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Weakly nonlinear thermohaline convection in a sparsely packed porous medium due to horizontal magnetic field

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Abstract

Thermohaline convection in a sparsely packed porous medium is studied due to horizontal magnetic field, using both linear and weakly nonlinear stability analyses. The Darcy–Lapwood–Brinkman (DLB) model is employed as the momentum equation. In the linear stability analysis, the normal mode technique is used to find the thermal critical Rayleigh number which is a function of q, Da, \(\Lambda \), \(R_2\) and L. In the weakly nonlinear analysis, a nonlinear two-dimensional Landau–Ginzburg (LG) equation is derived at the onset of stationary convection and the secondary instabilities and heat transport by convection are studied. Coupled one-dimensional LG equations are derived at the onset of oscillatory convection, and the stability regions of steady state, standing waves and travelling waves are studied.

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Data Availability Statement

This manuscript has associated data in a data repository. [Authors’ comment: The data used to support the findings of this study are included within the article (https://doi.org/10.1615/JPorMedia.v10.i8.70)].

Abbreviations

\(\overline{V}\) :

Vector fluid velocity

u,  v,  w :

Velocity components

\(\overline{H}\) :

Vector magnetic field

\(H_x,H_y,H_z\) :

Magnetic field components

\(\Delta T\) :

Temperature difference

\(\Delta S\) :

Salinity difference

P :

Pressure

g :

Acceleration due gravity

\(k_T\) :

Thermal diffusivity

\(k_S\) :

Saline diffusivity

k :

Permeability

\(A_{1L}\) :

Amplitude of left travelling waves

\(A_{1R}\) :

Amplitude of right travelling waves

A :

Complex amplitude

Da :

Darcy number

L :

Lewis number

\(\Lambda \) :

Brinkman number

M :

Non-dimensional heat capacity

Nu :

Nusselt number

\(Pr_1\) :

Thermal Prandtl number

\(Pr_2\) :

Magnetic Prandtl number

q :

Wavenumber

Q :

Chandrasekhar number

\(R_1\) :

Thermal Rayleigh number

\(R_2\) :

Magnetic Rayleigh number

\(\alpha \) :

Thermal expansion coefficient

\(\beta \) :

Solute expansion coefficient

\(\eta \) :

Magnetic diffusivity

\(\mu \) :

Fluid viscosity

\(\mu _e\) :

Effective fluid viscosity

\(\mu _m \) :

Magnetic permeability

\(\phi \) :

Porosity

\(\rho \) :

Fluid density

\(\nu \) :

Kinematic viscosity

\(\omega \) :

Vorticity

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Author information

Authors and Affiliations

  1. Department of Mathematics, National Institute of Technology Warangal, Warangal, 506004, India

    A. Benerji Babu & S. G. Tagare

  2. Department of HBS, Godavari Institute of Engineering and Technology, Rajahmundry, Andhra Pradesh, 533296, India

    N. Venkata Koteswara Rao

Authors
  1. A. Benerji Babu
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  2. N. Venkata Koteswara Rao
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  3. S. G. Tagare
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Corresponding author

Correspondence to A. Benerji Babu.

Appendix

Appendix

In linear stability analysis, the thermal Rayleigh number \(R_1\) coefficients are

