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International Journal of Geophysics
Volume 2011 (2011), Article ID 207123, 17 pages
http://dx.doi.org/10.1155/2011/207123
Research Article

Nonlinear Magnetoconvection in a Sparsely Packed Porous Medium

1Department of Mathematics, National Institute of Technology Warangal, Warangal 506004, India
2Disha Institute of Management and Technology, Satya Vihar, Vidhan Sabha-Chandrakhuri Marg, Raipur 492101, India

Received 2 July 2011; Revised 6 September 2011; Accepted 7 September 2011

Academic Editor: Steve Milan

Copyright © 2011 A. Benerji Babu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Linear and weakly nonlinear properties of magnetoconvection in a sparsely packed porous medium are investigated. We have obtained the values of Takens-Bogdanov bifurcation points and codimension two bifurcation points by plotting graphs of neutral curves corresponding to stationary and oscillatory convection for different values of physical parameters relevant to magnetoconvection in a sparsely packed porous medium near a supercritical pitchfork bifurcation. We have derived a nonlinear two-dimensional Ginzburg-Landau equation with real coefficients by using Newell-Whitehead (1969) method. The effect of the parameter values on the stability mode is investigated and shown the occurrence of secondary instabilities namely, Eckhaus and Zigzag instabilities. We have studied Nessult number contribution at the onset of stationary convection. We have also derived two nonlinear one-dimensional coupled Ginzburg-Landau-type equations with complex coefficients near the onset of oscillatory convection at a supercritical Hopf bifurcation and discussed the stability regions of standing and travelling waves.

1. Introduction

Magnetoconvection in a porous medium uniformly heated from below is of considerable interest in geophysical fluid dynamics, as this phenomena may occur within the mushy layer of Earth's outer core. Earth's outer core consists of molten iron and lighter alloying element, sulphur in its molten form. This lighter alloying element present in the liquid phase is released as the new iron freezes due to supercooling onto the solid inner core. Hence we get mushy layer near the inner core boundary where the problem becomes convective instability in a porous medium [1]. The effect of geomagnetic field on the magnetoconvection instability is of interest in geophysics, particular in the study of Earth's interior where the molten liquid Iron is electrically conducting, which can become convectively unstable as a result of differential diffusion.

Magnetoconvection in an electrically conducting fluid in a nonporous medium has been studied extensively [28]. However, magnetoconvection in a porous medium has not received any attention inspite of its application in geophysical fluid dynamics problems. Palm et al., [9] investigated Rayleigh-Benard convection problem in a porous medium. Brand and Steinberg [10, 11] investigated convecting instabilities in binary liquid in a porous medium; However, Plam et al. [9] and Brand et al. have made use of Darcy's law (𝜈2𝑉 is replaced by 𝐾𝑉 where 𝐾 is the permeability of a porous medium. for nonporous medium 𝐾 is infinity). They have also not considered usual convective nonlinearity. It is well known that Darcy's law breaks down in situations where in other effects like viscous shear and inertia come into play. In fact Darcy's law is applicable to densely packed porous medium. An alternative to Darcy's equation is Brinkman equation and is of the form𝜌𝜌𝜇𝑔=𝐾𝑉+𝜇𝑒2𝑉,(1) where 𝜇 is the fluid viscosity and 𝜇𝑒 is the effective fluid viscosity. Brinkman model is valid for a sparsely packed porous medium wherein there is more window fluid to flow so that the distortion of velocity give rise to the usual shear force. Lapwood [13] was the first to suggest the inclusion of convective term (𝑉)𝑉 in the momentum equation and study the Rayleigh-Benard convection in a sparsely packed porous medium. Recently, Tagare and Benerji [14] have investigated the problem of nonlinear convection in a sparsely packed porous medium due to thermal and compositional buoyancy.

In this paper we investigate the problem of magnetoconvection in a sparsely packed porous medium. The multiplicity of control parameters makes this system an interesting one for the study of hydrodynamic stability, bifurcation and turbulence [15]. Rudraiah [16] and Rudraiah and Vortmeyer [17] have studied both linear and steady nonlinear magnetoconvection in a sparsely packed porous medium using Brinkman model but they have taken effective viscosity 𝜇𝑒 same as fluid viscosity 𝜇. However, experiments show that the ratio of effective viscosity 𝜇𝑒 to fluid viscosity 𝜇 takes the value ranging from 0.5 to 10.9 [18]. In Section 2, we write basic dimensionless equations in Boussinesq approximation for magnetoconvection in a sparsely packed medium by using a momentum equation with effective viscosity different from fluid viscosity. In Section 3, we study linear stability analysis. In Section 4.1, by using multiple-scale analysis of Newell and Whitehead [19], we derive two-dimensional nonlinear Ginzburg-Landau equation in complex amplitude 𝐴(𝑋,𝑌,𝑇) with real coefficients near the super critical pitchfork bifurcation. In Section 4.2, we show the occurrence of secondary instabilities such as Eckhaus instability and Zigzag instability. We have also considered the effect of Nusselt number on heat transport by magnetoconvection in a sparsely packed porous medium. In Section 5, we derive two nonlinear one-dimensional coupled Ginzburg-Landau type equations with complex coefficients near the onset of oscillatory convection at a supercritical Hopf bifurcation. Following Matthews and Rucklidge [20], we have dropped slow space dependence in 𝑋 and obtained two coupled ordinary differential equations in 𝐴1𝑅 and 𝐴1𝐿 and discussed the stability regions of travelling and standing waves. By obtaining a one-dimensional Ginzburg-Landau equation in complex amplitude 𝐴(𝑋,𝑌,𝑇) with complex coefficients near a supercritical Hopf bifurcation, we have shown the condition for occurrence of Benjamin-Feir-type instability [21] for travelling and standing waves. In Section 6, we write conclusions of the paper.

