Hall Current Effect on the MHD Flow of Newtonian Fluid through a Porous Medium

Transcription

1 Hall Current Effect on the MHD Flow of Newtonian Fluid through a Porous Mediu Pudhari Srilatha Departent of Matheatics, Instute of Aeronautical Engineering, Hyderabad, TS, India. Abstract In this paper, we discuss the hall current effect on the pulsatile flow of a viscous incopressible fluid through a porous ediu in a fleible channel under the influence of transverse agnetic field using Brinkan s odel. The non-linear equations governing the flow are solved using perturbation technique. Assuing long wavelength approiation, the velocy coponents and stresses on the wall are calculated up to order in and the behavior of the aial and transverse velocies as well as the stresses is discussed for different variation in the governing paraeters. The shear stresses on the wall are calculated throughout the cycle of oscillation at different points whin a wavelength and the flow separation is analyzed. Keywords: unsteady flows, hall current effects, pulsatile flows, porous ediu INTRODUCTION The flow caused by a pulsatile pressure gradient through a porous channel or a porous pipe has been investigated in view of s applications in technological and physiological and physiological probles. In physiological fluid dynaics this odel plays a significant role in eplaining the dialysis of blood in artificial kidneys, vasootor of sall blood vessel such as arterioles, venues and capillaries. In ost of the above investigations the boundary surface of the channel or pipe is assued to have unifor cross section and an aial pressure gradient is aintained along the channel, which induces an unidirectional flow. In any bioedical probles one encounters the flow bounded by fleible boundaries and flow through unifor gap is only an approiation. Notable aong the is the blood flow through veins. A part fro this, the non-newtonian fluids are frequently encountered in food iing, chye oveent in the intestine, blood flow at low shear rate, the flow of nuclear slurries, liquid etals and alloys. There are also suations where agnetic field characteristics in nonnewtonian fluids are significant. For instance, flow of ercury aalgas and lubrication wh heavy oil and grease [ 9]. Current advances in the subject of Hall current involve Hall accelerators, nuclear power reactors, flight MHD and MHD generators. The Hall current IC swch or sensors are etreely useful to detects the presence or absence of a agnetic field and gives a digal signal for on and off. Large values of Hall current paraeter in the presence of heavy-duty agnetic fields corresponds to Hall current, which is how, shakes the current densy and allows one to understand the ipact of Hall current on the flow []. Hayat et al. [] studied effects of Hall current on peristaltic flow of a Mawell fluid in a porous ediu. The effects of Hall current and heat transfer on MHD flow of a Burgers fluid under the pull of eccentric rotating disks has been eained by Siddiqui et. al []. In another paper Gad [], eained the effect of Hall currents on interaction of pulsatile and peristaltic transport induced flows for a particle-fluid suspension. Recently, studies [ ] of heat and ass transfer in peristalsis have been considered by soe researchers due to s applications in bioedical sciences. Heat transfer involves any coplicated processes such as evaluating skin burns, destruction of undesirable cancer tissues, dilution technique in eaining blood flow, paper aking, food processing, vasodilation, etabolic heat generation and radiation between surface and s environent, etabolic heat generation and radiation between surface and s environent. It is clear fro reviewing the eisting lerature, that no uch attention has been given to the peristaltic flows wh heat generation and Hall Currents, especially such attepts being further narrowed down for the case of non-newtonian fluids. Recently, Krishna and M.G.Reddy [] discussed the MHD free convective rotating flow of visco-elastic fluid past an infine vertical oscillating plate. Krishna and G.S.Reddy [5] discussed the unsteady MHD convective flow of second grade fluid through a porous ediu in a rotating parallel plate channel wh a teperature-dependent source.krishna and Swarnalathaa [6] discussed the peristaltic MHD flow of an incopressible and electrically conducting Williason fluid in a syetric planar channel wh heat and ass transfer under the effect of an inclined agnetic field. Swarnalathaa and Krishna [7] discussed the theoretical and coputational study of the peristaltic heodynaic flow of couple stress fluids through a porous ediu under the influence of a agnetic field wh wall slip condion. Krishna and M.G. Reddy [8] discussed the unsteady MHD free convection in a boundary layer flow of an electrically conducting fluid through porous ediu subject to unifor transverse agnetic field over a oving infine vertical plate in the presence of heat source and cheical reaction. Krishna and G.S. Reddy [9] have investigated the siulation on the MHD forced convective flow through stupy pereable porous ediu (oil sands, sand) using Lattice Boltzann ethod. Krishna and K.Jyothi [] discussed the Hall effects on MHD Rotating flow of a visco-elastic fluid through a porous ediu over an infine oscillating porous plate wh heat source and cheical reaction. B.S.K. Reddy et al.[] investigated MHD flow of viscous incopressible nano-fluid through a saturating porous ediu. 67