$$\begin{aligned} \mathcal{K}&=\left( \delta ^4 +\frac{\phi ^2}{L^2}\omega ^2\right) ^{-1}\left( M^2 \delta ^4 +\phi ^2 \frac{Pr_2^2}{Pr_1^2}\omega ^2\right) ^{-1}, \nonumber \\ c_0&=M \delta ^8\left( Q m^2 \delta ^2+\frac{1}{Da}+\Lambda \delta ^6 +\frac{R_2}{L}q^2\right) ,\nonumber \\ c_1&=\Big [M(m^2 Q \delta ^2 + \frac{\delta ^4}{Da}+\Lambda \delta ^6)+\frac{\delta ^6}{Pr_1}\left( \frac{1}{\phi }-m^2 Q\phi Pr_2 \delta ^2\right) +\frac{M q^2 R_2}{L^2}(L-\phi )\Big ]\delta ^6,\nonumber \\ c_2&= \frac{\delta ^{10}}{\phi Pr_1}-\frac{q^2 \delta ^4 \phi R_2}{L}\left( \frac{M}{L}+\frac{\phi }{M}\frac{Pr_2^2}{Pr_1^2}\right) -\delta ^8 \phi ^2 M \left( \frac{1}{L^2}+\frac{1}{M^2}\frac{Pr_2^2}{Pr_1^2}\right) \left( \frac{1}{Da}+\delta ^2 \Lambda \right) \nonumber \\&\quad - m^2 Q \delta ^6 \phi (\frac{M\phi }{L^2}+\frac{Pr_2}{Pr_1}), \nonumber \\ c_3&=\frac{q^2 \delta ^2 \phi ^2}{L^2 M}\frac{Pr_2^2}{Pr_1^2}R_2 (\phi -L)-\Big [ M \delta ^6 \phi ^2 \left( \frac{1}{Da}+\delta ^2 \Lambda \right) -\frac{\phi \delta ^8}{Pr_1}\Big ]\left( \frac{1}{M^2}\frac{Pr_2^2}{Pr_1^2}+\frac{1}{L^2}\right) \nonumber \\&\quad -\frac{m^2 Q\delta ^4 \phi ^2}{L^2}(M-\phi \frac{Pr_2}{Pr_1}), \nonumber \\ c_4=&\frac{\delta ^4 \phi ^4}{L^2 M}\frac{Pr_2^2}{Pr_1^2}\left( \frac{1}{Da}+\Lambda \delta ^2\right) +\frac{q^2 \phi ^3}{L^2M}\frac{Pr_2^2}{Pr_1^2} R_2-\frac{\delta ^6 \phi }{Pr_1}\left( \frac{1}{L^2}+\frac{1}{M^2}\frac{Pr_2^2}{Pr_1^2}\right) +\frac{m^2 Q\delta ^6 \phi ^3}{L^3}\frac{Pr_2}{Pr_1}, \nonumber \\ c_5&=\frac{\delta ^2 \phi ^4}{L^2 M}\frac{Pr_2^2}{Pr_1^2}\left( \frac{1}{Da}+\Lambda \delta ^2+\frac{\delta ^2}{M\phi Pr_1}\right) , \nonumber \\ c_6&=\frac{\delta ^2 \phi ^3 Pr_2^2}{L^2 M^2 Pr_1^3}. \end{aligned}$$
(66)

At marginal stability analysis, when \(R_1\) is an independent variable, the following are the coefficients of polynomial in p,

$$\begin{aligned} A&=\frac{\delta ^4 \phi Pr_2}{L M^2 Pr_1^2} ,\nonumber \\ B&= \frac{\delta ^4 Pr_2}{M^2 Pr_1^2}\left( 1+\frac{\phi }{L}\right) +\frac{\delta ^2 \phi ^2 Pr_2}{L M Pr_1}\left( \Lambda +\frac{1}{Da}\right) +\frac{\delta ^4}{L M Pr_1} ,\nonumber \\ C&=\frac{\phi \delta ^2}{L}\left( m^2 Q+\frac{\delta ^2}{Da}\right) +\frac{\delta ^4 \phi Pr_2}{M Pr_1}\left( 1+\frac{\phi }{L}\right) \left( \Lambda \phi \delta ^2 + \frac{1}{Da}\right) +\frac{\delta ^6}{M Pr_1}\left( \frac{1}{\phi }+\frac{Pr_2}{M Pr_1}\right) \nonumber \\&\quad +\frac{\delta ^6}{L}\left( \Lambda \phi +\frac{1}{MPr_1}\right) +\frac{q^2 \phi Pr_2}{L M Pr_1}(R_1 - R_1 \phi ) ,\nonumber \\ D&=q^2 \delta ^2 \left[ \frac{\phi Pr_2}{M Pr_1}\left( R_1 +\frac{R_2}{L}\right) +\frac{1}{L}(R_1 -\phi R_1) \right] +m^2 Q \delta ^4 \left( 1+\frac{\phi }{L}\right) +\frac{\delta 6}{Da}(1+\frac{\phi }{L} \nonumber \\&\quad +\frac{\phi Pr_2}{M Pr_1})+ \delta ^8 \left[ \Lambda \left( 1+\frac{\phi }{L}\right) +\frac{1}{MPr_1}\left( \Lambda \phi Pr_2 +\frac{1}{\phi }\right) \right] ,\nonumber \\ E&=\delta ^4 \left[ m^2 Q\delta ^2 +\delta ^4 (\Lambda \delta ^2 )+\frac{1}{Da}-q^2 \left( R_1-\frac{R_2}{L})\right) \right] . \end{aligned}$$
(67)

These are coefficients of coupled LG equation derived at the onset of oscillatory convection.

