International Journal of Mathematical Archive-3(4), 2012, Page: Available online through ISSN

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1 International Journal of Matheatical Archive-3(4),, Page: 7-7 Available online through ISSN HALL EFFECT ON THERMAL INSTABILITY OF COMPRESSIBLE COUPLE-STRESS FLUID IN PRESENCE OF SUSPENDED PARTICLES V. Singh* & Shaily Dixit** *Departent of Applie Science, Moraaba Institute of Technology, Moraaba (U.P.) Inia **Departent of Matheatics, Hinu College, Moraaba (U.P.) Inia **E ail: shailyixit5@gail.co (Receive on: 9-3-; Accepte on: 5-4-) ABSTRACT Effect of Hall current on the the theral instability of a copressible couple stress flui with suspene particles is consiere in this paper. It has been foun that suspene particles have estabilizing effect on copresse couple stress flui. Also ispersion relations incluing the effect of Hall Current an copressibility on the theral instability of couple stress flui in the presence of suspene particles are analyze. Stability of the syste an oscillatory oes are consiere an it is foun that ue to the presence of agnetic fiel an suspene particles oscillatory oes are prouce which were non existent in their absence.. INTRODUCTION The theral instability of a flui layer with aintaine averse teperature graient by heating the unersie plays an iportant role in geophysics, interior of the Earth, Oceanography an atospheric physics etc. A etaile account of the theoretical an experiental stuy of the onset of Benar convection in Newtonian Fluis, uner varying assuptions of hyroynaics, has been given by Chanrasekhar []. The use of Boussinesq approxiation has been ae through out which states that the ensity changes are isregare in all other ters in the equation of otion except the external force ter. Shara [5] has consiere the effect of rotation an agnetic fiel on the theral instability in copressible fluis. The flui has been consiere to be Newtonian in all the above stuies while Scanlon an Segel [6] have consiere the effect of suspene particles on the onset of Benar convection an foun that the critical Rayleigh nuber was reuce solely because of the heat capacity of the pure flui. With the growing iportance of non-newtonian fluis in oern technologies an inustries, the investigations on such fluis are esirable. Stokes [7] propose an postulate the theory of couple-stress fluis. One of the applications of the couple-stress flui is its use to the stuy of the echanis of lubrication of synovial joints, which has becoe the object of scientific research. A huan joint is a ynaically loae bearing which has articular cartilage as the bearing an synovial flui as the lubricant. When a flui fil is generate, squeeze fil action is capable of proviing consierable protection to the cartilage surface. The shouler, knee, hip an ankle joints are the loae bearing synovial joints of the huan boy an these joints have a low-friction coefficient an negligible wear. Noral synovial flui is clear or yellowish an is viscous, non-newtonian flui. Accoring to the theory of Stokes [7], couple-stresses are foun to appear in noticeable agnitue in fluis with very large olecules. Since the long chain of hylauronic aci olecules are foun as aitives in synovial flui, Walicki an Walicka [7] oele synovial flui as a couple-stress flui in huan joints. Goel et al. [] have stuie the hyroagnetic stability of an unboune couple-stress binary flui ixture having vertical teperature an concentration graients with rotation. Shara an Shara [8] have stuie the couple stress flui heate fro below in porous eiu. An electrically conucting couple stress flui heate fro below in porous eiu in the presence of unifor horizontal agnetic fiel has also been stuie by Shara an Shara [9]. The use of agnetic fiel is being ae for the clinical purposes in etection an cure of certain iseases with the help of agnetic fiel evices / instruents. Shara an Thakur [] have stuie the theral convection in couple stress flui in porous eiu in hyroagnetics. Environental pollution is the ain cause of ust to enter into huan boy. The etal ust which filters into the bloo strea of those working near furnace causes extensive aage to the chroozoes an genetic utations so observe are likely to bree cancer as alforations in the coing progeny. Therefore, it is very essential to stuy the bloo flow with *Corresponing author: Shaily Dixit**, *E-ail: shailyixit5@gail.co International Journal of Matheatical Archive- 3 (4), April 7

