Research Article Some New Parallel Flows in Weakly Conducting Fluids with an Exponentially Decaying Lorentz Force

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1 Hindawi Pblising Corporation Matematical Problems in Engineering Volme 2007, Article ID 8784, 4 pages doi:0.55/2007/8784 Researc Article Some New Parallel Flows in Weakly Condcting Flids wit an Exponentially Decaying Lorentz Force Asterios Pantokratoras Received 8 Marc 2007; Revised 27 May 2007; Accepted 9 September 2007 Recommended by Merdad Massodi We investigate te flly developed flow between two parallel plates and te film flow over a plate in an electrically condcting flid nder te action of a parallel Lorentz force. Te Lorentz force varies exponentially in te vertical direction de to low-flid electrical condctivity and te special arrangement of te magnetic and electric fields at te lower plate. Exact analytical soltions are derived for velocity, flow rate, and wall sear stress at te plates. Te velocity reslts are presented in figres. All tese flows are new and are presented for te first time in te literatre. Copyrigt 2007 Asterios Pantokratoras. Tis is an open access article distribted nder te Creative Commons Attribtion License, wic permits nrestricted se, distribtion, and reprodction in any medim, provided te original work is properly cited.. Introdction Magnetoydrodynamics (MHD) is te stdy of te interaction between magnetic fields and moving condcting flids. Te simplest MHD flow is te flow between two infinite, orizontal, parallel plates nder te action of an external vertical magnetic field and a orizontal electric field (Figre.). Tis type of flow was first investigated by Hartmann and Lazars [] and is called te Hartmann flow (Davidson [2, page 53]). In 96 Gailitis and Lielasis [3] introdced te idea of sing a Lorentz force to control te flow of an electrically condcting flid over a flat plate. Tis is acieved by applying an external electromagnetic field (see Figre.2) by a stripwise arrangement of fls monted electrodes and permanent magnets of alternating polarity and magnetization. A similar arrangement was proposed by Rice [4]. Te Lorentz force, wic acts parallel to te plate, can eiter assist or oppose te flow. Tis idea was later abandoned and only recently attracted new attention (Henoc and Stace [5], Crawford and Karniadakis [6], O Sllivan and Biringen [7], Berger et al. [8], Kim and Lee [9], D and Karniadakis [0], D et al. [], Breer et al. [2], Lee and Sng [3], Spong et al. [4]). In addition, in te last

2 2 Matematical Problems in Engineering B Upper plate E Lower plate E = orizontal electric field B = vertical magnetic field Figre.. Te flow configration in te classical Hartmann flow. years mc investigation on flow control sing te Lorentz force is being condcted at te Rossendorf Institte and at te Institte for Aerospace Engineering in Dresden, Germany (Weier et al. [5], Posdziec and Grndmann [6], Weier et al. [7], Weier and Gerbet [8], Weier [9], Mtscke et al. [20], Albrect and Grndmann [2], Satrov and Gerbet [22]). Te work of [22] is concerned wit trblent flow in a cannel wit a Lorentz force. As a special case, te laminar Poiseille flow is treated wit a Lorentz force only for a flow rate eqal to 4/3. Te widt of bot electrodes and magnets is assmed to be eqal to a. Detote crossing electric and magnetic field lines, te Lorentz force acts in te streamwise direction. It is known tat wen a condcting flid moves inside a magnetic field, an electric crrent is prodced and tis electric crrent inflences te Lorentz force. In te present work, te flid electrical condctivity is assmed to be small and terefore te indced electric crrent is negligible. Tis means tat te flid motion as no inflence on te Lorentz force. Te Lorentz force depends only on te external electric and magnetic field. Assming ard ferromagnetic properties, te magnetic field of te cain of magnets can be easily calclated analytically. Te crrent distribtion of te electrode array can be fond in closed form. As a reslt, apart from inomogeneities near te magnet corners, te Lorentz force decays exponentially in te y direction. After averaging over te spanwise direction z, te Lorentz force can be calclated [9]. Altog te Lorentz force as been applied in many flow configrations, some flows nder te inflence of tis force ave not been investigated ntil now. In te present paper, we will investigate te classical flow between two parallel, infinite plates wit a Lorentz force created at te lower plate according to te arrangement sown in Figre.2. Te present problem is eqivalent to te classical Hartmann flow wit a different arrangement of te magnetic and electric fields, except tat we will investigate te problem of a film flowing over a plate wit a Lorentz force created at te plate according to te arrangement sown in Figre.2.

