Thermal Convection in a Couple-Stress Fluid in the Presence of Horizontal Magnetic Field with Hall Currents

Transcription

1 Available at Appl. Appl. Math. ISSN: Vol. 8, Issue (June 3), pp Applications and Applied Mathematics: An International Journal (AAM) hermal Convection in a Couple-Stress Fluid in the Presence of Horizontal Magnetic Field with Hall Currents Jaimala and Vikrant Department of Mathematics Charan Singh University (Campus) Meerut (Uttar Pradesh), India jaimalabishnoi@gmail.com ; vikrant_jawla@yahoo.com Vivek Kumar College of Engineering Studies University of Petroleum & Energy Studies Dehradun-87, (Uttarakhand), India vivek_shrawat@yahoo.co.in ; vivek.shrawat@gmail.com Abstract Received: July 3, ; Accepted: February 3, 3 he thermal stability of a couple-stress fluid is considered to include the effects of uniform horizontal magnetic field and Hall currents. he analysis is carried out within the framework of linear stability theory and normal made technique. For the case of stationary convection, magnetic field has stabilizing effect whereas Hall currents are found to have destabilizing effect on the system. he couple stress, however, has dual character in contrast to its stabilizing effect in the absence of Hall currents. Oscillatory modes are not allowed in the system. Graphs in each case have been plotted by giving numerical values to the parameters, depicting the stability characteristics. Keywords: hermal convection, couple-stress fluid, magnetic field, Hall currents MSC No.: 76E6, 76E7, 76E5, 76U5. Introduction he growing importance of the use of non-newtonian fluids in modern technology and industries has led various researchers to attempt diverse flow problems related to several non-newtonian fluids. One such fluid that has attracted the attention of research workers during the last four 6

2 6 Jaimala et al. decades is the couple stress fluid. he theory of couple stress fluids initiated by Stokes (966) is a generalization of the classical theory of viscous fluids, which allows for the presence of couple stresses and body couples in the fluid medium. he concept of couple stresses arises due to the way in which mechanical interactions in the fluid medium are modeled. In this theory the rotational field is defined in terms of velocity field itself and the rotation vector equals half the curl of the velocity vector. he stress tensor here is no longer symmetric. An ecellent introduction to this theory is available in the monograph heories of Fluids with Microstructure - An Introduction written by Stokes (98) himself. One of the applications of couple-stress fluid is its use in the study of the mechanisms of lubrication of synovial joints which has become the object of scientific research. A human joint is a dynamically loaded bearing which has articular cartilage as the bearing and synovial fluid as the lubricant. When a fluid film is generated, squeeze-film action is capable of providing considerable protection to the cartilage surface. he shoulder, hip, knee and ankle joints are the loaded-bearing synovial joints of the human body and these joints have a low friction coefficient and negligible wear. Normal synovial fluid is a viscous, non-newtonian fluid and is generally clear or yellowish. he synovial fluid has been modeled as a couple-stress fluid in human joints by Walicki and Walicka (999). In the recent past instability in either a couple stress fluid layer or couple stress fluid saturated porous layer heated from below has been investigated including the eternal constraints such as magnetic field and/or rotation. Goel et al. (999) have studied the hydromagnetic stability of an unbounded couple-stress binary fluid miture under rotation with vertical temperature and solute concentration gradients. A layer of couple-stress fluid saturating a porous medium heated from below in the presence of rotation has been studied by Sunil et al. () and condition for the onset of convection is obtained. he stability of superposed couple-stress fluid in porous medium in hydromagnetics has been studied by Sunil et al. (a). Sunil et al. (b) have investigated the effect of magnetic field and rotation on a layer of couple-stress fluid heated from below in a porous medium, while Sunil et al. () have considered the effect of suspended particles on the stability of a couple-stress fluid layer heated and soluted from below in a porous medium. Sharma and Sharma () have studied the onset of convection in a couple-stress fluid saturated porous layer in the presence of rotation and magnetic field. Siddheshwar and Pranesh () have made an analytical study of linear and non-linear convection in a couple-stress fluid layer. Malashettty and Basavaraja (5) have studied the effect of thermal and gravity modulation on the onset of Rayleigh-Benard convection in a couple stress fluid layers. Malashetty et al. (6) have studied the Soret effect on double diffusive convection in a couple stress liquid using both linear and non-linear analyses. Gaikwad et al. (7) have studied linear and non-linear double diffusive convection with Soret and Dufour effects in couple-stress liquids. Shivakumara (9) has discussed the effect of various non-uniform basic temperature gradients on the onset of convection in a couple-stress fluid saturated porous layer. Malashetty et al. (9) investigated the local thermal non-equilibrium effect on the onset of convection in a couple stress fluid saturated porous layers. Malashetty et al. () analyzed the double diffusive instability of a couple stress fluid saturated porous layer using linear and weakly non-linear theories. Malashetty and Kollur () investigated the double-diffusive convection in a horizontal couple stress fluid saturated anisotropic porous layer.