2 In this chapter we discuss the hall current effect on the pulsatile flow of a viscous incopressible fluid through a porous ediu in a fleible channel under the influence of transverse agnetic field using Brinkan s odel. FORMULATION AND SOLUTION OF THE PROBLEM Consider the unsteady fully developed pulsatile flow of viscous incopressible fluid through a porous ediu in a fleible channel under the influence of transverse agnetic field of strength H. At t > the fluid is driven by a constant O pressure gradient parallel to the channel walls. Choosing the Cartesian coordinate syste O(, y) the upper and lower walls of the channel are given by y as Where, a is the aplude, is the wavelength and s is an arbrary * function of the noralized aial co-ordinate. The entire flow is subjected to strong unifor transverse agnetic field noral to the plate in s own plane. Equation of otion along -direction the -coponent current densy μej y H o and the y-coponent current densy μ ej H o. The equations governing the two diensional flow of viscous incopressible fluid through a porous ediu under the influence of transverse agnetic field, using Brinkan s odel are u v y u u u u v t y p u u μ e J yh u y k (.) (.) the respective equations of continuy are trivially satisfied. When the strength of the agnetic field is very large, the generalized Oh s law is odified to include the Hall current, so that ωeτ e J J H σ (E μe q H) (.) H Where, q is the velocy vector, H is the agnetic field intensy vector, E is the electric field, J is the current densy vector, is the cyclotron frequency, is the electron e collision tie, is the fluid conductivy and μe is the agnetic pereabily. In equation (.), the electron pressure gradient, the ion-slip and thero-electric effects are neglected. We also assue that the electric field E= under assuptions reduces to where J y e J σμ H v (.5) J y e J σμ H u (.6) ω τ e is the hall paraeter. e On solving equations (.5) and (.6) we obtain σμeh ( v u) (.7) J σμeh ( v u) (.8) J y Using the equations (.7) and (.8) the equations of the otion wh reference to frae are given by u u u u v t y (.9) p u u σμe H ( v u) u y ρ( ) k e v v v u v t y p v v μ e J H v y y k (.) Where ( u, v) are the velocy coponents along O(, y) directions respectively. is the densy of the fluid, p is the fluid pressure, k is the pereabily of the porous ediu, µ e the agnetic pereabily, the coefficient of kineatic viscosy and H is the applied agnetic field. Since the o plates etends to infiny along and y directions, all the physical quanties ecept the pressure depend on z and t alone. Hence u and v are function of z and t alone and hence v v v p u v t y y v v μe H ( v u) v y ρ( ) k (.) Eliinating p fro equations (.9) and (.), the governing the flow in ters of appropriate strea function reduces to ( ) t y y Where H e ( ) k is the laplacian operator (.) 68

3 The relevant condions on are u, v (.) y The relevant boundary condions are, on y (.) yy y, k e on s y (.) An oscillatory tie dependent flu is iposed on the flow resulting in a pulsatile flow we assue that oscillatory flu across the channel is k e. Where f is the f characteristic flu, k is s aplude and the frequency of oscillation. We define a characteristic velocy q corresponding to the characteristic flu, so that is q a. We introduce the following non-diensional variables. *, y * y, t a * a * t,, *, q a f f Substuting the above non-diensional variables into the equation (.), the governing equation in ters of non-diensional paraeter (on dropping the asterisks) reduces to R ( ) S S Where y yyyy y yy yyy M y yy D tyy M t q a D yy (.5) aq R is the Reynolds nuber, S is the oscillatory a paraeter, D is the inverse Darcy paraeter, k e H a M is the Hartann nuber (Magnetic field Paraeter), ω τ e is the hall paraeter. e Equation (.5) is highly non-linear and is not aenable for eact solution. However assuing the slope of the fleible channel sall (<<). We take ay be given asyptotic epansion in the for k e ke (.6) We are aking use of transforation y (.7) s() And the boundary condions at y = s(), Now to be satisfied at. Substuting equation (.6) in the non-diensional equation (.5) and equating like powers of, the equations corresponding to the zeroth and first order steady and unsteady coponents are, Zeroth order, M y y First Order, M D D y y M D y y R y y M D y y (.8) (.9) (.) R (.) y y y y y y Substuting (..7) in equations (..8) and (..) we obtain M D s M D s Rs The corresponding boundary condions to be satisfied are, on (.) (.) (.),,, on (.5) 69