2 SUSPENDED PARTICLES/ IJMA- 3(4), April-, Page: 7-7 ust particles. Consiering bloo as couple-stress flui an ust particles as icro-organiss, Ratho an Thippesway [4] have stuie the gravity flow of pulsatile bloo through close rectangular incline channel with icro-organiss. If an electric fiel is applie at right angles to the agnetic fiel, the whole current will not flow along the electric fiel. This tenency of the electric current to flow across an electric fiel in the presence of agnetic fiel is calle Hall Effect. The Hall Current is likely to be iportant in flows of laboratory plasas as well as in any geophysical an astrophysical situations. Sheran an Sutton [] have consiere the effect of Hall currents on the efficiency of a agnetohyro-flui ynaic (MHD) generator while Gupta [3] stuie the effect of Hall currents on the theral instability of electrically conucting flui in the presence of unifor vertical agnetic fiel. Shara an Gupta [] investigate the effect of Hall currents on therosolutal instability of a rotating plasa an establishe the estabilizing influence of Hall currents. For copressible fluis, the equations governing the syste becoe quite coplicate. Spiegel an Veronis [3] have siplifie the set of equations governing the flow of copressible fluis assuing that the epth of the flui layer is uch saller than the scale height as efine by the an the otions of infinitesial aplitue are consiere. Shara [4] investigate the theral instability of copressible fluis in the presence of rotation an agnetic fiel while Shara an Gupta [5] stuie the effect of finite Laror raius on theral instability of copressible rotating plasa. Theral instability of copressible finite Larar raius Hall plasa has been stuie by Shara an Sunil [6] in a porous eiu. Shara an Shara have stuie the Effect of suspene particles on couple-stress flui heate fro below in the presence of rotation an agnetic fiel. Keeping in in the iportance in paper inustry, petroleu inustry, geophysics an bio-echanics an various applications entione above, Hall effect on the theral instability of a copressible couple-stress flui with suspene particles has been consiere in the present paper.]. FORMULATION OF THE PROBLEM Consier an infinite, horizontal, copressible couple-stress flui layer of thickness, heate fro below so that, the teperature an ensity at the botto surface z = are T, o respectively an at the upper surface z = are T, an T that a unifor averse teperature graient enote β = is aintaine. Let, P, T an v( uvw,, ) Z respectively the ensity, pressure, teperature an velocity of the flui, v ( xt, ) an N( xt, ) enote the velocity an nuber ensity of suspene particles respectively. µ e is the agnetic pereability, x= ( xyz,, ). The layer is acte upon by the gravity force g(,, g) an unifor vertical agnetic fiel H = (,, H). Copressible Couple stress flui z = z axis g (,, g) H = (,, H) z = x axis T o Heate fro below Geoetrical Configuration y axis Then the oentu balance, ass balance equations of the couple stress flui in presence of suspene particles an Hall currents are, IJMA. All Rights Reserve 7

3 SUSPENDED PARTICLES/ IJMA- 3(4), April-, Page: 7-7 v P δ µ + (. ) = ν. v = + µ e ( ) v v g v KN v v + H H 4π Where K = 6 π µ η', η being particle raius, is the Stokes rag coefficient. Assuing unifor particle size, spherical shape an sall relative velocities between the flui an particles, the presence of particles as an extra force ter, in the equation of otion () proportional to the velocity ifference between particles an flui. The force exerte by the flui on the particles is equal an opposite to that exerte by the particles on the flui. Inter particle reactions are ignore for we assue that the istances between the particles are large copare with their iaeters. If N is the ass of particles per unit volue, then the equations of otion an continuity for the particles are: v N + ( v. ) v = KN[ v v] N +.( Nv ) = Let C v, C pt enote the heat capacity of the flui at constant volue an the heat capacity of the particles. Assuing that the particles an flui are in theral equilibriu, the equation of heat conuction gives. () () (3) (4) T NC + (. ) + +. = C t pt v T v T κ T v (5) The kineatic viscosityν, couple stress viscosity µ, theral iffusivity κ an coefficient of theral expansion α are all assue to be constants. The Maxwell s equations in the presence of Hall currents yiel. H = ( v H) + η H [( H) H] t 4π Ne. H = where η, N an e enote, respectively, the resistivity, the electron nuber ensity an the charge of an electron. Spiegel an Veronis (96) efine ƒ as any of the state variables (pressure (P), ensity () or teperature (T)) an expresse these in the for (6) (7) ƒ (x, y, z, t) = ƒ + ƒ o (z) + ƒ (x, y, z, t) (8a) Where ƒ is the constant space average of ƒ, ƒ o is the variation in the absence of otion an ƒ is the fluctuation resulting fro the otion. The initial state is, therefore, a state in which the ensity, pressure, teperature an velocity in the flui are given by = = = = v =,,, H =,, H, N = N ( z), P P( z), T T( z), v (,,) where following Spiegel an Veronis [3], we have z = ( + ) P z P g z a constant (8b) (8b), IJMA. All Rights Reserve 73

