<h1>Boehmite (AlOOH) nanostrips and their growth mechanism</h1>
			<ul class="item-list">
	        	        <li>
	            Farooq Ahmad Dar	            	            	            	            <sup aria-label="Affiliated with National Institute of Technology, Srinagar, Kashmir, 190006, IN">
	                1	            </sup>
	            	        </li>
	        	        <li>
	            Ashaq Hussain Sofi	            	            <abbr title="This is the corresponding author for this article">*</abbr>
	            	            	                <a href="mailto:shifs237@gmail.com" class="tiny-icon email-link mx-1" title="Email Ashaq Hussain Sofi">
	                    Email
	                </a>
	            	            	            <sup aria-label="Affiliated with National Institute of Technology, Srinagar, Kashmir, 190006, IN">
	                1	            </sup>
	            	        </li>
	        	        <li>
	            Mohammad Ashraf Shah	            	            <abbr title="This is the corresponding author for this article">*</abbr>
	            	            	                <a href="mailto:shah@nitsri.net" class="tiny-icon email-link mx-1" title="Email Mohammad Ashraf Shah">
	                    Email
	                </a>
	            	            	            <sup aria-label="Affiliated with National Institute of Technology, Srinagar, Kashmir, 190006, IN">
	                1	            </sup>
	            	        </li>
	        	    </ul>
	    	    <ul class="affiliations" aria-hidden="true">
	        <li>National Institute of Technology, Srinagar, Kashmir, 190006, IN</li>	    </ul>
	    
<h2>Abstract</h2>
<p>Crystalline
 nanostrips of AlOOH have been prepared at 
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 through a fast route. Powder X-ray diffraction studies reveal that the as-prepared nanostrips are highly crystalline in nature and by morphological investigations using FESEM, it was revealed that the strips have average length of 210 nm and width of 60 
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 20 nm. A plausible theory is proposed which reveals the growth mechanism of nanostrips.</p><hr/><section><h2>Introduction</h2>
<p>Boehmite (AlOOH) nanostructures are of immense significance for use in advanced catalysts, absorbents, composite materials and ceramics [
<a href="#CR1"><sup>1</sup></a>
–
<a href="#CR3"><sup>3</sup></a>
]. Numerous studies on Boehmite have been undertaken recently and considerable efforts have been directed towards the preparation of nanostructures of Boehmite having different morphologies such as nanopowder [
<a href="#CR4"><sup>4</sup></a>
], rods and flakes [
<a href="#CR5"><sup>5</sup></a>
], nanotubes [
<a href="#CR6"><sup>6</sup></a>
] and well-crystallized 1-D nanostructures by employing various techniques [
<a href="#CR7"><sup>7</sup></a>
]. We have previously reported a procedure for Al
<span id="IEq3"><mml:math id="IEq3_Math"><mml:msub><mml:mrow/><mml:mn>2</mml:mn></mml:msub></mml:math><tex-math id="IEq3_TeX"><![CDATA[\documentclass[12pt]{minimal}
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O
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 nanorods and nanoflakes without surfactants and additives [
<a href="#CR8"><sup>8</sup></a>
, 
<a href="#CR9"><sup>9</sup></a>
]. For decades, researchers consistently have been developing efficient synthetic routes to well-defined nanostructures. Experimental investigations reveal the growth parameters like surface energy, growth rate, reaction temperature and time to be critical in determining the behavior of nanostructures. In our view point, the facile, inexpensive and mass preparation of Boehmite (AlOOH) nanostructures still remains blank. Herein, we report the preparation of crystalline AlOOH nanostrips using soft method in which de-ionized water was used as solvent as well as source of oxygen. The method is based on a reaction without using any catalysts or harmful chemicals. This process is unique for its simplicity, high efficiency and its potential to be operated at large scale. In addition, to supplement the mechanism behind growth, a mathematical model has been proposed for the first time.</p></section>
<section><h2>Experimental</h2>
<section><h2>Materials and synthesis</h2>
<p>Pure aluminum AR grade (Ranbaxy) with diameter of about 10 
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m was used as starting material. In the experiment, 5 mg of aluminum metal powder and 30 ml of de-ionized water have been well sonicated in a glass vial for around 10 min. After sonication, the reaction mixture was transferred to teflon-lined stainless steel chamber of 100 ml capacity and has been kept at 
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 in an oven for 3 h. After the desired time, the system was allowed to cool naturally. The reaction mixture was centrifuged to reclaim the precipitated sample and washed several times with DI water. The final product was air dried for few hours.</p></section>
<section><h2>Characterization of samples</h2>
<p>The morphology of the product was carried out using field emission scanning electron microscope (FEI SEM, NNL 200, Japan). Phase structure and the purity of the as-prepared sample were characterized by powder X-ray diffraction (XRD) taken on a Philips (X’Pert PRO PW-3710) diffractometer with 2
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 ranging from 10
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–80
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, using Cu K
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 (
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 nm) radiation operated at 40 kV and 30 mA.</p></section></section>
<section><h2>Results and discussion</h2>
<section><h2>Structural studies</h2>
<p>The XRD pattern of the as-prepared sample, synthesized at 
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 is shown in Fig. 
<a href="#Fig1"><sup>1</sup></a>
. The diffraction pattern reveals the well-defined peaks corresponding to orientation of the (020), (120), (031), (200) and (002) planes. The most intense diffraction peak is (020) lattice plane according to standard pattern [
<a href="#CR12"><sup>12</sup></a>
]. Diffraction peaks of AlOOH with lattice parameters 
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 = 4.76 
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 and 
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 = 12.99 
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 (corresponding to JCPDS No. 46-1212) are identified unambiguously. The relative broad peaks suggest high crystallinity of the samples. The result is quite different from the traditional process in which only amorphous phase can be obtained from the precipitates derived by sol–gel process before calcinations and further higher temperature heat treatment is normally required to induce crystallization. Thus, this method, the soft option of hydrothermal treatment may be regarded as an alternative to calcinations for promoting the crystallization.
<figure id="Fig1"><h3>Fig. 1</h3>
<figcaption><p>XRD pattern of as-prepared sample</p></figcaption>
<img src="40089_2015_138_Fig1_HTML.gif" /></figure></p></section>
<section><h2>Morphology examinations</h2>
<p>The as-prepared sample was directly transferred to FESEM chamber for examination. Figure 
<a href="#Fig2"><sup>2</sup></a>
a low--resolution and Fig. 