3 AAM: Intern. J., Vol. 8, Issue (June 3) 63 Effect of rotation on thermal convection in a couple-stress fluid saturated rotating rigid porous layer has been investigated by Shivakumara et al. (). Rudraiah et al. () investigated linear stability of electro hydrodynamic purely conducting couple stress fluid flowing through a porous channel. Hall currents are effects whereby a conductor carrying an electric current perpendicular to an applied magnetic field develops a voltage gradient which is transverse to both the current and the magnetic field. It was discovered by Hall in 879 while he was working on his doctoral degree at Johns Hopkins University at Baltimore, Maryland. he Hall effect has again become an active area of research with the discovery of the quantized Hall effect by Klaus von Klitzing for which he was bestowed with Nobel Prize of Physics in 985. In ionized gases (plasmas), where the magnetic field is very strong and effects the electrical conductivity, cannot be Hall currents. he effect of rotation and Hall currents on free convection and mass transfer flow has been studied by Raptis and Ram (98). Sharma and Rani (988) investigated the Hall effects on thermosolutal instability of plasma. Sunil et al. (5) analyzed the effect of Hall currents on thermosolutal instability of compressible Rivlin Ericksen fluid. Rani and omar ( a, b) investigated thermal and thermosolutal convection problem of micropolar fluid subjected to Hall current. Kumar () eamined the effect of Hall currents on thermal instability of compressible dusty viscoelastic fluid saturated in a porous medium subjected to vertical magnetic field. During the survey it has been noticed that Hall effects are completely neglected from the studies of couple stress fluid. Keeping in mind the importance of couple-stress fluid, convection in fluid layer heated from below, magnetic field and Hall current effects, the present paper attempts to studied the effect of Hall current on thermal instability of a couple-stress fluid in the presence of horizontal magnetic field.. Formulation of the Problem Notations a Dimensionless wave number c Speed of light d Depth of layer D Derivative with respect to z ( d / dz) e Charge of an electron F Couple stress parameter d o g (,, g) Acceleration due to gravity field H ( H,,) Uniform magnetic field ( h, hy, h z) Perturbations in magnetic field k Wave number in -direction k Wave number in y-direction y

4 6 Jaimala et al. k Resultant wave number k hermal diffusivity M Hall current parameter N Electron number density n Growth rate p Fluid pressure y k k k ch Ne p Prandtl number k p Magnetic Prandtl number q Velocity of fluid (u, v, w) Perturbations in fluid velocity Q Chandrasekhar number R Rayleigh number R g d k Critical Rayleigh number t ime coordinate emperature (, y, z) Space coordinates Greek Symbols eh d Electrical resistivity Coefficient of thermal epansion Uniform temperature gradient Perturbation in temperature p Perturbation in pressure p Fluid density Perturbation in density Kinematic viscosity Couple stress viscoelasticity e Magnetic permeability Del operator Curly operator n

