Effects of MHD Laminar Flow Between a Fixed Impermeable Disk and a Poros otating Disk Hemant Poonia * * Asstt. Prof., Deptt. of Math, Stat & Physics, CCSHAU, Hisar-54.. C. Chadhary etd. Professor, Deptt. of Maths, Uni. of ajasthan, Jaipr-3. Abstract: e formlate a mathematical model that goerns operations of many engineering systems particlarly the Ceiling fan to explain the flid flow between the fixed impermeable and the poros rotating disks in the presence of a transerse magnetic field. The model is based on the continity and the Naier-stokes eqations which are redced into set of copled ordinary differential eqations throgh transformation by similarity ariables. The copled ordinary differential eqations are soled sing pertrbation techniqes. The reslts for the elocity profiles and the flid pressre distribtion are displayed graphically showing the effects of arios parameters. The graphical reslts of the shear are presented and discssed. Keywords: Impermeable disk, Poros rotating disk, Incompressible iscos flid, Shear flows, MHD. MSC: 76D3, 76D5, 76F, 76U5, 765. INTODUCTION The problem of laminar flow between two parallel disks stemming from both practical interests (e. g. ocean circlation motion and trbo machinery applications) and theoretical interests (e. g. exact soltions of the Naierstokes eqations in certain geometric limiting cases) has receied mch attention oer the years. In modern times, the theory of flow throgh conergent- diergent channels has many applications in aerospace, chemical, ciil, enironmental, mechanical and bio-mechanical engineering as well as in nderstanding riers and canals. De to the complexity of a real flid flow, certain assmptions are made to simplify the mathematical conenience. Some of the basic assmptions are, the flid is ideal (i.e. withot iscosity for mathematical conenience). In sitations where the effect of iscosity is small, this assmption often yields reslts of acceptable accracy althogh where iscosity plays a major part (e. g. in bondary layers), the assmption is clearly ntenable. The flow is steady (i.e. the flow parameters do not change with time). The flid is incompressible. e consider the flow between a fixed impermeable disk and a poros rotating disk (Fig.(a)) both being immersed in a large body of flid. Motion of the flid is indced by the rotation of the poros disk. This stdy is interesting in its own right and also based on its applicability. In the seqel, the following notation will be sed: L= distance between the two disks. r = adis of each disk. = Anglar speed of the rotating disk. = Measre of the anglar speed or momentm of the rotating poros disk. = Sction elocity at which flid is withdrawn from the rotating disk (injection if is Negatie).,, w = Velocity components in the directions r, and z respectiely. f, g, p are the similarity ariables sed to redce the Naier-stokes non-linear partial differential eqations into a system of non-linear ordinary differential eqations. For fixed impermeable disk, =, w=, = while for poros rotating disk, =, w=, =rthe disks are separated by a distance L which is small compared to their radii. The direction of flid flow as shown by the arrows in the figre (a) is heading towards the poros disk which is rotating with a constant anglar speed. The sction elocity is assmed to be constant and eqal to. The rotation speed is gien by, where is the anglar speed of the rotating disk and the parameter is a reglator which controls rotation of the disk ( ). If =, then there is no rotation bt for >, rotating occrs. The qestion of existence and niqeness of soltions in the similarity formlation has been raised [6]. Moreoer, a qestion on the behaior of the flow between the two disks and in particlar near the stationary disk as the speed of the Fig.(a) Physical model representing laminar flow between parallel disks. rotating disk approaches infinity has not been satisfactorily addressed in this type of formlation. The simplified eqations that goern the flow oer an infinite rotating disk, is stdied by Kiken [] and Kelson et al. []. Theoretical work on this class of flows has been 736