4 SUSPENDED PARTICLES/ IJMA- 3(4), April-, Page: 7-7 ( z) = ( T T ) + K ( ) α T( z) = β z+ T, α = (=α, say), (8c) T k = P, where P an stan for a constant space istribution of P an while o an T o stan for ensity an teperature of the flui at the lower bounary. Following the assuptions given by Spiegel an Veronis [3] an results for copressible fluis, the flow equations are foun to be the sae as those of incopressible fluis except that the static teperature graient β is replace by its excess over the aiabatic (β g / c p ), c p being specific heat of the flui at constant pressure. 3. PERTURBATION EQUATIONS Assue sall perturbations aroun the basic solution an let δ P, δ, N, θ, v (u, v, w), v (l, r, s), h h, h, h enote the perturbations in flui pressure, ensity, particle nuber ensity N, teperature, couple ( x y z) stress flui velocity, particle velocity, agnetic fiel H respectively. The equation of state is = α T T where α is the coefficient of theral expansions as ensity variations arise ainly ue to teperature variations. Thereafter the change in ensity δ cause ainly by the perturbation θ in teperature is given by = α θ () Then the linearize hyroagnetic perturbation equations for theral convection in a copressible couple stress flui uner speigel an Veronis assuptions are (9) v µ δ KN µ = e δp + ν v + g + ( v v) + [( h) H] 4π. v = ( N ) v = KN v v () () (3) θ t g β κ θ C ( + h) = ( w + hs) + Where κ p v C = an v N C h = C pt an also α = T = α, say theral iffusivity respectively.. h = v µ ν = an g/c p,, (4) νκ stan for the aiabatic graient, kineatic viscosity an (5), IJMA. All Rights Reserve 74

5 SUSPENDED PARTICLES/ IJMA- 3(4), April-, Page: 7-7 h (6) (. ) η = H v + h h H 4π Ne Eliinating v between equations () (3), No/ µ + v = δp+ ( ν ) v + τ + δ µ e g ( h) H + 4π (7) Where τ = K Writing the scalar coponent of equation (7) No/ u µ µ e hz + = δp+ ν u H + τ x 4π x hx (8) N / v µ µ h hy δp ν v H τ t y + 4π y o e z + = + No / w µ + = δ + αθ + ν τ + P g w (9) () using equation () an on solving (8), (9) an () we get No/ µ 4 µ eh hz + w= ν w+ gα + θ + () τ x y + 4π z No / ζ µ µ eh ξ () + = ν ζ + + τ 4π Equation (4) an (6) can be written as H + τ g H t κ θ β = w C p + τ t Where + h = H w H ξ η hz = H z 4π Ne (3) (4) ζ H η ξ = H + z 4π Ne v u where ζ = x y ( hz ) is the z coponent of vorticity, IJMA. All Rights Reserve 75 (5)

6 ξ V. Singh* & Shaily Dixit**/ HALL EFFECT ON THERMAL INSTABILITY OF COMPRESSIBLE COUPLE-STRESS FLUID IN PRESENCE OF SUSPENDED PARTICLES/ IJMA- 3(4), April-, Page: 7-7 h y = x hx is the z coponent of current ensity. y 4. NORMAL MODE ANALYSIS METHOD AND DISPERSION RELATION Analyze the perturbation quantities into noral oes by seeking solutions in the for. [w, θ, h z ξ, ζ ] = [ w (z), Θ (z), K (z), X (z), Z (z)] exp (ik x x + ik y y + nt) (6) where k x an k y are the wave nubers along x an y irections an the resultant wave nuber is given by k = k x + k y an n is the growth rate. Using expression (6) equation () (5), in non iensional for becoes ƒ σ + + F( D a ) ( D a ) ( D a ) W + τσ = gαaθ µ eh ( D a ) DK ν + 4πν (7) ƒ µ eh σ + + F( D a ) ( D a ) Z = DX (8) + τσ 4πν ( ) ( H + τσ ) ( + τσ) β G D a H Pσ Θ = W κ G (9) H H D a P K = DW DX η + 4πNe η ( σ ) H H D a P X DZ D D a K η 4πNe η ( σ ) = ( ) (3) (3) Where we have non iensionalize various paraeters as follows a = k, σ = p D ν =, k z n τν No, τ =, ƒ= v µ F p ν C G p β g =, + h = H, =, ν = η = is the iensionless copressibility. After eliinating Θ, Z, X an K fro equation (7)-(3) we obtain ƒ σ + + F( D a ) ( D a ) ( D a ) W ( + τσ ) G ( H + τσ ) ƒ Ra + σ + G ( + τσ )( D a H Pσ ) ( + τσ ) F( D a ) ( D a )}( D a P σ ) + QD, IJMA. All Rights Reserve 76

