Thermomagnetic convection in stratified ferrofluids permeated with dusty particles through a porous medium

Transcription

1 Kumar et al., 6 J. asic appl. Res (): 6-5 Thermomagnetic convection in stratified ferrofluids permeated with dust particles through a porous medium Kapil Kumar *, Rajan Singh, nand Kumar Department of pplied Mathematics, Government Poltechnic, Pilihit, India. Department of Mathematics, I.F.T.M. Universit, Moradaad, India. Department of Mathematics, Ramaai Govt. Girls P.G. College karpur, medkarnagar, India. *corresponding author: kkchaudhar@gmail.com Received: --6 Revised: Pulished: -5-6 Kewords: Ferrofluids, Magnetic field, Porous medium, Suspended particles. stract: In the present note, the stailit prolem of an incompressile dust stratified ferromagnetic fluid is investigated through a porous medium when the fluid laer is sujected to vertical magnetic field intensit. The governing nonlinear equations are linearized using perturation technique and the cases of exponentiall varing stratifications for various phsical parameters are discussed. The sstem is found to e stale for stale stratification in oth the asence and presence of magnetic field. For unstale stratification, the sstem has oth stailizing and destailizing effects in the presence of magnetic field under certain conditions, whereas in the asence of magnetic field, the sstem has onl destailizing effect. The variations in the growth rate with respect to kinematic viscosit, medium porosit, medium permeailit, square of lfvén velocit and suspended particle parameter are also shown analticall.. INTRODUCTION The asic concepts to understand various fascinating and diverse applications of fluid mechanics have een given in Bansal () and Gupta and Gupta (). rigorous and elegant overview aout hdrodnamic stailit prolem of an incompressile Newtonian fluid has een well addressed in Chandrasekhar (98) and Drazin and Reid (98). The ehaviour and flow characteristic of non-newtonian fluids are significantl different with those of Newtonian fluids. Ferrofluids are electricall non-conducting colloidal suspensions of solid ferromagnetic particles in a carrier fluid such as kerosene. Ferrofluids manifest simultaneousl oth liquid and paramagnetic properties. The stud of stratified fluids has produced great interest in recent ears due to its numerous industrial and technological applications such as thermal stratification of reservoirs and oceans, densit, temperature and gravitational stratification of the atmosphere, salinit stratification in rivers, oceans and estuaries, laer stratification in the earth s interior and several heterogeneous mixtures in food processing industr. Stratified fluids are aundant and the understanding of the ehaviour and dnamics of these fluids are mandator for scientific and industrial purposes. During the last few decades, the stud on non-newtonian fluids has attracted several researchers and investigators to stud such industriall important fluids. Rosensweig (985) and Odenach () discussed aout the fundamental concepts ehind the use of ferrofluids and also provided a comprehensive and detailed account of ferrohdrodnamics and its applications in various commercial usages such as novel zero-leakage rotar shaft seals used in computer disk drives (Baile, 98); semiconductor manufacturing (Moskowitz, 975); pressure seals for compressor and lowers (Rosensweig, 985) and more. Other applications include the uses and interesting effects of ferrofluids in a wide range of technological and io-medical purposes such as vacuum technolog, instrumentation, lurication mechanism, acoustics theor, recover of metals, detection of tumours, drug deliver to a target site, magnetic fluid earings, non-destructive testing, sensors and actuators, sorting of industrial scrap metals such as titanium, aluminium and zinc, tracer of lood flow in non-invasive circulator measurements (Newower, 97) and in loudspeakers to conduct heat awa from speakers coil (Hathawa, 979). The stailit of ferrofluids intended for medical use is a current topic of frontier research and also attractive from a theoretical point of view. Thus, the overall field of ferrofluid research has a highl interdisciplinar character, ringing phsicists, engineers, chemists and mathematicians together. Finlason (97) discussed the convective instailit of ferromagnetic fluid laer heated from elow under the effect of a uniform vertical magnetic field and analzed the instailit with or without considering 6

