Oscillatory Hydromagnetic Couette Flow in a Rotating System with Induced Magnetic Field *

Transcription

1 CHAPTER-4 Oscillator Hdroagnetic Couette Flow in a Rotating Sste with Induced Magnetic Field * 4. Introduction Lainar flow within a channel or duct in the absence of agnetic field is a phenoenon which is eperienced in practice ver infrequentl. It has becoe a convention in engineering that lainar flow within the channel or duct is a kind of abstraction which can occur onl, ver rare, special cases. However, the situation is totall different for the fluid flow of an electricall conducting fluid in the presence of a agnetic field. It is found that agnetic field iproves significantl the hdrodnaic stabilit of the flow or else suppresses an turbulence alread present. This eans that the eistence of lainar flow in the flow paths of MHD devices is highl likel. Due to this reason several theoretical and eperiental investigations of lainar hdroagnetic flows of an electricall conducting fluid in the presence of electroagnetic fields are carried out b an researchers under different conditions and configurations to stud the nature of various natural phenoena and to find its application in science and engineering. There are a nuber of natural phenoena and engineering probles susceptible to agnetohdrodnaic analsis. It is well known that uch of the universe is filled with widel spaced charged particles and * Journal of Magnetohdrodnaics, Plasa and Space Research, 8 (No. ) (To appear)

2 84 pereated b agnetic fields. Due to this reason, investigation of hdroagnetic fluid flow probles in astrophsics is of uch significance. Geophsicists investigated an hdroagnetic phenoena which occur due to interaction of electricall conducting fluids and agnetic fields that are present in and around heavenl bodies. Engineers appl hdroagnetic principles in the design of heat echangers using liquid etal coolants, MHD pups, MHD accelerators and MHD flow-eters, in solving space vehicle propulsion, control and re-entr probles, in creating novel power generating sstes i.e. MHD energ generators and in developing confineent schees for controlled nuclear fusion. In general, governing equations of hdroagnetic fluid flow probles are inherentl non-linear. Siplified odels are, therefore, studied in literature with a view to discuss different aspects of fluid flow behaviors. Of these odels, the one corresponding to hdroagnetic Couette flow is known to lead to the equations for which analtical solution can be obtained in principle. The stud of unstead hdroagnetic Couette flow of a viscous, incopressible and electricall conducting fluid is of uch significance fro practical point of view because fluid transient a be epected at the start-up tie of an MHD devices, nael, MHD energ generators, MHD pups, MHD accelerators, MHD flow-eters, nuclear reactors using liquid etal coolants etc. Keeping in view this fact, unstead hdroagnetic Couette flow of a viscous, incopressible and electricall conducting fluid in the presence of a unifor transverse agnetic field is investigated b a nuber of researchers considering different aspects of the proble. Mention a be ade of research studies of Katagiri (96), Muhuri (963), Singh and Kuar (983), Vajravelu (988) and Seth et al. (0c). In recent ears there has been considerable interest in the probles of hdroagnetic flow of rotating fluid due to its possible applications to geophsical and astrophsical probles and in fluid engineering viz. nuclear engineering control (Narasihan, 963), plasa aerodnaics (Takenouchi, 985), echanical engineering anufacturing processes (Barin and Uspenskii, 986), astrophsical fluid dnaics (Takhar and Ra, 99) and MHD energ sstes (Hardianto et al., 008). Keeping in view this fact Seth et al. (98, 988, 00a,b, 0b, 0a,b,c), Chandran et al. (993), Singh et al. (994), Singh (000), Haat et al. (004a,b), Das et al. (009a) and