5 AAM: Intern. J., Vol. 8, Issue (June 3) 65 Consider an infinite layer of an incompressible, finitely conducting (electrically and thermally both) couple-stress fluid, confined between two horizontal planes situated at z and z d, acted upon by a uniform horizontal magnetic field H ( H,, ) and gravity field g (,, g ). he fluid layer is heated from below such that a steady adverse temperature gradient, d where and are the constant temperatures of the lower and upper boundaries with are maintained. Figure. Geometrical configuration he hydromagnetic equations [Stokes (966), Chandrasekhar (98)] relevant to the physical model are q t ' q q g q H H () e (. ) p v ( ),. q, () H c ( q. ) H ( H. ) q H ( H) H, (3) t Ne. H, () ( q. ) k, (5) t [ ( )], (6) where q,, p, and denote, respectively the fluid velocity, density, pressure and temperature and, ', k,, e, N, e, c, and stand for the kinematic viscosity, couple stress viscoelasticity, thermal diffusivity, coefficient of thermal epansion, magnetic permeability, electron number density, charge of an electron, speed of light and electrical resistivity. he suffi zero refers to the values at the reference level z =.

6 66 Jaimala et al. 3. Basic State and Perturbation Equations In the undisturbed state, let the fluid be at rest. Constants temperatures are maintained in the fluid and a constant horizontal magnetic field is applied, therefore, the steady state solution is given by with q (,, ), H ( H,, ), ( z ), ( z ), p p() z () z z and [ ( o)]. (7) o use linearized stability theory and normal mode technique let q( u, v, w), h ( h, h, h ),, p y z and are respectively the perturbations in the fluid velocity, magnetic field, density, pressure and temperature and are functions of space as well as time. hen linearized hydromagnetic perturbation equations for couple-stress fluid become q t ' p e g v q ( h) H, (9). q, () H c ( H. ) q h ( h) H, () t Ne. h, () ( q. ) k. (3) t he change in density caused by the perturbation in temperature is given by. () Analyzing the perturbations into normal modes, we assume that the perturbation quantities are of the form ik ikyy nt w,, h,, W( z), ( z), L( z), Z( z), X( z ) e, (5) z / where k and k y are the wave numbers in and y directions respectively, k ( k k y ) is the resultant wave number of propagation and n is the frequency of any arbitrary disturbance which v u is, in general, a comple constant. y and hy h are the z-components of the y vorticity and current density respectively.

7 AAM: Intern. J., Vol. 8, Issue (June 3) 67 We eliminate the physical quantities after using the non-dimensional parameters a kd, give nd, p, k p, * D D, and dropping (*) is for convenience, equations (9) to () d ( ) ( ) ( ) D a F D a D a W ga d ehd ik ( D a ) L, (6) ehd D a F D a Z ik X, (7) ikhd ikch, (8) ( ) ( ) D a p X Z D a L Ne ik Hd ik chd Ne D a p L W X, d k D a p W, (9) () where F is the couple stress parameter. On eliminating various physical parameters from d equations (6) to (), we obtain the final stability governing equation as ( D a ) F( D a ) D a p Qk d D a p Mk d ( D a ) F( D a ) ( D a ) ( D a ) F( D a ) D a p D a W Ra W Qk d ( D a ) F( D a ) D a p Qk d D a p D a W, () where M R ch Ne g d k is the Rayleigh number, e Q Hd is the Chandrasekhar number and is the non-dimensional number accounting for Hall currents. We now consider the case where both the boundaries are free as well as perfect conductors of heat. Since both the boundaries are maintained at constant temperature, the perturbation in the temperature are zero at the boundaries therefore, the appropriate boundary conditions are W, D W, DZ, at z and z

8 68 Jaimala et al. with DX, K. () herefore, the proper solution of equation () characterizing the lowest mode is W W sin z, (3) where W is constant. Using equation (3), equation () gives R ( ) ( ) F ( ) i ( i p ) Q cos ( ) F ( ) i ( i p ) Q cos ( i p )( ) ( ) F ( ) i ( i p ) Q cos ( i p ) M cos ( ) F ( ) i ( ), () R a Q where k k cos [Chandrasekhar (98)], R, i,, F F and Q. Equation () is the required dispersion relation including the parameters characterizing the Hall currents, magnetic field and couple-stress.. Results and Discussion.. Stationary Convection When the instability sets in as stationary convection ( ) equation (), reduces to F ( ) ( ) R Q F Q ( ) cos ( ) ( ) cos F ( ) ( ) Q cos M cos F ( ) ( ). (5) he above relation epresses the modified Rayleigh number R as a function of the parameters M, Q, F and dimensionless wave number. For the case of stationary convection, to study the dr effect of Hall currents, magnetic field and couple-stress, we eamine the nature of, dm dr df analytically. dr dq and

