DOCSLIB.ORG

Rayleigh-Bénard Chandrasekhar Convection in an Electrically Conducting Fluid Using Maxwell- Cattaneo Law with Temperature Modulation of the Boundaries

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Rayleigh-Bénard Chandrasekhar Convection in an Electrically Conducting Fluid Using Maxwell- Cattaneo Law with Temperature Modulation of the Boundaries International Journal of Engineering Research & Technology (IJERT) ISSN: 2278-0181 Vol. 4 Issue 10, October-2015 Rayleigh-Bénard Chandrasekhar Convection in An Electrically Conducting Fluid using Maxwell- Cattaneo Law with Temperature Modulation of the Boundaries S. Pranesh R.V. Kiran Department of Mathematics, Department of Mathematics, Christ University, Hosur Road, Christ Junior College, Bangalore 560 029, India. Hosur Road, Bangalore 560 029, India. Abstract - The effect of imposed time-periodic boundary dQ temperature (ITBT, also called temperature modulation) and Q T magnetic field at the onset of convection is investigated by dt making a linear analysis. The classical Fourier heat law is where, Q is the heat flux, is a relaxation time and is the replaced by the non-classical Maxwell-Cattaneo law. The classical approach predicts an infinite speed for the propagation heat conductivity. This heat conductivity equation together of heat. The adopted non-classical theory involves wave type of with conservation of energy equation introduces the heat transport and does not suffer from the physically hyperbolic equation, which describes heat propagation with unacceptable drawback of infinite heat propagation speed. The finite speed. Puri and Jordan [12, 13] and Puri and Kythe Venezian approach is adopted in arriving at the critical Rayleigh [14, 15] have studied other fluid mechanics problems by number, correction Rayleigh number and wave number for employing the Maxwell-Cattaneo heat flux law. small amplitude of ITBT. Three cases of oscillating temperature field are examined: (a) symmetric, so that the wall temperatures Thomson [16] and Chandrasekhar [17] studied are modulated in phase, (b) asymmetric, corresponding to out-of theoretically and Nakagawa [18, 19] and Jirlow [20] studied phase modulation and (c) only the lower wall is modulated. The experimentally, the thermal stability in a horizontal layer of temperature modulation is shown to give rise to sub-critical fluid with magnetic field and found that vertical magnetic motion. The shift in the critical Rayleigh number is calculated as a function of frequency and it is found that it is possible to field delays the onset of thermal convection. The application advance or delay the onset of convection by time modulation of of magnetoconvection is primarily found in astrophysics and the wall temperatures. It is shown that the system is most stable in particular by the observation of sunspots. Motivated by the when the boundary temperatures are modulated out-of-phase. It above applications several authors investigated the is also found that the results are noteworthy at short times and suppression of convection by strong magnetic fields. the critical eigenvalues are less than the classical ones. One of the effective mechanisms of controlling the convection is by maintaining a non-uniform temperature Keywords: Time periodic boundary temperature, Rayleigh-Bénard gradient across the boundaries. Such a temperature gradient Chandrasekhar convection, Magnetic field, Maxwell-Cattaneo may be generated by (i) an appropriate heating or cooling at law. the boundaries, (ii) injection/suction of fluid at the I. INTRODUCTION boundaries, (iii) an appropriate distribution of heat source, and (iv) radiative heat transfer. These methods are mainly The Classical Fourier law of heat conduction concerned with only space-dependent temperature gradients. expresses that the heat flux within a medium is proportional However in practice, the