3 85 Seth and Singh (0) investigated unstead hdroagnetic Couette flow of a viscous, incopressible and electricall conducting fluid in a rotating sste in the presence of unifor transverse agnetic field considering different aspects of the proble. In all these investigations induced agnetic field produced b fluid otion is neglected in coparison to the applied one. This assuption is valid because agnetic Renolds nuber is ver sall for liquid etals and partiall ionized fluids (Craer and Pai, 973). However, there are several astrophsical, geophsical and engineering probles in which induced agnetic field plas an iportant role in deterining flow-feature of the proble. Taking into consideration this fact Ghosh (993), Ansari et al. (0) and Seth et al. (0d) investigated unstead Hartann flow of a viscous, incopressible and electricall conducting fluid in a rotating channel taking induced agnetic field into account when fluid flow within the channel is induced due to an oscillating pressure gradient. Objective of the present chapter is to stud unstead hdroagnetic Couette flow of a viscous, incopressible and electricall conducting fluid in a rotating channel with nonconducting walls in the presence of a unifor transverse agnetic field taking induced agnetic field into account. Fluid flow within the channel is induced due to non-torsional oscillations of upper plate of the channel in its own plane. Eact solution of governing equations is obtained in closed for. Epressions for shear stress at both the plates due to priar and secondar flows and ass flow rate in the priar and secondar flow directions are also derived. Matheatical forulation of the proble, in non-diensional for, contains four pertinent flow paraeters, nael, agnetic interaction paraeter, Ekan nuber E, agnetic Prandtl nuber P and frequenc paraeter. Solution valid in the liit of vanishing agnetic Prandtl nuber P is also obtained and asptotic behavior of the solution for large values of frequenc paraeter is analzed to gain soe phsical insight into the flow pattern. It is found that there eist two odes of oscillations in the flow-field. These two odes correspond to odified hdroagnetic Stokes flow and are confined to thin boundar laers of thicknesses O 3 and 4 O. These boundar laers a be identified as the hdroagnetic Stokes-Ekan boundar laers. The thickness of these boundar laers decreases on increasing agnetic interaction

4 86 paraeter whereas it increases on increasing Ekan nuber. In the absence of agnetic field these two boundar laers a be recognized as Stokes-Ekan boundar laers. It is also noticed that there is no flow of fluid outside hdroagnetic Stokes-Ekan boundar laer region. For large, the thickness of and laers, which can be identified as agnetic diffusion laers, tends to infinit ipling thereb that the agnetic diffusion region etends up to the central line of the channel just as it happens in the liit 0 and P 0. The nuerical values of priar and secondar fluid velocities and priar and secondar induced agnetic fields are displaed graphicall versus channel width variable for various values of, E and whereas those of shear stress due to priar flow, shear stress due to secondar flow, ass flow rate in the priar flow direction and ass flow rate in the secondar flow direction are presented in tabular for for different values of, E and. 4. Forulation of the Proble and its Solution Consider unstead flow of a viscous, incopressible and electricall conducting fluid between two electricall non-conducting plates z 0 and z L in the presence of a unifor transverse agnetic field B 0 which is applied in a direction parallel to z - ais. Fluid and channel rotate in unison with unifor angular velocit about z-ais. Fluid flow within the channel is induced due to non-torsional haronic oscillations of upper plate z L in its own plane in - direction and lower plate z 0 is kept fied. Geoetr of the proble is presented in figure 4.. Since plates of the channel are of infinite etent along and directions all phsical quantities, ecept pressure, depend on z and t onl. Therefore, fluid velocit q and induced agnetic field B a be assued as,,0 and,, 0 q u v B B B B, (4.) which are in agreeent with fundaental equations of Magnetohdrodnaics in a rotating frae of reference.

5 87 z Secondar Flow UPPER PLATE z L B0 Priar Flow LOWER PLATE z 0 Fig. 4. Geoetr of the proble Under the above assuptions equations of otion and agnetic induction for a viscous, incopressible and electricall conducting fluid in a rotating frae of reference reduce to u u B B t z z, (4.) v 0 v e v B B 0 u t z e z, (4.3) p 0, (4.4) z B B u B 0 t z z (4.5) B B v B 0 t z z (4.6)

6 88. where p p B B B0 e u, v,,,, and p are, respectivel, fluid velocit in -direction, fluid velocit in e -direction, kineatic coefficient of viscosit, agnetic pereabilit, densit, agnetic diffusivit and fluid pressure including centrifugal force. Equation (4.4) shows constanc of odified pressure p along the ais of rotation (i.e. along z - ais) and absence of p in equation (4.3) iplies that there is a net cross flow in - direction. It is appropriate to ention here that the fluid flow is induced due to nontorsional haronic oscillations of the upper plate in - direction, therefore, pressure gradient ter p is not taken into account in equation (4.). Boundar conditions for fluid flow proble are given b u v 0 at z 0, (4.7a) it it u e e, v 0 at z L, (4.7b) B B 0 at z 0, (4.7c) B B 0 at z L. (4.7d) where is frequenc of oscillations. Equations (4.), (4.3), (4.5) and (4.6), in non-diensional for, becoe u u b E v E, (4.8) T v v b E u E T, (4.9) b b u P E E T, (4.0)

