Effect of Hall current and rotation on heat transfer in MHD flow of oscillating dusty fluid in a porous channel

Transcription

1 Indian Journal of Pure & Alied Physics Vol. 5 October Effect of Hall current and rotation on heat transfer in MHD flow of oscillating dusty fluid in a orous channel Khem Chand K D Singh & Shavnam Sharma 3 3 Deartment of Mathematics & Statistics Deartment of Mathematics (ICDEOL H P University Shimla India khemthakur99@gmail.com kdsinghshimla@gmail.com 3 Shavnamsharma3@gmail.com Received 4 August 0; revised 8 February 03; acceted 6 June 03 he effect of Hall current and rotation on heat transfer in MHD flow of a dusty viscous incomressible and electrically conducting fluid under the influence of an oscillating ressure gradient has been studied. he arallel lates of the channel are assumed to be orous and subjected to a uniform suction from above and injection from the below. A uniform magnetic field is alied in the direction normal to the lanes of the lates. he lates of the channel are ket at different temerature and the temerature of one of the late is varied eriodically. he entire system rotates about an axis normal to the lanes of the lates with uniform angular velocity. he governing equations are solved to yield the velocity and temerature distributions for both the fluid and dust articles hase. he effects of various arameters on the velocity and temerature rofiles for both fluid and articles are shown grahically and discussed. Keywords: Hall current Heat transfer oscillating dusty fluids Rotating orous channel Introduction he study of flow in rotating orous media is motivated by its ractical alications in geohysics and engineering. Among the alications of rotating flow in orous media to engineering discilines one can find the food rocessing industry chemical rocess industry centrifugation filtration rocesses and rotating machinery. Also the hydrodynamic rotating flow of electrically conducting viscous incomressible fluids has gained considerable attention because of its numerous alications in hysics and engineering. In geohysics it is alied to measure and study the osition and velocities with resect to fixed frame of references on the surface of the earth which rotate with resect to an inertial frame in the resence of its magnetic field. he subject of geohysical dynamics now-a-days has become an imortant branch of fluid dynamics due to the increasing interest to study environment. In astrohysics it is alied to study the stellar and solar structure interlanetary and inter stellar matter solar storms etc. In engineering it finds its alications in MHD generators ion roulsions MHD boundary layer control of reentry vehicles etc. he effect of rotation and Hall current on mixed convection MHD flow through a orous medium in a vertical channel in the resence of thermal radiation was analyed by Singh and Kumar. he Hall current effect on heat and mass transfer in the flow of an oscillating viscoelastic fluid through orous medium in sli flow regime has been studied by Chand and Kumar. he effect of Hall current on the velocity and temerature distribution of Couette flow with variable roerties and uniform suction and injection has been studied by Attia 3. On the other hand the flow and heat transfer of dusty fluids between the arallel lates have been analyed by number of researchers due to its imortant alications in the fields of fluidiation etroleum industry air cool conditioners and urification of crude oil electrostatic reciitation olymer technology and aint sraying. he ossible resence of solid articles such as ash or soot in combustion MHD generators and lasma. MHD accelerators and their effect on the erformance of such devices lead to study of the articulates susensions in conducting fluid in the resence of magnetic field. For examle in an MHD generator coal mixed with seed mixture is fed into a combustor. he coal and seed mixture is burned in oxygen and the combustion gas exands through nole before it enters the generator section. he gas mixture flowing through the MHD channels consists of a condensable vaour (slag and non-condensable gas mixed with seeded coal combustion roducts. Both the slag and