non-uniform temperature gradients to the local temperature gradient in the system. A well find its origin in transient heating or cooling at the known consequence of this law is that heat perturbations boundaries. Hence, basic temperature profile depends propagate with an infinite velocity. This drawback of the explicitly on position and time. This problem, called the classical law motivated Maxwell [1], Cattaneo [2], Lebon and thermal modulation problem, involves solution of the energy Cloot [3], Dauby et al. [4], Straughan [5], Siddheshwar [6], equation under suitable time-dependent boundary conditions. Pranesh [7], Pranesh and Kiran [8, 9, 10] and Pranesh and This temperature profile which is a function of both space and Smita[11] to adopt a non-classical Maxwell-Cattaneo heat time can be used as an effective mechanism to control the flux law in studying Rayleigh-Bénard/Marangoni convection convective flow by proper adjustment of its parameters, to get rid of this unphysical results. This Maxwell-Cattaneo namely, amplitude and frequency of modulation. It can be heat flux law equation contains an extra inertial term with used to control the convection in material processing respect to the Fourier law. applications to achieve higher efficiency and to advance IJERTV4IS100189 www.ijert.org 174 (This work is licensed under a Creative Commons Attribution 4.0 International License.) International Journal of Engineering Research & Technology (IJERT) ISSN: 2278-0181 Vol. 4 Issue 10, October-2015 convection in achieving major enhancement of heat, mass and Maxwell – Cattaneo heat flux law: momentum transfer. . There are many studies available in the literature Q 1 Q Q T, (4) concerning how a time-periodic boundary temperature affects the onset of Rayleigh-Bénard convection. Since the problem Magnetic induction equation: of Taylor stability and Bénard stability are very similar, H 2 Venezian [21] investigated the thermal analogue of (q.)H (H.)q m H, (5) Donnelly’s experiment [22] for small amplitude temperature t modulations. Many researchers, under different conditions, Equation of state: have investigated thermal instability in a horizontal layer with o[1 (T To )]. (6) temperature modulation. Some of them are Gershuni and Zhukhovitskii [23], Rosenblat and Tanaka [24], Siddheshwar where, q is the velocity, H is the magnetic field, T is the and Pranesh [25, 26], Mahabaleswar [27], Bhatia and temperature, P is the hydromagnetic pressure, is the Bhadauria [28, 29], Malashetty and Mahantesh Swamy [30], Bhadauria [31], Siddheshwar and Annamma [32], Bhadauria density, o is the density of the fluid at reference and Atul [33], Pranesh and Sangeetha [34, 35], Pranesh [36] 1 and Pranesh and Riya[37]. temperature T To , m , m is magnetic m The objective of the present problem is to investigate permeability, g is the acceleration due to gravity, is the the combined effect of magnetic field and temperature thermal conductivity, is the coefficient of thermal modulation on the onset of Raleigh-Bénard 1 magnetoconvection using Maxwell-Cattaneo law. expansion, is the dynamic viscosity, q, Q is 1 2 II. MATHEMATICAL FORMULATION the heat flux vector and is the constant relation time. Consider an infinite horizontal layer of a Boussinesquian, electrically conducting fluid, of depth‘d’. The lower wall z 0 and upper wall z d are Cartesian co-ordinate system is taken with origin in the lower subjected to the temperatures boundary and z-axis vertically upwards. Let T be the 1 temperature difference between the lower and upper T T0 T[1 cos(t)], (7) boundaries. (See Fig. 1.) 