7 89 b b v P E E T, (4.) where z L, u u L, v v L, T t, b B B L, b B B L. E L, de dh B0 e 0 e 0 and P e are, respectivel, Ekan nuber, agnetic interaction paraeter and agnetic Prandtl nuber. Here de is Ekan depth and d H B0 is Hartann depth. Therefore presents the ratio of Ekan to Hartann depth. This ratio is independent of kineatic coefficient of viscosit and fro its position in the equations (4.8) and (4.9), easures the strength of the electroagnetic bod force relative to Coriolis force. It a be noted that for phsical situations of interest both E and P are saller than unit. Boundar conditions (4.7a) to (4.7d), in non-diensional for, becoe u v 0 at 0, (4.a) it it u e e, v 0 at, (4.b) b b 0 at 0, (4.c) b b 0 at. (4.d) where is frequenc paraeter. Cobining equations (4.8) and (4.0) with equations (4.9) and (4.) respectivel, we obtain F F B E if E T, (4.3) B B F P E E T, (4.4) where F u iv and B b ib.

8 90 Boundar conditions (4.a) to (4.d) are epressed in copact for as F 0, B 0 at 0, (4.5a) it it F e e, B 0 at. (4.5b) Since fluid flow within the channel is induced due to non-torsional haronic oscillations of upper plate in - direction so fluid velocit and induced agnetic field are assued in the following for it, F T F e F e it, (4.6a) it, B T B e B e it. (4.6b) Equations (4.3) and (4.4), with the help of (4.6a) and (4.6b), reduce to d F db E i F E, (4.7) d d d F db E i F E, (4.8) d d d B df E i P B E, (4.9) d d d B df E i P B E. (4.0) d d Making use of (4.6a) and (4.6b) in boundar conditions (4.5a) and (4.5b), we obtain F F 0, B B 0 at 0, (4.a) F F, B B 0 at. (4.b) Solution of equations (4.7) to (4.0), subject to the boundar conditions (4.a) and (4.b), is obtained and the solution for fluid velocit F, T B, T is epressed in the following for and induced agnetic field

9 9 k F, T C cosh cosh C sinh sinh e + it * k A cosh cosh A sinh sinh * e it, (4.) 3/ ie k 3 P B, T C sinh sinh C cosh cosh / ie i C sinh sinh C cosh P 3/ k it ie 3 3 cosh e A sinh sinh P * / 3 k 3 ie i A cosh * cosh A sinh P * k it sinh A cosh * cosh e, (4.3) where, E k,,, E k,, (4.4a) / /,,, k n n q k n n q, (4.4b) n, n i P, q, q i P, (4.4c) k, E i, k, E i * *,,,,, (4.4d) C cosh cosh k sinh sinh k sinh sinh k sinh sinh k sinh sinh k cosh cosh, (4.4e)

10 9 C k sinh sinh k k k, (4.4f) sinh sinh sinh sinh cosh cosh A cosh cosh * * * * k sinh sinh k sinh sinh * * * * * * k sinh sinh k sinh sinh k cosh cosh, (4.4g) and A * * * k sinh sinh * * * * * * k k k. 4.4h sinh sinh sinh sinh cosh cosh 4.. Oscillator Hdroagnetic Couette Flow in a Rotating Sste in the Liit of Vanishing Prandtl Nuber (i.e. P 0 ) Setting P 0 in (4.) to (4.4), we obtain / k i, 0, (4.5a) / k i, 0, (4.5b) n i, q 0, (4.5c) n i, q 0, (4.5d) k E i, 0, (4.5e) k * E * i, 0. (4.5f) / / sinh E k sinh E k it F, T e e / / sinh E k sinh E k it, (4.6)