2. Basic Equations

We consider an electrically and thermally conducting fluid saturating an infinite horizontal layer of a sparsely packed isotropic porous medium of depth 𝑑 with a uniform magnetic field 𝐻0 in the vertical 𝑧-direction. This layer is heated from below, the upper and lower bounding surfaces of the layer are assumed to be stress-free. Physical properties of the fluid are assumed to be constant, except for the density in the buoyancy term, so that the Boussinesq approximation is valid. The temperature difference across the stress-free boundaries is Δ𝑇 and the flow in the sparsely packed porous medium is governed by the Darcy-Lapwood-Brinkman model. The relevant basic equations are𝑉=0,𝐻=0,(2)𝜌01𝜙𝜕𝑉+1𝜕𝑡𝜙2𝑉𝜇𝑉𝑚𝐻4𝜋0𝜕𝐻𝜕𝑧+𝐻𝜇𝐻=𝑝+𝑚||𝐻8𝜋||2+𝜇𝑚𝐻04𝜋2𝐻𝑧+𝜌𝜇𝑔𝐾𝑉+𝜇𝑒2𝑉,(3)𝑀𝜕𝑇+𝜕𝑡𝑉𝑇=𝜅2𝑇,(4)𝜙𝜕𝐻𝜕𝑡=×𝑉×𝐻0̂𝑒𝑧+×𝑉×𝐻+𝜂2𝐻.(5) The fluid density 𝜌 is described by𝜌=𝜌0𝑇1𝛼𝑇𝑏,(6) where 𝛼=𝜌01(𝜕𝜌/𝜕𝑇) is thermal expansion coefficient and 𝜌0 is mean fluid density. Here 𝑝 is pressure, 𝑉 is mean fluid velocity, 𝑇 is temperature, 𝐻 is magnetic field, 𝜙 is porosity, 𝑔 is acceleration due to gravity, 𝐾 is permeability of porous medium, 𝜇𝑒 is coefficient of effective fluid viscosity, 𝜅 is thermal diffusivity, 𝜇𝑚 is magnetic permeability, and 𝜂 is magnetic diffusivity. Equation (3) is known as Darcy-Lapwood-Brinkman equation and is valid for 0.8<𝜙<1. Givler and Altobelli [18] shown that the range of Λ=𝜇𝑒/𝜇 varies from 0.5 to 10.9. 𝑀 is dimensionless heat capacity and is defined as the ratio of the effective heat capacity of the porous medium to the heat capacity (𝜌𝐶𝑝)𝑓 of the fluid. In a nonporous medium, 𝜙=𝑀=Λ=1 and 𝐾 and (3) reduces to Navier-Stokes equation. In this paper, for sparsely packed porous medium, we consider 𝑀=0.9,𝜙=0.9. The conduction state is characterized by𝑉𝑠=0,𝑇𝑠=𝑇0Δ𝑇𝑑𝑧,(7) and we take the temperature perturbation as 𝜃=𝑇𝑇𝑠. We use the scaling𝑥𝑥=𝑑𝑦,𝑦=𝑑𝑧,𝑧=𝑑,𝑡=𝑡𝑀𝑑2,/𝜅𝑢=𝑢𝜅/𝑀𝑑,𝑣=𝑣𝜅/𝑀𝑑,𝑤=𝑤,𝜅/𝑀𝑑𝜃=𝜃Δ𝑇,𝑃=𝑃𝜌0𝑀2𝜅2𝑑2,𝐻=𝐻𝜅𝐻0./𝜂(8) Here 𝑀𝑑2/𝜅 is thermal diffusion time in a porous medium. Using (6) and (8), we can write basic dimensionless equations for magnetoconvection in a porous medium as𝑉=0,𝐻=0,(9)1𝑀2𝜙Pr1𝜕𝑉+1𝜕𝑡𝜙𝑉𝑉𝑄Pr2Pr1𝐻𝜕𝐻𝑄𝐻𝑃𝜕𝑧=𝑀Pr1+𝑄2Pr2Pr1||𝐻||2+𝑄𝐻𝑧1𝑀𝐷𝑎𝑉+Λ𝑀2𝑉+𝑅𝜃̂𝑒𝑧,(10)𝜕𝜃+1𝜕𝑡𝑀𝑤𝑉𝜃=𝑀+2𝜃,(11)𝜙Pr2Pr1𝜕𝐻𝜕𝑡𝑀2𝐻=×𝑉×̂𝑒𝑧+Pr2Pr1×𝑉×𝐻.(12) The dimensionless parameters required for the description of the motion are Rayleigh number 𝑅=𝑔𝛼Δ𝑇𝑑3/𝜅𝜈, thermal Prandtl number Pr1=𝜈/𝜅, magnetic Prandtl number Pr2=𝜈/𝜂, Chandrasekhar number 𝑄=𝜇𝑚𝐻20𝑑2/4𝜋𝜌0𝜈𝜂, and Darcy number 𝐷𝑎=𝜅/𝑑2. The Curl of (10) gives1𝑀2𝜙Pr1𝜕+1𝜕𝑡𝑀𝐷𝑎Λ𝑀2𝜕𝜔𝑄𝐽𝜕𝑧𝑅×𝜃̂𝑒𝑧=𝑄Pr2Pr1×𝐻𝐻1𝑀2𝜙2Pr1×𝑉𝑉,(13) where vorticity 𝜔=×𝑉, current 𝐽=×𝐻 and ×𝑉𝑉=𝑉𝜔𝜔𝑉,×𝐻𝐻=𝐻𝐽𝐽𝐻.(14) The Curl of (13) in turn gives, after use of (9),1𝑀2𝜙Pr1𝜕+1𝜕𝑡𝑀𝐷𝑎Λ𝑀22𝑉𝑅2𝜃̂𝑒𝑧𝜕𝜃𝜕𝜕𝑧𝑄𝜕𝑧2𝐻=1𝑀2𝜙2Pr1×𝑉𝜔𝜔𝑉𝑄Pr2Pr1×𝐻𝐽𝐽𝐻.(15) Now taking the scalar product of (12), (13), and (15) with ̂𝑒𝑧, we get,𝜙Pr2Pr1𝜕𝜕𝑡𝑀2𝐻𝑧𝜕𝑤=𝜕𝑧Pr2Pr1̂𝑒𝑧×𝑉×𝐻.(16)1𝑀2𝜙Pr1𝜕+1𝜕𝑡𝑀𝐷𝑎Λ𝑀22𝜔𝑧+𝑄𝜕𝐽𝑧𝜕𝑧𝑅𝜕𝜃𝜕𝑥=𝑄Pr2Pr1̂𝑒𝑧×𝐻1𝐻𝑀2𝜙2Pr1̂𝑒𝑧×𝑉𝑉,(17)1𝑀2𝜙Pr1𝜕+1𝜕𝑡𝑀𝐷𝑎Λ𝑀22𝑤𝑅2𝜕𝜃𝑄𝜕𝑧2𝐻𝑧=1𝑀2𝜙2Pr1̂𝑒𝑧×𝑉𝜔𝜔𝑉𝑄Pr2Pr1̂𝑒𝑧×𝐻𝐽𝐽𝐻.(18) Geophysically acceptable velocities of propagating instabilities corresponding to geometric scalar variations occur only Pr2/Pr1>1 (where instabilities develop in ohmic diffusion timescale 𝑑2/𝜂), Pr2/Pr1=2 and 5, when the turbulent is present in the Earth's outer core. In the case of Pr2/Pr11 the instabilities are extremely slow depending on the thermal diffusion timescale 𝑑2/𝜅. Using (11), (18), and (16) can be reduced to a form𝑤=𝒩,(19) where𝒟=𝜙𝒟Pr1𝜕𝑄2𝜕𝑧2𝒟2𝑅𝑀2𝒟𝜙,(20)𝒩=𝑄𝒟2Pr2Pr1𝜕𝜕𝑧𝐻𝑤𝐻𝑉𝑧+𝒟𝒟𝜙̂𝑒𝑧1𝑀2𝜙2Pr1×𝑉𝜔𝜔𝑉𝑄Pr2Pr1×𝐻𝐽𝐽𝐻𝑅𝑀2𝒟𝜙𝑉𝜃,(21) here𝜕𝒟=𝜕𝑡2,𝒟𝜙=𝜙Pr2Pr1𝜕𝜕𝑡𝑀2,𝒟Pr1=1𝑀2𝜙Pr1𝜕+1𝜕𝑡𝑀𝐷𝑎Λ𝑀2,2=𝜕2𝜕𝑥2,2=𝜕2𝜕𝑥2+𝜕2𝜕𝑧2.(22)

Boundary Conditions
We assume that fluid is contained between 𝑧=0 and 𝑧=1, where 𝑧=0 corresponds to boundary of solid iron core with Earth's mushy layer and 𝑧=1 corresponds to boundary of Earth's mushy layer with Earth's outer liquid core. For perfectly conducting boundary with temperature, we have 𝜃=0,𝐻𝑧=0on𝑧=0,𝑧=1𝑥,𝑦.(23) Also the normal component of the velocity would vanish on 𝑧=0, 𝑧=1, that is, 𝑤=0on𝑧=0,𝑧=1𝑥,𝑦.(24) However, there are two more conditions to be imposed on velocity depending on the nature of the surface. In this paper we consider free-free boundary conditions, that is, on surfaces the tangential stresses vanish, which is equivalent to 𝑃𝑥𝑧=𝜇𝜕𝑢+𝜕𝑧𝜕𝑤𝑃𝜕𝑥=0,𝑦𝑧=𝜇𝜕𝑣+𝜕𝑧𝜕𝑤𝜕𝑦=0,(25) where 𝜇=𝛾𝜌0 is dynamic viscosity. Since 𝑤 vanishes for 𝑥,𝑦 on 𝑧=0, 𝑧=1, it follows that 𝜕𝑢/𝜕𝑧=𝜕𝑣/𝜕𝑧=0 on a free surface 𝑧=0,𝑧=1. Hence from equation of continuity we have 𝜕2𝑤/𝜕𝑧2=0 on 𝑧=0,𝑧=1 for all 𝑥, 𝑦. In this paper we have considered only the idealized stress-free conditions on the surface and vanishing of temperature fluctuations. Thus 𝑤=𝐷2𝑤=𝐷4𝑤=0 at 𝑧=0,1. 𝑤 and its even derivatives vanish at 𝑧=0 and 𝑧=1.