ndertaken mainly in the framework of similarity soltions becase the assmption of self-similarity to redce Naierstokes eqations from partial to ordinary differential eqations greatly simplifies the analysis. Start [3] showed that the effect of sction is to thin the bondary layer by decreasing the radial and azimthal components of the elocity while at the same time increasing the axial flow towards the disk at infinity. Trkyilmazogl [4] extended the classical on-karman problem of flow oer a rotating disk to accont for compressibility effects with inslated and isothermal wall conditions. He sed an exponentially decaying series method to find the soltion of the steady laminar flow of an incompressible, iscos, electrically-condcting flid oer a rotating disk in the presence of a niform transerse magnetic field. The stdy of the motion of a iscos incompressible rotating flid is of considerable interest in recent years de to its wide applications in cosmical and geophysical flid dynamics. The research papers dealing with rotating flid hae appeared, for example, Vidyanidhi and Nigam [6], Pri [], Jana and Dtta [8]. In a recent paper, Mazmder [5] and in a note, Ganapathy [6] gae an approximate soltion of the oscillatory coette flow between two infinite parallel plates in a rotating system nder bondary layer approximations. Pearson [9], Lance and ogers [3] obtained the nmerical soltion of this problem. Gar [7] discssed the same problem by considering the effect of porosity. Khare [] stdied the problem of axisymmetric steady flow of a iscos incompressible flid between two co-axial circlar disks, one rotating and the other stationary with niform sction at stationary disk for electrically condcting iscos flid in the presence of a transerse magnetic field. Prohit and Patidar [] stdied the steady flow and heat transfer of a iscos incompressible flid between two infinite rotating disks for small eynolds nmber, where rate of sction of one disk is different from the rate of injection at the other disk. ecently Das and Aziz [4] extended the problem inestigated by Prohit and Patider [] inclding transerse magnetic field where rate of sction in one disk is different to that the rate of injection at other. Sibanda and Makinde [] inestigated the heat transfer characteristics of steady MHD flow in a iscos electrically condcting incompressible flid with Hall crrent past a rotating disk with ohmic heating and iscos dissipation. They fond that the magnetic field retards the flid motion de to the opposing Lorentz force generated by the magnetic field. Trkyilmazogl [4] conclded the exact soltions for the incompressible iscos magnetohydrodynamic flid of a rotating disk flow. The effects of an axial magnetic field applied to the flid with Hall effects are stdied by Attia and Abol- Hassan []. Maleqe and Sattar [4] hae considered the effects of ariable flid properties on laminar bondary layers, namely the density, the iscosity and the thermal condctiity to flow de to a poros rotating disk. The effect of temperatre dependent iscosity on the flow and heat transfer along a niformly heated implsiely rotating disk in a poros medim is discssed by Attia []. Osalsi and Sibanda [7] stdied the effects of ariable properties with magnetic effect. Barik et al [3] considered hall effects on nsteady MHD flow between two rotating disks with non-coincident parallel axes. Osalsi [8] analyzed the combined effects of slip and thermal radiation to the MHD flow and thermal fields oer a rotating single disk. The nsteady Coette flow between two infinite horizontal parallel plates in a rotating system nder bondary layer approximation is stdied by Das et al. [5]. Kaenke et al. [9] considered the modeling laminar flow between a fixed impermeable disk and a poros rotating disk both being immersed in a large body of the flid. Model and Analysis: Fig.