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Penetrative phototactic bioconvection in an isotropic scattering suspension

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10.1063/1.4813402

### Abstract

Phototaxis is a directed swimming response dependent upon the light intensity sensed by micro-organisms. Positive (negative) phototaxis denotes the motion directed towards (away from) the source of light. Using the phototaxis model of Ghorai, Panda, and Hill [“Bioconvection in a suspension of isotropically scattering phototactic algae,” Phys. Fluids22, 071901 (Year: 2010)]10.1063/1.3457163, we investigate two-dimensional phototactic bioconvection in an absorbing and isotropic scattering suspension in the nonlinear regime. The suspension is confined by a rigid bottom boundary, and stress-free top and lateral boundaries. The governing equations for phototactic bioconvection consist of Navier–Stokes equations for an incompressible fluid coupled with a conservation equation for micro-organisms and the radiative transfer equation for light transport. The governing system is solved efficiently using a semi-implicit second-order accurate conservative finite-difference method. The radiative transfer equation is solved by the finite volume method using a suitable step scheme. The resulting bioconvective patterns differ qualitatively from those found by Ghorai and Hill [“Penetrative phototactic bioconvection,” Phys. Fluids17, 074101 (Year: 2005)]10.1063/1.1947807 at a higher critical wavelength due to the effects of scattering. The solutions show transition from steady state to periodic oscillations as the governing parameters are varied. Also, we notice the accumulation of micro-organisms in two horizontal layers at two different depths via their mean swimming orientation profile for some governing parameters at a higher scattering albedo.

© 2013 AIP Publishing LLC

Received 05 February 2013
Accepted 17 June 2013
Published online 17 July 2013

Acknowledgments: M. K. Panda wishes to thank National Board for Higher Mathematics (NBHM), India, for providing a Post Doctoral Fellowship and Dr. S. Ganesan for providing computing facility. Also, the authors are grateful to the referees for their valuable comments and suggestions which have helped in better exposition of the paper.

Article outline:

I. INTRODUCTION

II. PHOTOTAXIS WITH ABSORPTION AND SCATTERING

III. MATHEMATICAL FORMULATION

A. Geometry of the problem

B. Governing equations

C. Boundary conditions

D. Scaling of the equations

E. Average orientation

IV. NUMERICAL SCHEME

A. Finite volume method for RTE

B. Solution procedure

V. CODE VALIDATION

A. Flow in a driven cavity

B. Natural convection in three dimensions

C. Computation of critical Rayleigh number

VI. NUMERICAL RESULTS

A. *V* _{ c } = 10

B. *V* _{ c } = 15 and *V* _{ c } = 20

VII. SUMMARY OF RESULTS

VIII. CONCLUSIONS

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2013-07-17

2014-04-16

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