<a href="#Fig2"><sup>2</sup></a>
b high-resolution FESEM images of nanostrips obtained by the reaction of aluminum metal powder with DI water at 
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 for 3 h.
<figure id="Fig2"><h3>Fig. 2</h3>
<figcaption><p>Typical 
<strong>a</strong>
 low- and 
<strong>b</strong>
 high-resolution images of nanostrips obtained by the reaction of aluminum powder with water at 
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 for 3 h</p></figcaption>
<img src="40089_2015_138_Fig2_HTML.jpg" /></figure></p>
<p>The FESEM images confirm that the nanostrips are grown in a very high density. It was observed that the grown product has a shape of nanostrips with an average length of 210  nm and a width of 60 
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 20 with an average width of 72 nm as shown in Fig. 
<a href="#Fig3"><sup>3</sup></a>
.
<figure id="Fig3"><h3>Fig. 3</h3>
<figcaption><p>Histograms showing 
<strong>a</strong>
 average length of nanostrips, 
<strong>b</strong>
 average width of nanostrips</p></figcaption>
<img src="40089_2015_138_Fig3_HTML.gif" /></figure></p>
<p>The nanostrips crystallized were composed of aluminum and oxygen only and the unit cells are very close to orthorhombic structure [
<a href="#CR10"><sup>10</sup></a>
, 
<a href="#CR11"><sup>11</sup></a>
].</p></section>
<section><h2>Formation mechanism</h2>
<p>The formation of AlOOH nanostructures from the reagents of aluminum and water can be explained by the following facile reaction:
<section id="Equ1"><mml:math display="block" id="Equ1_Math"><mml:mrow><mml:mtable columnspacing="0.5ex"><mml:mtr><mml:mtd columnalign="right"><mml:mrow><mml:mn>2</mml:mn><mml:mi mathvariant="normal">Al</mml:mi><mml:mo>+</mml:mo><mml:mn>6</mml:mn><mml:msub><mml:mi mathvariant="normal">H</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mi mathvariant="normal">O</mml:mi><mml:munder accentunder="true"><mml:mi mathvariant="normal">Δ</mml:mi><mml:mo stretchy="false">→</mml:mo></mml:munder><mml:mn>2</mml:mn><mml:msub><mml:mrow><mml:mi mathvariant="normal">Al</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="normal">OH</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mn>3</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mn>3</mml:mn><mml:msub><mml:mi mathvariant="normal">H</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="normal">gas</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><tex-math id="Equ1_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\begin{aligned} 2\mathrm{Al} + 6\mathrm{H}_2 \mathrm{O} {\underrightarrow{\Delta}} 2\mathrm{Al(OH)}_3 + 3\mathrm{H}_2 \mathrm{(gas)} \end{aligned}$$\end{document}]]></tex-math><graphic specific-use="web" mime-subtype="GIF" xlink:href="40089_2015_138_Article_Equ1.gif"/></section>
<section id="Equ2"><mml:math display="block" id="Equ2_Math"><mml:mrow><mml:mtable columnspacing="0.5ex"><mml:mtr><mml:mtd columnalign="right"><mml:mrow><mml:mn>2</mml:mn><mml:msub><mml:mrow><mml:mi mathvariant="normal">Al</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="normal">OH</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mn>3</mml:mn></mml:msub><mml:mo stretchy="false">→</mml:mo><mml:mn>2</mml:mn><mml:mi mathvariant="normal">AlOOH</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mi mathvariant="normal">H</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mi mathvariant="normal">O</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><tex-math id="Equ2_TeX"><![CDATA[\documentclass[12pt]{minimal}
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Now, our primary emphasis is to analyze the formation of Boehmite nanostructures for that we would use tensor notations and Green’s functions [
<a href="#CR14"><sup>14</sup></a>
]. As the conditions are inert, the effect due to impurities can be neglected and the mixture can be treated as a binary mixture of aluminum particles and de-ionized water. Firstly, on heating this mixture at a temperature of 
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 for 3 h leads to the formation of homogenous mixture of aluminum particles and DI water. Secondly, in this mixture, the aluminum crystals are broken down to nanosize under high pressure. Thirdly, due to difference in concentration and temperature there arise concentration and temperature gradients leading to convection, due to which a cell structure occurs which remains even after drying DI water [
<a href="#CR14"><sup>14</sup></a>
, 
<a href="#CR15"><sup>15</sup></a>
]. Now, for a ternary mixture, we have
<section id="Equ3"><mml:math display="block" id="Equ3_Math"><mml:mrow><mml:mtable columnspacing="0.5ex"><mml:mtr><mml:mtd columnalign="right"><mml:mrow><mml:mi mathvariant="italic">ρ</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi mathvariant="italic">ρ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:msub><mml:mi>B</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mi>C</mml:mi><mml:mo>+</mml:mo><mml:msubsup><mml:mi>B</mml:mi><mml:mi>c</mml:mi><mml:mo>′</mml:mo></mml:msubsup><mml:msup><mml:mi>C</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>+</mml:mo><mml:msub><mml:mi>B</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mi>T</mml:mi><mml:mo stretchy="false">]</mml:mo></mml:mrow></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><tex-math id="Equ3_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\begin{aligned} \rho = \rho _0 [ 1 + B_c C + B_c' C' + B_t T ] \end{aligned}$$\end{document}]]></tex-math><graphic specific-use="web" mime-subtype="GIF" xlink:href="40089_2015_138_Article_Equ3.gif"/></section>
where 
<span id="IEq21"><mml:math id="IEq21_Math"><mml:msup><mml:mi>C</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:math><tex-math id="IEq21_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$C'$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40089_2015_138_Article_IEq21.gif"/></span>
 is the concentration of impurities. Since the conditions are inert we have 
<span id="IEq22"><mml:math id="IEq22_Math"><mml:mrow><mml:msup><mml:mi>C</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math><tex-math id="IEq22_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$C' = 0$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40089_2015_138_Article_IEq22.gif"/></span>
 and hence for a binary mixture we can write