3. Linear Stability Analysis

We perform a linear stability analysis of the problem by substituting𝑤=𝑊(𝑧)𝑒𝑖𝑞𝑥+𝑝𝑡,(26) into linearized version of (19) is 𝑤=0, and obtaining an equation𝐷2𝑞2𝑝𝑀𝐷2𝑀𝑞2𝑝𝜙Pr2Pr1𝐷2𝑞2×Λ𝑀𝐷2𝑞21𝑀𝐷𝑎𝑝𝑀2𝜙Pr1𝑊=𝑅𝑞2𝑀𝑀𝐷2𝑀𝑞2𝑝𝜙Pr2Pr1𝐷+𝑄2𝑞2𝐷2𝑞2𝐷𝑝2𝑊.(27) We consider stress-free boundary conditions, then 𝑊=𝐷2𝑊=0 on 𝑧=0, 𝑧=1 for all 𝑥,𝑦. Thus we can assume 𝑊=sin𝜋𝑧.

Substituting 𝑊=sin𝜋𝑧 and 𝑝=𝑖𝜔 into (27), we get𝑀𝑅=𝑞2𝐴1𝐴+𝑖𝜔2𝜔2+𝐴3,(28)𝐴1=𝒦𝑀𝛿61𝐷𝑎+Λ𝛿2+𝑀𝛿4×𝑄𝜋2𝜔2Pr2𝑀2Pr21𝜔2𝑀𝜙Pr1𝜔2𝜙Λ𝑀Pr2Pr1+𝜔2Pr2Pr1Pr2Pr1𝛿4𝜙2Λ𝑀𝜔2𝑀𝜙Pr21×𝛿41Λ𝜙+𝑀Pr1+𝑄𝜋2𝜙+𝛿𝜙2Pr2𝑀𝐷𝑎Pr1,(29)𝐴2𝛿=𝒦2Pr2Pr12𝜙2Λ𝑀+𝜙𝑀2Pr1+𝜙2Pr22𝑀𝐷𝑎Pr21,(30)𝐴3𝛿=𝒦41𝑀Λ+𝜙Pr1+𝑀𝛿2𝐷𝑎+𝑄𝜋2𝑀𝜙Pr2Pr1,(31) where 𝒦=𝛿2(𝑀4𝛿4+𝜔2𝜙2Pr22/Pr21)1 and 𝛿2=(𝜋2+𝑞2), from relation equation (30), 𝐴2>0.

3.1. Stationary Convection (𝜔=0)

Substituting 𝜔=0 in (28), we get𝑅𝑠=𝛿2𝑠𝑞2𝑠𝛿2𝑠1𝐷𝑎+𝛿2𝑠Λ+𝑄𝜋2,(32) here 𝑅𝑠 is the value of the Rayleigh number for stationary convection. The minimum value of 𝑅𝑠 is obtained for 𝑞𝑠=𝑞sc. where𝑞2Λ𝜋6+13Λ+𝜋2𝐷𝐷𝑎𝑞𝜋4𝑄=Λ+𝜋2+1𝜋2𝐷𝐷𝑎.(33) The wave number is identical to that for the single component fluid, while the threshold for the onset of stationary convection at pitchfork bifurcation is given by (34) with 𝑞𝑠=𝑞sc,𝑅sc=𝛿2sc𝑞2sc𝛿2sc1𝐷𝑎+𝛿2scΛ+𝑄𝜋2,(34) where 𝛿2sc=𝜋2+𝑞2sc. Thus the magnetic field inhibits the onset of stationary convection.

3.2. Oscillatory Convection (𝜔2>0)

For the oscillatory convection (𝜔0) and from (28), 𝑅 will be complex. But the physical meaning of 𝑅 requires it to be real. The condition that 𝑅 is real implies that imaginary part of (28) is zero, that is,𝐴2𝜔2+𝐴3=0,(35) where 𝐴2 and 𝐴3 are given by (30) and (31). For oscillatory convection 𝜔2=𝐴3/𝐴2>0 since 𝐴2>0, for oscillatory convection 𝐴3<0. For 𝐴3=0, (35) implies that 𝜔=0 is a double zero corresponding to Takens-Bogdanov bifurcation point. For oscillatory convection, we have𝜔2=𝛿4𝑜𝑀Λ+1/𝜙Pr1+𝑀𝛿2𝑜/𝐷𝑎+𝑄𝜋2𝑀𝜙Pr2/Pr1𝛿2𝑜Pr2/Pr12𝜙2Λ/𝑀+𝜙/𝑀2Pr1+𝜙2Pr22/𝑀𝐷𝑎Pr21,(36)where 𝛿2𝑜=𝜋2+𝑞2𝑜. A necessary condition for 𝜔2>0 isPr2Pr1>1𝜙.(37) However, this is not sufficient condition and one must have in addition𝛿𝑄>4𝑜𝑀Λ+1/𝜙Pr1+𝑀𝛿2𝑜/𝐷𝑎𝜋2𝜙Pr2/Pr1.𝑀(38) At Takens-Bogdanov bifurcation point 𝑅𝑜(𝑞𝑜)=𝑅𝑠(𝑞𝑠)=𝑅𝑐(𝑞𝑐), 𝑞𝑜=𝑞𝑠=𝑞𝑐, and 𝜔2=0 is a double zero at 𝑄=𝑄𝑐(𝑞𝑐) where𝛿𝑄=4𝑐𝑀Λ+1/𝜙Pr1+𝑀𝛿2𝑐/𝐷𝑎𝜋2𝜙Pr2/Pr1𝑀,𝑞=𝑞𝑐.(39) The Takens-Bogdanov bifurcation point occurs when the neutral curves for Hopf and pitchfork bifurcation meet and only a single wave number is present, namely, 𝑞𝑜=𝑞𝑠=𝑞𝑐. If 𝑞𝑐>𝑞sc then for all 𝑞<𝑞𝑐 the first instability to set in is an oscillatory convection. The asymptotic values of 𝑞𝑐 and 𝑞sc for large Chandrasekhar number (𝑄) are𝑞𝑐(𝜙Pr2𝑀Pr1)𝑄𝜋2(𝑀ΛPr1+1/𝜙)1/4,𝑞sc𝑄𝜋4𝑀2Λ1/6.(40) From the monotonic dependence of 𝑞𝑐 and 𝑞sc on 𝑄, we may conclude that for Pr2>𝑃r1, there exists a 𝑄(Pr1,Pr2,𝑀,Λ,𝜙,𝐷𝑎) such that for 𝑄<𝑄(Pr1,Pr2,𝑀,Λ,𝜙,𝐷𝑎) the onset of first instability will be stationary convection at pitchfork bifurcation while for 𝑄>𝑄(Pr1,Pr2,𝑀,Λ,𝜙,𝐷𝑎) it will be oscillatory convection at Hopf bifurcation. 𝑄(Pr1,Pr2,𝑀,Λ,𝜙,𝐷𝑎) and for 𝑄=𝑄(Pr1,Pr2,𝑀,Λ,𝜙,𝐷𝑎), we have𝑅ct=𝑅oc𝑞oc=𝑅sc𝑞scbut𝑞oc𝑞sc,(41) above condition (41) gives codimension-two bifurcation point. However, there is no simple formula to give 𝑄(Pr1,Pr2,𝑀,Λ,𝜙,𝐷𝑎) at the codimension-two bifurcation point by assuming 𝑅 as an independent variable, such kind of interesting result is not available in Chandrasekhar [2]. In Figures 1 and 2, each solid line stands for stationary convection (pitchfork bifurcation) and dotted line stands for oscillatory convection (Hopf bifurcation). In Figures 1 and 2, we have showed the effect of several physical parameters, like 𝑄, Pr1, Pr2, Λ, 𝑀, 𝜙, and 𝐷𝑎 on the onset of both stationary convection and oscillatory convection when a physical parameter increases for the remaining fixed parameters, the onset of instabilities increases, that is, the onset of stationary convection and oscillatory convection inhibit when a parameter increases with the remaining fixed parameters.