2 Kumar et al., 6, J. asic appl. Res (): 6-5 the effect of od force (gravit force). He also concluded that convection can e induced in a ferromagnetic fluid due to variation in magnetization which depends upon the strength of magnetic field, temperature gradient and densit of fluid and is known as ferroconvection, which is ver similar to Bénard convection (Chandrasekhar, 98). Magneto-hdrodnamics theor of electricall conducting fluids has several scientific and practical applications in astrophsics, geophsics, space sciences etc. Magnetic field is also used in several clinical areas such as neurolog and orthopaedics for proing and curing the internal organs of the od in several diseases like tumours detection, heart and rain diseases, stroke damage etc. Sunil et al. (, 5) considered, theoreticall, the thermal and thermosolutal convection prolems for ferromagnetic fluid to include the effects of magnetic field and dust particles saturating a porous medium. Sharma et al. (6) pointed out the comined effects of magnetic field and rotation on the stailit of stratified visco-elastic Walters (model B') fluid through a porous medium and concluded that the sstem is found to e unstale at stale stratification, whereas for unstale stratification, magnetic field is found to stailize the small wavelength perturations. It is also shown that the growth rate increases with the increase in kinematic viscosit and permeailit, whereas it decreases with the increase in kinematic viscoelasticit. Kumar et al. (, 5a, ) discussed thermal convection prolem for Oldrodian, couple-stress and ferrofluid under the effects of magnetic field, rotation, compressiilit, variale gravit, suspended particles and heat source strength through a Darc as well as Brinkman porous medium. The flow through a porous medium is of fundamental importance in solidification, chemical processing industr, geophsical fluid dnamics, petroleum industr, filtering equipment, recover of crude oil from earth s interior etc. detailed stud of convection through porous medium has een given Nield and Bejan (6) in his famous monograph. In man ranches of sanitar work, notal in the stud of factor conditions, the enumeration of the actual numer of dust particles present is quite as important as the determination of the total weight of dust.dust comes from a wide variet of sources, including soil, vegetation (pollens and fungi), sea salt, fossil fuel comustion, urning of iomass, and industrial activities. In geophsical context, the fluid is often not pure ut ma instead e permeated with dust particles. The effects of suspended particles on the stailit of superposed fluids have industrial and scientific importance in geophsics, chemical engineering and astrophsics. Scanlon and Segel (97) have considered the effect of suspended particles on the onset of Bénard convection and found that the critical Raleigh numer was reduced solel ecause the heat capacit of the pure fluid was supplemented that of the particles. The governing hdrodnamic equations of motion are solved using a regular perturation technique. The intention is to investigate theoreticall the stailit of a stratified ferrofluid in the presence of magnetic field and suspended particles through a porous medium using linear stailit theor. The use of Boussinesq approximation has een made throughout in the equations of motion which states that the densit variations occurs onl in the external force term or gravitational force term. The purpose and practical relevance of the present investigation is in determining the influence of the impurities and magnetic strength in a stratified ferrofluid in thermal convection phenomena.. MTHEMTICL FORMULTION ND GOVERNING EQUTIONS The phsical configuration considered here consists of a ferromagnetic fluid of variale densit, kinematic viscosit, medium porosit, medium permeailit N, particle numer densit and thermometric conductivit, arranged in a rectangular channel ounded two infinite horizontal stratum separated an altitude d apart in a porous medium. uniform vertical magnetic field H,, H k pervades the sstem with gravit acting verticall downward. The Cartesian axes are chosen with the z-axis verticall upward and the x- axis in the direction of applied horizontal temperature gradient. It is also assumed that the flow in the porous medium is governed the Darc s law in the equation of motion. 7

3 Kumar et al., 6, J. asic appl. Res (): 6-5 Z-axis H (,, H) z=d Dust Ferrofluid Darc Porous Medium g (,,-g) z= o X-axis Y-axis Heated from elow Fig. Geometrical sketch of the phsical prolem The governing equations of conservation of mass and momentum alance for an incompressile magnetized ferrofluid in a porous medium are as follows q. p q q. t Xi M H q k (.) KN qd - q e H H. q (.) q. (.) t where, in the aove equations, the smols, t,, q, q, p, N,,, H,B and X g denote, respectivel, the densit of ferromagnetic fluid, the time, co-efficient of viscosit, velocit of fluid particles, velocit of suspended particles, pressure gradient for ferromagnetic fluid, suspended particles numer densit, magnetic permeailit of medium, magnetic permeailit of vacuum, magnetic field strength, magnetic induction and the external force due to gravit. The term K 6, where eing the particle radius, represent the Stokes drag co-efficient. The equation for energ alance which oes Fourier s law of heat conduction is T cv scs cv q. T mnc pt q d. T kt T (.) t t where,, c, c, T and k denote, respectivel, the densit of solid material, heat capacit of solid material, s s v T specific heat at constant volume, the temperature and the thermal conductivit of fluid particles. d e i i The Maxwell s equations of electromagnetism are H q H t. H where, the electrical resistivit is taken as zero. (.5) (.6) If mn is the mass of particles per unit volume, then the equations of motion and continuit for the dust particles are 8