2 670 INDIAN J PURE & APPL PHYS VOL 5 OCOBER 03 the non-condensable gas are electrically conducting and was studied by Chamkha 4. he hydrodynamics flow of dusty fluid has been studied by number of researchers. Unsteady flow of an electrically conducing dusty viscous liquid between two arallel lates has been investigated by Saxena et al 5. Attia 6 studied unsteady MHD Couette flow and heat transfer of dusty fluid between arallel lates with variable hysical roerties. Gireesha et al 7. investigated the ulsatile flow of an unsteady dusty fluid through rectangular Channel. In these studies the Hall current term was ignored in alying Ohm s law as they have no marked effect for small and moderate value of magnetic field. However the current trend for the alication of magnetohydrodynamic is towards a strong magnetic field so that the effect of electromagnetic force is noticeable under these condition and the Hall current is imortant and it has a marked effect on the magnitude and direction of current density and consequently on the magnetic force term Crammer and Pai 8. he Hall current effect on the flow and heat transfer of dusty conducting fluid in a rectangular channel has been investigated by Aboul Hassan and Attia 9. he effect of variable roerties on the unsteady Hartmann flow with heat transfer considering Hall effect has been studied by Attia and Aboul-Hassan 0. he effect of variable viscosity and Hall current on unsteady laminar flow of dusty conducting fluid between two arallel late through orous medium with temerature deendent viscosity and thermal radiation has recently been studied by Singh and Yadav. he effect of Hall current on Couette flow with heat transfer of dusty conducting fluid in the resence of uniform suction and injection has been investigated by Attia. Unsteady hydromagnetic Couette flow of dusty fluid with temerature deendent viscosity and thermal conductivity under exonential decaying ressure gradient was investigated by Attia 3. Ghosh et al 4. has investigated the Hall effect on MHD flow in rotating system with heat transfer characteristics. Shawky 5 has investigated the Pulsatile flow with heat transfer of dusty magnetohydrodynamic Ree-Erying fluid through a channel. In the resent study the effect of Hall current and rotation on heat transfer in the oscillating flow of an electrically conducting viscous incomressible dusty fluid has been investigated. he article hase is assumed to incomressible electrically nonconducting dusty and ressure less. he orous lates of the channel are ket stationary. he fluid is flowing between two infinite electrically insulating orous lates maintained at two constant but different temeratures. Descrition of the Problem he dusty fluid is assumed to be flowing between two infinite horiontal orous lates located at d ± lanes as shown in Fig.. he fluid is injected with constant velocity w 0 through the lower orous late and simultaneously removed with same suction velocity through the uer orous late. hus the -comonent of the velocity of the fluid is constant and denoted by w 0. he dust articles are assumed to be electrically non-conducting sherical and uniformly distributed in the fluid and are big enough so that they are not umed out through the orous lates and have no -comonent of velocity. he two-lates are assumed to be electrically nonconducting and ket at two different temeratures for the lower late and for the uer late with >. A eriodic ressure gradient varying with time is alied in x-direction. A uniform magnetic field B 0 is alied in the -direction. his is only magnetic field in the roblem as the induced magnetic field is neglected by assuming very small magnetic Reynolds number Crammer and Pai 8. he whole system is rotating with constant angular velocity about the -axis. It is required to obtain the time varying velocity distribution for both the fluid and the dust articles. Since the lates are infinite in x and y directions all hysical quantities for this fully develoed flow deends only on and t coordinate excet the ressure. 3 Basic Equations he governing equations for this study are based on the conversation laws of masses linear momentum and energy of both hases. Fig. Physical configuration of the roblem

3 CHAND et al.: EFFEC OF HALL CURREN AND ROAION ON HEA RANSFER Velocity distribution u u v w Let ( and u ( u v w be the fluid and dust articles velocities resectively. he magnetic field and angular velocity for the resent roblem are B( 00 B0 and Ω (00 Ω0 resectively. he flow of the fluid is governed by the following momentum equation: u ( u. u Ω u ρ KN µ u J B u u ρ ρ ( ( where ρ is the density of the clean fluid µ the viscosity of the clean fluid J the current density B the magnetic flux density vector the ressure distribution Ω the constant angular velocity of the channel N the number of dust article er unit volume K 6πµ a is the Stokes constant and a is the average radius of the dust articles. he first three terms in the RHS of Eq. ( are the ressure gradient viscosity and the Lorent force terms resectively. he last term reresents the force due to the relative motion between fluid and dust articles. It is assumed that the Reynolds number of the relative velocity is small. In such a case the force between fluid and dust articles is roortional to the relative velocity as suggested by Saffman 6. If the Hall term is retained the current density J from the generalied Ohm s law is given by: J σe u B ˆ ˆ 0k ( J B0k β ( where σ is the electric conductivity of the fluid and is the Hall arameter. Solving Eq. ( for J gives: σ B0 J B ( ( 0 u mv i v - mu j m (3 where m σβ B0 is Hall current arameter. hus in term of Eq. (3 the two comonents of the equation read: u u w 0 Ω 0v µ u ρ x ( σ B0 ( KN u u u mv ρ ρ( m (4 v v w 0 Ω 0u µ v ρ y ( σ B0 ( KN v v v mu ρ ρ( m (5 he motion of the dust articles is governed by Newton s second law of motion alied in x and y- directions. he two comonents of this equation read as: u m Ω 0v KN( u u v m Ω 0u KN( v v... (6 (7 Here m is the average mass of the dust articles. Boundary conditions relevant to the roblem are given by: d u v u v 0 and d u v u v 0. (8 3. emerature distribution Heat transfer takes lace from the uer hot late to the lower cold late by conduction through the fluid. Since the hot late is above there is no natural convection; however there is a forced convection due to the suction and injection. In addition to the heat transfer there is a heat generation due to both the Joule and viscous dissiations. he dust articles gain heat from the fluid by conduction through their sherical surface. Since the roblem deals with twohase flow therefore two-energy equations are required (Crammer and Pai 8 Schlichting 7. he energy equations describing the temerature distributions for the fluid and dust articles neglecting the viscous dissiation and the Joule dissiation are given by: ρ c ρc ρc ω k s v v 0 ( γ ( γ (9 (0