7 SUSPENDED PARTICLES/ IJMA- 3(4), April-, Page: 7-7 Where µ eh 4πνη 4 ƒ σ + + F D a D a ( { D a P + τσ ) ( ) ( ) ( σ) } ( σ ) + MD D a + D D a P Q QD D a W = Q = is the Chanrasekhar nuber R gαβ ν k = is the theral Rayleigh nuber, H M = is the non-iensional nuber accounting for Hall currents. 4πNe η Since both the bounaries are aintaine at constant teperature, the perturbation in the teperature are zero at the bounaries. The case of two free bounaries is little artificial but it enables us to fin analytical solutions an to ake soe qualitative conclusions. The appropriate bounary conitions with respect to which equations (7) to (3) ust be solve are W =, DZ =, Θ = at z = an. K = an DX =, on perfectly conucting bounaries (33) an h x, h y, h z are continuous. Since the coponents of agnetic fiel are continuous an the tangential coponents are zero outsie the flui, we have DK =, on the bounaries (34) The constitutive equations for the couple stress flui are τ ij = (µ µ ) e ij, (3) e ij v v i j = + xj xi The conitions on a free surface are u w x v w + = y τ xz = (µ µ ) + =, τ yz = (µ µ ) (35) Fro the equation of continuity () ifferentiate with respect to z, we conclue that w µ µ + + = x y (36) which iplies that 4 w w =, =, at z = an 4 (37) Using equation (6), the bounary conitions (37) in non iensional for transfor to D W = D 4 W = at z = an (38), IJMA. All Rights Reserve 77

8 SUSPENDED PARTICLES/ IJMA- 3(4), April-, Page: 7-7 Using bounary conitions (33), (34) an (38), we can see that all the even orer erivatives of W ust vanish for z = an. Therefore the proper solution of W characterizing the lowest oe is W = W sin π z (39) where W is a constant. substituting the proper solution W = W sinπz in equation (3) an letting Q a σ π π π Q =, x=, iσ =, F = π F R R = 4 π We obtain the ispersion relation G + x R = ( + x + ih σp) G x ƒ iσ + + F ( + x) + ( + x) + Q + iστπ ƒ iσ + F + ( + x) + ( + x) ( + x+ iσp) + Q + iστπ ƒ iσ + + F ( + x) + ( + x) ( + x+ iσp) + M( + x) + iστπ ( + iστπ ) + Q( + x+ iσ P) ( H + iστπ ) { } Equation (4) is the require ispersion relation incluing the effects of Hall currents an copressibility on the theral instability of couple stress flui in the presence of suspene particles. 5. STATIONARY CONVECTION: Let us consier the case when instability sets in the for of stationary convection. For stationary convection σ = an the ispersion relation (4) reuces to G R = G 3 {( + x) F + ( + x) } 3 Q{ ( + x) F+ ( + x) + Q} {( ) ( )}{ } + x F+ + x + x+ M + Q which expresses the oifie Rayleigh nuber R as a function of iensionless wave nuber x an the paraeters Q, M, F, G an H. Let the non-iensional nuber G accounting for copressibility effect is kept as fixe, then we get G Rc = R, (4) c G Where R c an R c enote, respectively, the critical Rayleigh nubes in the presence an absence of copressibility. Thus the effect of copressibility is to postpone the onset of theral instability. The cases G < an G = correspon to negative an infinite value of Rayleight nuber which are not relevant in the present stuy. Hence, copressibility has a stabilizing effect on theral convection. (4) (4), IJMA. All Rights Reserve 78