fig1
Figure 1: Numerically calculated marginal stability curves are plotted in (𝑅,𝑞)-plane for Pr1=1, Pr2=1.5, 𝐷𝑎=1500, Λ=0.85, 𝜙=0.9, and 𝑀=0.9,  (a)  𝑄=104, (b)  𝑄=106, (c)  𝑄=108, and (d)  𝑄=1010, then the onset of stationary convection and the onset of oscillatory convection increase (stationary convection represented by solid lines and oscillatory convection represented by dotted lines).
fig2
Figure 2: Neutral curves for the stationary bifurcation (solid lines) and for the Hopf bifurcation (dashed lines) near the codimension two point for 𝑄=2000, 𝐷𝑎=1500, Λ=0.85, 𝜙=0.9, Pr1=1, and 𝑀=0.9, (a)  Pr2=1.27, (b)  Pr2=1.3, (c)  Pr2=1.35𝑥-axis wave number, 𝑦-Rayleigh numbers 𝑅𝑠,𝑅𝑜.

4. Onset of Stationary Convection at Supercritical Pitchfork Bifurcation

4.1. Derivation of Two-Dimensional Nonlinear Ginzburg-Landau Equation Using Newell-Whitehead [19] Method

In this section the evolution of a general pattern is developed by means of a multiple scale analysis used by Newell and Whitehead [19]. A small amplitude convection cell is imposed on the basic flow. If this amplitude is of the size 𝑂(𝜖) then the interaction of the cell with itself forces a second harmonic and mean state correction of size 𝑂(𝜖2) and then in turn drives an 𝑂(𝜖3) correction to the fundamental component of the imposed roll. A solvability criteria for this correction yields the one-dimensional nonlinear Ginzburg-Landau equation of the complex valued amplitude 𝐴(𝑋,𝑌,𝑇) of the imposed disturbance with real coefficients. To simplify the problem we assume the formulation of cylindrical rolls with axis parallel to 𝑦-axis, so that 𝑦-dependence disappears from (19). The 𝑧-dependence is contained entirely in the sine and cosine functions, which ensures that stress-free boundary conditions are satisfied. We use the expansion parameter 𝜖 as𝜖2=𝑅𝑅sc𝑅sc.(42) For the values of 𝑅 close to threshold value 𝑅sc that is, 𝜖1, the structure of the slow length scales will be insensitive to 𝜖, but a slow modulation in space and time is possible by making use of the band of the unstable solutions and linear growth rate is likely to saturate due to nonlinear effects. This behavior can be analyzed by writing solutions of (9)–(12) in power series 𝜖 as𝑓=𝜖𝑓0+𝜖2𝑓1+𝜖3𝑓2+,(43) where 𝑓=𝑓(𝑢,𝑣,𝑤,𝜃,𝐻𝑥,𝐻𝑦,𝐻𝑧) with the first approximation is given by the eigenvector of the linearized problem:𝑢0=𝑖𝜋𝑞sc𝐴(𝑋,𝑌,𝑇)𝑒𝑖𝑞sc𝑥,𝑣cos𝜋𝑧c.c.0𝑤=0,0=𝐴(𝑋,𝑌,𝑇)𝑒𝑖𝑞sc𝑥𝜃sin𝜋𝑧+c.c.,0=1𝑀𝛿2sc𝐴(𝑋,𝑌,𝑇)𝑒𝑖𝑞sc𝑥,𝐻sin𝜋𝑧+c.c.𝑥0=𝑖𝜋2𝑀𝑞sc𝛿2sc𝐴(𝑋,𝑌,𝑇)𝑒𝑖𝑞sc𝑥,𝐻sin𝜋𝑧c.c.𝑦0𝐻=0,𝑧0=𝜋𝑀𝛿2sc𝐴(𝑋,𝑌,𝑇)𝑒𝑖𝑞sc𝑥,cos𝜋𝑧+c.c.(44) where 𝛿2sc=𝜋2+𝑞2sc, here C.C. stands for complex conjugate, 𝑒𝑖𝑞sc𝑥sin𝜋𝑧 is the critical mode for the linear problem at 𝑅=𝑅sc and 𝑞=𝑞sc. The complex amplitude 𝐴(𝑋,𝑌,𝑇) depends on the slow variables 𝑋, 𝑌, 𝑍, and 𝑇 to be scaled by introducing multiple scales𝑋=𝜖𝑥,𝑌=𝜖1/2𝑦,𝑍=𝑧,𝑇=𝜖2𝑡,(45) and these formally separate the fast and slow dependent variables in 𝑓. It should be noted that the difference in scaling in the two directions reflects the inherent symmetry breaking of instability which was chosen here with wave vector in 𝑥-direction. The differential operators can be expressed as𝜕𝜕𝜕𝑥𝜕𝜕𝑥+𝜖,𝜕𝜕𝑋𝜕𝑦𝜖1/2𝜕,𝜕𝜕𝑌𝜕𝜕𝑧,𝜕𝜕𝑍𝜕𝑡𝜖2𝜕𝜕𝑇(46) with the assumption (46), the operators (20) and (21) are transformed into a set of linear in homogeneous equations. The solvability conditions for the latter yields the amplitude equation using (44) in the linear operator (20) can be written as=0+𝜖1+𝜖22+,(47) where0=Λ8+1𝐷𝑎6+𝑄4𝜕2𝜕𝑧2+𝑅sc2𝜕2𝜕𝑥2,(48)1=2𝜕2+𝜕𝜕𝑥𝜕𝑋2𝜕𝑌2×3𝐷𝑎44Λ6+2𝑄2𝜕2𝜕𝑧2+𝑅sc2+𝑅sc𝜕2𝜕𝑥2,(49)2=𝜕1𝜕𝑇Λ+𝑀𝜙Pr1Λ+𝜙𝑀Pr2Pr161𝐷𝑎+𝜙Pr2Pr11𝑀𝐷𝑎4𝑄2𝜕2𝜕𝑧2𝑅sc𝑀𝜙Pr2Pr1𝜕2𝜕𝑥2+𝜕2𝜕𝑋2×3𝐷𝑎44Λ6+2𝑄2𝜕2𝜕𝑧2+𝑅sc2+𝑅sc𝜕2𝜕𝑥2+4𝜕4𝜕𝑥2𝜕𝑋2+12𝜕2𝜕𝑌2×6Λ4+3𝐷𝑎2𝜕+𝑄2𝜕𝑧2+𝑅sc.(50) Similarly nonlinear term 𝒩 is given by𝒩=𝜖2𝒩0+𝜖3𝒩1+,(51) substituting (47), (51), and (43) into (19), we get by equating the coefficients of 𝜖,𝜖2,𝜖3;0𝑤0=0,(52)0𝑤1+1𝑤0=𝒩0,(53)0𝑤2+1𝑤1+2𝑤0=𝒩1.(54) Equation (48) gives the critical Rayleigh number for the onset of stationary convection𝑅sc=𝛿2sc𝑞2sc𝛿4sc1Λ+𝐷𝑎𝛿2sc+𝑄𝜋2.(55) In (53), 𝒩0=0, 1𝑤0=0 and hence 𝑤1=0. From equation of continuity we find that 𝑢1=0. The relevant equations for 𝜃1 and 𝐻𝑧1 are𝜕𝜕𝑡2𝜃1=𝑤1𝑀1𝑀𝑢0𝜕𝜃0𝜕𝑥+𝑤0𝜕𝜃0𝜕𝑧,(56) form (56) and (44), we get𝜃11=2𝜋𝑀2𝛿2sc||𝐴||2sin2𝜋𝑧.(57) Equation (12) gives relevant equation for 𝐻𝑧1 as𝜙Pr2Pr1𝑀2𝐻𝑧1=𝜕𝑤1+𝜕𝑧Pr2Pr1𝜕𝑤𝜕𝑥0𝐻𝑥0𝑢0𝐻𝑧0.(58) From (58) and (44), we get𝐻𝑧1=Pr2Pr1𝜋24𝛿2sc𝑞2sc𝐴2𝑒2𝑖𝑞sc𝑥.+c.c.(59) Similarly we have 𝐻𝑥1=0,𝐻𝑦1=0. The solvability criterion of (54) gives the amplitude equation which can be written as𝜆0𝜕𝐴𝜕𝑇𝜆1𝜕𝑖𝜕𝑋2𝑞sc𝜕2𝜕𝑌22𝜆2𝐴+𝜆3||𝐴||2𝐴=0,(60) where𝜆0=1Λ+𝑀𝜙Pr1+𝜙Pr2Pr1Λ𝑀𝛿6sc+1𝐷𝑎+𝜙Pr2Pr11𝑀𝐷𝑎𝛿4sc+𝑄𝜋2𝛿2sc𝑅sc𝑀Pr2Pr1𝑞2sc𝜆𝜙,1=4𝑞2sc6Λ𝛿4sc+3𝐷𝑎𝛿2sc+𝑄𝜋2𝑅sc,𝜆2=𝑅sc𝑞2sc𝛿2sc,𝜆3=𝑄Pr22Pr21𝜋4𝑀2𝑞2sc𝑞2sc𝜋2+𝑅sc2𝑀2𝑞2sc.(61) Equation (60) is two-dimensional, nonlinear time-dependent Ginzburg-Landau equation describing the effect of magneticfield in a sparsely packed porous medium near the onset of stationary convection at supercritical pitchfork bifurcation. Here 𝜆0 is always positive for Pr2/Pr1<1/𝜙 and for any 𝑄 but if Pr2/Pr1>1/𝜙 then 𝜆0 is positive only if 𝑄<𝑄𝑐. Thus for supercritical pitchfork bifurcation 𝜆0 is always positive. For Pr2/Pr1>1/𝜙, 𝜆0 decreases as 𝑄 increases and becomes zero at 𝑄=𝑄𝑐. 𝜆1 and 𝜆2 are always positive. 𝜆3 is positive only if𝑅𝑄<sc𝑞4sc2𝜋4𝜋2𝑞2scPr21Pr22.(62) The pitchfork bifurcation is supercritical if 𝜆3>0 and subcritical if 𝜆3<0. At 𝜆3=0, we get tricritical bifurcation point [22] (see Figure 3). Dropping the time dependence from (60), we getd2𝐴d𝑋2+𝜆2𝜆1𝜆13𝜆2||𝐴||2𝐴=0,(63) since 𝜆1>0, the solution of (63) is given by𝐴(𝑋)=𝐴0𝑋tanhΛ1,(64) where𝐴0=𝜆2𝜆31/2,Λ1=2𝜆1𝜆21/2.(65)