11 93 It a be noted that the agnetic Renolds nuber is ver sall in the liit of vanishing agnetic Prandtl nuber P. Therefore, induced agnetic field B, T produced b fluid otion is negligible in coparison to the applied one (Craer and Pai, 973). When frequenc paraeter is large and both Ekan nuber E and agnetic interaction paraeter are of sall orders of agnitude i.e. E O and O, boundar laer tpe flow is epected. For boundar laer flow adjacent to the oscillating plate, introducing boundar laer variable, we obtain the epressions for the priar velocit u, T and secondar velocit v, T fro (4.6) as 3 4 u, T e cost 3 e cost 4, (4.7) 3 4 v, T e sin T 3 e sin T 4, (4.8) where / / 3, 3 E E, (4.9a) / / 4, 4 E E. (4.9b) The epressions in (4.7) to (4.9) deonstrate the eistence of two odes of oscillations in the flow-field. These two odes correspond to the odified hdroagnetic Stokes flow and are confined to thin boundar laers of thicknesses O 3 and O 4. These boundar laers a be identified as hdroagnetic Stokes-Ekan boundar laers. The thickness of these boundar laers decreases on increasing agnetic interaction paraeter whereas it increases on increasing Ekan nuber E. In the absence of agnetic field these two boundar laers a be recognized as Stokes-Ekan boundar laers. The eponential ters in (4.7) and (4.8) dap out quickl as increases. When i.e. outside the boundar laer region, equations (4.7) and (4.8) reduce to 4

12 94 u 0, v 0. (4.30) It is evident fro the epressions in (4.30) that there is no flow of fluid outside the hdroagnetic Stokes-Ekan boundar laer region. It is worth to ention here that in the liit of P 0, and becoe zero which iplies that for large the thickness of and laers, which can be identified as agnetic diffusion laers, tends to infinit ipling thereb that the agnetic diffusion region etends up to the central line of the channel just as it happens in the liit 0 and P Shear Stress at the Plates Non-diensional shear stress coponents and at oscillating and stationar plates, due to priar and secondar flow respectivel, are given b k i C sinh sinh C cosh cosh e * k A sinh sinh A cosh * cosh e * k i T k i C 0 e A * e 4.4 Mass Flow Rates it it it, (4.3). (4.3) Non-diensional ass flow rates respectivel are epressed as Q and Q in the priar and secondar flow direction sinh sinh cosh k cosh it Q iq C C e * sinh sinh sinh k sinh it A A e *, (4.33)

13 Results and Discussion In order to analze the effects of agnetic field, rotation and oscillations on the velocit field and induced agnetic field the nuerical values of fluid velocit and induced agnetic field, coputed fro the analtical solution (4.) to (4.4), are displaed graphicall in figures 4. to 4.7 for various values of agnetic interaction paraeter, Ekan nuber E and frequenc paraeter taking T / and P 0.7 (ionized hdrogen). Figure 4. depicts the influence of agnetic field on the priar velocit u and secondar velocit v. It is evident fro figure 4. that both the priar velocit u and secondar velocit v decrease on increasing agnetic interaction paraeter. This iplies that agnetic field tends to retard fluid flow in both the priar and secondar flow directions. Figure 4.3 illustrates the effects of rotation on the priar and secondar velocities. Figure 4.3 reveals that u and v decrease on increasing Ekan nuber E. Since Ekan nuber represents the ratio of viscous force to Coriolis force. Ekan nuber E decreases when angular velocit increases. This iplies that rotation tends to accelerate fluid flow in both the priar and secondar flow directions. Figure 4.4 deonstrates the effects of oscillations on both the priar and secondar fluid velocities. It is evident fro figure 4.4 that u and v increase on increasing frequenc paraeter. This iplies that oscillations tend to accelerate fluid flow in both the priar and secondar flow directions. Figure 4.5 displas the influence of agnetic field on the priar induced agnetic field b and secondar induced agnetic field b. It is revealed fro figure 4.5 that b and b decrease on increasing. This iplies that agnetic field tends to reduce both the priar and secondar induced agnetic fields. Figure 4.6 presents the effects of rotation on both the priar and secondar induced agnetic fields. Figure 4.6 shows that both b and b decrease on increasing E. This iplies that rotation tends to enhance both the priar and secondar induced agnetic fields. Figure 4.7 displas the influence of oscillations on induced agnetic fields. Figure 4.7 reveals that b and b increase on increasing. This iplies that oscillations tend to enhance both the