9 SUSPENDED PARTICLES/ IJMA- 3(4), April-, Page: 7-7 To stuy the effects of agnetic fiel, Hall currents, couple-stress an suspene particles, we exaine the natures of R R R R,, an Q M F H analytically. Fro equation (4) we have R G + x = Q G xh ( + x) F + ( + x)( + x+ M) { ( + x) F + ( + x) + Q} + Q { ( x) F ( x) }( x M) Q (43) Fro equation [43] it follows that agnetic fiel has a stabilizing effect on theral convection of a copressible couple stress flui. R G ( + x) = {( + xf ) + } M G xh 3 Q{( + x) F+ ( + x) } + Q [{( + x) F+ ( + x)}( + x+ M) + Q] (44) Fro equation [44] we see Hall currents have estabilizing effects on theral convection. 3 R G ( + x) MQ = ( + x) F G xh [{( + x) F+ ( + x)}( + x+ M) + Q] (45) Fro equation [45] shows that couple stress has both stabilizing an estabilizing effect if ( + x) > or < MQ [{( + x) F + ( + x)}( + x+ M) + Q ] R G x Q = {( + x) F+ ( + x) + Q} {( ) H G xh + x F + ( + x) } + [{( + x ) F+ ( + x )}( + x+ M ) + Q ] (46) Equation [46] follows that suspene particles have estabilizing effect on copressible couple-stress flui. Stability of the syste an oscillatory oes To eterine the possibility of oscillatory oes we ultiply equation (7) by W *, the coplex conjugate of W an using equation (8) (3) together with the bounary conition (33), (34) an (38) we obtain. ƒ µ eη gακa G σ + I+ FI+ I3 + ( I6 + Pσ * I7) + τσ 4πν νβ G + τσ * µη e ( I4 + H Pσ * I5) + ( I8 + PσI9) + H + τσ * 4πν σ ƒ * τσ * I FI I + + = +, IJMA. All Rights Reserve 79 (47)

10 SUSPENDED PARTICLES/ IJMA- 3(4), April-, Page: 7-7 Where I = DW + a W z ( ) I = DW + a D W + a DW + a W 3 I = D W + a DW + a W z 4 I = DΘ + a Θ z I 5 6 = Θ z 4 I = D K + a DK + a K 7 I = DK + a K z 8 I = DX + a X z I9 = X z I = Z z 4 I = D Z + a DZ + a Z z I = DZ + a Z z Where integrals I, I.. I, I are all positive efinite. Putting σ = iσ i an equating iaginary part of equation (47), we get f µ eη gακa G i + I PI 7+ + τσ i 4πν νβ G σ τ ( H ) H + τσ i µη I + H PI + PI H + τσ i H + τσ i 4πν f + I = τσ + i (48) Fro equation (48) σ i ay be zero or non zero, eaning that the oes ay be non oscillatory or oscillatory. The oscillatory oes are introuce ue to the presence of agnetic fiel (hence Hall current) an suspene particles, which were non existence in their absence. REFERENCE: [] Chanrasekhar S., Hyroynaic an Hyroagnetic Stability. Dover Publication. New York. 98. [] Goel A.K., Agrawal S. C. an Agrawal G. S., Inian J. pure appl. Math., 3 (999), 99. [3] Gupta, A.S., Hall Effects on Theral Instability, Rev. Rou. Math. Pures AppI., (967), pp [4] Ratho V.P. an Thippeswai G., Math. Eu., 33 (999), [5] Shara R. C., Theral instability on copressible fluis in the presence of rotation an agnetic fiel. J. Math. Anal. Appl., USA, 6 (977), 7 35., IJMA. All Rights Reserve 7

11 SUSPENDED PARTICLES/ IJMA- 3(4), April-, Page: 7-7 [6] Scanlon J.W. an Segel L.A., Soe effects of suspene particles on the onset of Benar Convection. Phys. Fluis. 6 (973), [7] Stokes V. K. Phys. Fluis, 9 (966), [8] Shara R.C. an Shara S., Inian J. Phys., 75B (), [9] Shara R.C. an Shara S., Int. J. Appl Mech. Engng., 6 (), [] Shara R.C. an Thakur K.D., Czech. J. Phys., 5 () [] Sheran, A., Sutton, G. W., Magnetohyroynaics, Northwestern University Press, Evanston, IL, 96 [] Shara, K.C., Gupta, D., Hall Effects on Therosolutal Instability of a Rotating Plasa, IL Nuovo Ceento, D () (99), pp [3] Spiegal, E.A, Veronis, G., On the Boussinesq Approxiation for a Copressible Flui, Astrophys. J., 3 (96), pp. 44. [4] Shara, R.C., Theral Instability in Copressible Fluis in the Presence of Rotation an Magnetic Fiel, J. Math. Anal. Appl., 6 (977), pp [5] Shara, K.C., Gupta, U., Finite Laror Raius Effect on Theral Instability of a Copressible Rotating Plasa, Astrophysics an Space Science, 67 (99), pp [6] Shara, R.C., Sunil, Theral Instability of Copressible Finite-Laror-Raius Hall Plasa in Porous Meiu, Journal of Plasa Physics, 55() (996), pp [7] Walicki, E. an Walicka A., Int. J. Appl. Mech. Engng., 4 (999), ************************, IJMA. All Rights Reserve 7