207123.fig.003
Figure 3: Above figure is plotted for 𝐷𝑎=1500, Λ=0.85, 𝜙=0.9, 𝑀=0.9, and Pr2=1.𝜆3 is the nonlinear coefficient of Ginzburg-Landau equation at the onset of stationary convection. The pitchfork bifurcation is supercritical if 𝜆3>0, subcritical if 𝜆3<0 and 𝜆3=0 on the curve.
4.2. Long Wavelength Instabilities (Secondary Instabilities)

The secondary Instabilities arising in nonequilibrium systems do not exhibit strict symmetries but may show spatially slow deformations of the cellular structures. Further, there are secondary instabilities like Eckhaus and Zigzag instabilities, such phenomena are studied using evolution equations for amplitudes which are slowly varying in time as well as in space. These envelope equations can be derived by the method of Newell and Whitehead [19]. The two-dimensional Ginzburg-Landau equation (60), can be written in fast variables 𝑥,𝑦,𝑡, and 𝐴(𝑋,𝑌,𝑇)=𝐴(𝑥,𝑦,𝑡)/𝜖, as𝜆0𝜕𝐴𝜕𝑡𝜆1𝜕𝑖𝜕𝑥2𝑞sc𝜕2𝜕𝑦22𝐴𝜖2𝜆2𝐴+𝜆3||𝐴||2𝐴=0.(66) In order to study the properties of a structure with a given phase winding number 𝛿𝑘, we substitute𝐴(𝑥,𝑦,𝑡)=𝐴1(𝑥,𝑦,𝑡)𝑒𝑖𝛿𝑘𝑥,(67) into (66) and we obtain𝜆0𝜕𝐴1=𝜖𝜕𝑡2𝜆2𝜆1(𝛿𝑘)2𝐴1+2𝑖𝜆1𝜕𝛿𝑘𝑖𝜕𝑥2𝑞sc𝜕2𝜕𝑦2𝐴1+𝜆1𝜕𝑖𝜕𝑥2𝑞sc𝜕2𝜕𝑦22𝐴1𝜆3||𝐴1||2𝐴1=0.(68) The steady-state uniform solution of (68) is𝐴1=𝐴1𝑜=𝜖2𝜆2𝜆1(𝛿𝑘)2𝜆311/2.(69) Let ̃̃𝑢(𝑥,𝑦,𝑡)+𝑖𝑣(𝑥,𝑦,𝑡) be an infinitesimal perturbation from a uniform steady-state solution 𝐴1𝑜 given by (69). Now substituting 𝐴1=𝐴1𝑜=𝜖2𝜆2𝜆1(𝛿𝑘)2𝜆311/2̃+̃𝑢+𝑖𝑣,(70) into (68) and equating real and imaginary parts, we obtain 𝜆0𝜕̃𝑢=𝜖𝜕𝑡22𝜆2𝜆1(𝛿𝑘)2+𝜆1𝜕2𝜕𝑥2+𝛿𝑘𝑞sc𝜕2𝜕𝑦214𝑞2sc𝜕4𝜕𝑦4̃𝑢2𝜆1𝜆𝛿𝑘1𝑞sc𝜕2𝜕𝑦2𝜕̃𝑣,𝜆𝜕𝑥0𝜕̃𝑣=𝜕𝑡2𝜆1𝜆𝛿𝑘1𝑞sc𝜕2𝜕𝑦2𝜕̃𝑢𝜕𝑥+𝜆1𝜕2𝜕𝑥2+𝛿𝑘𝑞sc𝜕2𝜕𝑦214𝑞2sc𝜕4𝜕𝑦4̃𝑣.(71) We analyze (71) by using normal modes of the form̃𝑢=𝑈𝑒𝑆𝑡𝑞cos𝑥𝑥𝑞cos𝑦𝑦,̃𝑣=𝑉𝑒𝑆𝑡𝑞sin𝑥𝑥𝑞cos𝑦𝑦.(72) Putting (72) in (71) we get, 𝜆0𝜖𝑆+22𝜆2𝜆1(𝛿𝑘)2+𝜒1𝑈+𝜆1𝑞𝑥𝜒2𝜆𝑉=0,1𝑞𝑥𝜒2𝜆𝑈+0𝑆+𝜒1𝑉=0.(73) Here 𝜒1=𝜆1[𝑞2𝑥+(𝑞2𝑦𝛿𝑘)/𝑞sc+𝑞4𝑦/4𝑞2sc], 𝜒2=(2𝛿𝑘+𝑞2𝑦/𝑞sc). On solving (73) we get,𝜆20𝑆2+2𝑆2𝜆0𝜖2𝜆2𝜆1(𝛿𝑘)2+𝜆0𝜒1+2𝜖2𝜆2𝜆1(𝛿𝑘)2+𝜒1𝜓1𝑞2𝑥𝜆1𝜒2=0,(74) whose roots (𝑆±) are real. Here (𝑆±) is defined as1(𝑆±)=𝜆202𝜆0𝜖2𝜆2𝜆1(𝛿𝑘)2+𝜆0𝜒1±2𝜆0𝜖2𝜆2𝜆1(𝛿𝑘)22+𝜆21𝑞2𝑥𝜒221/2.(75) Solution 𝑆() is clearly negative, thus the corresponding mode is stable and if 𝑆(+) is positive then rolls can be unstable. Symmetry considerations help us to restrict the study of 𝑆(+) to a domain 𝑞𝑥0,𝑞𝑦0.