2 and H0 Z 1 T T T[1 cos(t )]. (8) 1 0 T T0 T[1 cos(t )] 2 Z =d 2 respectively, where is the small amplitude, is the frequency and is the phase angle. III. BASIC STATE Y The basic state of the fluid being quiescent is described by: Z = 0 1 qb 0, P Pb (z,t), b (z,t), O T T0 T[1 cos(t)] 2 X Q 0,0,Qb (z), T Tb (z,t), , (9) Figure 1. Schematic diagram of the Rayleigh – Bénard situation. ˆ Hb H0k The basic governing equations are: The temperature Tb , pressure Pb , Qb heat flux and density Continuity equation: b satisfy .q 0, (1) T 2T Conservation of linear momentum: b b , (10) t z2 q ˆ o (q.)q P gk P t , (2) b g, (11) z b 2 q m H.H T Q b , (12) Conservation of energy: b z T qT .Q, (3) b 0[1 (Tb T0)]. (13) t IJERTV4IS100189 www.ijert.org 175 (This work is licensed under a Creative Commons Attribution 4.0 International License.) International Journal of Engineering Research & Technology (IJERT) ISSN: 2278-0181 Vol. 4 Issue 10, October-2015 The solution of “(10)” that satisfies the thermal boundary 0T. (25) conditions of “(7)” and “(8)” is Operating divergence on the “(23)” and substituting T T (z) T (z,t), (14) b s 1 in “(22)”, on using “(20)”, we get The steady temperature field Ts and an oscillating field T1 T 1 Tb 2 are given by 1 1 W where, t t 2 z , (26) T T T (d 2z), (15) 2 Tb s 0 T 1 W 2d t z z z where, d d it 1 T1 Re a()e a()e e . (16) The perturbation “(21)”, “(24)” and “(26)” are non- dimensionalised using the following definitions: In the “(16)” (x, y,z) q 1 (x*,y*,z*) , q* , d2 2 d (1 i) , (17) d 2 * W * t W , t , , (27) 2 T ei e d d a() . (18) 2 e e T H T* , H* and Re stands for the real part. T Ho We now superpose infinitesimal perturbation on this basic state and study the stability of the system. Using “(25)” in “(21)”, operating curl twice on the resulting equation and non-dimensionalising the resulting IV. LINEAR STABILITY ANALYSIS equation and also “(24)” and “(26)”, using “(27)”, we get: Let the basic state be disturbed by an infinitesimal thermal perturbation. We now have 1 2 2 4 ( W) R1 T W Pr t q qb q , P Pb P , , (28) Pr 2 Hz b , Q Qb Q, , (19) Q Pm z ˆ T Tb T, Hb H0k H Hz W Pr 2 The primes indicate that the quantities are H , (29) z z Pm z infinitesimal perturbations and subscript ‘b’ indicates basic state value. T T0 2 2 1 2C C W T Substituting “(19)” in to “(1)” to “(6)” and using the t t z basic state solutions “(10)” to “(13)”.
Load more

Recommended publications
  • UC Santa Cruz 2010 International Summer Institute for Modeling in Astrophysics
    UC Santa Cruz 2010 International Summer Institute for Modeling in Astrophysics Title Taming jets in magnetised fluids Permalink https://escholarship.org/uc/item/5ff523wv Authors Kosuga, Yusuke Brummell, Nicholas Publication Date 2010-09-01 eScholarship.org Powered by the California Digital Library University of California Taming Jets in magnetized fluids Y. Kosuga1 and N. H. Brummell2 1)Center for Astrophysics and Space Sciences and Department of physics, University of California at San Diego, 9500 Gilman Drive, La Jolla, CA 92093 2)Department of Applied Mathematic and Statistics, University of California at Santa Cruz, 1156 High Street, Santa Cruz, CA 95064 Effects of a uniform horizontal magnetic field on jets dynamics in 2D Boussinesq turbulence, i.e. Howard-Krishnamurti problem are studied with a numerical simula- tion. For a fixed fluid and magnetic diffusivity, it is shown that as the imposed field strength becomes larger jets start behaving in a more organized way, i.e. achieve sta- tionary state and are finally quenched. The time evolution of total stress, Reynolds stress, Maxwell stress is examined and all the stresses are shown to vanish when jets are quenched. The quenching of jets is confirmed for different values of magnetic diffusivity, albeit the required field strength increases. It is also shown that the in- clusion of overstable modes reinforces jets where Maxwell stress overcomes Reynolds stress. For a larger imposed field jets are shown to quench. A possible mechanism for the transition to the reinforcement of jets by Maxwell stress is discussed based on the transition in the most unstable mode in the underlying turbulence.
