov A. V. (2010). The onset of nanofluid bioconvection in a suspension containing both nanoparticles and gyrotactic microorganisms. International Communications in Heat and Mass Transfer, Vol. 37, No. 10, pp. 1421-1425. http://doi.org/10.1016/j.icheatmasstransfer.2010.08.015
Kuznetsov A. V., Avramenko A. A. (2003). The effect of deposition and declogging on the critical permeability in bioconvection in a porous medium. Acta Mechanica, Vol. 160, No. 1, pp. 113-125. http://doi.org/10.1007/s00707-002-0978-x
Kvernvold O. (1979). On the stability of non-linear convection in a Hele-Shaw cell. International Journal of Heat and Mass Transfer, Vol. 22, No. 3, pp. 395-400. http://doi.org/10.1016/0017-9310(79)90006-1
Nield D. A., Kuznetsov A. V. (2014a). Thermal instability in a porous medium layer saturated by a nanofluid: A revised model. International Journal of Heat and Mass Transfer, Vol. 68, pp. 211-214. https://doi.org/10.1016/j.ijheatmasstransfer.2013.09.026
Nield D. A., Kuznetsov A. V. (2009). Thermal instability in a porous medium layer saturated by a nanofluid. International Journal of Heat and Mass Transfer, Vol. 52, No. 25, pp. 5796-5801. http://doi.org/10.1016/j.ijheatmasstransfer.2013.09.026
Nield D. A., Kuznetsov A. V. (2010). The onset of convection in a horizontal nanofluid layer of finite depth. European Journal of Mechanics-B/Fluids, Vol. 29, No. 3, pp. 217-223. http://doi.org/10.1016/j.euromechflu.2010.02.003
Nield D. A., Kuznetsov A. V. (2014b). The onset of convection in a horizontal nanofluid layer of finite depth: A revised model. International Journal of Heat and Mass Transfer, Vol. 77, pp. 915-918. http://doi.org/10.1016/j.euromechflu.2010.02.003
Pedley T. J., Hill N., Kessler J. O. (1988). The growth of bioconvection patterns in a uniform suspension of gyrotactic micro-organisms. Journal of Fluid Mechanics, Vol. 195, No. 1, pp. 223. https://doi.org/10.1017/S0022112088002393
Pedley T. J., Kessler J. O. (1987). The orientation of spheroidal microorganisms swimming in a flow field. Proceedings of the Royal Society of London B: Biological Sciences, Vol. 231, No. 1262, pp. 47–70. http://doi.org/10.1098/rspb.1987.0035
Saini S., Sharma Y. D. (2018). A bio-thermal convection in water-based nanofluid containing gyrotactic microorganisms: Effect of vertical throughflow. Journal of Applied Fluid Mechanics, Vol. 11, No. 4, pp. 273-280.
Saini S., Sharma Y. D. (2018). Numerical study of nanofluid thermo-bioconvection containing gravitactic microorganisms in porous media: Effect of vertical throughflow. Advanced Powder Technology. http://doi.org/10.1016/j.apt.2018.07.021
Sokolov A., Goldstein R. E., Feldchtein F. I., Aranson I. S. (2009). Enhanced mixing and spatial instability in concentrated bacterial suspensions. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, Vol. 80, No. 3. https://doi.org/10.1103/PhysRevE.80.031903
Sparrow E. M., Goldstein R. J., Jonsson V. K. (1964). Thermal instability in a horizontal fluid layer: effect of boundary conditions and non-linear temperature profile. Journal of Fluid Mechanics, Vol. 18, No. 4, pp. 513-528. http://doi.org/10.1017/S0022112064000386
Tham L., Nazar R., Pop I. (2013). Mixed convection flow over a solid sphere embedded in a porous medium filled by a nanofluid containing gyrotactic microorganisms. International Journal of Heat and Mass Transfer, Vol. 62, pp. 647-660. http://doi.org/10.1016/j.ijheatmasstransfer.2013.03.012
Tzou D. Y. (2008). Thermal instability of nanofluids in natural convection. International Journal of Heat and Mass Transfer, Vol. 51, No. 11, pp. 2967-2979. http://doi.org/10.1016/j.ijheatmasstransfer.2007.09.014
Wooding R. A. (1960). Rayleigh instability of a thermal boundary layer in flow through a porous medium. Journal of Fluid Mechanics, Vol. 9, No. 2, pp. 183-192. http://doi.org/10.1017/S0022112060001031
Yadav D., Lee J. (2016). Onset of convection in a nanofluid layer confined within a Hele-Shaw cell. Journal of Applied Fluid Mechanics, Vol. 9, No. 2, pp. 519-527. http://doi.org/10.18869/acadpub.jafm.68.225.24433