4 Again substuting (.7) in the equations (.9) and (.) we obtain, M D s M D s Rs The corresponding boundary condions to be satisfied. (.6). (.7), on (.8),,, on (.9) Solving the equations (.) and (.) subjects to the boundary condions (.) and (.5) we obtain. C ) (.) sinh( C C sinh( ) C C5 sinh( ) (..) C6 cosh( ) C7 sinh( ) Siilarly solving equations (.6) and (.7) subjected to the corresponding boundary condions (.8) and (.9). 8 ) 9 C sinh( C C sinh( ) C C sinh( ) C cosh( ) C sinh( ) (.) (.) Substuting equations (.), (.), (.) and (.) in the equation (.6), we obtain C sinh( ) C k e C sinh( ) C C sinh( c 8 9 ) C5 sinh( ) C6 cosh( ) C7 sinh( ) ke C sinh( ) C C sinh( ) C cosh( ) C sinh( ) (.) Using the equation (.), the aial velocy and the transverse velocy are given by The aial velocy, u s( ) ( C sinh( ) C ) k e ( C sinh( ) ) 8 C9 sinh( ) C C sinh( ) C cosh( ) sinh( ) ( C 5 6 C7 C ) C C sinh( ) C cosh( ) sinh( ) ke sinh( C and the transverse velocy, v C sinh( ) C ) k e ( C sinh( ) C ) ( 8 9 k e ( C sinh( ) C C5 sinh( ) C6 cosh( ) C7 sinh( ) C ) C C sinh( ) C cosh( ) C sinh( ) sinh( Shear stress at the wall y=s() is given by y ( s ) ( yy ( s ) ) s Where ) y, s( ) sin ( yy, yy y 6

5 There are substuting for yy etc., we obtained the non-diensional shear stress. s A ( ( sin ) A [ ( sin ). sin ) s ke sinh( s).[ A C C cos C5 s 6 ] s cosh( s).(c cos C 5 6 ( s)) C cos.sinh( s) C s [ 5 ( k e 7 sinh( s) ] )( sin ) { cos sinh( s)[cosh( s) ( s ) s sinh( s) cosh( s)] s cos cosh( s)(cosh( s) s sinh( s) s cos }] RESULTS AND DISCUSSION: The flow governed by the non-diensional paraeters R the Reynolds nuber, D - inverse Darcy paraeter, the aplude of the boundary wave, k the aplude of oscillatory flu, M the agnetic paraeter (Hartan nuber), S the oscillatory paraeter and hall paraeter. The aial, transverse velocies and the stresses are evaluated coputationally for different variations in the governing paraeters R, D -,, k, M, S and. For coputational purpose we chose the boundary wave s() = + sin in the non-diensional for. The figures (-6) represent the velocy coponents u and v for different variations of the governing paraeters being the other paraeters fied. We observe that for all variations in the governing paraeters, the aial velocy u attains s aiu on the id plane of the channel. We notice that the agnude of the aial velocy u enhances and the transverse velocy v reduces wh increasing the Reynolds paraeter R. The behavior of transverse velocy v is oscillatory wh s agnude decreasing on R increases through sall values less than and later reduces for further increase in R. The resultant velocy also enhances wh increasing the Reynolds paraeter R (Fig and ). Fro figures ( and ) we concluded that both the velocy coponents u and v reduces wh increase in the inverse Darcy paraeter D -. Here we observe that higher the pereabily of the porous ediu larger the aial velocy along the channel and rate of increase is sufficiently high. Siilarly, the resultant velocy reduces wh increasing in the inverse Darcy paraeter D -. It is evident that the agnude of u, v and the resultant velocy increase wh increasing the paraeters k, and (5, 6, 9,, and ). The agnude of the aial velocy u enhances and the transverse velocy v decreases wh increasing in the aplude of the boundary wave. The resultant velocy also reduces wh increasing the paraeter (Fig 7 and 8). We notice that the agnude of the velocy coponents u and v reduces wh increasing the intensy of the agnetic field M. The resultant velocy also decreases wh increasing the Hartann nuber M (Fig and ). The agnude of the aial velocy u enhances and the transverse velocy v reduces wh increasing the oscillatory paraeter S. The behavior of transverse velocy v is oscillatory wh s agnude decreasing on S increases through sall values less than and later reduces for further increase in S. The resultant velocy also enhances wh increasing the oscillatory paraeter S (Fig 5 and 6). The shear stress on the upper wall wh S= is evaluated through the entire cycle of oscillation at different points whin a wave length for different sets of paraeter. we choose the boundary wave as s() = + sin in nondiensional for. As already stated the crerion for the recognion of separation is the eistence of a point at which the shear stresses at the wall less than or equal to zero throughout the entire cycle of oscillation. The influence of porosy in checking separation ay be observed fro table () for arbrary values of R, D, k,, M and. This is evident that the shear stress on the upper wall does not vanish or becoe negative at any point in a wave length range throughout the entire cycle of oscillation. The agnudes of the stresses enhance wh increasing R, D, k, and R and reduces wh increasing and agnetic paraeter M being fied S. 6