(a) depicts a system of poros rotating disk and a stationary disk both being immersed in a large flid body. Flid motion is set p by both rotation of the poros disk and sction (or injection) of the flid itself. e se the cylindrical polar coordinates (r,, z) and the corresponding elocity components by (,, w). Howeer, the angle will not appear in or analysis becase of rotational symmetry. The plane z = L rotates abot the z-axis with constant anglar elocity and the sction elocity is gien by. In order to neglect the end effects, we assme that the gap L is ery small compared to the radii of the disks that is, L<< r ( <L<< r). The eqations goerning the motion of an incompressible iscos flid arise from conseration of mass principle and the momentm principle. From the conseration of mass principle, we hae r w r r z () From the momentm principle and ignoring graity we hae the Naier-stokes eqations, one for each co-ordinate direction P B w r z r r r r z r () B r z r r r z r (3) w w w w w w P w r z r r r z z (4) The small gap L between the two disks allows s to neglect the behaior of the flow arond the edges. Therefore the bonding conditions to be specified are those applicable to elocity components at both disk srfaces and not the edges. 737
For the elocity, it is assmed that the no slip condition applies at the srface of the disks. The bondary conditions are,,,,, r, r, w r, at z r L r L r w r L at z L (5) Since the flid is incompressible, it is possible to define the stream fnction from the goerning continity eqation in two dimensions. This enables s to sole the continity eqation () in the familiar way by setting,, w r z r r (6) and by defining a dimensionless normal distance from the disk. z z L (7) e assme a similarity transformation of the form, r n r f (8) Eqations (6) and (8) yield the expression for the radial and tangential elocities rf, w f (9) For the axial elocity and pressre ariables, we assme that, r g, p r A P () The corresponding bondary conditions for the fnctions f and g are f, f, g f, f, g () Sbstitting (9) and () into eqations () to (4), we obtain f f f g A f Mf here, the primes denote differentiation with respect to he parameter is an arbitrary constant, is the eynolds nmber and A M B is magnetic parameter, is the electrical condctiity, B is the magnetic indction and is the density. Differentiating eqation () with respect to f f g g f Mf (5) Eqation (3) can be written as f g f g g Mg (6) Or main focs is on soling the two copled nonlinear ordinary differential eqations (5) and (6), sbject to the bondary conditions f, f, g f, f, g (7) Thogh the transformation has proided a set of ordinary differential eqations, a closed form soltion is still not possible. Conseqently, we apply the pertrbation techniqe. Treating as a pertrbation parameter, sbstitting the expansions into f f f f... g g g g... (8) eqations (5) and (6) transforms the intractable original problem into a seqence of simple ones. By collecting terms of the same order, we hae f (9) g () f f g g f Mf () f g f g g Mg (3) P f f f L (4) 4 () f g f g g Mg () Sbject to the bondary conditions 738
f f g, f, f, g f f g, f f g (3) The soltion of (9) to () nder the gien bondary conditions yields the following 3 3 f (4) g (5) f (6) g 6 5 3M 7 6 3 M B B 7 6 5 4 3 M 3 M (7) 5 4 6 6 3 M 6 3 3, 7 5 8 M 6 3M 7 3 here, B M B Flid pressre distribtion: Integrating eqation (4) from to n and applying the bondary conditions, we obtain,, P r P r f f f f L (8) Pr P r,, P f f (9) Shear : The action of elocity in the flid adjacent to the disks tends to set a tangential shear which opposes the rotation of the disk. The tangential es at the rotating poros disk are gien by r z g z (3) z (3) r M r zr f z (3) r 3 M 6 3M zr 3 B B 5 4 6 (33) eslt and Discssion: Insight into the physical occrrences within the flowing flid can be obtained by a stdy of elocity profiles. The distribtions of the normal elocity (w), the radial elocity () and azimthal elocity () are plotted as a fnction of eta (). Also, the pressre and shear ariation are represented graphically. In figre, and 3, the radial elocity profile is plotted against for different ales of sction elocity () (or anglar speed of the poros rotating disk ()), the radis of disk (r) and eynolds nmber () respectiely. In this