<section id="Equ4"><mml:math display="block" id="Equ4_Math"><mml:mrow><mml:mtable columnspacing="0.5ex"><mml:mtr><mml:mtd columnalign="right"><mml:mrow><mml:mi mathvariant="italic">ρ</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi mathvariant="italic">ρ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:msub><mml:mi>B</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mi>C</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mi>B</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mi>T</mml:mi><mml:mo stretchy="false">]</mml:mo></mml:mrow></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><tex-math id="Equ4_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\begin{aligned} \rho = \rho _0 [ 1 + B_c C + B_t T ] \end{aligned}$$\end{document}]]></tex-math><graphic specific-use="web" mime-subtype="GIF" xlink:href="40089_2015_138_Article_Equ4.gif"/></section>
where 
<span id="IEq23"><mml:math id="IEq23_Math"><mml:mi mathvariant="italic">ρ</mml:mi></mml:math><tex-math id="IEq23_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\rho $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40089_2015_138_Article_IEq23.gif"/></span>
 is the density of homogenous mixture of almunium particles and de-ionized water (which depends on both the temperature of this mixture and concentration of aluminum particles in de-ionized water), 
<span id="IEq24"><mml:math id="IEq24_Math"><mml:mrow><mml:msub><mml:mi>B</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi mathvariant="italic">α</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mi mathvariant="normal">Δ</mml:mi><mml:mi>C</mml:mi></mml:mrow></mml:math><tex-math id="IEq24_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$B_c = \alpha _c \Delta C$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40089_2015_138_Article_IEq24.gif"/></span>
, 
<span id="IEq25"><mml:math id="IEq25_Math"><mml:mrow><mml:msub><mml:mi>B</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi mathvariant="italic">α</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mi mathvariant="normal">Δ</mml:mi><mml:mi>T</mml:mi></mml:mrow></mml:math><tex-math id="IEq25_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$B_t = \alpha _t \Delta T$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40089_2015_138_Article_IEq25.gif"/></span>
 and 
<span id="IEq26"><mml:math id="IEq26_Math"><mml:msub><mml:mi mathvariant="italic">α</mml:mi><mml:mi>c</mml:mi></mml:msub></mml:math><tex-math id="IEq26_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\alpha _c$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40089_2015_138_Article_IEq26.gif"/></span>
 and 
<span id="IEq27"><mml:math id="IEq27_Math"><mml:msub><mml:mi mathvariant="italic">α</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:math><tex-math id="IEq27_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\alpha _t$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40089_2015_138_Article_IEq27.gif"/></span>
 are the solute and thermal expansion coefficients, To write momentum balance equation, we first define 
<span id="IEq28"><mml:math id="IEq28_Math"><mml:mrow><mml:mi>D</mml:mi><mml:msub><mml:mi>v</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi mathvariant="italic">∂</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:msub><mml:mi>v</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:msup><mml:mi>v</mml:mi><mml:mi>j</mml:mi></mml:msup><mml:msub><mml:mi mathvariant="italic">∂</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:msub><mml:mi>v</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:math><tex-math id="IEq28_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$ D v_i = \partial _t v_i + v^j \partial _j v_i$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40089_2015_138_Article_IEq28.gif"/></span>
 (with 
<span id="IEq29"><mml:math id="IEq29_Math"><mml:mrow><mml:msup><mml:mi mathvariant="italic">∂</mml:mi><mml:mi>i</mml:mi></mml:msup><mml:msub><mml:mi>v</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math><tex-math id="IEq29_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\partial ^i v_i = 0$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40089_2015_138_Article_IEq29.gif"/></span>
, i.e., divergenceless vector field), called substantial derivative, which represents the time rate of change of a physical quantity subjected to space- and time-dependant velocity fluid in continuum mechanics. Now, momentum balance equation can be written as:
<section id="Equ5"><mml:math display="block" id="Equ5_Math"><mml:mrow><mml:mtable columnspacing="0.5ex"><mml:mtr><mml:mtd columnalign="right"><mml:mrow><mml:msub><mml:mi mathvariant="italic">ρ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mi>D</mml:mi><mml:msub><mml:mi>v</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mo>-</mml:mo><mml:msub><mml:mi mathvariant="italic">∂</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mi>p</mml:mi><mml:mo>+</mml:mo><mml:mi mathvariant="italic">μ</mml:mi><mml:msup><mml:mi mathvariant="italic">∂</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msub><mml:mi>v</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>-</mml:mo><mml:mi mathvariant="italic">ρ</mml:mi><mml:mi>g</mml:mi><mml:msub><mml:mi mathvariant="italic">λ</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><tex-math id="Equ5_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
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				\begin{document}$$\begin{aligned} \rho _0 D v_i = - \partial _i p + \mu \partial ^2 v_i - \rho g \lambda _i. \end{aligned}$$\end{document}]]></tex-math><graphic specific-use="web" mime-subtype="GIF" xlink:href="40089_2015_138_Article_Equ5.gif"/></section>
where 
<span id="IEq30"><mml:math id="IEq30_Math"><mml:mi mathvariant="italic">μ</mml:mi></mml:math><tex-math id="IEq30_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\mu $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40089_2015_138_Article_IEq30.gif"/></span>
 is a constant viscosity and other symbols have their usual meanings. we also require equations 
<span id="IEq31"><mml:math id="IEq31_Math"><mml:mrow><mml:mi>D</mml:mi><mml:mi>T</mml:mi><mml:mo>=</mml:mo><mml:mi mathvariant="italic">κ</mml:mi><mml:msup><mml:mi mathvariant="italic">∂</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mi>T</mml:mi></mml:mrow></mml:math><tex-math id="IEq31_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$DT = \kappa \partial ^2 T$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40089_2015_138_Article_IEq31.gif"/></span>
 and 
<span id="IEq32"><mml:math id="IEq32_Math"><mml:mrow><mml:mi>D</mml:mi><mml:mi>C</mml:mi><mml:mo>=</mml:mo><mml:mspace width="4pt"/><mml:msub><mml:mi>d</mml:mi><mml:mn>11</mml:mn></mml:msub><mml:msup><mml:mi mathvariant="italic">∂</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mi>C</mml:mi></mml:mrow></mml:math><tex-math id="IEq32_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$DC = \ d_{11} \partial ^2 C$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40089_2015_138_Article_IEq32.gif"/></span>
, where 