3 Asterios Pantokratoras 3 y x z F N S a + S N S N S + N Electrode Magnet Figre.2. Arrangement of electrodes and magnets for te creation of an EMHD Lorentz force F in te flow along a flat plate (Weier [9]). 2. Te matematical model Consider te flow between two orizontal, infinite, parallel plates wit and v denoting, respectively, te velocity components in te x and y directions, were x is te coordinate along te plates and y is te coordinate perpendiclar to x. It is assmed tat an electromagnetic field exists at te lower plate and terefore a Lorentz force, parallel to te plates, is prodced. Te flid is forced to move de to te action of te Lorentz force. For steady, two-dimensional flow, te bondary layer eqations wit constant flid properties are [9, 23] continity eqation: momentm eqation: x + v y = 0, x + v y = p ρ x + ν 2 y 2 + πj 0M 0 exp 8ρ ( π a y ), (2.) were p is te pressre, v is te flid kinematic viscosity, j 0 (A/m 2 ) is te applied crrent density in te electrodes, M 0 (Tesla) is te magnetization of te permanent magnets, a is te widt of magnets and electrodes, and ρ is te flid density. Te last term in te momentm eqation is te Lorentz force wic decreases exponentially wit y and is independent of te flow. For flly developed conditions, te flow is parallel, te transverse velocity is zero, and te flow is described only by te following momentm eqation: momentm eqation: 3. Reslts and discssion ρ dp dx + ν 2 y 2 + πj ( 0M 0 exp π ) 8ρ a y = 0. (2.2) 3.. Te classical Coette flow wit Lorentz force. Te first viscos flid flow treated in te classical book by Wite [24] is te steady flow between a fixed and a moving plate (Coette flow) and tis appens in almost all flid mecanics books becase tis flow

4 4 Matematical Problems in Engineering is te simplest in flid mecanics (Liggett [25, page 56], Kleinstreer [26, page 2], Panton [27, page 32]). Tis flow is called Coette flow in onor of te frenc Coette [28] wo performed experiments on te flow between a fixed and moving concentric cylinder. Te bondary conditions for tis case are at y = 0: = 0, as y = : = 2, (3.) were is te distance between te plates and 2 is te velocity of te pper plate. Here we will investigate tis flow nder te action of a Lorentz force prodced at te lower plate. Te momentm (2.2), witot te pressre gradient, wit bondary conditions (3.) as te following exact analytical soltion: = y [ ( 2 + Z exp π ) a y y ( ( a. (3.2) Te first term on te rigt is de to te motion of te pper plate (classical Coette flow) and te second term de to te action of te Lorentz force. Te parameter Z is defined as Z = j 0M 0 a 2 8πμ 2, (3.3) were μ is te flid dynamic viscosity. Te parameter Z is dimensionless, expresses te balance between te electromagnetic forces to viscos forces, and is eqivalent to sqare of te classical Hartmann nmber or to Candrasekar nmber (Brr et al. [29, page 23], Arno and Olson [30, page 284]). Tis nmber is sed in te analysis of te bondary layer flow over a flat plate sitated in a free stream and tere is caracteristic velocity sed as te free stream velocity [9]. We see tat tis nmber appears also in te Coette flow wit caracteristic velocity of te moving plate. Te dimensionless velocity given by (3.2) dependson bot Z and te ratio /a. Te above combination of te classical Coette flow wit Lorentz forces is presented ere for te first time in te literatre. Te dimensionless flow rate M between te plates is M = dy (3.4) 2 0 and is obtained by integrating te velocity fnction. Ts, we ave M = 2 + Z 2π [( π +2 a ) ( exp π ) a + π 2 a ]. (3.5) Te wall sear stresses at te two plates are τ = μ 2 π 2Z[ + μ a ( ( a, τ 2 = μ [ 2 + μ 2Z ( π + a + ) ( exp π )] a. (3.6)