    [Show full text]
  • Canonical Models of Geophysical and Astrophysical Flows: Turbulent Convection Experiments in Liquid Metals
    Metals 2015, 5, 289-335; doi:10.3390/met5010289 OPEN ACCESS metals ISSN 2075-4701 www.mdpi.com/journal/metals Article Canonical Models of Geophysical and Astrophysical Flows: Turbulent Convection Experiments in Liquid Metals Adolfo Ribeiro 1;*, Guillaume Fabre 1;2, Jean-Luc Guermond 3 and Jonathan M. Aurnou 1 1 Department of Earth, Planetary and Space Sciences, University of California, Los Angeles, CA 90095, USA; E-Mail: [email protected] 2 Laboratoire de Physique, École Normale Supérieure, Lyon 69007, France; E-Mail: [email protected] 3 Department of Mathematics, Texas A & M University, College Station, TX 77843, USA; E-Mail: [email protected] * Author to whom correspondence should be addressed; E-Mail: [email protected]; Tel.: +1-310-467-6102. Academic Editor: Enrique Louis Received: 25 July 2014 / Accepted: 9 February 2015 / Published: 9 March 2015 Abstract: Planets and stars are often capable of generating their own magnetic fields. This occurs through dynamo processes occurring via turbulent convective stirring of their respective molten metal-rich cores and plasma-based convection zones. Present-day numerical models of planetary and stellar dynamo action are not carried out using fluids properties that mimic the essential properties of liquid metals and plasmas (e.g., using fluids with thermal Prandtl numbers P r < 1 and magnetic Prandtl numbers P m 1). Metal dynamo simulations should become possible, though, within the next decade. In order then to understand the turbulent convection phenomena occurring in geophysical or astrophysical fluids and next-generation numerical models thereof, we present here canonical, end-member examples of thermally-driven convection in liquid gallium, first with no magnetic field or rotation present, then with the inclusion of a background magnetic field and then in a rotating system (without an imposed magnetic field).
    [Show full text]
  • Frequency Power Spectra of Global Quantities in Magnetoconvection
    Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 5 March 2021 Article Frequency power spectra of global quantities in magnetoconvection Sandip Das and Krishna Kumar* [1] Department of Physics, Indian Institute of Technology Kharagpur, Kharagpur-721302, India * Correspondence: [email protected] 1 Abstract: We present the results of direct numerical simulations of power spectral densities for 2 kinetic energy, convective entropy and heat flux for unsteady Rayleigh-Bénard magnetoconvection 3 in the frequency space. For larger values of frequency, the power spectral densities for all the −2 4 global quantities vary with frequency f as f . The scaling exponent is independent of Rayleigh 5 number, Chandrasekhar’s number and thermal Prandtl number. 6 Keywords: Magnetoconvection, Turbulent flow, Power spectral densities (PSD) 7 1. Introduction 8 The temporal fluctuations of spatially averaged (or, global) quantities are of interest 9 in several fields of research including turbulent flows [1–4], nanofluids [5], biological 10 fluids [6,7], geophysics [8,9], phase transitions [10,11]. The probability density function 11 (PDF) of the temporal fluctuations of thermal flux in turbulent Rayleigh-Bénard con- 12 vection (RBC) was found to have normal distribution with slight asymmetries at the 13 tails. The direct numerical simulations (DNS) of the Nusselt number Nu, which is a 14 measure of thermal flux, also showed the similar behaviour in presence of the Lorentz 15 force [12]. The power spectral density (PSD) of the thermal flux in the frequency ( f ) −2 16 space [3,12,13] was found to vary as f . In this work, we present the results obtained 17 by DNS of temporal signals of global quantities: spatially averaged kinetic energy per Citation: Das, S.; Kumar, K.