case, rotation of poros disk is constant, it is obsered that the radial elocity profile remains negatie throghot the region between the fixed impermeable disk and poros rotating disk and ale becomes zero at the both disks. It decreases exponentially with increasing from the fixed impermeable disk, reaches its minimm ale and increases towards the poros rotating disk. It is also conclded that the radial elocity profile decreases with increasing the sction elocity () and the radis of the disk (r) in figre and respectiely, where as, in figre 3, an increasing in eynolds nmber cases increase in the radial elocity. -.5..4.6.8 -. -.5 -. -.5 -.3 =., Ω=.4 =., Ω=.6 =.3, Ω=3.6 Fig.. adial elocity for arios ales of sction elocity () & anglar speed of disk () -...4.6.8 -. -.3 -.4 -.5 -.6 -.7 r=. -.8 r=. -.9 -. r=.3 Fig.. adial elocity for arios ales of radis of disk (r) 739
-.5..4.6.8 -. -.5 -. -.5 -.3 -.35 -.4 =.4, Ω=.5 =.5, Ω=.4 =.6, Ω=.33 Fig.3. adial elocity for arios ales of eynolds nmber () & anglar speed of disk ().4.35.3.5..5..5 =., Ω=.4 =., Ω=.6 =.3, Ω=3.6 -.5..4.6.8 Fig.6. Azimthal elocity for arios ales of sction elocity () & anglar speed of disk () -.5..4.6.8 -. M= M= -.5 M= -. -.5 -.3 -.35 Fig.4. adial elocity for arios ales of magnetic parameter (M) Figre 4 and 5, represent the radial elocity profile for different ales of the magnetic parameter (M) and the rotation of the poros disk () respectiely. It is obsered that in both figres 4 and 5, the radial elocity profile decreases in the region.5 with increasing M and and after that it increases towards the poros rotating disk. In figre 6, 7, and 8, the azimthal elocity profile () is plotted against for different ales of the sction parameter (), the radis of the disk (r), and the rotation of the poros disk () respectiely. It is obsered that the azimthal elocity profile -5..4.6.8 - -5 - -5-3 -35-4 Fig.5. adial elocity for arios ales of rotation of the disk () increases continosly with increasing towards the poros rotating disk. In fig. 6 and 7, it decreases with increasing sction elocity and the radis of the disk respectiely in the region.4 and after that it increases. Similarly in figre 8, the azimthal elocity profile decreases with increasing the rotation of the poros disk in the region.8 and then it increases..4...8.6.4. r=. r=. r=.3 -...4.6.8 Fig.7. Azimthal elocity for arios ales of radis of the disk (r) 6 5 4 3 -..4.6.8-5 Fig.8. Azimthal elocity for arios ales of rotation of the disk () Figre 9, represents the azimthal elocity profile against for different ales of eynolds nmber. It is obsered that an increase in eynolds nmber cases a decrease in the azimthal elocity profile. In figre, the azimthal elocity is plotted against for different ales of the magnetic parameter (M). The azimthal elocity increases continosly with increasing and reaches its maximm ale at the poros rotating disk (. The ale of remains positie for 74
M= and for M>, the ale of remains negatie at the fixed impermeable disk and tends to positie ale towards the poros rotating disk. It is also conclded that an increase in M reslts a decrease in. Figre, shows the normal elocity profile (w) against for different ales of sction elocity (or anglar speed of the poros rotating disk ()). It is seen that the normal elocity profile increases with increasing and reaches maximm ale at the poros rotating disk. It is also obsered that an increase in the sction elocity reslts increase in the normal elocity profile..6.5.4.3.. =.4, Ω=.5 =.5, Ω=.4 =.6, Ω=.33 -...4.6.8 Fig.9. Azimthal elocity for arios ales of eynolds nmber () & anglar speed of disk ().6.4. -...4.6.8 -.4 -.6 -.8 M= M= M= Fig.. Azimthal elocity for arios ales of magnetic parameter (M) w.35.3.5..5..5 =., Ω=.4 =., Ω=.6 =.3, Ω=3.6..4.6.8 Fig.. Normal elocity for arios ales of sction elocity () & anglar speed of disk () w...8.6.4. M= M= M=..4.6.8 Fig.. Normal elocity for arios ales of magnetic parameter (M) In fig. and 3, the normal elocity profile is plotted against for different ales of the magnetic parameter and the rotation of the poros disk. In