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Effects of second diffusing component and cross diffusion on primary and secondary thermoconvective instabilities in couple stress liquids

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Abstract

Linear and weakly nonlinear analyses are made for the Rayleigh-B´enard convection in two-component couple-stress liquids with the Soret effect. Conditions for pitchfork, Hopf, Takens-Bogdanov, and codimension-two bifurcations are presented. The Lorenz model is used to study the inverted bifurcation. Positive values of the Soret co-efficient favor a pitchfork bifurcation, whereas negative values favor a Hopf bifurcation. Takens-Bogdanov and codimension-two bifurcations are not as much influenced by the Soret coefficient as pitchfork and Hopf bifurcations. The influence of the Soret coefficient on the inverted bifurcation is similar to the influence on the pitchfork bifurcation. The in-fluence of other parameters on the aforementioned bifurcations is also similar as reported earlier in the literature. Using the Newell-Whitehead-Segel equation, the condition for occurrence of Eckhaus and zigzag secondary instabilities is obtained. The domain of ap-pearance of Eckhaus and zigzag instabilities expands due to the presence of the Soret coefficient for positive values. The Soret coefficient with negative values enhances heat transport, while positive values diminish it in comparison with heat transport for the case without the Soret effect. The dual nature of other parameters in influencing heat and mass transport is shown by considering positive and negative values of the Soret coefficient.

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Abbreviations

A,B,C,D,E :

amplitudes

β T :

thermal expansion coefficient

β S :

solutal expansion coefficient

C S :

couple stress parameter

D S :

Soret parameter

D m :

solutal diffusivity

D 1 :

Soret coefficient

d :

depth

g :

acceleration due to gravity

κ :

thermal diffusivity

Le :

Lewis number

µ :

dynamic viscosity

µ1:

couple-stress viscosity

P :

pressure

Pr :

Prandtl number

q :

wave number

R T :

thermal Rayleigh number

R S :

solutal Rayleigh number

ρ :

density

S :

concentration

T :

temperature

t :

time

τ :

scaled time

Θ:

non-dimensional temperature

(u, v,w):

vertical velocity components

V (u', v',w'):

velocity vector

X, Y,Z :

scaled coordinate

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Acknowledgements

The second author (C. KANCHANA) is grateful to the University Grants Commission (UGC), New Delhi, India for supporting her research work with a Rajiv Gandhi National Fellowship. The authors are grateful to the two reviewers for their educative comments that helped us refine the paper to the present form.

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Authors and Affiliations

  1. Department of Humanities and Sciences, National Institute of Technology Goa, Goa, 403401, India

    R. Ravi

  2. Department of Mathematics, Jnana Bharathi Campus, Bangalore University, Bangalore, 560056, India

    C. Kanchana & P. G. Siddheshwar

Authors
  1. R. Ravi
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  2. C. Kanchana
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  3. P. G. Siddheshwar
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Correspondence to P. G. Siddheshwar.

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Ravi, R., Kanchana, C. & Siddheshwar, P.G. Effects of second diffusing component and cross diffusion on primary and secondary thermoconvective instabilities in couple stress liquids. Appl. Math. Mech.-Engl. Ed. 38, 1579–1600 (2017). https://doi.org/10.1007/s10483-017-2280-9

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  • DOI: https://doi.org/10.1007/s10483-017-2280-9

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