9 AAM: Intern. J., Vol. 8, Issue (June 3) 69 From equation (5), we have dr dm Q ( ) cos F ( ) ( ) Q cos F ( ) F ( ) ( ) Q cos M( ) cos F ( ), (6) which is negative, therefore Hall currents have a destabilizing effect on the thermal convection in a couple-stress fluid. From equation (5), we have dr dq F ( ) ( ) Q cos M cos ( ) F ( ) F ( ) cos ( ) Q ( )cos Q M cos ( ) F ( ) 3 6 F ( ) ( ) Q cos M cos ( ) F ( ), (7) which shows that magnetic field has a stabilizing effect on the thermal convection in a couplestress fluid. Furthermore, equation (5) yields dr df ( ) 3 ( ) F ( ) ( ) Q cos M cos ( ) F ( ) Q M cos F ( ) ( ) Q cos M cos ( ) F ( ), (8) 3 6 which shows that couple-stress has stabilizing or destabilizing effect according as F ( ) ( ) Q cos M cos ( ) F ( ) or Q M cos ( ) 3 6. It is to be noted that in the absence of Hall currents, equation (8) provides dr df ( ), (9) which clearly shows that couple-stress has a stabilizing effect. hus couple stress has a dual character, in the absence of Hall currents it has stabilizing effect while in their presence it may destabilize the system... Oscillatory Convection * Multiplying equation (6) with W (the comple conjugate of W), integrating over the range of z and making use of equations (7) to () together with the boundary condition (), we get

10 7 Jaimala et al. where g k a d I FI I I p I I p I I p I * e * e 3 [ 5] [ 6 7] 8 9 * d I FI I, (3) I D W a DW a W dz, 3 6 I D W 3 a D W 3 a DW a W dz, I3 DW a W dz, I6 D K a DK a K dz, I8 DX a X dz, I D a dz, I9 X dz, I D Z a Z a DZ dz, I 5 dz, I7 DK a K dz, I DZ a Z dz, I Z dz. All the integrals I - I are positive definite. Putting r i i in equation (3) and then equating the real and imaginary parts, we get r g k a d I p I p I p I d I e e g k a d I FI I I I d I FI (3) e e 6 8 and i g k a d I p I p I p I d I. (3) e e It may be inferred from equation (3) that r may be positive or negative which means that the system may be unstable or stable while equation (3) predicts that i necessarily because all the terms in the bracket are positive definite, which implies that oscillatory modes are not allowed in the system. 5. Numerical Computations For the stationary convection critical thermal Rayleigh number for the onset of instability is dr determined for critical wave number obtained by the condition and analyzed numerically d using Newton-Raphson method.

11 AAM: Intern. J., Vol. 8, Issue (June 3) 7 Critical Rayleigh number R is plotted against Hall currents parameter M for fied values of 5, Q 5 and F, 5,. he critical Rayleigh number R decreases with increase in Hall currents parameter M which shows that Hall currents have a destabilizing effect on the system. Further R increases with increasing couple stress which shows the stabilizing character of couple stress (see Figure ). Critical Rayleigh number R is plotted against Hall currents parameter M for fied value of 5, F and Q,, 3. he critical Rayleigh number R decreases with increase in Hall currents parameter M and increases with increase inq which shows that Hall currents have a destabilizing effect on the system while magnetic field has a stabilizing effect on it (see Figure 3). Critical Rayleigh number R is plotted against magnetic field parameter Q for fied value of 5, F and M 5, 5, 5. he critical Rayleigh number R increases with increase in magnetic field parameter Q as well as Hall current parameter M which shows that magnetic field and Hall currents both have stabilizing effect on the system (see Figure ). Critical Rayleigh number R is plotted against magnetic field parameter Q for fied value of 5, M and F 5, 5, 5. he critical Rayleigh number R increases with increase in magnetic field parameter Q and couple stress parameter F which shows that magnetic field and couple stress both have stabilizing effect on the system (see Figure 5). Critical Rayleigh number R is plotted against couple-stress parameter F for fied value of 5, M and Q,, 3. he critical Rayleigh number R increases with increase in couple-stress parameter F and magnetic field parameter Q which shows that couple-stress and magnetic field both have stabilizing effect on the system (see Figure 6). Critical Rayleigh number R is plotted against couple-stress parameter F for fied value of 5, M andq,, 3. he critical Rayleigh number R increases with increase in couple-stress parameter F and rotation parameter Q which shows that couple-stress and rotation both have stabilizing effect on the system in the absence of Hall currents (see Figure 7). Critical Rayleigh number R is plotted against couple-stress parameter F for fied value of 5, Q 5 and M, 3, 5. he critical Rayleigh number R decreases up to certain values of F and gradually which shows that couple-stress has both destabilizing and stabilizing effect on the system (see Figure 8).