    [Show full text]
  • Thermal Convection Experiments in Liquid Metal Flows with and Without
    Thermal convection experiments in liquid metal flows with and without magnetic field Dissertation zur Erlangung des akademischen Grades Doktoringenieur (Dr.-Ing.) vorgelegt der Fakultät für Maschinenbau der Technischen Universität Ilmenau von Herrn M.Sc. Till Zürner geboren am 1. Oktober 1990 in Dresden, Deutschland 1. Gutachter: Univ.-Prof. Dr. rer. nat. habil. Jörg Schumacher 2. Gutachter: Dr. rer. nat. Sven Eckert 3. Gutachter: Prof. Dr. Alban Pothérat Tag der Einreichung: 2. Juli 2019 Tag der wissenschaftlichen Aussprache: 12. November 2019 urn:nbn:de:gbv:ilm1-2019000433 Abstract The interaction of electrically conducting fluid flows with magnetic fields appears in numerous natural phenomena and technical applications. Since the relevant fluids – such as liquid metals and plasmas – are generally very hot, the flows are often accompanied or even driven by thermal convection. The study of this so-called magnetoconvection is thus of interest for a number of physical systems. Two aspects are investigated in this thesis. The first concerns the case when an imposed magnetic field does not alter the fluid flow. The second case explores the changes of the flow structure and global transport properties in the presence of strong magnetic fields. The first point is relevant for inductive measurement techniques, which are required to probe the flow without disturbing it. Here, the size of the fluid volume affected by a localised magnetic field is of major importance. This topic is investigated theoretically by deriving an algorithm to calculate the penetration depth of the magnetic field into the medium. This allows the prediction of a magnetic field strength, above which a flow is significantly disturbed.
    [Show full text]
  • Experiments on Rayleigh–Bénard Convection, Magnetoconvection
    J. Fluid Mech. (2001), vol. 430, pp. 283–307. Printed in the United Kingdom 283 c 2001 Cambridge University Press Experiments on Rayleigh–Benard´ convection, magnetoconvection and rotating magnetoconvection in liquid gallium By J. M. AURNOU AND P. L. OLSON Department of Earth and Planetary Sciences,† Johns Hopkins University, Baltimore, MD 21218, USA (Received 4 March 1999 and in revised form 12 September 2000) Thermal convection experiments in a liquid gallium layer subject to a uniform rotation and a uniform vertical magnetic field are carried out as a function of rotation rate and magnetic field strength. Our purpose is to measure heat transfer in a low-Prandtl-number (Pr =0.023), electrically conducting fluid as a function of the applied temperature difference, rotation rate, applied magnetic field strength and fluid-layer aspect ratio. For Rayleigh–Benard´ (non-rotating, non-magnetic) convection 0.272 0.006 we obtain a Nusselt number–Rayleigh number law Nu =0.129Ra ± over the range 3.0 103 <Ra<1.6 104. For non-rotating magnetoconvection, we find that the critical× Rayleigh number× RaC increases linearly with magnetic energy density, and a heat transfer law of the form Nu Ra1/2. Coherent thermal oscillations are detected in magnetoconvection at 1.4Ra∼C . For rotating magnetoconvection, we find that the convective heat transfer is∼ inhibited by rotation, in general agreement with theoretical predictions. At low rotation rates, the critical Rayleigh number increases linearly with magnetic field intensity. At moderate rotation rates, coherent thermal oscillations are detected near the onset of convection. The oscillation frequencies are close to the frequency of rotation, indicating inertially driven, oscillatory convection.
    [Show full text]
  • The Dimnum Package
    The dimnum package Miguel R. Clemente [email protected] v1.0.1 from 2021/04/05 Note: Prandtl number is redefined from the amsmath package. 1 Introduction This package simplifies the calling of Dimensionless Numbers in math or text mode. In Table 1 you can find all available Dimensionless Numbers. 2 Usage A Dimensionless number is composed of four items: the command, the symbol, the name, its identifier. You can call a Dimensionless Number in three distinct ways: by its symbol { using the command (i.e. \Ar { Ar). by its name (short version) { appending [s] to the command (i.e. \Bi[s] { Biot). by its name and identifier (long version) { appending [l] to the command (i.e. \Kn[l] { Knudsen number). Symbol, short and long versions, all work in math or text mode without the need of further commands. 1 Besides the comprehensive list of included Dimensionless Numbers, this pack- age also introduces a command to create new Dimensionless Numbers. Creating a Dimensionless Number is achieved by using \newdimnum{\command}{symbol}{name}{identifier} for example, to add the Morton number we write \newdimnum{\Mo}{Mo}{Morton}{number} The identifier can be left empty, such as in the case of Drag Coefficient \newdimnum{Cd}{\ensuremath{C_d}}{Drag Coefficient}{} in this example we also introduce an important command. When the Dimension- less Number symbol is always expressed in math mode { either by definition or the use of subscripts or superscripts { we add \ensuremath{} to encompass the symbol, ensuring a proper representation of the Dimensionless Number. You can add your own Dimensionless Numbers to your projects.