5 Asterios Pantokratoras 5 /a = y/ 0.4 Z = 0 Z =.28 Z = 2 Z = Z = / 2 Figre 3.. Velocity profiles for Coette flow wit Lorentz force for /a = and different vales of te Candrasekar nmber. Te case Z = 0 corresponds to Coette flow wit pper plate moving. Te wall sear stress τ becomes zero wen te Candrasekar nmber takes te vale [ Z = π ( a + exp π )] a (3.7) wile τ 2 becomes zero wen [ ( ) ( π Z = a + exp π )] a. (3.8) In Figre 3., some velocity profiles are presented for different vales of te Candrasekar nmber and /a =. Te profile corresponds to zero sear stress at te lower plate and tis appens wen Z = Profile 3 corresponds to zero sear stress at te pper plate and tis appenswen Z = Profile 2 corresponds to Coette flow. Wen te lower plate, were te electromagnetic field is prodced, is moving and te pper plate is motionless, te velocity is given by te following eqation: = y [ ( + Z exp π ) a y y ( ( a, (3.9) were is te velocity of te lower plate and te Candrasekar nmber is based on velocity of te lower plate. Now te flow rate is defined as M = 0 dy (3.0)

6 6 Matematical Problems in Engineering /a = Z = 0 y/ 0.4 Z = 2 Z = Z = Z = / Figre 3.2. Velocity profiles for Coette flow wit Lorentz force for /a = and different vales of te Candrasekar nmber. Te case Z = 0 corresponds to Coette flow wit lower plate moving. and (3.5) is valid also for tis case. Te wall sear stresses at te two plates are τ = μ π Z[ + μ a ( ( a, τ 2 = μ [ + μ Z ( π + a + ) ( exp π )] a. (3.) Te wall sear stress τ becomes zero wen te Candrasekar nmber takes te vale [ ( π Z = a +exp π )] a, (3.2) wile τ 2 becomes zero wen [ ( ) ( π Z = + a + exp π )] a. (3.3) In Figre 3.2, some velocity profiles are presented for different vales of te nmber and /a =. Profile corresponds to zero sear stress at te pper plate and tis appens wen Z = Profile 3 corresponds to zero sear stress at te lower plate and tis appens wen Z = Profile 2 corresponds to Coette flow Te classical Poiseille flow wit Lorentz forces. Anoter kind of flow between parallel plates is te Poiseille flow (Poiseille [3]) wic is cased by a constant pressre gradient along te plates wile te plates are motionless. Tis flow is also inclded in

7 Asterios Pantokratoras 7 Flid Mecanics books ([25, page 57], [27, page 25], [24, page 06]). Te bondary conditions are at y = 0: = 0, as y = : = 0. (3.4) Te analytical soltion of (2.2) wit bondary conditions (3.4)is ( = 2 dp y y 2μ dx ) + j 0M 0 a 2 8πμ [ ( exp π ) a y y ( ( a. (3.5) Te first term on te rigt is de to te pressre gradient (classical Poiseille flow) and te second term de to te action of te Lorentz force. In te above eqation, tere are two caracteristic velocities, te first one de to pressre gradient and te second de to Lorentz forces as follows: P = 2 dp 2μ dx, Z = j 0M 0 a 2 8πμ. Taking into accont tese caracteristic velocities, (3.5) becomes ( y = P y ) [ ( + Z exp π ) a y y ( exp( π ))] a (3.6) (3.7) and in dimensionless form, = ( P y y ) [ ( + exp π ) Z Z a y y ( ( a. (3.8) Te ratio P / Z is a new dimensionless nmber wic expresses te balance between te pressre forces to electromagnetic forces: Pa p = P. (3.9) Z Te dimensionless velocity given by (3.8) isafnctionofpa p and /a. Teabovecombination of te classical Poiseille flow wit Lorentz forces is a new kind of parallel flow. We define te dimensionless flow rate as M = dy (3.20) Z 0 and M is M = Pa p 6 + [( π +2 a ) ( exp π ) 2π a + π 2 a ]. (3.2)