<span id="IEq33"><mml:math id="IEq33_Math"><mml:msub><mml:mi>d</mml:mi><mml:mn>11</mml:mn></mml:msub></mml:math><tex-math id="IEq33_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$d_{11}$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40089_2015_138_Article_IEq33.gif"/></span>
 is the aluminum diffusion coefficient and 
<span id="IEq34"><mml:math id="IEq34_Math"><mml:mi mathvariant="italic">κ</mml:mi></mml:math><tex-math id="IEq34_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\kappa $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40089_2015_138_Article_IEq34.gif"/></span>
 is the thermal diffusivity of the fluid.</p>
<p>Using transformation 
<span id="IEq35"><mml:math id="IEq35_Math"><mml:mrow><mml:msup><mml:mi>x</mml:mi><mml:mi>i</mml:mi></mml:msup><mml:mo stretchy="false">→</mml:mo><mml:mi>l</mml:mi><mml:msup><mml:mi>x</mml:mi><mml:mi>i</mml:mi></mml:msup><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant="italic">∂</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy="false">→</mml:mo><mml:msup><mml:mi>l</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mi mathvariant="italic">∂</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>v</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy="false">→</mml:mo><mml:mi>k</mml:mi><mml:msup><mml:mi>l</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mi>v</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">→</mml:mo><mml:msup><mml:mi>l</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mi>k</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>t</mml:mi><mml:mo>,</mml:mo></mml:mrow></mml:math><tex-math id="IEq35_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$x^i \rightarrow l x^i, \partial _i \rightarrow l^{-1} \partial _i, v_i \rightarrow k l^{-1} v_i, t \rightarrow l^2 k^{-1} t,$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40089_2015_138_Article_IEq35.gif"/></span>
 and 
<span id="IEq36"><mml:math id="IEq36_Math"><mml:mrow><mml:mi>p</mml:mi><mml:mo stretchy="false">→</mml:mo><mml:mi>p</mml:mi><mml:mi mathvariant="italic">μ</mml:mi><mml:mi>k</mml:mi><mml:msup><mml:mi>l</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msub><mml:mi mathvariant="italic">ρ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>l</mml:mi><mml:mo>-</mml:mo><mml:msup><mml:mi mathvariant="italic">λ</mml:mi><mml:mi>i</mml:mi></mml:msup><mml:msub><mml:mi>r</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math><tex-math id="IEq36_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
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				\usepackage{amssymb}
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				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$p \rightarrow p \mu k l^{-2} + \rho _0 g (l- \lambda ^i r_i) $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40089_2015_138_Article_IEq36.gif"/></span>
, satisfying the condition 
<span id="IEq37"><mml:math id="IEq37_Math"><mml:mrow><mml:msup><mml:mi mathvariant="italic">∂</mml:mi><mml:mi>i</mml:mi></mml:msup><mml:msub><mml:mi>v</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math><tex-math id="IEq37_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
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				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\partial ^i v_i = 0$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40089_2015_138_Article_IEq37.gif"/></span>
, results in non-dimensional momentum conservation and continuity equations as under:
<section id="Equ6"><mml:math display="block" id="Equ6_Math"><mml:mrow><mml:mtable columnspacing="0.5ex"><mml:mtr><mml:mtd/><mml:mtd columnalign="left"><mml:mrow><mml:mrow><mml:mi>P</mml:mi><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>D</mml:mi><mml:msub><mml:mi>v</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow><mml:mo>=</mml:mo><mml:mo>-</mml:mo><mml:msub><mml:mi mathvariant="italic">∂</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mi>p</mml:mi><mml:mo>+</mml:mo><mml:msup><mml:mi mathvariant="italic">∂</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msub><mml:mi>v</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mi>C</mml:mi><mml:msub><mml:mi mathvariant="italic">λ</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mi>T</mml:mi><mml:msub><mml:mi mathvariant="italic">λ</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="right"><mml:mrow/></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mrow><mml:mi>D</mml:mi><mml:mi>C</mml:mi></mml:mrow><mml:mo>=</mml:mo><mml:mi>L</mml:mi><mml:msubsup><mml:mi>e</mml:mi><mml:mn>11</mml:mn><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:msup><mml:mi mathvariant="italic">∂</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mi>C</mml:mi><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="right"><mml:mrow/></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mrow><mml:mi>D</mml:mi><mml:mi>T</mml:mi></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mi mathvariant="italic">∂</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mi>T</mml:mi><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><tex-math id="Equ6_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
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				\begin{document}$$\begin{aligned} &{Pr^{-1} D v_i }= - \partial _ i p + \partial ^2 v_i+ R_c C \lambda _i +R_t T \lambda _i , \nonumber \\ &{D C }= Le^{-1}_{11} \partial ^2 C, \nonumber \\ &{D T}= \partial ^2 T, \end{aligned}$$\end{document}]]></tex-math><graphic specific-use="web" mime-subtype="GIF" xlink:href="40089_2015_138_Article_Equ6.gif"/></section>
where 
<span id="IEq38"><mml:math id="IEq38_Math"><mml:mrow><mml:mi>P</mml:mi><mml:mi>r</mml:mi><mml:mo>=</mml:mo><mml:mi>k</mml:mi><mml:msub><mml:mi mathvariant="italic">ρ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:msup><mml:mi mathvariant="italic">μ</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math><tex-math id="IEq38_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
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				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$ Pr = k \rho _0 \mu ^{-1}$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40089_2015_138_Article_IEq38.gif"/></span>
 is the Prandtl number, 
<span id="IEq39"><mml:math id="IEq39_Math"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mi mathvariant="italic">μ</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mi>k</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mi>B</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:msub><mml:mi mathvariant="italic">ρ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mi>g</mml:mi><mml:msup><mml:mi>l</mml:mi><mml:mn>3</mml:mn></mml:msup></mml:mrow></mml:math><tex-math id="IEq39_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