4 67 INDIAN J PURE & APPL PHYS VOL 5 OCOBER 03 where is the temerature of the fluid the temerature of the dust articles c the secific heat caacity of the fluid at constant volume c s the secific heat caacity of the articles at constant volume k is the thermal conductivity of the fluid ρ is the mass of the dust articles er unit volume of the fluid γ is the temerature relaxation time. he last term on the right-hand side of Eq. (9 reresents the heat conduction between the fluid and dust articles resectively. he temerature relaxation time deends in general on the geometry. Here 3Pγ c γ c r s sa where γ ρ is the velocity relaxation time 9µ c Pr µ 3ρ ρ is the Prandtl number and ρs is 3 k 4π a N the density of the dust articles. Boundary conditions relevant to the roblem are given by: d 0 P 0 and d P 0 cosω t ( Eqs (4-( can be made dimensionless by introducing the following non-dimensional variables and arameters: ( x y ( u v d w ( x y ( u v ( u v mν P P w0 KNd 0 0 ω t w0 Ω0d w0d ( u v G d ω t Ω w d ν ν 0 KNd σ ρd R M B0d L0 ρw µ µ µγ 0 where G is article mass arameter R article concentration arameter ω frequency of oscillation M magnetic field arameter suction arameter L 0 temerature relaxation time arameter. 0 In terms of these dimensionless variables and arameters the Eqs (4-( transform to: u u Ω u R v u u x M ( u mv ( m ( m ( v v Ωu v R v v y M ( v mu u Ω v ( u u G v Ω u ( v v G t Pr 3Pr R ( L0 ( ( he transformed boundary conditions become: (... (3 (4 (5 (6 (7 u v u v 0 0 P 0 and u v u v 0 cos ωt P cos ωt (8 We assume that the fluid flow under the ressure gradient along the x-axis varying with time is of the form: Ae x it and y 0 (9 where A Amlitude of the ressure gradient. Eqs ( & (3 and (4 & (5 can be combined into following equations:

5 CHAND et al.: EFFEC OF HALL CURREN AND ROAION ON HEA RANSFER 673 F F iω F F Ae R M ( im ( F F F ( m (0 F iω F ( F F ( G e ( t e e e e n n n n e e L n n n n 0 ( t n n n n L0 iω e e n n n n e e e e e (6 (7 where F u iv and F u iv are comlex fluid and articles velocity resectively. he boundary conditions reduce to: ; F F 0; and ; F F 0 ( 4 Method of Solution o solve the Eqs (6 (7 (0 and (; we assume the solution of the form: ( φ ( ( ψ ( F t e F t e and ( t θ ( e ( t ξ ( e (3 he resulting equations are solved under the transformed boundary conditions and we obtain the following exressions for the fluid velocity dust article velocity fluid temerature and dust articles temerature. e A F ( t e F ( t A ( ig( ω Ω e e sinh e sinh sinh( (4 sinh e sinh sinh( (5 5 Mathematical Exression in Limiting Cases 5. Case I When ω M Ω O(. In this asymtotic case the mathematical exressions for fluid velocity article velocity fluid temerature and article temerature reduces to: { 3 ω 4} ( ( ω F t W i t W P e ( ( ω sinh - e sinh ( - sinh { 5 ω 8} F t S i t S ( t P ( t e sinh e sinh ( sinh ( n n n n e e e e e e n n n n ( L0 iω L n n n n e e e e e e n n n n... (8 (9 (30 (3 5. Case II Ω ω M O (. In this asymtotic case the mathematical exressions for fluid velocity article velocity fluid temerature and article temerature reduces to:

6 674 INDIAN J PURE & APPL PHYS VOL 5 OCOBER 03 ( { Ω } F t K V V P e ( ( Ω F t K V V ( t P ( t sinh e sinh ( sinh ' 5 6 e sinh e sinh ( sinh K n n n n e e e e e e n n n n ( L0 L0 ( K n n n n e e e e e e n n n n (3 (33... (34... ( Case III M ω Ω O(. In this asymtotic case the mathematical exressions for fluid velocity article velocity fluid temerature and article temerature reduces to the following form: ( ( 7 8 F t K V M V P e sinh e sinh ( sinh ''' ( 3 ( 7 8 F t K U V M V e sinh e sinh ( sinh ''' (36 (37 ( t P ( t K n n n n e e e e e e n n n n K n n n n e e e e e e n n n n (38... (39 where the constants used above have been listed in the aendix. 6 Results and Discussion he following discussion brings out the effects of some ertinent arameters such as the article concentration arameter (R the magnetic field arameter (M the suction arameter ( the frequency of oscillation (ω the amlitude of the ressure gradient (A the Hall arameter (m and the article mass arameter (G on the fluid and dust articles velocity. he numerical calculations have been carried out for small (Ω5 and large (Ω0 values of the rotation arameter (Ω. he values of Prandtl number are chosen as (P r 0.7 and (P r 7 which reresent air and water-the most common form of fluid on earth. he results have been grahically exressed for both the fluid and dust articles velocities and temerature rofiles. he variations in the fluid velocities rofiles are shown in Figs (-8. he study of these Figs (-8 shows that the fluid velocity decreases with the increasing value of the article concentration arameter (R the magnetic field arameter (M the frequency of the oscillation (ω and the Hall arameter (m whereas it increases with the increasing value of the amlitude of the ressure gradient (A the suction arameter ( and the article mass arameter (G for both small (Ω5 and large (Ω0 value of rotation arameter (Ω. It is also clear from these Figs (-8 that the fluid velocity is maximum at the centre of the channel. Figure 9 shows that the temerature of the fluid decreases with the increasing value of the suction arameter ( Prandtl number (P r frequency of the oscillation (ω and the article concentration arameter (R. he dust articles velocity variations with these arameters have been shown in Figs (0-6. he study of these Figs (0-6 shows

7 CHAND et al.: EFFEC OF HALL CURREN AND ROAION ON HEA RANSFER 675 Fig. Variation of fluid velocity for M R 0.5 ω5 A5 G m and t0 Fig. 3 Variation of fluid velocity for R 0.5 ω5 A5 G m and t0 Fig. 4 Variation of fluid velocity for (M R 0.5 ω5 A5 G m and t0

8 676 INDIAN J PURE & APPL PHYS VOL 5 OCOBER 03 Fig. 5 Variation of fluid velocity for R M 0.5 A5 G m and t0 Fig. 6 Variation of fluid velocity for R M 0.5 ω5 G m and t0 Fig. 7 Variation of fluid velocity for R M 0.5 ω5 G A5 and t0

9 CHAND et al.: EFFEC OF HALL CURREN AND ROAION ON HEA RANSFER 677 Fig. 8 Variation of fluid velocity for R M 0.5 ω5 m A5 and t0 Fig. 9 Variation of fluid temerature for R G A5 and t0 Fig. 0 Variation of dust articles velocity for M 0.5 ω5 m A5 G and t0

10 678 INDIAN J PURE & APPL PHYS VOL 5 OCOBER 03 Fig. Variation of dust articles velocity for R 0.5 ω5 m A5 G and t0 Fig. Variation of dust articles velocity for R M ω5 m A5 G and t0 Fig. 3 Variation of dust articles velocity for R M 0.5 m A5 G and t0

11 CHAND et al.: EFFEC OF HALL CURREN AND ROAION ON HEA RANSFER 679 Fig. 4 Variation of dust articles velocity for R M 0.5 ω5 G m and t0 Fig. 5 Variation of dust articles velocity for R M 0.5 ω5 m A5 and t0 Fig 6 Variation of dust articles velocity for R M 0.5 ω A5 G and t0

12 680 INDIAN J PURE & APPL PHYS VOL 5 OCOBER 03 Fig. 7 Variation of dust articles temerature for R 0.5 P r 0.7 A5 G and t0 Fig. 8 Variation of fluid velocity for different value of rotation arameter Ω Fig. 9 Variation of dust article velocity for different value of rotation arameter Ω