8 8 Matematical Problems in Engineering /a = y/ 0.4 Pap = 2.84 Pap =.5 Pap = 0.82 Pap = Pap = / Z Figre 3.3. Velocity profiles for Poiseille flow wit Lorentz force for /a = and different vales of te Pa p parameter. Te case Pa p = 0 corresponds to zero pressre gradient. Te wall sear stresses at te two plates are τ = μ P π Z[ + μ a τ 2 = μ P + μ Z ( ( a, [ + ( π a + ) ( exp π )] a. Te wall sear stress τ becomes zero wen te qantity Pa p takes te vale [ Pa p = π ( a + exp π )] a (3.22) (3.23) wile τ 2 becomes zero wen [ ( ) ( π Pa p = + a + exp π )] a. (3.24) In Figre 3.3, we present some velocity profiles for /a = anddifferent Pa p vales. Profile corresponds to zero sear stress at te lower plate and tis appens wen Pa p = Profile 3 corresponds to zero sear stress at te pper plate and tis appens wen Pa p = Profile 4 corresponds to zero pressre gradient. Profiles, 2, and 3 are S saped and eac of tem as an inflection point Flow between parallel plates de to Lorentz force only. If bot plates are motionless and te pressre gradient is zero, we ave a flow cased by te Lorentz force only.

9 Asterios Pantokratoras 9 From (3.5) wegettevelocityoftisflowbypttingtepressregradientzero.ts, we ave = j 0M 0 a 2 [ ( exp π ) 8πμ a y y ( ( a (3.25) and in a dimensionless form, [ ( = exp π ) Z a y y ( ( a. (3.26) Tis flow is completely new, it is presented ere for te first time in te literatre. Te dimensionless flow rate is defined as M = dy (3.27) Z 0 and te flow rate is M = [( π +2 a ) ( exp π ) 2π a + π 2 a ]. (3.28) Te wall sear stresses at te two plates are [ π τ = μ Z a ( ( exp π )] a, [ τ 2 = μ Z ( π + a + ) ( exp π )] a. (3.29) In Figre 3.4, velocity profiles are sown for different vales of /a. Tis flow as some special caracteristics. Wen /a, te maximm dimensionless velocity tends to, and te velocity profile tends to compose from two straigt lines: one of tem orizontal and te oter wit inclination eqal to 45 degrees. Wen /a 0, te dimensionless velocity tends to 0, and te velocity profile tends to become symmetric wit its maximm at te centerline between te plates. We see also tat te velocity maximm moves to te centerline as /a decreases Film flow de to Lorentz forces. Anoter kind of simple parallel flow is tat of a film falling down an inclined wall de to gravity and de to te action of a constant sear stress on te free srface (Bird et al. [32, page 45], [27, page 35]). Here we will treat te motion of a film de to Lorentz force ignoring gravity and retaining te action of te srface sear stress. Te bondary conditions for tis case are at y = 0: = 0, as y = : μ y = τ 2, (3.30)

10 0 Matematical Problems in Engineering y/ 0.4 /a = 0. /a = 0.5 /a = 0.2 /a = 2 /a = 0 0 /a = / Z Figre 3.4. Velocity profiles for flow de to Lorentz force only for different vales of /a. were τ 2 is known and constant. Te analytical soltion of (2.2), witot pressre gradient, wit bondary conditions (3.30) is = j 0M 0 a 2 [ ( exp π )] [ 8πμ a y τ2 + μ j ( 0M 0 a 8μ exp π )]y a (3.3) and in dimensionless form, [ ( = exp π )] [ Z a y τ2 + π ( μ Z a exp π )] y a. (3.32) Te new dimensionless nmber is Pa f = τ 2 μ Z (3.33) and (3.32) becomes [ ( = exp π )] [ Z a y + Pa f π ( a exp π )] y a. (3.34) Te dimensionless flow rate is defined as M = dy (3.35) Z 0

11 Asterios Pantokratoras /a = Pa f = Pa f = 2 Pa f = 0 Pa f = y/ / Z Figre 3.5. Velocity profiles for film flow for /a = and different vales of te Pa f nmber. Te case Pa f = 0 corresponds to zero srface sear stress. and te flow rate is M = a [ ( exp π ) ] π a + [ Pa f π ( 2 a exp π )] a +. (3.36) Te sear stress at te plate is τ = j [ ( 0M 0 a exp π )] 8 a + τ 2. (3.37) Te wall sear stress τ becomes zero wen te qantity Pa f takes te vale Pa f = π [ ( exp π ) ] a a. (3.38) In Figre 3.5, velocity profiles are sown for different vales of Pa f nmber and /a =. Crve corresponds to zero wall sear stress and tis appens wen Pa f = wile crve 2 corresponds to zero srface sear stress (Pa f = 0), tat is, te flow is prodced by te Lorentz force only Film flow de to Lorentz force only. If te sear stress at te srface is zero we ave a film flow cased by te Lorentz force only. From (3.34) we get te velocity of tis flow by ptting Pa f = 0. Ts, we ave [ ( = exp π )] [ ( π Z a y a exp π )] y a. (3.39)