    [Show full text]
  • SASO-ISO-80000-11-2020-E.Pdf
    SASO ISO 80000-11:2020 ISO 80000-11:2019 Quantities and units - Part 11: Characteristic numbers ICS 01.060 Saudi Standards, Metrology and Quality Org (SASO) ----------------------------------------------------------------------------------------------------------- this document is a draft saudi standard circulated for comment. it is, therefore subject to change and may not be referred to as a saudi standard until approved by the boardDRAFT of directors. Foreword The Saudi Standards ,Metrology and Quality Organization (SASO)has adopted the International standard No. ISO 80000-11:2019 “Quantities and units — Part 11: Characteristic numbers” issued by (ISO). The text of this international standard has been translated into Arabic so as to be approved as a Saudi standard. DRAFT DRAFT SAUDI STANDADR SASO ISO 80000-11: 2020 Introduction Characteristic numbers are physical quantities of unit one, although commonly and erroneously called “dimensionless” quantities. They are used in the studies of natural and technical processes, and (can) present information about the behaviour of the process, or reveal similarities between different processes. Characteristic numbers often are described as ratios of forces in equilibrium; in some cases, however, they are ratios of energy or work, although noted as forces in the literature; sometimes they are the ratio of characteristic times. Characteristic numbers can be defined by the same equation but carry different names if they are concerned with different kinds of processes. Characteristic numbers can be expressed as products or fractions of other characteristic numbers if these are valid for the same kind of process. So, the clauses in this document are arranged according to some groups of processes. As the amount of characteristic numbers is tremendous, and their use in technology and science is not uniform, only a small amount of them is given in this document, where their inclusion depends on their common use.
    [Show full text]
  • Magnetoconvection
    Magnetoconvection In lecture 8, we considered how the flow of a conducting fluid affects the magnetic field. Now we will study how the magnetic field affects the flow. We will extend our linear stability analysis for rotating convection in a plane layer (lecture 6) by adding a magnetic field. A Lorentz force must be added to the Navier Stokes equation and the induction equation must be solved along with the other equations. ∂u ρ ( + u ⋅ ∇u) + 2 ρ Ω e × u + ∇p = η ∇ 2u + ρ g' + µ −1 (∇ × B) × B ∂t z o Inertia Coriolis pressure viscosity mod. gravity Lorentz Scaling length, time, etc. as in 3.3 and using a characteristic magnetic field strength Bo : 1 ∂u 2 Ch Pr ( + u ⋅∇u) + e × u + ∇p = ∇ 2u + Ra T e + (∇ × B) × B Pr ∂t E z z Pm ∂T ∂B Pr + u ⋅ ∇T = ∇ 2 T = ∇ × (v × B) + ∇ 2 B ∇ ⋅u = 0 ∇ ⋅ B = 0 ∂t ∂t Pm B2 D2 The Chandrasekhar number is a measure Ch = o Chandrasekhar number for the relative importance of Lorentz forces µo ρνλ to viscous forces. The ratio q = Pm/Pr is ν Pm = magnetic Prandtl number sometimes called the Roberts number. We λ abbreviate Ch‘ = ChPr/Pm Symbols (index R - rotating reference frame, I – inertial frame): Ω – rotation frequency, g‘ – modified gravity, U – characteristic Christensen: Fluid dynamics of Earth and Planetary Interiors, Kyoto, November 2006 9.1 Magnetoconvection: Linear stability Solving only the induction equation for a given u is the kinematic dynamo problem. Solving the full set of equations on 9.1, with boundary conditions for B that imply no external source, is the magnetohydrodynamic dynamo problem (if boundary conditions for u imply no external driving of the flow, it is a convection-driven dynamo).