12 7 Jaimala et al. 6. Conclusion In the present paper, we have investigated the effect of Hall currents on an electrically conducting couple-stress fluid layer heated from below in the presence of horizontal magnetic field. Dispersion relation governing the effects of Hall currents, magnetic field and couple-stress is derived. he main results obtained are as follows: (i) For the case of stationary convection, Hall currents have a destabilizing effect on the thermal convection as can be seen from equation (6) and graphically from Figure and Figure 3. (ii) In the absence or presence of Hall currents, magnetic field has a stabilizing effect on the thermal convection as can be seen from equation (7) and graphically from Figure and Figure 5. (iii) In the presence of Hall currents, couple-stress has destabilizing or stabilizing effect on the thermal convection under certain conditions as can be seen from equation (8) and graphically from Figure 6 and Figure 8, while in their absence couple-stress has a stabilizing effect on the thermal convection as can be seen from equation (9) and graphically from Figure 7, in this way dual character of couple-stress is recognized. (iv) Oscillatory modes are not allowed in the system and Principle of Echange of Stabilities is satisfied in the present problem. 6 F= F=5 F= R M Figure. Variations of critical Rayleigh number R with M for fied value of 5, Q 5 and F,5,

13 AAM: Intern. J., Vol. 8, Issue (June 3) 73 6 Q= Q= Q=3 R M Figure 3: Variations of critical Rayleigh number R with M for fied value of F and Q,,3 5, 5 R 5 M=5 M=5 M=5 3 5 Q Figure. Variations of critical Rayleigh number R with Q for fied value of F and M 5,5, 5 5,

14 7 Jaimala et al. 3 8 R Q F=5 F=5 F=5 Figure 5. Variations of critical Rayleigh number R with Q for fied value of M and F 5,5,5 5, 3 R F Q= Q= Q=3 Figure 6. Variations of critical Rayleigh number R with F for fied value of M and Q,,3 5,

15 AAM: Intern. J., Vol. 8, Issue (June 3) 75 5 M= M=3 M=5 R F Figure 7. Variations of critical Rayleigh number R with F for fied value of Q 5 and M,3,5 5, R F Q= Q= Q=3 Figure 8. Variations of critical Rayleigh number R with F for fied value of 5, M and Q,,3

