On the upper limits for complex growth rate in rotatory electrothermoconvection in a dielectric fluid layer saturating a sparsely distributed porous medium
{"title":"On the upper limits for complex growth rate in rotatory electrothermoconvection in a dielectric fluid layer saturating a sparsely distributed porous medium","authors":"Jitender Kumar, Chitresh Kumari, Jyoti Prakash","doi":"10.2478/sgem-2024-0008","DOIUrl":null,"url":null,"abstract":"\n <jats:p>It is proved analytically that the complex growth rate <jats:italic>n</jats:italic> = <jats:italic>n<jats:sub>r</jats:sub>\n </jats:italic> + <jats:italic>in<jats:sub>i</jats:sub>\n </jats:italic> (<jats:italic>n<jats:sub>r</jats:sub>\n </jats:italic> and <jats:italic>n<jats:sub>i</jats:sub>\n </jats:italic> are the real and imaginary parts of <jats:italic>n</jats:italic> , respectively) of an arbitrary neutral or unstable oscillatory disturbance of growing amplitude in rotatory electrothermoconvection in a dielectric fluid layer saturating a sparsely distributed porous medium heated from below, for the case of free boundaries, is located inside a semicircle in the right half of the <jats:italic>n<jats:sub>r</jats:sub>n<jats:sub>i</jats:sub>\n </jats:italic> − plane, whose centre is at the origin and radius = \n<jats:inline-formula>\n <jats:alternatives>\n <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_sgem-2024-0008_ineq_001.png\"/>\n <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n <m:mrow>\n <m:msqrt>\n <m:mrow>\n <m:mo>max</m:mo>\n <m:mfenced>\n <m:mrow>\n <m:msub>\n <m:mi>T</m:mi>\n <m:mi>a</m:mi>\n </m:msub>\n <m:msubsup>\n <m:mi>P</m:mi>\n <m:mi>r</m:mi>\n <m:mn>2</m:mn>\n </m:msubsup>\n <m:mo>,</m:mo>\n <m:mfrac>\n <m:mrow>\n <m:msub>\n <m:mi>R</m:mi>\n <m:mrow>\n <m:mi mathvariant=\"italic\">ea</m:mi>\n </m:mrow>\n </m:msub>\n <m:msub>\n <m:mi>P</m:mi>\n <m:mi>r</m:mi>\n </m:msub>\n </m:mrow>\n <m:mi>A</m:mi>\n </m:mfrac>\n </m:mrow>\n </m:mfenced>\n </m:mrow>\n </m:msqrt>\n </m:mrow>\n </m:math>\n <jats:tex-math>\\sqrt {\\max \\left( {{T_a}P_r^2,{{{R_{ea}}{P_r}} \\over A}} \\right)} </jats:tex-math>\n </jats:alternatives>\n </jats:inline-formula>\n, where <jats:italic>T<jats:sub>a</jats:sub>\n </jats:italic> is the modified Taylor’s number, <jats:italic>P<jats:sub>r</jats:sub>\n </jats:italic> is the modified Prandtl number, <jats:italic>R<jats:sub>ea</jats:sub>\n </jats:italic> is electric Rayleigh number and <jats:italic>A</jats:italic> is the ratio of heat capacities. The upper limits for the case of rigid boundaries are derived separately. Furthermore, similar results are also derived for the same configuration when heated from above.</jats:p>","PeriodicalId":44626,"journal":{"name":"Studia Geotechnica et Mechanica","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Studia Geotechnica et Mechanica","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.2478/sgem-2024-0008","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MECHANICS","Score":null,"Total":0}
引用次数: 0
Abstract
It is proved analytically that the complex growth rate n = nr + ini (nr and ni are the real and imaginary parts of n , respectively) of an arbitrary neutral or unstable oscillatory disturbance of growing amplitude in rotatory electrothermoconvection in a dielectric fluid layer saturating a sparsely distributed porous medium heated from below, for the case of free boundaries, is located inside a semicircle in the right half of the nrni − plane, whose centre is at the origin and radius =
maxTaPr2,ReaPrA\sqrt {\max \left( {{T_a}P_r^2,{{{R_{ea}}{P_r}} \over A}} \right)}
, where Ta is the modified Taylor’s number, Pr is the modified Prandtl number, Rea is electric Rayleigh number and A is the ratio of heat capacities. The upper limits for the case of rigid boundaries are derived separately. Furthermore, similar results are also derived for the same configuration when heated from above.
It is proved analytically that the complex growth rate n = nr + ini (nr and ni are the real and imaginary parts of n , respectively) of an arbitrary neutral or unstable oscillatory disturbance of growing amplitude in rotatory electrothermoconvection in a dielectric fluid layer saturating a sparsely distributed porous medium heated from below, for the case of free boundaries, is located inside a semicircle in the right half of the nrni − plane, whose centre is at the origin and radius = max T a P r 2 , R ea P r A \sqrt {\max \left( {{T_a}P_r^2,{{{R_{ea}}{P_r}} \over A}} \right)} , where Ta is the modified Taylor’s number, Pr is the modified Prandtl number, Rea is electric Rayleigh number and A is the ratio of heat capacities.对刚性边界的上限进行了单独推导。此外,还推导出从上方加热相同构造时的类似结果。
期刊介绍:
An international journal ‘Studia Geotechnica et Mechanica’ covers new developments in the broad areas of geomechanics as well as structural mechanics. The journal welcomes contributions dealing with original theoretical, numerical as well as experimental work. The following topics are of special interest: Constitutive relations for geomaterials (soils, rocks, concrete, etc.) Modeling of mechanical behaviour of heterogeneous materials at different scales Analysis of coupled thermo-hydro-chemo-mechanical problems Modeling of instabilities and localized deformation Experimental investigations of material properties at different scales Numerical algorithms: formulation and performance Application of numerical techniques to analysis of problems involving foundations, underground structures, slopes and embankment Risk and reliability analysis Analysis of concrete and masonry structures Modeling of case histories