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				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$R_c = \mu ^{-1} k^{-1}B_c \rho _0 g l^3 $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40089_2015_138_Article_IEq39.gif"/></span>
 is the concentration Rayleigh number, 
<span id="IEq40"><mml:math id="IEq40_Math"><mml:mrow><mml:msub><mml:mi>R</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mi mathvariant="italic">μ</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mi>k</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mi>B</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:msub><mml:mi mathvariant="italic">ρ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mi>g</mml:mi><mml:msup><mml:mi>l</mml:mi><mml:mn>3</mml:mn></mml:msup></mml:mrow></mml:math><tex-math id="IEq40_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
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				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$R_t = \mu ^{-1} k^{-1}B_t \rho _0 g l^3 $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40089_2015_138_Article_IEq40.gif"/></span>
 is the temperature Rayleigh number, and 
<span id="IEq41"><mml:math id="IEq41_Math"><mml:mrow><mml:mi>L</mml:mi><mml:msub><mml:mi>e</mml:mi><mml:mn>11</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mi>k</mml:mi><mml:msubsup><mml:mi>d</mml:mi><mml:mn>11</mml:mn><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:math><tex-math id="IEq41_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
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				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$Le_{11} = k d^{-1}_{11 }$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40089_2015_138_Article_IEq41.gif"/></span>
 are the Lewis numbers.</p>
<p>Now, using the boundary conditions 
<span id="IEq42"><mml:math id="IEq42_Math"><mml:mrow><mml:msup><mml:mi mathvariant="italic">λ</mml:mi><mml:mi>i</mml:mi></mml:msup><mml:msub><mml:mi>v</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math><tex-math id="IEq42_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\lambda ^i v_i = 0 $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40089_2015_138_Article_IEq42.gif"/></span>
, 
<span id="IEq43"><mml:math id="IEq43_Math"><mml:mrow><mml:msup><mml:mi mathvariant="italic">λ</mml:mi><mml:mi>i</mml:mi></mml:msup><mml:msup><mml:mi mathvariant="normal">Λ</mml:mi><mml:mi>j</mml:mi></mml:msup><mml:msub><mml:mi mathvariant="italic">∂</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:msub><mml:mi>v</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math><tex-math id="IEq43_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$ \lambda ^i \Lambda ^j \partial _i v_j = 0 $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40089_2015_138_Article_IEq43.gif"/></span>
, where 
<span id="IEq44"><mml:math id="IEq44_Math"><mml:mrow><mml:msup><mml:mi mathvariant="normal">Λ</mml:mi><mml:mi>i</mml:mi></mml:msup><mml:msub><mml:mi mathvariant="italic">λ</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math><tex-math id="IEq44_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$ \Lambda ^i \lambda _i = 0$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40089_2015_138_Article_IEq44.gif"/></span>
, and 
<span id="IEq45"><mml:math id="IEq45_Math"><mml:mrow><mml:mi>C</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mi>T</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math><tex-math id="IEq45_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
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				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$C(0) =T(0) = 1$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40089_2015_138_Article_IEq45.gif"/></span>
, 
<span id="IEq46"><mml:math id="IEq46_Math"><mml:mrow><mml:mi>C</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mi>T</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math><tex-math id="IEq46_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
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				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$C(1)= T(1) =0$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40089_2015_138_Article_IEq46.gif"/></span>
, 
<span id="IEq47"><mml:math id="IEq47_Math"><mml:mrow><mml:mi>p</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math><tex-math id="IEq47_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
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				\usepackage{amssymb}
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				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$ p(1) =0$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40089_2015_138_Article_IEq47.gif"/></span>
, 
<span id="IEq48"><mml:math id="IEq48_Math"><mml:mrow><mml:msub><mml:mi mathvariant="italic">λ</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math><tex-math id="IEq48_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
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				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\lambda _i = (0, 0, 1)$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40089_2015_138_Article_IEq48.gif"/></span>
 and 
<span id="IEq49"><mml:math id="IEq49_Math"><mml:mrow><mml:msub><mml:mi>r</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mi>z</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math><tex-math id="IEq49_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
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				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$r_i = (x, y, z)$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40089_2015_138_Article_IEq49.gif"/></span>
, and adding small perturbations to steady-state solutions and defining 
<span id="IEq50"><mml:math id="IEq50_Math"><mml:mrow><mml:msubsup><mml:mi>D</mml:mi><mml:mi>b</mml:mi><mml:mi>a</mml:mi></mml:msubsup><mml:mo>=</mml:mo><mml:msup><mml:mi>a</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mi mathvariant="italic">∂</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>-</mml:mo><mml:msup><mml:mi>b</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mi mathvariant="italic">∂</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:math><tex-math id="IEq50_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
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				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$D^a_b = a^{-1} \partial _t - b^{-1} \partial ^2$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40089_2015_138_Article_IEq50.gif"/></span>
, we get