6 Figure : The velocy profile for u against R k =, S=, D - =, M=, =, t,. Figure : The velocy profile for v against R k =, S=, D - =, M=, =, t,. 6

7 Figure : The velocy profile for u against D - R=, S=, k =, M=, = t,. Figure : The velocy profile for v against D - R=, S=, k =, M=, = t,. 6

8 Figure 5: The velocy profile for u against k =, S=, D - =, M=, t, R=,. Figure 6: The velocy profile for v against k =, S=, D - =, M=, t, R=,. 6

9 Figure 7: The velocy profile for u against k =, S=, D - =, R=, M=, t, = Figure 8: The velocy profile for v against k =, S=, D - =, R=, M=, t, = 65

10 Figure 9: The velocy profile for u against R=, S=, D - =, k =, M=, =, t,. Figure : The velocy profile for v against R=, S=, D - =, k =, M=, =, t,. 66

11 Figure : The velocy profile for u against M k =, S=, D - =, =, t, R=,. Figure : The velocy profile for v against M k =, S=, D - =, =, t, R=,. 67

12 Figure : The velocy profile for u against k =, S=, D - =, M=, t, R=,. Figure : The velocy profile for v against k =, S=, D - =, M=, t, R=,. 68

13 Figure 5: The velocy profile for u against S k =, D - =, M=, = t, R=,. Figure 6: The velocy profile for v against S k =, D - =, M=, =, t, R=,. 69

14 Table : The shear stress at the upper wall wh S= CONCLUSIONS. The resultant velocy also enhances wh increasing the Reynolds paraeter R.. Higher the pereabily of the porous ediu larger the aial velocy along the channel and rate of increase is sufficiently high. The resultant velocy reduces wh increasing in the inverse Darcy paraeter D -.. The resultant velocy increase wh increasing the paraeters k, and.. The resultant velocy also reduces wh increasing the paraeter. 5. The resultant velocy also decreases wh increasing the Hartann nuber M. 6. The agnude of the aial velocy u enhances and the transverse velocy v reduces wh increasing the oscillatory paraeter S. The resultant velocy also enhances wh increasing the oscillatory paraeter S. 7. The shear stresses at the wall less than or equal to zero throughout the entire cycle of oscillation. 8. The influence of porosy in checking separation ay be observed. 9. The shear stress on the upper wall does not vanish or becoe negative at any point in a wave length range throughout the entire cycle of oscillation.. The agnude of the stresses enhances wh increasing R, D, k, and and reduces wh increasing and agnetic paraeter M being fied S. REFERENCES []. Latha TW, Fluid otion in a peristaltic pup, MIT Cabridge MA, 966. []. Shapiro AH, Jaffrin MY, Weinberg SL, Peristaltic puping wh long wavelengths at low Reynolds nuber. J. Fluid Mech. 969; 7: []. Eldabe NTM, El-Sayed MF, Ghaly AY, Sayed HM, Peristaltically induced transport of a MHD biviscosy fluid in a non-unifor tube. Physica A. 7; 8:5 66. doi:.6/j.physa []. Noreen S, Hayat T, Alsaedi A, Qasi M, Mied convection heat and ass transfer in the peristaltic flow wh cheical reaction and inclined agnetic field. Indian Journal of Physics. ; 87: doi:.7/s [5]. Noreen S, Ahad B, Hayat T, Mied Convection Flow of Nanofluid in Presence of an Inclined Magnetic Field. PloS One. ; 8:e78. doi:.7/journal.pone.78 PMID: 8676 [6]. Kothandapani M, Srinivas S, Peristaltic transport of a Jeffrey fluid under the effect of agnetic field in an asyetric channel. Int. J. Non-Linear Mech. 8; :95 9. doi:.6/j.ijnonlinec [7]. Mehood OU, Mustapha N, Shafie S, Peristaltically Induced Flow of Fourth Grade Fluid wh Viscous Dissipation. WASJ. ; :6,. [8]. Mehood OU, Mustapha N, Shafie S, Heat transfer on peristaltic flow of fourth grade fluid in inclined asyetric channel wh partial slip. Appl. Math. Mech. -Engl. Ed. ; : 8. [9]. Tripathi D, Pandey SK, Das S, Peristaltic flow of viscoelastic fluid wh fractional Mawell odel through a channel. Appl. Math. Coput. ; 5: []. Nowar K, Peristaltic flow of a nanofluid under the effect of Hall current and porous ediu. Math. Prob. Engg. ; pii 8958 (5). []. Hayat T, Ali N and Asghar S, Hall effects on peristaltic flow of a Mawell fluid in a porous 65

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