the both figres, the normal elocity profile increases continosly with increasing. The normal elocity profile increases with increasing the magnetic parameter and the rotation of the poros disk respectiely. w 6 5 4 3..4.6.8 Fig.3. Normal elocity for arios ales of rotation of the disk () P *.3.5..5..5 =., Ω=.4 =., Ω=.6 =.3, Ω=3.6 -.5..4.6.8 Fig.4. Flid pressre distribtion for arios ales of sction elocity () & anglar speed of disk () Figre 4 and 5, show that the flid pressre distribtion (P * ) is plotted against for different ales of the sction elocity and the eynolds nmber respectiely. In both figres, the flid pressre distribtion increases rapidly from the fixed impermeable disk with increasing and reaches its maximm ale and falls down towards the poros rotating disk. In figre 4, the flid pressre distribtion increases in the region.96 with increasing the sction elocity and after that the flid pressre distribtion decreases towards the poros rotating disk, whereas, in figre 5, the flid pressre distribtion 74
decreases throgh ot the region with increasing the eynolds nmber. P *.45.35.5.5.5 =.4, Ω=.5 =.5, Ω=.4 =.6, Ω=.33.3.5..5..5 all shear adial shear..4.6.8 -.5..4.6.8 Fig.5. Flid pressre dist. for arios ales of eynolds nmber () & anglar speed of disk () P *.35.3.5..5..5 M= M= M= -.5..4.6.8 Fig.6. Flid pressre dist. for arios ales of magnetic parameter (M) In figre 6 and 7, the flid pressre distribtion is plotted against for different ales of the magnetic parameter and the rotation of the poros disk respectiely. In both figres, it increases sharply with increasing, reaches its maximm ale and falls down rapidly towards the poros rotating disk. It is also obsered that the flid pressre distribtion increases from the fixed impermeable disk with increasing the magnetic parameter and the rotation of the poros disk respectiely in both figres, bt after >.48, the flid pressre distribtion decreases towards the poros rotating disk in both figres. P * 4 3 - -..4.6.8 Fig.7. Flid pressre dist. for arios ales of rotation of the disk () Fig.8. Effects of rotation of the disk () on all and adial shear Figre 8 and 9, show the effects of and M on the wall shear and the radial shear respectiely. In both figres, the wall shear increases steadily and the radial shear decreases slightly as increasing and M. Figre and, show the effects of and on the wall shear and the radial shear respectiely. In figre, the wall shear decreases slightly and the radial shear decreases sharply at =. and after this, it decreases slightly as increasing, whereas, in figre, both the wall shear and the radial shear increase exponentially as increasing..3.5..5..5 all shear adial shear 3 4 5 6 M Fig.9. Effects of magnetic parameter (M) on all and adial shear 7 6 5 4 3 all shear adial shear...3.4.5 Fig.. Effects of eynolds nmber () on all and adial shear 74
35 3 5 5 5 all shear adial shear...3.4.5 Fig.. Effects of sction elocity () on all and adial shear.4..8.6.4. all shear adial shear...3.4.5 r Fig.. Effects of radis of the disk (r) on all and adial shear Figre, shows the effects of r on the wall shear and the radial shear. The wall shear increases slowly and the radial shear increases sharply as increasing r. CONCLUSIONS: The presence of flid in flow between the two disks garantees the absence of cooling and in general, cooling is felt otside the region and in front of the poros rotating disk de to sction and flid ot flow. This effect can easily be obsered from a ceiling fan operation and other fan systems sed in or homes. This stdy presents a mathematical model in which the electrically condcting flid is flowing between the fixed impermeable disk and the poros rotating disk in the presence of transerse magnetic field. The main findings can be smmarized as:. The radial elocity profile decreases with increasing and r, whereas, it increases with increasing.. In positie sense, the azimthal elocity profile increases and in negatie sense, decreases with increasing, r and whereas, it decreases as increases in and M. 3. Increasing, M and leads to an increase in the normal elocity profile. 