13 CHAND et al.: EFFEC OF HALL CURREN AND ROAION ON HEA RANSFER 68 that the fluid velocity decreases with the increasing value of the article concentration arameter (R the magnetic field arameter (M the frequency of the oscillation (ω the article mass arameter (G and the Hall arameter (m. he fluid and dust article velocity decrease with the increasing value of M the Hartmann number hysically it means that the flow is dragged backward due to the increasing strength of magnetic field (Lorent force. he fluid and dust article velocity decreases with the increasing value of article concentration arameter (R hysically; it means that as the number of articles increases the interaction between the fluid and dust article molecules increases which reduce the velocity of the fluid and the dust articles. On the other hand dust article velocity increases with the increasing value of the amlitude of the ressure gradient (A and the suction arameter ( for both small (Ω5 and large (Ω0 value of rotation arameter (Ω. Figure 7 shows that the temerature of the dust articles decreases with the increasing value of the suction arameter ( and the article concentration arameter (R Prandtl number (P r and frequency of the oscillation (ω whereas temerature of the dust articles increases with the increasing value of temerature relaxation time (L 0. It is also found that the temerature is far less in case of water (P r 7.0 than the temerature in the case of air (P r 7.0. he variation in the fluid and dust article velocities with rotation arameter is shown in Figs (8 and (9 resectively for fixed value of other arameters i.e. (M R 0.5 ω5 A5 G m and t0. It is observed that the fluid and the dust article velocity rofiles decrease with increasing value of rotation arameter and for the large value of rotation arameter velocity rofiles tend to flatten in the central core region and the maxima of rofiles exist near the walls of the channel which indicate the formation of boundary layers adjacent to the walls of the channel. 7 Conclusions he fluid and dust articles velocities are significantly enhanced with the amlitude of the ressure gradient and the suction arameter. All other arameters diminish the fluid and the dust articles velocity.he fluid and the dust article velocity decreases with the increasing value of rotation arameter and the Hall current arameter. Fluid and the dust article velocity are maximum along the centre of the channel. emerature of the dust articles are enhanced with the temerature relaxation time. References Singh K D & Kumar R Indian J Pure & Al Phys 47 ( Chand K & Kumar R Indian J Pure & Al Phys 50 ( Attia H A Comutional & Al Maths 8( ( Chamkha A J International J Heat & Fluid Flow ( Saxena S & Sharma G C Indian J Pure & Al Math 8( ( Attia H A Al Math & Comutational 77 ( Gireesha B J Bagewadi C S & Prasannakumara B C Communication in Nonlinear Scis & Numerical Simulation 4 (5 ( Crammer K R & Pai S I Magnetofluiddynamics for Engineer & Scientists (McGraw Hill New York Aboul Hassan A L & Attia H A Canadian J Phys 80 ( Attia H A & Aboul Hassan A L Al Mathematical Modelling 7(7 ( Singh H & Yadav S S Int J Maths rends & ech 3(4 (0 3. Attia H A African J Math Phys ( ( Attia H A Communication in Nonlinear Sciences & Numerical Simulation 3 ( Ghosh S Anwar Beg O & Narahari M Meccanica 44(6 ( Shawky H M Heat & Mass ransfer 45(0 ( Saffman P G J Fluid Mechanics 3 ( Schlichting Boundary Layer heory (McGraw Hill New York 968. Aendix ν G Kd m ( ω ( ( ( iω m i m M im R m R ig Ω m ( ω m 3 RG. 4 m 3 m 4G R. 8G 5 RG ( 4G 6 ( 4G.

14 INDIAN J PURE & APPL PHYS VOL 5 OCOBER 03 RG m ( G 7 m RG 8... GR m ( G ( G ( G 0 4RG ( G 9 ( 3 ( ( RG... 3 RG G m 3 4 m m ( 4( W W ω ( W W 4 ω ( V V 4 Ω ( V V 4 Ω ( V V 4 7 Ω 8 ( V V 4 7 Ω 8 n iω R P 3( 0 r L i ω ( 4 r r n n ( 4 r r n n n i ω r RL0 3 i r R L 3 n ω L r 4 L ω n n r 4 L ω n i r RL0 R L3 3 3 r 4L3 n n r 4L3 W 3 i4 W A W W W i ( 3 4 A W 4 S G 3 G S 4G S S ( 4G... 8G 3 G ( 4G ( 4G ( ( ( S S S S A S is W 5 S A S is W 6 S7 A S3 is4 W L L0t L0 i r R L 3 i r RL0 R L3 3 3 ( (.. K ( V ( i K it it A K V i 7 8 V i V4 3 i4 V5 V3V V6 V4V V3V V7 i V 8 3 i4 G G G G 4G G G G ( G ( 3 ( ( 3 U G ( ( U G G... U 3 U iu.