<section id="Equ7"><mml:math display="block" id="Equ7_Math"><mml:mrow><mml:mtable columnspacing="0.5ex"><mml:mtr><mml:mtd/><mml:mtd columnalign="left"><mml:mrow><mml:msubsup><mml:mi>D</mml:mi><mml:mn>1</mml:mn><mml:mrow><mml:mi>P</mml:mi><mml:mi>r</mml:mi></mml:mrow></mml:msubsup><mml:msub><mml:mi>u</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi mathvariant="italic">∂</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mi mathvariant="italic">ϕ</mml:mi><mml:mo>-</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:msub><mml:mi mathvariant="italic">θ</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:msub><mml:mi mathvariant="italic">θ</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:msub><mml:mi mathvariant="italic">λ</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="right"><mml:mrow/></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:msubsup><mml:mi>D</mml:mi><mml:mrow><mml:mi>L</mml:mi><mml:mi>e</mml:mi><mml:mn>11</mml:mn></mml:mrow><mml:mn>1</mml:mn></mml:msubsup><mml:msub><mml:mi mathvariant="italic">θ</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo>-</mml:mo><mml:msup><mml:mi mathvariant="italic">λ</mml:mi><mml:mi>i</mml:mi></mml:msup><mml:msub><mml:mi>u</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="right"><mml:mrow/></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:msubsup><mml:mi>D</mml:mi><mml:mn>1</mml:mn><mml:mn>1</mml:mn></mml:msubsup><mml:msub><mml:mi mathvariant="italic">θ</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>-</mml:mo><mml:msup><mml:mi mathvariant="italic">λ</mml:mi><mml:mi>i</mml:mi></mml:msup><mml:msub><mml:mi>u</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><tex-math id="Equ7_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\begin{aligned} &D^{Pr}_1 u_i + \partial _i \phi - ( R_c \theta _c + R_t \theta _t ) \lambda _i= 0, \nonumber \\ &D^1_{Le11} \theta _c - \lambda ^i u_i= 0, \nonumber \\ &D^1_1 \theta _t - \lambda ^i u_i= 0. \end{aligned}$$\end{document}]]></tex-math><graphic specific-use="web" mime-subtype="GIF" xlink:href="40089_2015_138_Article_Equ7.gif"/></section>
Now,
<section id="Equ8"><mml:math display="block" id="Equ8_Math"><mml:mrow><mml:mtable columnspacing="0.5ex"><mml:mtr><mml:mtd columnalign="right"><mml:mrow><mml:mi>E</mml:mi><mml:msup><mml:mi>a</mml:mi><mml:mi>i</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mi mathvariant="italic">∂</mml:mi><mml:mi>i</mml:mi></mml:msup><mml:msup><mml:mi mathvariant="italic">∂</mml:mi><mml:mi>p</mml:mi></mml:msup><mml:msub><mml:mi>a</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mo>-</mml:mo><mml:msup><mml:mi mathvariant="italic">∂</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mi>a</mml:mi><mml:mi>i</mml:mi></mml:msup><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><tex-math id="Equ8_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\begin{aligned} E a^i = \partial ^i \partial ^p a_p - \partial ^2 a^i. \end{aligned}$$\end{document}]]></tex-math><graphic specific-use="web" mime-subtype="GIF" xlink:href="40089_2015_138_Article_Equ8.gif"/></section>
where 
<span id="IEq51"><mml:math id="IEq51_Math"><mml:mi>E</mml:mi></mml:math><tex-math id="IEq51_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$E$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40089_2015_138_Article_IEq51.gif"/></span>
 is the operator taking twice a curl of any vector field 
<span id="IEq52"><mml:math id="IEq52_Math"><mml:msup><mml:mi>a</mml:mi><mml:mi>i</mml:mi></mml:msup></mml:math><tex-math id="IEq52_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$a^i$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40089_2015_138_Article_IEq52.gif"/></span>
 and as we know the gradient of a scalar field vanishes when its curl is taken twice. Further, 
<span id="IEq53"><mml:math id="IEq53_Math"><mml:mrow><mml:mi>E</mml:mi><mml:msup><mml:mi mathvariant="italic">∂</mml:mi><mml:mi>i</mml:mi></mml:msup><mml:mi mathvariant="italic">ϕ</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math><tex-math id="IEq53_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
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				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$ E \partial ^i \phi = 0 $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40089_2015_138_Article_IEq53.gif"/></span>
, as 
<span id="IEq54"><mml:math id="IEq54_Math"><mml:msub><mml:mi>u</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:math><tex-math id="IEq54_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$u_i$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40089_2015_138_Article_IEq54.gif"/></span>
 is divergence-less, and 
<span id="IEq55"><mml:math id="IEq55_Math"><mml:mrow><mml:msup><mml:mi mathvariant="italic">λ</mml:mi><mml:mi>i</mml:mi></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>E</mml:mi><mml:msub><mml:mi mathvariant="italic">λ</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mi mathvariant="italic">ϕ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mi mathvariant="italic">λ</mml:mi><mml:mi>i</mml:mi></mml:msup><mml:msub><mml:mi mathvariant="italic">∂</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:msup><mml:mi mathvariant="italic">λ</mml:mi><mml:mi>i</mml:mi></mml:msup><mml:msub><mml:mi mathvariant="italic">∂</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mi mathvariant="italic">ϕ</mml:mi><mml:mo>-</mml:mo><mml:msup><mml:mi mathvariant="italic">∂</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mi mathvariant="italic">ϕ</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mover accent="true"><mml:mi mathvariant="italic">∂</mml:mi><mml:mo stretchy="false">~</mml:mo></mml:mover><mml:mn>2</mml:mn></mml:msup><mml:mi mathvariant="italic">ϕ</mml:mi></mml:mrow></mml:math><tex-math id="IEq55_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$ \lambda ^i (E \lambda _i \phi ) = \lambda ^i\partial _i \lambda ^i \partial _j \phi - \partial ^2 \phi = \tilde{\partial }^2 \phi $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40089_2015_138_Article_IEq55.gif"/></span>
. Now, acting by 
<span id="IEq56"><mml:math id="IEq56_Math"><mml:mi>E</mml:mi></mml:math><tex-math id="IEq56_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$E$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40089_2015_138_Article_IEq56.gif"/></span>
, contracting by 
<span id="IEq57"><mml:math id="IEq57_Math"><mml:msub><mml:mi mathvariant="italic">λ</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:math><tex-math id="IEq57_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\lambda _i$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40089_2015_138_Article_IEq57.gif"/></span>
, using Green’s function and applying the boundary conditions (derivatives of 
<span id="IEq58"><mml:math id="IEq58_Math"><mml:mrow><mml:msup><mml:mi mathvariant="italic">λ</mml:mi><mml:mi>i</mml:mi></mml:msup><mml:msub><mml:mi mathvariant="italic">∂</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:math><tex-math id="IEq58_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\lambda ^i \partial _i$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40089_2015_138_Article_IEq58.gif"/></span>