14 96 priar and secondar induced agnetic fields. It is also noticed fro figures 4.5 to 4.7 that secondar induced agnetic field lower half of the channel. The nuerical values of priar shear stress b oves in different directions in the upper and and secondar shear stress at oscillating and stationar plates, coputed fro analtical epressions (4.3) and (4.3), are displaed in tabular for in tables 4. to 4.4 for various values of, E and while those of priar ass flow rate Q and secondar ass flow rate Q, coputed fro analtical epression (4.33), are presented in tables 4.5 and 4.6 for different values of, E and taking T / and P 0.7. It is evident fro table 4. that the priar shear stress at the oscillating plate i.e. decreases on increasing whereas secondar shear stress at the oscillating plate i.e. increases on increasing. increases, attains a aiu and then decreases on increasing E whereas decreases on increasing E. This iplies that agnetic field tends to reduce priar shear stress at the oscillating plate whereas it has reverse effect on secondar shear stress at oscillating plate. Rotation tends to enhance secondar shear stress at oscillating plate. It is found fro table 4. that priar shear stress at stationar plate i.e. 0 and secondar shear stress at stationar plate i.e. increase on increasing 0 whereas and 0 0 decrease on increasing E. This iplies that agnetic field and rotation tend to enhance both priar and secondar shear stress at stationar plate. It is observed fro tables 4.3 and 4.4 that and increase whereas 0 and 0 decrease on increasing. This iplies that oscillations tend to enhance priar shear stress at the oscillating and stationar plates whereas these have reverse effect on secondar shear stress at the oscillating and stationar plates. It is noticed fro tables 4.5 and 4.6 that priar ass flow rate Q and secondar ass flow rate Q decrease on increasing either or E whereas Q and Q increase on increasing. This iplies that agnetic field tends to

15 97 reduce priar and secondar ass flow rates whereas rotation and oscillations have reverse effects on it. 4.6 Conclusions Present investigation deals with the theoretical stud of oscillator MHD Couette flow in a rotating channel taking induced agnetic field into account. The significant results are suarized below:. For large values of frequenc paraeter, there arise thin boundar laers of thicknesses O 3 and O 4 near oscillating plate of the channel. These boundar laers a be identified as the hdroagnetic Stokes-Ekan boundar laers. There is no flow of fluid outside the hdroagnetic Stokes-Ekan boundar laer region.. Magnetic field tends to retard fluid flow in both the priar and secondar flow directions whereas rotation and oscillations have reverse effect on it. 3. Magnetic field tends to reduce both the priar and secondar induced agnetic fields whereas rotation and oscillations have reverse effect on it. Secondar induced agnetic field oves in different directions in the upper and lower half of the channel. 4. Magnetic field tends to reduce priar shear stress at the oscillating plate whereas it has reverse effect on secondar shear stress at oscillating plate. Rotation tends to enhance secondar shear stress at oscillating plate. Magnetic field and rotation tend to enhance priar and secondar shear stress at stationar plate. Oscillations tend to enhance priar shear stress at oscillating and stationar plates whereas these have reverse effect on secondar shear stress at oscillating and stationar plates. Magnetic field tends to reduce priar and secondar ass flow rates whereas rotation and oscillations have reverse effects on it.

16 u v 0.5 5, 6, 7 u, -v Fig. 4. Priar and secondar velocit profiles when E=0.3 and =3 0.0 u v 0.5 E 0.3, 0.5, 0.7 u, -v Fig. 4.3 Priar and secondar velocit profiles when =6 and =3

17 u v 0.0 u, -v , 4, Fig. 4.4 Priar and secondar velocit profiles when =6 and E= b b , 6, 7 b, -b Fig. 4.5 Priar and secondar induced agnetic field profiles when E=0.3 and =3

18 E 0.3, 0.5, 0.7 b b b, -b Fig. 4.6 Priar and secondar induced agnetic field profiles when =6 and = b b 0.03 b, -b , 4, Fig. 4.7 Priar and secondar induced agnetic field profiles when =6 and E=0.3

19 0 Table 4. Priar and secondar shear stress at the oscillating plate when =3 E Table 4. Priar and secondar shear stress at the stationar plate when =3 0 0 E Table 4.3 Priar and secondar shear stress at the oscillating plate when E= Table 4.4 Priar and secondar shear stress at the stationar plate when E=

20 0 Table 4.5 Priar and secondar ass flow rates when =3 Q Q E Table 4.6 Priar and secondar ass flow rates when E=0.3 Q Q *****