4 Kumar et al., 6, J. asic appl. Res (): 6-5 qd mn t N. Nq d t The densit equation of state is T T q. q KN q -q d d d (.7) (.8) (.9) where,, and T denote, respectivel, the coefficient of thermal expansion, reference densit and the temperature at the lower oundar. Maxwell s equations for an electricall non-conducting fluid with no displacement currents, ecome. B, H (.) ccording to Penfield and Haus (967), the magnetic induction (B), magnetization (M) and the intensit of magnetic field (H) are coupled the relation B H + M) ( (.) Now, for the prolem under consideration, it is assumed that the magnetization (M) does not depend on the magnetic field (H) and is a function of temperature (T) onl. So, as an initial approximation, we consider the form M M [ T T ] T and M (.) where, are the reference temperature and reference magnetization, respectivel and M M T H stands for the promagnetic co-efficient with H H, M M and M M T. Now, the stailit of the asic state defined elow is analzed using the regular perturation technique.,,, p p z p g z q, T, z T z H H,, H, M M,,. z N N z z z (.) Let the conduction state descried Eq. (.) e slightl pertured assuming perturations of the form p p z p T T z hx h hz z, N N N, M M z mm, m, m. q = q + q,,, H H h,,, x z m (.) The changes in densit and magnetization caused perturations θ and in temperature and concentration, respectivel, are given m, m M (.5) ssuming these perturation quantities to e ver small, the relevant linearized perturation equations for the magnetized ferrofluid ecome q p g i M H M. h q t k (.6) KN qd - q e h H. q (.7) wd (.8) t 9

5 Kumar et al., 6, J. asic appl. Res (): 6-5 E w s t h. Hq t. h m q q K t d N. Nq d t (.9) (.) (.) (.) (.) where, w and sdenote, respectivel, the vertical fluid velocit and suspended particle velocit and C mnc pt k d E s s,,, T, dt, i,,, D. C dz v Cv Cv dz Now, considering an exponential solution with a dependence on x, and t of the form exp ikxx ik nt (.) k k where, kx and are the wave numers along x and directions, respectivel and k x k is the overall horizontal wave numer and n is the growth rate of harmonic disturance, which is, in general, a complex quantit. Equations (.6) - (.) ield e n ikxp H u MDhx u D ikx (.5) k hx hz e n ikp H v MDh D ik (.6) k v h hz nw D p g i MDh z M DH w (.7) k ik u ik v Dw x n ne wd D nh H Du x nh H Dv nh H Dw z ik h ik h Dh x s x z w n n N. N q d n (.8) (.9) (.) (.) (.) (.) (.) (.5) (.6) m where,, E E, n n mn n (.7) K Multipling Eq. (.5) ik and Eq. (.6) ik and adding, we otain x e n D k p H w MD h D D k z w hz k (.8) 5

6 Kumar et al., 6, J. asic appl. Res (): 6-5 Now, sutracting Eq. (.7) (after multipling k ) from Eq. (.8) and also using Eqs. (), () and (7), an expression is otained as n n DD k DmN D mn k w w n w w gk D w e D k w D k D n k n H (.9) where, mn i. e. mass of particles per unit volume Equation (.9) represents the general dispersion relation for stratified ferromagnetic fluid in terms of magnetic field and dust particle parameters in a porous medium.. EXPONENTILLY VRYING STRTIFICTIONS Mk DH n w MH D k D ne D n w w Considering the stratifications in various phsical parameters of the forms,,, k, N,, E,,,, k, N, H, E, H e z where,,,, k, N, H, E, and all are constants. (.) On using stratification expression (.) in relation (.9) and otain n mnn ne D nd k w n k gk V MH n n n ne D nw ne D M k H n D k D w B nw e The oundar conditions (for the case of free oundaries) are defined as satisfing the oundar condition (.) can e proposed as where, are constants and for the lowest mode. On using solution (.), Eq. (.) ields H where, V is the square of lfvén velocit named after Hannes lfvén with B. w D w at z and d. Now, a proper solution for w w sin l z w and l l w (.) (.) (.) n mnn n k ne l l k n gk V MH n n n ne l n ne l M k H n l k l B n Simplifing Eq. (.5), a polnomial of degree four is otained as (.5) 5

7 Kumar et al., 6, J. asic appl. Res (): 6-5 a n a n a n a n a where, the coefficients a ES. DISCUSSION ND RESULTS all are constants and defined as (.6). Equation (.6) is i-quadratic in the growth The effect of various emedded parameters responsile for the stailit/instailit of the sstem is discussed and some more important results are also otained. Stale stratification cases: Case: Let,, and e the four roots of Eq. (.6) then for the conditions,,, B and g E M H, all the coefficients will e positive and therefore Eq. (.6) does not admit an positive root of n.so, the a product of roots. a This indicates that the sstem is stale for disturances of all wave numers. Case:In the asence of magnetic field Unstale stratification cases: Case. If a, a, a, a and a E S a mn E S l S k a a S E l mn l S V MH k E l S k ge M H ls V MH l S gk k E l M k H B a V MH l S g k l where, in the aove coefficients, it is assumed S l k rate n and therefore it must give four roots., the sstem ma have stailizing effect under V MH l S gk then all the coefficients a, a, a, a and a will e positive and therefore Eq. (.6) does not admit an positive root. So, the sstem has stailizing effect for disturances of all wave numers. Case: If and V MH l S gk then the constant term a in Eq. (.6) will e negative and therefore Eq. (.6) has at least one positive root there impling the instailit of the sstem. Case:In the asence of magnetic field..,, the term a will e negative for. Hence the sstem is unstale for all wave numers. a, a, a, a and a the conditions, B and. i e H V..,,,, B, g E M H i e H V and 5