4.2.1. Longitudinal Perturbations and Eckhaus Instability

Inserting 𝑞𝑦=0 into (75), we get𝜆20𝑆2+2𝑆2𝜆0𝜖2𝜆2𝜆1(𝛿𝑘)2+𝜆0𝜆1𝑞2𝑥+𝜆1𝑞2𝑥2𝜖2𝜆23𝜆1(𝛿𝑘)2+𝑞2𝑥=0,(76) since the roots are real and their sum always negative, the pattern is stable as long as both roots are negative, that is, their product is positive. The cell pattern becomes unstable when the product is negative, that is, when 𝑞2𝑥23𝜆1𝛿𝑘2𝜖2𝜆2,(77) for this requires |𝛿𝑘|(𝜖2𝜆2/3𝜆1), this condition defines the domain of Eckhaus instability. The above condition implies that the most unstable wave vector tends to zero, when |𝛿𝑘|(𝜖2𝜆2/3𝜆1).

4.2.2. Transverse Perturbations and Zigzag Instability

Let us consider 𝑞𝑥=0 into (75), we get𝜆20𝑆2+2𝑆2𝜆0𝜖2𝜆2𝜆1(𝛿𝑘)2+𝜆0𝜒𝑦1+2𝜖2𝜆2𝜆1(𝛿𝑘)2+𝜒𝑦1𝜒𝑦1=0,(78) where 𝜒𝑦1=𝜆1(𝑞2𝑦𝛿𝑘/𝑞sc+𝑞4𝑦/4𝑞2sc). The two eigenmodes are uncoupled and we have 𝑆(),𝜖𝑆()=22𝜆2𝜆1(𝛿𝑘)2𝜆1𝑞sc𝛿𝑘𝑞2𝑦𝜆14𝑞2sc𝑞2𝑦<0,(79) for one of them. The other is amplified when 𝑆(+)=𝜆1𝑞2𝑦𝑞𝛿𝑘+2𝑦4𝑞sc>0.(80) This implies that 𝛿𝑘<0, the above condition defines the domain of the Zigzag Instability. When 𝛿𝑘0 from below the wave vector 𝑞𝑦 of the instability also tends to zero while the growth rate varies as 𝑞2𝑦. We have studied the effect of magnetic field on long wavelength instabilities. We have observed that Eckhaus instability and Zigzag instability regions increases when 𝑄 increases (see Figure 4).

207123.fig.004
Figure 4: Numerically computed secondary instability regions of Eckhaus instability (E), Zigzag instability (Z), and stable regions (S) are plotted in (𝜆2/𝜆1,𝛿𝑞𝑠)-plane for 𝑄=2000, 𝐷𝑎=1500, Λ=0.85, 𝜙=0.9, 𝑀=0.9, Pr1=1, and Pr2=2. As |𝛿𝑞𝑠| increases then the secondary instability regions increases.
4.3. Heat Transport by Convection

The maximum of steady amplitude 𝐴 is denoted by |𝐴max| which is given as||𝐴max||=𝜖2𝜆2𝜆31/2.(81) Equation (81) is obtained from (64) with tanh(𝑋/Λ1)=1. We use |𝐴max| to calculate Nusselt number Nu. To discuss the heat transfer near the neutral region, we express it through the Nusselt number is defined as Nu=𝐻𝑑/𝜅Δ𝑇, which is the ratio of the heat transported across any layer to the heat which would be transported by conduction alone. Here 𝐻 is the rate of heat transfer per unit area and is defined as𝐻=𝜕𝑇total𝜕𝑧𝑧=0.(82) In (82), angular brackets correspond to a horizontal average. The Nusselt number Nu can be calculated in terms of amplitude 𝐴 and is given as𝜖Nu=1+2𝛿2sc||𝐴max||2.(83) From (83), we get conduction for 𝑅𝑅sc and convection for 𝑅>𝑅sc. Since the amplitude equation is valid for 𝜆3>0, which is possible for 𝑅>𝑅sc (supercritical pitchfork bifurcation), thus we get Nu>1 for 𝑅>𝑅sc. We get convection for Nu>1 and conduction for Nu1. In stationary convection Nu increases implies that heat conducted by steady mode increases. In the problem of double diffusive convection in porous medium with magnetic field, Nu depends on Pr1,Pr2,Λ,𝑀,𝜙,𝐷𝑎, and 𝑄. We have computed Nu for different values of 𝑄, for some fixed values of other parameters and observed that Nu increases as 𝑄 decreases (see Figures 5(a) and 5(b)). This implies that magnetic field inhibits the heat transport. The parameters Pr1, Pr2, Λ, 𝑀, 𝜙, and 𝐷𝑎 show the same result as 𝑄 shows on Nu.

fig5
Figure 5: Graph (a) is plotted for 𝑄=1000 and graph (b) is plotted for 𝑄=3000 for the fixed values of 𝐷𝑎=1500, Λ=0.85, 𝜙=0.9, Pr2=1, Pr1=2, and 𝑀=0.9. in (Nu,𝑅/𝑅sc)-plane. In graphs (a) and (b), as 𝑅/𝑅sc increases then Nu increases.