 vanishes on 
<span id="IEq59"><mml:math id="IEq59_Math"><mml:mrow><mml:msup><mml:mi mathvariant="italic">λ</mml:mi><mml:mi>i</mml:mi></mml:msup><mml:msub><mml:mi>u</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:math><tex-math id="IEq59_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\lambda ^i u_i$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40089_2015_138_Article_IEq59.gif"/></span>
) we arrive at
<section id="Equ9"><mml:math display="block" id="Equ9_Math"><mml:mrow><mml:mtable columnspacing="0.5ex"><mml:mtr><mml:mtd columnalign="right"><mml:mrow><mml:msubsup><mml:mi>D</mml:mi><mml:mn>1</mml:mn><mml:mn>1</mml:mn></mml:msubsup><mml:msubsup><mml:mi>D</mml:mi><mml:mrow><mml:mi>L</mml:mi><mml:mi>e</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:msubsup><mml:msubsup><mml:mi>D</mml:mi><mml:mn>1</mml:mn><mml:mrow><mml:mi>P</mml:mi><mml:mi>r</mml:mi></mml:mrow></mml:msubsup><mml:msup><mml:mi mathvariant="italic">∂</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mi mathvariant="italic">λ</mml:mi><mml:mi>i</mml:mi></mml:msup><mml:msub><mml:mi>u</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:msup><mml:mover accent="true"><mml:mi mathvariant="italic">∂</mml:mi><mml:mo stretchy="false">~</mml:mo></mml:mover><mml:mn>2</mml:mn></mml:msup><mml:msubsup><mml:mi>D</mml:mi><mml:mn>1</mml:mn><mml:mn>1</mml:mn></mml:msubsup><mml:msup><mml:mi mathvariant="italic">λ</mml:mi><mml:mi>i</mml:mi></mml:msup><mml:msub><mml:mi>u</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:msup><mml:mover accent="true"><mml:mi mathvariant="italic">∂</mml:mi><mml:mo stretchy="false">~</mml:mo></mml:mover><mml:mn>2</mml:mn></mml:msup><mml:msubsup><mml:mi>D</mml:mi><mml:mrow><mml:mi>L</mml:mi><mml:mi>e</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:msubsup><mml:msup><mml:mi mathvariant="italic">λ</mml:mi><mml:mi>i</mml:mi></mml:msup><mml:msub><mml:mi>u</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><tex-math id="Equ9_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\begin{aligned} D^1_1 D^1_{Le} D^{Pr}_1 \partial ^2 \lambda ^iu_i = R_c \tilde{\partial }^2 D^1_1 \lambda ^i u_i + R_t \tilde{\partial }^2 D^1_{Le} \lambda ^i u_i (r'). \end{aligned}$$\end{document}]]></tex-math><graphic specific-use="web" mime-subtype="GIF" xlink:href="40089_2015_138_Article_Equ9.gif"/></section>
and its solution
<section id="Equ10"><mml:math display="block" id="Equ10_Math"><mml:mrow><mml:mtable columnspacing="0.5ex"><mml:mtr><mml:mtd columnalign="right"><mml:mrow><mml:msup><mml:mi mathvariant="italic">λ</mml:mi><mml:mi>i</mml:mi></mml:msup><mml:msub><mml:mi>u</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mi>A</mml:mi><mml:mo>sin</mml:mo><mml:mi>n</mml:mi><mml:mi mathvariant="italic">π</mml:mi><mml:mspace width="0.166667em"/><mml:mi mathvariant="normal">exp</mml:mi><mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mi>i</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>k</mml:mi><mml:mi>x</mml:mi></mml:msub><mml:mi>x</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mi>k</mml:mi><mml:mi>y</mml:mi></mml:msub><mml:mi>y</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mi mathvariant="italic">σ</mml:mi><mml:mi>t</mml:mi><mml:mo stretchy="false">]</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><tex-math id="Equ10_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
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				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\begin{aligned} \lambda ^i u_i = A \sin n\pi \, {\mathrm{exp}} [i(k_x x + k_y y) + \sigma t], \end{aligned}$$\end{document}]]></tex-math><graphic specific-use="web" mime-subtype="GIF" xlink:href="40089_2015_138_Article_Equ10.gif"/></section>
Thus, we have
<section id="Equ11"><mml:math display="block" id="Equ11_Math"><mml:mrow><mml:mtable columnspacing="0.5ex"><mml:mtr><mml:mtd columnalign="right"><mml:mrow><mml:mi>C</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>k</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="italic">σ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><tex-math id="Equ11_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\begin{aligned} C(k_n, k, \sigma ) =0, \end{aligned}$$\end{document}]]></tex-math><graphic specific-use="web" mime-subtype="GIF" xlink:href="40089_2015_138_Article_Equ11.gif"/></section>
where
<section id="Equ12"><mml:math display="block" id="Equ12_Math"><mml:mrow><mml:mtable columnspacing="0.5ex"><mml:mtr><mml:mtd columnalign="right"><mml:mrow><mml:mi>C</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>k</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="italic">σ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="italic">σ</mml:mi><mml:mo>+</mml:mo><mml:msubsup><mml:mi>k</mml:mi><mml:mi>n</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:mi>L</mml:mi><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>P</mml:mi><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi mathvariant="italic">σ</mml:mi><mml:mo>+</mml:mo><mml:msubsup><mml:mi>k</mml:mi><mml:mi>n</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="italic">σ</mml:mi><mml:mo>+</mml:mo><mml:msubsup><mml:mi>k</mml:mi><mml:mi>n</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:msubsup><mml:mi>k</mml:mi><mml:mi>n</mml:mi><mml:mn>2</mml:mn></mml:msubsup></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign="right"><mml:mrow/></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mo>-</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="italic">σ</mml:mi><mml:mo>+</mml:mo><mml:msubsup><mml:mi>k</mml:mi><mml:mi>n</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:mi>L</mml:mi><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:msup><mml:mi>k</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msub><mml:mi>R</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>-</mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="italic">σ</mml:mi><mml:mo>+</mml:mo><mml:msubsup><mml:mi>k</mml:mi><mml:mi>n</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:msup><mml:mi>k</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msub><mml:mi>R</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><tex-math id="Equ12_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\begin{aligned} C(k_n, k, \sigma )&= (\sigma + k_n^2 Le^{-1} ) (Pr^{-1}\sigma + k_n^2 ) (\sigma + k_n^2 ) k_n^2 \nonumber \\&- (\sigma + k_n^2 Le^{-1} ) k^2 R_t - (\sigma + k_n^2 ) k^2 R_c. \end{aligned}$$\end{document}]]></tex-math><graphic specific-use="web" mime-subtype="GIF" xlink:href="40089_2015_138_Article_Equ12.gif"/></section>
By substituting 