    [Show full text]
  • Astronomy & Astrophysics Magnetic Instability in a Sheared Azimuthal Flow
    A&A 388, 688–691 (2002) Astronomy DOI: 10.1051/0004-6361:20020510 & c ESO 2002 Astrophysics Magnetic instability in a sheared azimuthal flow A. P. Willis and C. F. Barenghi Department of Mathematics, University of Newcastle, Newcastle upon Tyne, NE2 7RU, UK Received 8 January 2002 / Accepted 3 April 2002 Abstract. We study the magneto-rotational instability of an incompressible flow which rotates with angular velocity Ω(r)=a + b/r2 where r is the radius and a and b are constants. We find that an applied magnetic field destabilises the flow, in agreement with the results of R¨udiger & Zhang (2001). We extend the investigation in the region of parameter space which is Rayleigh stable. We also study the instability at values of magnetic Prandtl number which are much larger and smaller than R¨udiger & Zhang. Large magnetic Prandtl numbers are motivated by their possible relevance in the central region of galaxies (Kulsrud & Anderson 1992). In this regime we find that increasing the magnetic Prandtl number greatly enhances the instability; the stability boundary drops below the Rayleigh line and tends toward the solid body rotation line. Very small magnetic Prandtl numbers are motivated by the current MHD dynamo experiments performed using liquid sodium and gallium. Our finding in this regime confirms R¨udiger & Zhang’s conjecture that the linear magneto-rotational instability and the nonlinear hydrodynamical instability (Richard & Zahn 1999) take place at Reynolds numbers of the same order of magnitude. Key words. accretion discs – instabilities – magnetohydrodynamics – turbulence 1. Motivation to liquid sodium and gallium (Pm ≈ 10−5) which are used in current MHD dynamo experiments (Tilgner 2000).
    [Show full text]
  • Linear and Non-Linear Magnetoconvection in a Porous Medium
    Proc. Indian Acad. Sci. (Math. Sci.), Vol. 93, Nos 2 & 3, December 1984, pp. 117-135 Printed in India. Linear and non-linear magnetoconvection in a porous medium N RUDRAIAH UGC-DSA Centre in Fluid Mechanics, Department of Mathematics, Central College, Bangalore University, Bangalore 560001, India Abstract. Linear and non-linear magnetoconvection in a sparsely packed porous medium with an imposed vertical magnetic field is studied. In the case of linear theory the conditions for direct and oscillatory modes are obtained using the normal modes. Conditions for simple- and Hopf-bifurcations are also given. Using the theory of self-adjoint operator the variation of critical eigenvalue with physical parameters and boundary conditions is studied. In the case of non-linear theory the subcritical instabilities for disturbances of finite amplitude is discussed in detail using a truncated representation of the Fourier expansion. The formal eigenfunction expansion procedure in the Fourier expansion based on the eigenfunctions of the correspond- ing linear stability problem is justified by proving a completeness theorem for a general class of non-self-adjoint eigenvalue problems. It is found that heat transport increases with an increase in Rayleigh number, ratio of thermal diffusivity to magnetic diffusivity and porous parameter but decreases with an increase in Chandrasekhar number. 1. Introduction The linear and non-linear magnetoconvection in an electrically conducting fluid has been extensively studied by many authors ([1], [2], [17], [18], [21], [22]) but the corresponding problem in a porous medium has not been given much attention, inspite of its various engineering and geophysical applications. Rudraiah [12], Rudraiah and Prabhamani [13], and Prabhamani and Rudraiah [9-11], have considered this problem using a universal stability analysis (i.e., stability for arbitrary disturbances) which gives, as in infinitesimal disturbances, only the criterion for the onset of convection but is silent about the prediction of heat transfer.