$$\begin{aligned} \Lambda _{0}&=\left[ \delta ^4 \left( 1+\frac{\phi }{L}\right) \left( m^2Q+\frac{\delta ^2}{Da}+\Lambda \delta ^2\right) -(\phi R_1-R_2)\frac{q^2 \delta ^2}{L}-\frac{q^2 \delta ^2 \phi R_{12}Pr_2}{M Pr_1} \right. \nonumber \\&\quad -\frac{3 \delta ^4 \omega ^2 Pr_2}{M^2 Pr_1^2}\left( 1+\frac{\phi }{L}\right) +\frac{\delta ^2 \phi Pr_2}{M Pr_1}(\frac{1}{+}{Da}+\Lambda \delta ^2)\left( \delta ^4-\frac{3 \phi ^2 \omega ^2}{L}\right) +\frac{\delta ^4}{M Pr_1}\nonumber \\&\quad \left( \frac{\delta ^4}{\phi }-\frac{3 \omega ^2}{L}\right) \Big ]+i \Big [\frac{2 \delta ^6 \omega }{L}\left( \Lambda \phi +\frac{1}{M Pr_1}\right) +\frac{2 \delta ^6 \omega }{M Pr_1}\left( \frac{1}{\phi }+\frac{Pr_2}{M Pr_1}\right) +\frac{2 \delta ^2 \phi \omega }{L}\nonumber \\&\quad \left( m^2Q+\frac{\delta ^2}{Da}\right) +\frac{2 \delta ^4 \phi \omega Pr_2}{M Pr_1}\left( \frac{1}{Da}+\Lambda \delta ^2\right) \left( 1+\frac{\phi }{L}\right) -\frac{4\delta ^2 \phi \omega ^3 Pr_2}{L M^2 Pr_1^2}\Big ], \nonumber \\ \Lambda _{1}&=\Big [\frac{q^2 \phi Pr_2}{L M Pr_1}(R_2 -\phi R_1)+\frac{\delta 6}{M Pr_1}\left( \frac{1}{L}+\frac{1}{\phi }+\frac{Pr_2}{Pr_1}\right) +\frac{\delta ^2 \phi }{L}\left( m^2 Q-\frac{6 \omega ^2Pr_2}{M^2 Pr_1^2}\right) \nonumber \\&\quad +\delta ^4 \phi \left( \frac{1}{Da}+\delta ^2 \Lambda \right) \left( \frac{1}{L}+\frac{Pr_2}{Pr_1}+\frac{\phi Pr_2}{L M Pr_1}\right) \Big ]+i \Big [\frac{3\delta ^2 \phi ^2 \omega Pr_2}{L M Pr_1}\left( \frac{1}{Da}+\delta ^2 \Lambda \right) \nonumber \\&\quad +\left. \frac{3\delta ^2 \omega }{M Pr_1}\left( \frac{1}{L}+\frac{Pr_2}{M Pr_1}+\frac{\phi Pr_2}{L M Pr_1}\right) \right] , \nonumber \\ \Lambda _2&=\Bigg [\frac{\delta ^2 \phi \omega ^2 Pr_2}{M Pr_1}\left( 1+\frac{\phi }{L}\right) \left( \frac{2}{Da}+3 \Lambda \delta ^2\right) + \frac{3 \delta ^4 \omega ^2}{M Pr_1}\left( \frac{1}{L}+\frac{1}{\phi }+\frac{Pr_2}{M Pr_1}\right) \nonumber \\&\quad -\left( \frac{\omega ^2}{M Pr_1}+\phi R_1 - R_2\right) +\delta ^2 \left( R_1 - \frac{R_2}{L}\right) (2 Qq^2 +\delta ^2)+\frac{\phi \omega ^2}{L}\left( m^2 Q+\frac{2\delta ^2}{Da}\right) \nonumber \\&\quad +\delta ^4\left( 3m^2Q+\frac{4 \delta ^2}{Da}+5 \Lambda \delta ^4-\frac{3 \omega ^2 \Lambda \delta ^4}{L}\right) \Bigg ]+i\Bigg [\frac{\phi ^2 \omega ^3 Pr_2}{L M Pr_1}\left( \frac{1}{Da}+2\delta ^2\right) \nonumber \\&\quad -2 \omega \delta ^2 \left( 1+\frac{\phi }{L}\right) \left( m^2 Q- \frac{\omega ^2Pr_2}{M^2 Pr_1^2}\right) -4 \delta ^6 \omega \left( \frac{\Lambda \phi }{ L}+\frac{1}{M \phi Pr_1}+\frac{\Lambda \phi Pr_2}{M Pr_1}\right) \nonumber \\&\quad + \delta ^2 \omega \left( \frac{2 \omega 62}{L M Pr_1}-\frac{3 \delta ^2 \phi }{Da L}-4 \delta ^4 \Lambda \right) +\phi \omega R_1 (q^2+\delta ^2)\left( \frac{1}{L}+\frac{Pr_2}{M Pr_1}\right) \nonumber \\&\quad + \omega \left( 1+\frac{\phi Pr_2}{M Pr_1}\right) \left( \frac{(q^2+\delta ^2)R_2}{L}+\frac{3\delta ^4}{Da}\right) , \nonumber \\ \Lambda _{3}&=q^2\left( \delta ^4-\frac{\phi ^2 \omega ^2 Pr_2}{L M Pr_1}\right) +i q^2 \delta ^2 \phi \omega \left( \frac{1}{L}+\frac{Pr_2}{M Pr_1}\right) , \nonumber \\ \Lambda _{4}&=-\frac{Q Pr_2}{4 M \pi ^2 Pr_1}e_1e_2e_3\left[ \frac{s_1}{e_3}\left( \frac{1}{e_6}\right) +\left( \frac{s_1}{2}+s_2\right) \left( \frac{1}{e_3^3}+\frac{1}{|e_3|^2}\right) \right] +\frac{R_1 q^2 }{4M^3}e_2e_3\left( \frac{1}{e_1}+\frac{1}{e_1^*}\right) \nonumber \\&\quad - \frac{R_2 q^2}{4 M^3L^3}e_1e_3\left( \frac{1}{e_2}+\frac{1}{e_2^*}\right) -\frac{Qm^2 \delta ^2}{4M}\left( \frac{Pr_2}{Pr_1}\right) ^2e_1e_2\left( \frac{1}{e_3}+\frac{1}{e_3^*}\right) , \nonumber \\ \Lambda _{5}&=-\frac{Q Pr_2}{Pr_1}e_1e_2e_3\left\{ \frac{1}{4 M \pi ^2}\left[ \frac{s_1}{e_3}\left( \frac{1}{e_3}+\frac{1}{e_3^*}\right) +(s_1+s_2)\left( \frac{1}{e_3^2}+\frac{1}{|e_3|^2}\right) \right] +\frac{1}{|e_3|^2e_6}\left( \frac{3s_1}{2}+s_2\right) \right\} \nonumber \\&\quad +\frac{R_1 q^2}{M}e_2e_3\left[ \frac{1}{4M^2}\left( \frac{1}{e_1}+\frac{1}{e_1^*}\right) +\frac{\pi ^2}{M^2e_1e_4}\right] -\frac{R_2 q^2}{M L}e_1e_3\left[ \frac{1}{4M^2 L^2}\left( \frac{1}{e_2}+\frac{1}{e_2^*}\right) +\frac{\pi ^2}{M^2 L^2 e_2e_5} \right] \nonumber \\&\quad -\frac{Q \delta ^2 Pr_2}{Pr_1}e_1e_2 \left[ \frac{m^2 Pr_2}{4 M Pr_1}\left( \frac{1}{e_3}+\frac{1}{e_3^*}\right) +\frac{m^2\pi ^2Pr_2}{Pr_1e_3e_6}\right] , \end{aligned}$$
(68)

where

$$\begin{aligned} e_1&=i \omega +\delta ^2, \quad e_4=i\omega +2\pi ^2, \quad s_1=-2m^2\pi ^4\frac{Pr_2}{Pr_1}, \\ e_2&=\frac{\phi }{L}\omega +\delta ^2, \quad e_5=\frac{\phi }{L}i\omega +2\pi ^2, \quad s_2=-l^2m^2\pi ^2\frac{Pr_2}{Pr_1}, \\ e_3&=\phi \frac{Pr_2}{Pr_1}i \omega +\delta ^2, \quad e_6=\phi \frac{Pr_2}{Pr_1}i \omega +2 M \pi ^2. \end{aligned}$$

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Babu, A.B., Rao, N.V.K. & Tagare, S.G. Weakly nonlinear thermohaline convection in a sparsely packed porous medium due to horizontal magnetic field. Eur. Phys. J. Plus 136, 795 (2021). https://doi.org/10.1140/epjp/s13360-021-01736-x

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