4. An increase in and cases an increase in the flid pressre distribtion for and a decrease for. 5. The wall shear increases as the increasing, M, and r, whereas, it decreases as the increasing. 6. Increasing, M and decreases the radial shear and increasing and r increases rapidly the radial shear. EFEENCES [] H. A. Attia, A. L. Abl-Hassan, On hydromagnetic flow de to a rotating disk, Appl. Math. Modelling, ol. 8, pp. 7-4, 4. [] H. A. Attia, Unsteady flow and heat transfer of iscos incompressible flid with temperatre-dependent iscosity de to a rotating disk in a poros medim, J. Phys. A: Math. Gen., ol. 39, pp. 979-99, 6. [3]. N. Barik, G. C. Dash and P. K. ath, Hall effects on nsteady MHD flow between two rotating disks with non-coincident parallel axes, Natl. Acad. Sci., India, Sect. A Phys. Sci., ol. 83, pp. -7, 3. [4] U. N. Das, A. Aziz, Hydromagnetic steady flow and heat transfer of an electrically condcting iscos flid between two rotating poros disks of different transpiration, Bll. Cal. Math. Soc., ol. 6, pp. 3-36, 998. [5] B. K. Das, M. Gria,. N. Jana, Unsteady coette flow in a rotating system, Meccanica, ol. 43, pp. 57-5, 8. [6]. Ganapathy, A note on oscillatory coette flow in a rotating system, Trans. ASME J. Appl. Mech., ol. 6, pp. 8-9, 994. [7] Y. N. Gar, Viscos incompressible flow between two infinite poros rotating disks, Indian J. Pre Appl. Math., ol. 4, p. 89, 97. [8]. N. Jana, N. Dtta, Coette flow and heat transfer in a rotating system, Acta Mech., ol. 6, pp. 3-36, 997. [9] D. P. Kaenke, E. Massawe, O. D. Makinde, Modeling laminar flow between a fixed impermeable disk and a poros rotating disk, AJMCS, ol., pp. 57-6, 9. [] N. Kelson, A. Desseasx, Note on poros rotating flow, ANZIAM J., ol. 4, pp. 837-855,. [] S. Khare, Steady flow between a rotating and a poros stationary disk in presence of transerse magnetic field, Indian J. Pre Appl. Math., ol. 8, p. 88, 977. [] H. K. Kiken, The effect of normal blowing on the flow near a rotating disk of infinite extent, J. Flid Mech., ol. 47, pp. 789-798, 97. [3] G. N. Lance, M. H. ogers, The axially symmetric flow of a iscos flid between two infinite rotating disks, Proc. Soc. London A, ol. 66, p. 9, 96. [4] A. K. Maleqe, A. M. Sattar, Steady laminar conectie flow with ariable properties de to a poros rotating disk, J. Heat Transfer, ol. 7, pp. 46-49, 5. [5] B. S. Mazmder, An exact soltion of oscillatory coette flow in a rotating system, Trans. ASME J. Appl. Mech., ol. 56, pp. 4-7, 99. [6] G. L. Mellor, P. J. Chappel, V. K. Stokes, On the flow between a rotating and a stationary disk, J. Flid Mech., ol. 3, pp. 95-, 968. [7] A. K. Osalsi, P. Sibanda, On ariable laminar conectie flow properties de to a poros rotating in a magnetic field, omanian J. of Phys., ol. 5, pp. 933-944, 6. [8] E. Osalsi, Effects of thermal radiation on MHD and slip flow oer a poros rotating disk with ariable properties, om. J. Phys., ol. 5, pp. 7-9, 7. [9] C. E. Pearson, Nmerical soltions for the time dependent iscos flow between two rotating co-axial disks, J. Flid Mech., ol., p. 63, 965. [] P. Pri, Flctating flow of a iscos flid on a poros plate in a rotating medim, Acta Mech., ol., pp. 53-58, 975. [] G. N. Prohit,. P. Patidar, Steady and flow and heat transfer between two rotating disks of different transpiration, Indian J. Math., ol. 34, p. 89, 99. [] P. Sibanda, O.D. Makinde, On steady MHD flow and heat transfer past a rotating disk in a poros medim with ohmic heating and iscos dissipation., Int. J. Nmer. Methods Heat Flid Flow, ol., pp. 69 85,. [3] J. T. Start, On the effect of niform sction on the steady flow de to a rotating disk, Q. J. Mech. Appl. Math., ol. 7, pp. 446-457, 954. [4] M. Trkyilmazogl, The MHD bondary layer flow de to a rogh rotating disk, ZAMM Z. Angew. Math. Mech., ol. 9, pp. 7 8,. [5] M. Trkyilmazogl, Exact soltions for the incompressible iscos magnetohydrodynamic flid of a rotating disk flow, Meccanica, ol. 46, pp. 85-9,. [6] V. Vidyanidhi, S. D. NIgam, Secondary flow in a rotating channel, J. Math. Phys. Sci., ol., pp.35-48, 967. 743