16 76 Jaimala et al. Acknowledgement he authors are grateful to the referees for their technical comments and valuable suggestions, resulting in a significant improvement of the paper. REFERENCES Chandrasekhar, S. (98). Hydrodynamic and Hydromagnetic Stability, Dover Pub., New York. Gaikwad. S. N., Malashetty, M. S. and Rama Prasad, K. (7). An analytical study of linear and non-linear double diffusive convection with Soret and Dufour effects in couple-stress liquids, Int. J. Non-linear Mech., Vol., No. 7, pp Goel, A. K. and Agrawal, S. C. (999). Hydromagnetic stability of an unbounded couple stress binary fluid miture having vertical temperature and concentration gradients with rotation, Int. J. Pure & Appl. Maths, Vol. 3, No., pp Kumar, P. (). Effects of hall currents on thermal instability of compressible dusty viscoelastic fluid saturated in a porous medium, Studia Geotechnica et Mechnica, Vol. XXXIII, No., pp Malashetty, M. S. and Basavaraja, D. (5). Effect of thermal gravity modulation on the onset of Rayleigh-Benard convection in a couple stress fluid, Int. J. rans. Phenom., Vol. 7, pp. 3-. Malashetty, M. S. and Kollur P. (). he onset of double diffusive convection in a couple stress fluid saturated anisotropic porous layer, ransp. Porous Med., Vol. 86, pp Malashetty, M. S., Dulala, P. and Premila, K. (). Double diffusive convection in a Darcy porous medium saturated with couple stress fluid, Fluid Dyn. Res., Vol., pp Malashetty, M. S., Gaikwad, S. N. and Swamy, M. (6). An analytical study of linear and nonlinear double diffusive convection with Soret effect in couple stress liquids, Int. J. herm. Sci., Vol. 5, No. 9, pp Malashetty, M. S., Shivakumara, I. S. and Kulkarni, S. (9). he onset of convection in a couple-stress fluid saturated porous layer using a thermal non-equilibrium model, Phys. Lett., Vol. A373, pp Rani, N., omar, S. K. ( a). hermal convection problem of micropolar fluid subjected to hall current, Applied Mathematical modeling, Vol. 3, pp Rani, N., omar, S. K. ( b). Double diffusive convection of micropolar fluid with Hall currents, Int. J. of Applied Math. and Mech., Vol. 6, No. 9, pp Raptis, A. and Ram, P. C. (98). Role of rotation and Hall currents on free convection and mass transfer flow through a porous medium, International communications in Heat and Mass ransfer, Vol., No., pp Rudraiah N., Shankar, B. M. and Ng Chiu-On (). Electrohydrodynamic stability of couple stress fluid flow in a channel occupied by a porous medium, Special opics Rev. Porous Media., Vol., pp. -.

17 AAM: Intern. J., Vol. 8, Issue (June 3) 77 Sharma, R. C. and Sharma, M. (). Effect of suspended particles on couple- stress fluid heated from below in the presence of rotation and magnetic field, Indian J. Pure and Appl. Math., Vol. 35, No. 8, pp Shivakumara, I.S. (9). Onset of convection in a couple stress fluid saturated porous medium: effects of non-uniform temperature gradients, Arch. Appl. Mec., Vol. 9, pp Shivakumara, J. S., Sureshkumar, S. and Devaraju, N. (). Coriolis effect on thermal convection in a couple-stress fluid-saturated rotating rigid porous layer, Arch. Appl. Math., Vol. 8, pp Siddheshwar, P. G. and Pranesh, S. (). An analytical study of linear and non-linear convection in Boussinesq-Stokes suspensions, Int. J. Non-linear Mech. Vol. 39, pp Stokes, V. K. (966). Couple-Stresses in fluids, Phys. Fluids, Vol. 9, pp Stokes, V. K. (98). heories of Fluids with Microstructure, Springer-Verleg, New-York. Sunil, R. C. Sharma, P. and Singh, M. ( a). On a Couple-stress Fluid Heated from below in a Porous Medium in the Presence of a Magnetic Field and Rotation, J Porous Media, Vol. 5, No., pp Sunil, Sharma, R. C. and Chandel, R. S. ( b). On superposed couple-stress fluid in a porous medium in hydromagnetic, Z. Naturforch., Vol. 57a, pp Sunil, Sharma, R. C. and Chandel, R. S. (). Effect of suspended particles on couple-stress fluid heated and soluted from below in porous medium, J. Porous Media, Vol. 7, No., pp Sunil, Sharma, R. C. and Mohinder, P. (). On couple-stress fluid heated from below in porous medium in the presence of rotation, App. Mech. Engg., Vol. 5, No., pp Sunil, Sharma, Y. D., Bharti, P. K. and Sharma, R.C. (5). hermosolutal instability of Compressible Rivlin-Ericksen fluid with hall currents, International journal of Applied Mechanics and Engineering, Vol., No., pp Sharma, R. C. and Rani, N. (988). Hall effects on thermosolutal instability of a plasma, Ind. J. of pure and Applied Mathematics, Vol. 9, No., pp. -7. Walicki, E. and Walicka, A. (999). Interia effect in the squeeze film of a couple-stress fluidsin biological bearings, Int. J. Appl. Mech. Engg., Vol., pp