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Penetrative phototactic bioconvection in an isotropic scattering suspension
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10.1063/1.4813402
/content/aip/journal/pof2/25/7/10.1063/1.4813402
http://aip.metastore.ingenta.com/content/aip/journal/pof2/25/7/10.1063/1.4813402

Figures

Image of FIG. 1.
FIG. 1.

Schematic diagram of the computational domain. The intensities incident on the non-reflecting top and bottom surfaces pass through them unchanged. The lateral walls are symmetric with respect to the intensities passing through them.

Image of FIG. 2.
FIG. 2.

Typical taxis functions with critical intensity (a) = 1 and (b) = 1.54.

Image of FIG. 3.
FIG. 3.

(a) Staggered grid geometry, (b) a typical control volume with control angle orientation, and (c) angular discretization in the upper hemisphere.

Image of FIG. 4.
FIG. 4.

(a)–(d) Schematic diagram of four sweeps to calculate diffused intensities in all directions.

Image of FIG. 5.
FIG. 5.

Neutral curve (solid line for the stationary mode and dashed line for the oscillatory mode) at the onset of bioconvection for . The taxis function with critical intensity = 1 [see Fig. 2(a) ] is given in Eq. (28) . The solutions obtained from the bioconvection code are marked on the lines using squares.

Image of FIG. 6.
FIG. 6.

Basic equilibrium concentration profiles for = 10: (a) τ = 0.5 and (b) τ = 1.0. The concentration profile becomes steeper for a higher value τ.

Image of FIG. 7.
FIG. 7.

Streamlines of steady solutions for = 10, τ = 0.5, and ω = 0.47. The peak of the basic equilibrium concentration profile for these parameters is near the midheight of the chamber. The concentration contours (dotted lines) are overplotted in (a) and the values on the concentration contours increase as we move towards the inner contours.

Image of FIG. 8.
FIG. 8.

(a) Concentration and (b) horizontal velocity component on the mid-horizontal line = 0.5 in the steady state. The spatial and angular grid systems × × × are 32 × 32 × 24 × 28 (solid line), 64 × 64 × 24 × 28 (solid line with triangles), and 32 × 32 × 48 × 56 (solid line with cross). The parameter values are = 10, τ = 0.5, ω = 0.47, and = 20 .

Image of FIG. 9.
FIG. 9.

Components of the average swimming orientation (solid contour lines) in the steady state: (a) horizontal component ⟨ ⟩ and (b) vertical component ⟨ ⟩. Critical intensity occurs on the thick zero (solid) contour line. The total light intensity is overplotted with swimming orientation components using dotted contour lines. The fixed parameter values are = 10, τ = 0.5, ω = 0.47, and = 1.5 .

Image of FIG. 10.
FIG. 10.

Streamlines of steady solutions for = 10, τ = 0.5, and ω = 0.39. The peak of the basic equilibrium concentration profile for these parameters is near the three-quarter height of the chamber.

Image of FIG. 11.
FIG. 11.

Components of the average swimming orientation (solid contour lines) in the final steady state: (a) horizontal component ⟨ ⟩ and (b) vertical component ⟨ ⟩. Critical intensity occurs on the thick zero (solid) contour line. The total light intensity is overplotted with swimming orientation components using dotted contour lines. The fixed parameter values are = 10, τ = 0.5, ω = 0.39, and = 1.5 .

Image of FIG. 12.
FIG. 12.

Snapshots of instantaneous streamlines during one cycle of the oscillation for = 10, τ = 0.5, ω = 0.39, and = 10 . The peak of the basic equilibrium concentration profile for these parameters is near the three-quarter height of the chamber. Time increases from (a) to (d).

Image of FIG. 13.
FIG. 13.

(a)–(d) Streamlines of steady solutions for = 10, τ = 1.0, and ω = 0.59. The peak of the basic equilibrium concentration profile for these parameters is near the midheight of the domain.

Image of FIG. 14.
FIG. 14.

(a)–(d) Streamlines of steady solutions for = 10, τ = 1.0, and ω = 0.55. The peak of the basic equilibrium concentration profile for these parameters is near the three-quarter height of the domain.

Image of FIG. 15.
FIG. 15.

Neutral curve showing stationary (solid line) and oscillatory (dashed line) branches. Fixed parameter values are = 20, τ = 1.0, and ω = 0.575.

Image of FIG. 16.
FIG. 16.

Variation of the central concentration, = (λ/2, 1/2), with time: (a) = 1.03 and (b) = 1.5 . The fixed parameter values are = 20, τ = 1.0, and ω = 0.575.

Image of FIG. 17.
FIG. 17.

(a)–(f) The instantaneous streamlines and concentration are plotted using solid and dotted lines, respectively, during one cycle of the oscillation for = 1.03 . The fixed parameter values are = 20, τ = 1.0, and ω = 0.575.

Image of FIG. 18.
FIG. 18.

(a) Basic equilibrium concentration profile and (b) the corresponding neutral curve for fixed parameters = 10, τ = 0.7, and ω = 0.9.

Image of FIG. 19.
FIG. 19.

Components of the average swimming orientation (solid contour lines) in the initial state: (a) horizontal component ⟨ ⟩ and (b) vertical component ⟨ ⟩. Critical intensity occurs on the thick zero (solid) contour line. The total light intensity is overplotted with swimming orientation components using dotted contour lines. The fixed parameter values are = 10, τ = 0.7, ω = 0.9, and = 1.05 .

Image of FIG. 20.
FIG. 20.

Components of the average swimming orientation (solid contour lines) in the final steady state: (a) horizontal component ⟨ ⟩ and (b) vertical component ⟨ ⟩. Critical intensity occurs on the thick zero (solid) contour line. The total light intensity is overplotted with swimming orientation components using dotted contour lines. The fixed parameter values are = 10, τ = 0.7, ω = 0.9, and = 1.05 .

Image of FIG. 21.
FIG. 21.

Streamlines of steady solution for = 10, τ = 0.7, ω = 0.9, and = 1.05 . The peak of the basic equilibrium concentration profile for these parameters is located near two depths, i.e., around = 0.9 and = 1.0 of the domain.

Tables

Generic image for table
Table I.

Stream-function (ψ) and vorticity (ζ) at the centre of primary vortex for the two-dimensional cavity problem.

Generic image for table
Table II.

Comparison of characteristic values for the natural convection in a side-heated cavity for Prandtl number = 0.71 and (thermal) Rayleigh number .

Generic image for table
Table III.

Critical Rayleigh numbers and critical wavelengths at the onset of phototactic bioconvection for = 10: (a) τ = 0.5, (b) τ = 1.0.

Generic image for table
Table IV.

Comparison of the characteristic values on different grids for = 10, τ = 0.5, ω = 0.47, and = 20 .

Generic image for table
Table V.

Summary of the numerical results. See the text for the explanation of the notations. The critical wavelengths are shown in the last row of the table.

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/content/aip/journal/pof2/25/7/10.1063/1.4813402
2013-07-17
2014-04-17
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Penetrative phototactic bioconvection in an isotropic scattering suspension
http://aip.metastore.ingenta.com/content/aip/journal/pof2/25/7/10.1063/1.4813402
10.1063/1.4813402
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