8 Kumar et al., 6, J. asic appl. Res (): 6-5 The variation in the growth rate parameter n for unstale stratification with respect to various parameters such k as kinematic viscosit, medium porosit, medium permeailit, square of lfven velocit and suspended particle parameter B has een examined analticall evaluating various derivatives i.e. dn dn dn dn dn,,, and respectivel. d d dk dv db Sn E n E l n l dn d a n a n a n a k E S n S ls k M HB n dn k d an an an a Sn E n E l n l dn k dk a n a n a n a E l k M H n k k l S E n E l n l dn dv a n a n a n a dn M k H n db a n a n a n a (.) (.) (.) (.) (.5) V From the aforementioned derivatives (.) (.5), it is clear that the growth rate increases with an increase in medium permeailit and suspended particles impling there the destailizing effects of medium permeailit and suspended particle parameter. The parameters kinematic viscosit and square of lfvén velocit oth have stailizing effects as the growth rate decreases under these parameters. The medium porosit has a dual character as the growth rate oth increases and decreases according as the numerator in Eq. (.) is negative or positive respectivel. REFERENCES Baile, R.L. (98). Lesser known applications of ferrofluids, Journal of Magnetism and Magnetic Materials, 9, pp Bansal, J.L. ().Viscous fluid dnamics, Oxford and IBH Pulishing Compan, Delhi, India. Chandrasekhar, S.C. (98). Hdrodnamic and Hdromagnetic Stailit, Dover, New York. Drazin, P.G. and Reid, W.H. (98). Hdrodnamic Stailit, Camridge Universit Press, Camridge. Finlason, B.. (97). Convective instailit in ferromagnetic fluids, Journal of Fluid Mechanics, (),pp Gupta, V. and Gupta, S.K. (). Fluid mechanics and its applications, New ge International Pvt. Ltd., New Delhi, India. Hathawa, D.B. (979). Use of ferrofluid in moving coil loudspeakers, db-sound Eng. Mag.,, pp.. Kumar, K., Singh, V. and Sharma, S. ().Stailit of an Oldrodian viscoelastic fluid permeated with suspended particles through a Brinkman porous medium with variale gravit field in hdromagnetics, merican Journal of Fluid Dnamics, (), pp doi:.59/j.ajfd... Kumar, K., Singh, V. and Sharma, S. (5a). On the onset of convection in a dust couple-stress fluid with variale gravit through a porous medium in hdromagnetics, Journal of pplied Fluid Mechanics, Vol. 8 (), pp

9 Kumar et al., 6, J. asic appl. Res (): 6-5 Kumar, K., Singh, V. and Sharma, S. (5).Linear stailit analsis for ferromagnetic fluids in the presence of magnetic field, compressiilit, internal heat source and rotation through a porous medium, Journal of Theoretical and pplied Mechanics, Vol. 5 (), pp Moskowitz, R. (975). Dnamic Sealing with Magnetic fluids, SLE Trans., 8 (), pp.5-. Newower, R.S. (97). new technique for circulator measurements emploing magnetic fluid tracers, Proceedings Biomedical Smp., San Diego. Nield, D.. and Bejan,. (6).Convection in porous media, Springer, New-York. Odenach, S. ().Magnetoviscous effects in ferrofluids, Springer-Verlag, Berlin. Penfield, P. and Haus, H.. (967).Electrodnamics of moving media, Institute of Technolog Press, Camridge. Rosensweig, R.E. (985). Ferrohdrodnamics, Camridge Universit Press, Camridge. Scanlon, J.W. and Segel, L.. (97).Effect of suspended particles on the onset of Bénard convection, Phsics Fluids, 6, pp Sharma, V., Sunil and Gupta, U. (6).Stailit of stratified elasticoviscous Walters (Model B') fluid in the presence of horizontal magnetic field and rotation in porous medium, rchive of Mechanics, 58 (), pp Sunil, Bharti, P.K. and Sharma, R.C. ().Thermosolutal convection in ferromagnetic fluid, rchive of Mechanics, 56 (), pp Sunil, Sharma, D. and Sharma, R.C. (5).Effect of dust particles on thermal convection in ferromagnetic fluid saturating a porous medium, Journal of Magnetism and Magnetic Materials, 88, pp