5. Oscillatory Convection at the Supercritical Hopf Bifurcation

The existence of a threshold (critical value of Rayleigh number for the onset of oscillatory convection 𝑅=𝑅oc) and a cellular structure (critical wave number 𝑞=𝑞oc) are main characteristics of the oscillatory convection. In this section we treat the region near the onset of oscillatory convection. Here the axis of cylindrical rolls is taken as 𝑦-axis, so that 𝑦-dependence disappears from equation 𝑤=𝒩. The 𝑧-dependence contained entirely in sine and cosine functions which ensure that the free-free boundary conditions are satisfied. The purpose of this section is to derive coupled one-dimensional nonlinear time-dependent Ginzburg-Landau type equations near the onset of oscillatory convection at supercritical Hopf bifurcation. We introduce 𝜖 as𝜖2=𝑅𝑜𝑅oc𝑅oc1.(84) We assume that𝑤0=𝐴1𝐿𝑒𝑖(𝑞oc𝑥+𝜔oc𝑡)+𝐴1𝑅𝑒𝑖(𝑞oc𝑥𝜔oc𝑡)+c.c.sin𝜋𝑧(85) is a solution to linearized equation 𝑤=0, which satisfies free-free boundary conditions. Here 𝐴1𝐿 denotes the amplitude of left travelling wave of the roll and 𝐴1𝑅 denotes the amplitude of right travelling wave of the roll, which depends on slow space and time variables [23]𝑋=𝜖𝑥,𝜏=𝜖𝑡,𝑇=𝜖2𝑡,(86) and assume that 𝐴1𝐿=𝐴1𝐿(𝑋,𝜏,𝑇), 𝐴1𝑅=𝐴1𝑅(𝑋,𝜏,𝑇). The differential operators can be expressed as𝜕𝜕𝜕𝑥𝜕𝜕𝑥+𝜖,𝜕𝜕𝑋𝜕𝜕𝑡𝜕𝜕𝑡+𝜖𝜕𝜏+𝜖2𝜕.𝜕𝑇(87) The solution of basic equations can be sought as power series in 𝜖,𝑓=𝜖𝑓0+𝜖2𝑓1+𝜖3𝑓2+,(88) where 𝑓=𝑓(𝑢,𝑣,𝑤,𝜃,𝐻𝑥,𝐻𝑦,𝐻𝑧) with the first approximation given by eigenvector of the linearized problem:𝑢0=𝑖𝜋𝑞oc𝐴1𝐿𝑒𝑖(𝑞oc𝑥+𝜔oc𝑡)+𝐴1𝑅𝑒𝑖(𝑞oc𝑥𝜔oc𝑡)𝑣c.c.cos𝜋𝑧,0𝐻=0,𝑦0𝜃=0,0=1𝑀1𝑒1𝐴1𝐿𝑒𝑖(𝑞oc𝑥+𝜔oc𝑡)+1𝑒1𝐴1𝑅𝑒𝑖(𝑞oc𝑥𝜔oc𝑡)𝐻+c.c.×sin𝜋𝑧,𝑥0=𝑖𝜋2𝑞oc1𝑒2𝐴1𝐿𝑒𝑖(𝑞oc𝑥+𝜔oc𝑡)+1𝑒2𝐴1𝑅𝑒𝑖(𝑞oc𝑥𝜔oc𝑡)𝐻c.c.×sin𝜋𝑧,𝑧01=𝜋𝑒2𝐴1𝐿𝑒𝑖(𝑞oc𝑥+𝜔oc𝑡)+1𝑒2𝐴1𝑅𝑒𝑖(𝑞oc𝑥𝜔oc𝑡)+c.c.×cos𝜋𝑧.(89) where 𝛿2oc=(𝜋2+𝑞2oc), 𝑒1=(𝛿2oc+𝑖𝜔oc), and 𝑒2=(𝑀𝛿2oc+𝑖𝜔oc𝜙Pr2/Pr1), here 𝑒1 and 𝑒2 are complex conjugate of 𝑒1 and 𝑒2.

We expand the linear operator and nonlinear term 𝒩 as the following power series=0+𝜖1+𝜖22+,𝒩=𝜖2𝒩0+𝜖3𝒩1+,(90) substituting (87) and (88) into 𝑤=𝒩, for each order of 𝜖, we get0𝑤0=0,(91)0𝑤1+1𝑤0=𝒩0,(92)0𝑤2+1𝑤1+2𝑤0=𝒩1.(93) Here0=𝒟𝜙𝒟Pr1𝜕𝑄2𝜕𝑧2𝒟2𝑅𝑀𝜕2𝜕𝑥2𝒟𝜙,1=𝜕𝜕𝜏1𝜕+22𝜕𝑥𝑋2,2=𝜕1+𝜕𝜕𝑇4𝜕𝑥2𝑋2𝑀2𝒟𝜙𝒟+𝑀𝒟Pr1+Λ𝒟2Λ𝑀𝒟𝒟𝜙+Λ𝑀𝒟𝜙2𝜕+𝑄2𝜕𝑧2+𝜕+𝑅22𝜕𝑋2𝜕+22𝜕𝜕𝑥𝑋×𝒟𝜕𝜏𝜙𝒟Pr1𝑀+𝜙Pr2Pr1𝒟Pr12𝜙Pr2Pr1Λ𝑀+1𝑀𝜙Pr1𝒟2+1𝑀2𝜙Pr1𝒟𝒟𝜙+𝜙Pr2Pr1𝒟𝒟Pr1Λ𝑀+1𝑀2𝜙Pr1𝒟𝜙2𝜕𝑄2𝜕𝑧2𝜙𝑅Pr2𝑀Pr1+𝜕2𝜕𝜏2𝜙Pr2Pr1𝒟Pr12+Pr2𝑀2Pr21𝒟2+1𝑀2𝜙Pr1𝒟𝜙2𝑅𝑀2𝒟𝜙,(94) where1=𝒟𝜙𝒟Pr1+𝜙Pr2Pr1𝒟𝒟Pr1+1𝑀2𝜙Pr1𝒟𝒟𝜙2𝑄2𝜕2𝜕𝑧2𝜙𝑅Pr2𝑀Pr12,2=𝒟𝒟𝜙𝒟𝜙2𝑀𝒟2𝒟Pr1Λ𝑀𝒟𝒟𝜙2+𝑄2𝜕2𝜕𝑧2𝜕𝑄𝒟2𝜕𝑧2𝑅𝑀𝒟𝜙+𝑅2.(95) Equation (91) is linear problem. We get critical Rayleigh number for the onset of oscillatory convection by using the zeroth-order solution 𝑤0 in (91). At 𝑂(𝜖2),𝒩0=0 and 1𝑤0=0 gives𝜕𝐴1𝐿𝜕𝜏𝑣𝑔𝜕𝐴1𝐿𝜕𝑋=0,𝜕𝐴1𝑅𝜕𝜏𝑣𝑔𝜕𝐴1𝑅𝜕𝑋=0,(96) where 𝑣𝑔=(𝜕𝜔/𝜕𝑞)𝑞=𝑞oc is the group velocity and is real. Hence from (92), we get 𝑤1=0. From equation of continuity we find that 𝑢1=0. Substituting the zeroth-order and first-order approximation into (56) and (58) we get,𝜃1=𝜋𝑀2||𝐴1𝐿||2+||𝐴1𝑅||2𝑡1+2𝑒1𝑒4𝐽1+2𝑒1𝑒4𝐽1𝑣sin2𝜋𝑧,1𝐻=0,𝑦1𝐻=0,𝑥1=𝑖𝜋2𝑀𝑞ocPr2Pr11𝑒21𝑒2||𝐴1𝐿||2||𝐴1𝑅||2𝐻sin2𝜋𝑧,𝑧1=2𝜋2Pr2Pr11𝑒2𝑒5𝐴21𝐿𝑒2𝑖(𝑞oc𝑥+𝜔oc𝑡)+1𝑒2𝑒5𝐴21𝑅𝑒2𝑖(𝑞oc𝑥𝜔oc𝑡)+14𝑀𝑞2oc1𝑒2+1𝑒2𝐴1𝐿𝐴1𝑅𝑒2𝑖𝑞oc𝑥,+c.c.(97) where 𝑡1=(1/4𝜋2)(1/𝑒1+1/𝑒1), 𝐽1=𝐴1𝐿𝐴1𝑅𝑒2𝑖𝜔oc𝑡, 𝑒4=(4𝜋2+2𝑖𝜔oc), and 𝑒5=(4𝑀𝑞2oc+2𝑖𝜙𝜔ocPr2/Pr1) and 𝑒4, 𝑒5 and 𝐽1 are complex conjugate of 𝑒4,𝑒5 and 𝐽1, respectively.