<span id="IEq60"><mml:math id="IEq60_Math"><mml:mrow><mml:mi mathvariant="italic">σ</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math><tex-math id="IEq60_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
				\usepackage{amsfonts}
				\usepackage{amssymb}
				\usepackage{amsbsy}
				\usepackage{mathrsfs}
				\usepackage{upgreek}
				\setlength{\oddsidemargin}{-69pt}
				\begin{document}$$\sigma =0$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40089_2015_138_Article_IEq60.gif"/></span>
, the critical behavior under temperature dependance is given by
<section id="Equ13"><mml:math display="block" id="Equ13_Math"><mml:mrow><mml:mtable columnspacing="0.5ex"><mml:mtr><mml:mtd columnalign="right"><mml:mrow><mml:mi>C</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>k</mml:mi><mml:mi>m</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><tex-math id="Equ13_TeX"><![CDATA[\documentclass[12pt]{minimal}
				\usepackage{amsmath}
				\usepackage{wasysym}
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				\begin{document}$$\begin{aligned} C(k_m, k, 0 ) =0, \end{aligned}$$\end{document}]]></tex-math><graphic specific-use="web" mime-subtype="GIF" xlink:href="40089_2015_138_Article_Equ13.gif"/></section>
where
<section id="Equ14"><mml:math display="block" id="Equ14_Math"><mml:mrow><mml:mtable columnspacing="0.5ex"><mml:mtr><mml:mtd columnalign="right"><mml:mrow><mml:mi>C</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>k</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mo>=</mml:mo><mml:msubsup><mml:mi>k</mml:mi><mml:mi>n</mml:mi><mml:mn>6</mml:mn></mml:msubsup><mml:msup><mml:mi>k</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi>L</mml:mi><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>-</mml:mo><mml:mi>L</mml:mi><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mi>R</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><tex-math id="Equ14_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\begin{aligned} C(k_n, k, 0)&= k_n^6 k^{-2}Le^{-1} - Le^{-1} R_t - R_c. \end{aligned}$$\end{document}]]></tex-math><graphic specific-use="web" mime-subtype="GIF" xlink:href="40089_2015_138_Article_Equ14.gif"/></section>
Now, the effective Rayleigh number is
<section id="Equ15"><mml:math display="block" id="Equ15_Math"><mml:mrow><mml:mtable columnspacing="0.5ex"><mml:mtr><mml:mtd columnalign="right"><mml:mrow><mml:mi>R</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>R</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:mi>L</mml:mi><mml:mi>e</mml:mi><mml:msub><mml:mi>R</mml:mi><mml:mi>c</mml:mi></mml:msub></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><tex-math id="Equ15_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\begin{aligned} R = R_t + Le R_c \end{aligned}$$\end{document}]]></tex-math><graphic specific-use="web" mime-subtype="GIF" xlink:href="40089_2015_138_Article_Equ15.gif"/></section>
Hence, 
<span id="IEq61"><mml:math id="IEq61_Math"><mml:mrow><mml:mi>R</mml:mi><mml:mo>=</mml:mo><mml:msubsup><mml:mi>k</mml:mi><mml:mi>n</mml:mi><mml:mn>6</mml:mn></mml:msubsup><mml:msup><mml:mi>k</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>.</mml:mo></mml:mrow></mml:math><tex-math id="IEq61_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$ R = k_n^6 k^{-2}.$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40089_2015_138_Article_IEq61.gif"/></span>
 For lowest value 
<span id="IEq62"><mml:math id="IEq62_Math"><mml:mrow><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math><tex-math id="IEq62_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$n =1$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40089_2015_138_Article_IEq62.gif"/></span>
, the instability starts at
<section id="Equ16"><mml:math display="block" id="Equ16_Math"><mml:mrow><mml:mtable columnspacing="0.5ex"><mml:mtr><mml:mtd columnalign="right"><mml:mrow><mml:mfrac><mml:mrow><mml:mi mathvariant="italic">∂</mml:mi><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="italic">∂</mml:mi><mml:msup><mml:mi>k</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math><tex-math id="Equ16_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$\begin{aligned} \frac{\partial R}{\partial k^2} = 0. \end{aligned}$$\end{document}]]></tex-math><graphic specific-use="web" mime-subtype="GIF" xlink:href="40089_2015_138_Article_Equ16.gif"/></section>
which gives 
<span id="IEq63"><mml:math id="IEq63_Math"><mml:mrow><mml:msup><mml:mi>k</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mi mathvariant="italic">π</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy="false">/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math><tex-math id="IEq63_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$ k^2 = \pi ^2/2 $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40089_2015_138_Article_IEq63.gif"/></span>
, and the value of 
<span id="IEq64"><mml:math id="IEq64_Math"><mml:mi>R</mml:mi></mml:math><tex-math id="IEq64_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$ R$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40089_2015_138_Article_IEq64.gif"/></span>
 will be 
<span id="IEq65"><mml:math id="IEq65_Math"><mml:mrow><mml:mi>R</mml:mi><mml:mo>=</mml:mo><mml:mn>657.5</mml:mn></mml:mrow></mml:math><tex-math id="IEq65_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$ R= 657.5 $$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40089_2015_138_Article_IEq65.gif"/></span>
. Thus, by maintaining the rate of evaporation to an extent wherein 
<span id="IEq66"><mml:math id="IEq66_Math"><mml:mrow><mml:mi>R</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>657.5</mml:mn></mml:mrow></mml:math><tex-math id="IEq66_TeX"><![CDATA[\documentclass[12pt]{minimal}
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				\begin{document}$$R > 657.5$$\end{document}]]></tex-math><inline-graphic specific-use="web" mime-subtype="GIF" xlink:href="40089_2015_138_Article_IEq66.gif"/></span>
, the formation of nanostructures takes place.</p></section></section>
<section><h2>Conclusion</h2>
<p>Highly crystalline nanostrips of Boehmite (AlOOH) have been prepared by a fast, an inexpensive and environmentally benign technique. The mechanism for the formation of nanostructures is briefly described in accordance with decomposition of metal with water giving out hydrogen. The formation mechanism has been explained by a mathematical model and it would be quite interesting to apply this technique in other metal and metal oxide nanostructures.</p></section><hr/><h2>Acknowledgments</h2>
<p>We are pleased to acknowledge KAU, KAUST and World Bank for characterization of samples and SEM. The authors are also highly thankful to Mir Faizal and Sofi Javaid Jameel for their immense help.</p>
<hr/><h2>References</h2>
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