    [Show full text]
  • Using Standard Syste
    RAPID COMMUNICATIONS PHYSICAL REVIEW E VOLUME 62, NUMBER 4 OCTOBER 2000 Effect of a vertical magnetic field on turbulent Rayleigh-Be´nard convection S. Cioni, S. Chaumat,* and J. Sommeria Ecole Normale Supe´rieure de Lyon, Laboratoire de Physique, CNRS URA1325, 46, Alle´e d’Italie, 69364 Lyon Cedex 07, France ͑Received 3 July 2000͒ The effect of a vertical uniform magnetic field on Rayleigh-Be´nard convection is investigated experimen- tally. We confirm that the threshold of convection is in agreement with linear stability theory up to a Chan- drasekhar number QӍ4ϫ106, higher than in previous experiments. We characterize two convective regimes influenced by MHD effects. In the first one, the Nusselt number Nu proportional to the Rayleigh number Ra, which can be interpreted as a condition of marginal stability for the thermal boundary layer. For higher Ra, a second regime NuϳRa0.43 is obtained. PACS number͑s͒: 47.27.Te, 47.65.ϩa, 47.27.Gs We consider the Rayleigh-Be´nard problem of a fluid layer ary conditions for velocity ͑free slip or no slip͒. Note also heated from below and cooled from above. Convection oc- that the threshold does not depend on the electric boundary curs for a Rayleigh number ͓1͔ higher than a critical value conditions ͑i.e., wall conductivity͒, whatever the value of Q. ͑ ͒ Rac . We use an electrically conducting fluid mercury and Our experimental cell ͓9,10͔ is a vertical cylinder with impose a uniform vertical magnetic field B, resulting in eddy aspect ratio ⌫ϭD/Hϭ1, suitable to reach high Rayleigh currents and flow damping by the Lorentz force.
    [Show full text]
  • AAPP | Atti Della Accademia Peloritana Dei Pericolanti Classe Di Scienze Fisiche, Matematiche E Naturali ISSN 1825-1242
    DOI: 10.1478/AAPP.941A5 AAPP j Atti della Accademia Peloritana dei Pericolanti Classe di Scienze Fisiche, Matematiche e Naturali ISSN 1825-1242 Vol. 94, No. 1, A5 (2016) LAMINAR HYDROMAGNETIC FLOWS IN AN INCLINED HEATED LAYER PAOLO FALSAPERLA,a ANDREA GIACOBBE,a SEBASTIANO LOMBARDO,a AND GIUSEPPE MULONE a∗ (communicated by Paolo V. Giaquinta) ABSTRACT. In this paper we investigate, analytically, stationary laminar flow solutions of an inclined layer filled with a hydromagnetic fluid heated from below and subjectto the gravity field. In particular we describe in a systematic way the many basic solutions associated to the system. This extensive work is the basis to linear instability and nonlinear stability analysis of such motions. 1. Introduction Laminar flows in fluid dynamics and in magneto-fluid dynamics are very important for many physical applications, for instance in geophysics and astrophysics (Alexakis et al. 2003; Batchelor 2000; Ferraro and Plumpton 1966). Many applications are possible in industry, e.g., in metallurgy (Branover, Mond, and Unger 1988) and in biology (Tao and Huang 2011). Usually the results related to laminar flow are applied to smooth flows ofa viscous liquid through tubes or pipes (Batchelor 2000; Joseph 1976), but laminar flows in layers are also studied (see, for example, Pai 1962; Rionero and Mulone 1991). Moreover, the variation of the temperature in the layer plays an important role in convective problems (Chandrasekhar 1961; Mulone 1991b). Generally the layer is orthogonal to the gravity field (horizontal layer), but also interesting are layers inclined with respect to the horizontal plane (see, e.g., Goldstein, Ullmann, and Brauner 2015; Guo and Kaloni 1995), including the vertical case (Donaldson 1961).
    [Show full text]