12 2 Matematical Problems in Engineering 0.8 /a = 0. /a = y/ /a = 0.4 /a = /a = 5 0 /a = / Z Figre 3.6. Velocity profiles for film flow de to Lorentz force only for different vales of /a. Te dimensionless flow rate is defined as M = dy (3.40) Z 0 and te flow rate is M = a [ ( exp π ) ] π a [ ( π 2 a exp π )] a +. (3.4) Te sear stress at te plate is τ = j [ ( 0M 0 a exp π )] 8 a. (3.42) In Figre 3.6, velocity profiles are sown for different vales of /a. All velocity profiles ave zero gradient at te srface and meet te srface vertically. Wen /a,temaximm dimensionless velocity, wic lies on te free srface, tends to and te velocity profile tends to compose from two straigt lines: one of tem orizontal and te oter vertical. Wen /a 0, te dimensionless velocity tends to Conclsions In tis paper some new kinds of parallel flows ave been presented and analyzed for weakly electrically condcting flids. Exact analytical soltions ave been given for velocity, flow rate, and wall sear stresses. Te ator believes tat te reslts of te present

13 Asterios Pantokratoras 3 work will enric te list wit te existing exact soltions of te Navier-Stokes eqations and may elp te investigation of flow of electrically condcting flids in MHD. References [] J. Hartmann and F. Lazars, Hg-dynamics II. Experimental investigations on te flow of mercry in a omogeneos magnetic field, Det Kongelige Danske Videnskabernes Selskabs Skrifter, vol. 5, no. 7, 937. [2] P. A. Davidson, An Introdction to Magnetoydrodynamics, Cambridge Texts in Applied Matematics, Cambridge University Press, Cambridge, UK, 200. [3] A. Gailitis and O. Lielasis, On a possibility to redce te ydrodynamic resistance of a plate in aelectrolyte, Applied Magnetoydrodynamics, vol. 2, pp , 96. [4] W. A. Rice, Proplsion system, US Patent no , 96. [5] C. Henoc and J. Stace, Experimental investigation of a salt water trblent bondary layer modified by an applied streamwise magnetoydrodynamic body force, Pysics of Flids, vol. 7, no. 6, pp , 995. [6] C. H. Crawford and G. E. Karniadakis, Reynolds stress analysis of EMHD-controlled wall trblence part I: treamwise forcing, Pysics of Flids, vol. 9, no. 3, pp , 997. [7] P. L. O Sllivan and S. Biringen, Direct nmerical simlations of low Reynolds nmber trblent cannel flow wit EMHD control, Pysics of Flids, vol. 0, no. 5, pp. 69 8, 998. [8] T. W. Berger, J. Kim, C. Lee, and J. Lim, Trblent bondary layer control tilizing te Lorentz force, Pysics of Flids, vol. 2, no. 3, pp , [9] S.-J. Kim and C. M. Lee, Investigation of te flow arond a circlar cylinder nder te inflence of an electromagnetic force, Experiments in Flids, vol. 28, no. 3, pp , [0] Y. Q. D and G. E. Karniadakis, Sppressing wall trblence by means of a transverse traveling wave, Science, vol. 288, no. 5469, pp , [] Y. Q. D, V. Symeonidis, and G. E. Karniadakis, Drag redction in wall-bonded trblence via a transverse travelling wave, Jornal of Flid Mecanics, vol. 457, pp. 34, [2] K. S. Breer, J. Park, and C. Henoc, Actation and control of a trblent cannel flow sing Lorentz forces, Pysics of Flids, vol. 6, no. 4, pp , [3] J.-H. Lee and H. J. Sng, Response of a spatially developing trblent bondary layer to a spanwise oscillating electromagnetic force, Jornal of Trblence, vol. 6, pp. 5, [4] E. Spong, J. A. Reizes, and E. Leonardi, Efficiency improvements of electromagnetic flow control, International Jornal of Heat and Flid Flow, vol. 26, no. 4, pp , [5] T. Weier, G. Gerbet, G. Mtscke, E. Platacis, and O. Lielasis, Experiments on cylinder wake stabilization in an electrolyte soltion by means of electromagnetic forces localized on te cylinder srface, Experimental Termal and Flid Science, vol. 6, no. -2, pp. 84 9, 998. [6] O. Posdziec and R. Grndmann, Electromagnetic control of seawater flow arond circlar cylinders, Eropean Jornal of Mecanics B, vol. 20, no. 2, pp , 200. [7] T. Weier, G. Gerbet, G. Mtscke, O. Lielasis, and G. Lammers, Control of flow separation sing electromagnetic forces, Flow, Trblence and Combstion, vol. 7, no. 4, pp. 5 7, [8] T. Weier and G. Gerbet, Control of separated flows by time periodic Lorentz forces, Eropean Jornal of Mecanics B, vol. 23, no. 6, pp , [9] T. Weier, Elektromagnetisce Strömngskontrolle mit wandparallelen Lorentzkräften in scwac leitfäigen Fliden, Dissertation, Tecnisce Universität, Dresden, Germany, [20] G. Mtscke, G. Gerbet, T. Albrect, and R. Grndmann, Separation control at ydrofoils sing Lorentz forces, Eropean Jornal of Mecanics B, vol. 25, no. 2, pp , [2] T. Albrect, R. Grndmann, G. Mtscke, and G. Gerbet, On te stability of te bondary layer sbject to a wall-parallel Lorentz force, Pysics of Flids, vol. 8, no. 9, Article ID 09803, 4 pages, 2006.