Equation (93) is solvable when 0𝑤0=0, one requires that its right-hand side be orthogonal to 𝑤0, which is ensured that if the coefficients of sin𝜋𝑧 in 𝒩12𝑤0 are equal to zero. This implies thatΛ0𝜕𝐴1𝐿𝜕𝑇+Λ1𝜕𝜕𝜏𝑣𝑔𝜕𝐴𝜕𝑋2𝐿Λ2𝜕2𝐴1𝐿𝜕𝑋2Λ3𝐴1𝐿+Λ4||𝐴1𝐿||2𝐴1𝐿+Λ5||𝐴1𝑅||2𝐴1𝐿Λ=0,0𝜕𝐴1𝑅𝜕𝑇+Λ1𝜕𝜕𝜏𝑣𝑔𝜕𝐴𝜕𝑋2𝑅Λ2𝜕2𝐴1𝑅𝜕𝑋2Λ3𝐴1𝑅+Λ4||𝐴1𝑅||2𝐴1𝑅+Λ5||𝐴1𝐿||2𝐴1𝑅=0,(98) whereΛ0=1𝑀2𝜙Pr1𝑒1𝑒2+𝑒2𝑒3+𝜙Pr2Pr1𝑒1𝑒3+𝑄𝜋2𝛿2oc𝑅oc𝑞2oc𝜙Pr2𝑀Pr1,Λ1=𝛿2oc𝑒3𝜙Pr2Pr1+𝑒1Pr2𝑀2Pr21+𝑒2𝑀2𝜙Pr1,Λ2=4𝑞2oc𝑒2𝑒3+𝑀𝑒3𝛿2oc+Λ𝑒1𝛿2oc+Λ𝑀𝑒1𝑒2+𝑀𝑒1𝑒3+Λ𝑀𝑒2𝛿2oc+𝑄𝜋2,Λ𝑅3=𝑅𝑀𝑞2oc𝑒2,Λ4𝜋=𝑄4𝑀Pr22Pr21𝑒11𝑒21𝑒22𝑄𝜋4Pr22Pr21𝑒1𝑒51𝛿2oc+𝜋23𝑞2oc𝑒2+𝑅𝑀3𝜋2𝑞2oc𝑡1𝑒2,Λ5𝜋=𝑄4𝑀Pr22Pr21𝑒11𝑒21𝑒2𝑄𝜋4𝑀𝑞2ocPr22Pr21𝑒1𝜋2𝑞2oc1𝑒2+1𝑒2+𝑅𝑀3𝜋2𝑞2oc𝑒2𝑡1+2𝑒1𝑒4.(99) Here 𝑒3=(𝑖𝜔/𝑀2𝜙Pr+(Λ/𝑀)𝛿2+1/𝑀𝐷𝑎). It should be noted that 𝐴1𝐿 and 𝐴1𝑅 are of order 𝜖 and 𝐴2𝐿 and 𝐴2𝑅 are of order 𝜖2. If 𝜔oc=0 in Λ0,Λ2,Λ3, and Λ4 then these expressions match with the coefficients 𝜆0, 𝜆1, 𝜆2, and 𝜆3 of Ginzburg-Landau equation at the onset of stationary convection. From (96), we get 𝐴1𝐿(𝜉,𝑇) and 𝐴1𝑅(𝜂,𝑇), where 𝜉=𝑣𝑔𝜏+𝑋,𝜂=𝑣𝑔𝜏𝑋. Equations (98) can be written as2𝑣𝑔Λ1𝜕𝐴2𝐿𝜕𝜂=Λ0𝜕𝐴1𝐿𝜕𝑇+Λ2𝜕𝐴1𝐿𝜕𝑋2+𝜆3𝐴1𝐿Λ4||𝐴1𝐿||2+Λ5||𝐴1𝑅||2𝐴1𝐿,(100)2𝑣𝑔Λ1𝜕𝐴2𝑅𝜕𝜂=Λ0𝜕𝐴1𝑅𝜕𝑇+Λ2𝜕𝐴1𝑅𝜕𝑋2+𝜆3𝐴1𝑅Λ4||𝐴1𝑅||2+Λ5||𝐴1𝐿||2𝐴1𝑅.(101)

Let 𝜉𝜖[0,𝑙1], 𝜂𝜖[0,𝑙2] where 𝑙1,𝑙2 are periods of 𝐴1𝐿,𝐴1𝑅, respectively. Expansion (88) remains asymptotic for times 𝑡=𝑂(𝜖2) only if an appropriate solvability condition holds. This condition obtained integrating (100) over 𝜂 and (101) over 𝜉, we getΛ0𝜕𝐴1𝐿𝜕𝑇=Λ2𝜕𝐴1𝐿𝜕𝑋2+𝜆3𝐴1𝐿Λ4||𝐴1𝐿||2+Λ5||𝐴1𝑅||2𝐴1𝐿,Λ(102)0𝜕𝐴1𝑅𝜕𝑇=Λ2𝜕𝐴1𝑅𝜕𝑋2+𝜆3𝐴1𝑅Λ4||𝐴1𝑅||2+Λ5||𝐴1𝐿||2𝐴1𝑅.(103)

5.1. Travelling Wave and Standing Wave Convection

To study the stability regions of travelling waves and standing waves, Coullet et al. [24]. we proceed as follows.

On dropping slow variable 𝑋 from (102) and (103), we get a pair of first ODE'sd𝐴1𝐿=Λd𝑇3Λ0𝐴1𝐿Λ4Λ0𝐴1𝐿||𝐴1𝐿||2Λ5Λ0𝐴1𝐿||𝐴1𝑅||2,(104)d𝐴1𝑅=Λd𝑇3Λ0𝐴1𝑅Λ4Λ0𝐴1𝑅||𝐴1𝑅||2Λ5Λ0𝐴1𝑅||𝐴1𝐿||2.(105) Put𝛽=Λ3Λ0,𝛾Λ=4Λ0,𝛿Λ=5Λ0.(106) Then (104) and (105) take the following formd𝐴1𝐿d𝑇=𝛽𝐴1𝐿+𝛾𝐴1𝐿||𝐴1𝐿||2+𝛿𝐴1𝐿||𝐴1𝑅||2,(107)d𝐴1𝑅d𝑇=𝛽𝐴1𝑅+𝛾𝐴1𝑅||𝐴1𝑅||2+𝛿𝐴1𝑅||𝐴1𝐿||2.(108) Consider 𝐴1𝐿=𝑎𝐿𝑒𝑖𝜙𝐿 and 𝐴1𝑅=𝑎𝐿𝑒𝑖𝜙𝑅 (we can write a complex number in the amplitude and phase form), where 𝑎𝐿=|𝐴1𝐿|, 𝜙𝐿=arg(𝐴1𝐿)=tan1(Im(𝐴1𝐿)/Re(𝐴1𝐿)) and 𝑎𝑅=|𝐴1𝑅|, 𝜙𝑅=arg(𝐴