14 4 Matematical Problems in Engineering [22] V. Satrov and G. Gerbet, Magnetoydrodynamic drag redction and its efficiency, Pysics of Flids, vol. 9, no. 3, Article ID 03509, 2 pages, [23] A. B. Tsinober and A. G. Stern, Possibility of increasing te flow stability in a bondary layer by means of crossed electric and magnetic fields, Magnetoydrodynamics, vol. 3, pp , 967. [24] F. Wite, Viscos Flid Flow, McGraw-Hill, New York, NY, USA, 3rd edition, [25] J. Liggett, Flid Mecanics, McGraw-Hill, New York, NY, USA, 994. [26] C. Kleinstreer, Engineering Flid Dynamics: An Interdisciplinary Systems Approac, Cambridge University Press, Cambridge, UK, 997. [27] R. L. Panton, Incompressible Flow, Jon Wiley & Sons, New York, NY, USA, [28] M. Coette, Etdes sr le frottement des liqides, Annales de Cimie et de Pysiqe, vol. 2, pp , 890. [29] U. Brr, L. Barleon, P. Jocmann, and A. Tsinober, Magnetoydrodynamic convection in a vertical slot wit orizontal magnetic field, Jornal of Flid Mecanics, vol. 475, pp. 2 40, [30] J. M. Arno and P. L. Olson, Experiments on Rayleig-Benard convection, magnetoconvection and rotating magnetoconvection in liqid gallim, Jornal of Flid Mecanics, vol. 430, pp , 200. [3] J. L. M. Poiseille, Recerces experimentelles sr le movement des liqides dans les tbes de tres petits diametres, Comptes Rends, vol., pp , 840. [32] R. Bird, W. Stewart, and E. Ligtfoot, Transport Penomena, Jon Wiley & Sons, New York, NY, USA, Asterios Pantokratoras: Scool of Engineering, Democrits University of Trace, 6700 Xanti, Greece address: apantokr@civil.dt.gr

Kragujevac J. Sci. 34 (2012) UDC 532.5: :537.63

Kragujevac J. Sci. 34 (2012) UDC 532.5: :537.63 5 Kragjevac J. Sci. 34 () 5-. UDC 53.5: 536.4:537.63 UNSTEADY MHD FLOW AND HEAT TRANSFER BETWEEN PARALLEL POROUS PLATES WITH EXPONENTIAL DECAYING PRESSURE GRADIENT Hazem A. Attia and Mostafa A. M. Abdeen