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International Journal of Engineering Research & Technology (IJERT) ISSN: 2278-0181 Vol. 4 Issue 10, October-2015 Rayleigh-Bénard Chandrasekhar in An Electrically Conducting Fluid using Maxwell- Cattaneo Law with Temperature Modulation of the Boundaries

S. Pranesh R.V. Kiran Department of Mathematics, Department of Mathematics, Christ University, Hosur Road, Christ Junior College, Bangalore 560 029, India. Hosur Road, Bangalore 560 029, India.

 Abstract - The effect of imposed time-periodic boundary dQ  temperature (ITBT, also called temperature modulation) and   Q  T magnetic field at the onset of convection is investigated by dt  making a linear analysis. The classical Fourier heat law is where, Q is the heat flux,  is a relaxation time and  is the replaced by the non-classical Maxwell-Cattaneo law. The classical approach predicts an infinite speed for the propagation heat conductivity. This heat conductivity equation together of heat. The adopted non-classical theory involves wave type of with conservation of energy equation introduces the heat transport and does not suffer from the physically hyperbolic equation, which describes heat propagation with unacceptable drawback of infinite heat propagation speed. The finite speed. Puri and Jordan [12, 13] and Puri and Kythe Venezian approach is adopted in arriving at the critical Rayleigh [14, 15] have studied other fluid mechanics problems by number, correction and wave number for employing the Maxwell-Cattaneo heat flux law. small amplitude of ITBT. Three cases of oscillating temperature field are examined: (a) symmetric, so that the wall temperatures Thomson [16] and Chandrasekhar [17] studied are modulated in phase, (b) asymmetric, corresponding to out-of theoretically and Nakagawa [18, 19] and Jirlow [20] studied phase modulation and (c) only the lower wall is modulated. The experimentally, the thermal stability in a horizontal layer of temperature modulation is shown to give rise to sub-critical fluid with magnetic field and found that vertical magnetic motion. The shift in the critical Rayleigh number is calculated as a function of frequency and it is found that it is possible to field delays the onset of thermal convection. The application advance or delay the onset of convection by time modulation of of magnetoconvection is primarily found in astrophysics and the wall temperatures. It is shown that the system is most stable in particular by the observation of sunspots. Motivated by the when the boundary temperatures are modulated out-of-phase. It above applications several authors investigated the is also found that the results are noteworthy at short times and suppression of convection by strong magnetic fields. the critical eigenvalues are less than the classical ones. One of the effective mechanisms of controlling the convection is by maintaining a non-uniform temperature Keywords: Time periodic boundary temperature, Rayleigh-Bénard gradient across the boundaries. Such a temperature gradient Chandrasekhar convection, Magnetic field, Maxwell-Cattaneo may be generated by (i) an appropriate heating or cooling at law. the boundaries, (ii) injection/suction of fluid at the I. INTRODUCTION boundaries, (iii) an appropriate distribution of heat source, and (iv) radiative heat transfer. These methods are mainly The Classical Fourier law of heat conduction concerned with only space-dependent temperature gradients. expresses that the heat flux within a medium is proportional However in practice, the non-uniform temperature gradients to the local temperature gradient in the system. A well find its origin in transient heating or cooling at the known consequence of this law is that heat perturbations boundaries. Hence, basic temperature profile depends propagate with an infinite velocity. This drawback of the explicitly on position and time. This problem, called the classical law motivated Maxwell [1], Cattaneo [2], Lebon and thermal modulation problem, involves solution of the energy Cloot [3], Dauby et al. [4], Straughan [5], Siddheshwar [6], equation under suitable time-dependent boundary conditions. Pranesh [7], Pranesh and Kiran [8, 9, 10] and Pranesh and This temperature profile which is a function of both space and Smita[11] to adopt a non-classical Maxwell-Cattaneo heat time can be used as an effective mechanism to control the flux law in studying Rayleigh-Bénard/Marangoni convection convective flow by proper adjustment of its parameters, to get rid of this unphysical results. This Maxwell-Cattaneo namely, amplitude and frequency of modulation. It can be heat flux law equation contains an extra inertial term with used to control the convection in material processing respect to the Fourier law. applications to achieve higher efficiency and to advance

IJERTV4IS100189 www.ijert.org 174 (This work is licensed under a Creative Commons Attribution 4.0 International License.) International Journal of Engineering Research & Technology (IJERT) ISSN: 2278-0181 Vol. 4 Issue 10, October-2015 convection in achieving major enhancement of heat, mass and Maxwell – Cattaneo heat flux law: momentum transfer.  .     There are many studies available in the literature Q 1 Q  Q  T, (4) concerning how a time-periodic boundary temperature affects   the onset of Rayleigh-Bénard convection. Since the problem Magnetic induction equation:  of Taylor stability and Bénard stability are very similar, H     2  Venezian [21] investigated the thermal analogue of  (q.)H  (H.)q  m H, (5) Donnelly’s experiment [22] for small amplitude temperature t modulations. Many researchers, under different conditions, Equation of state: have investigated thermal instability in a horizontal layer with   o[1 (T  To )]. (6) temperature modulation. Some of them are Gershuni and   Zhukhovitskii [23], Rosenblat and Tanaka [24], Siddheshwar where, q is the velocity, H is the magnetic field, T is the and Pranesh [25, 26], Mahabaleswar [27], Bhatia and temperature, P is the hydromagnetic pressure,  is the Bhadauria [28, 29], Malashetty and Mahantesh Swamy [30], Bhadauria [31], Siddheshwar and Annamma [32], Bhadauria density, o is the density of the fluid at reference and Atul [33], Pranesh and Sangeetha [34, 35], Pranesh [36] 1 and Pranesh and Riya[37]. temperature T  To , m  , m is magnetic m The objective of the present problem is to investigate permeability, g is the acceleration due to gravity,  is the the combined effect of magnetic field and temperature thermal conductivity,  is the coefficient of thermal modulation on the onset of Raleigh-Bénard  1   magnetoconvection using Maxwell-Cattaneo law. expansion,  is the dynamic ,   q, Q is 1 2 II. MATHEMATICAL FORMULATION the heat flux vector and  is the constant relation time. Consider an infinite horizontal layer of a Boussinesquian, electrically conducting fluid, of depth‘d’. The lower wall z  0 and upper wall z  d are Cartesian co-ordinate system is taken with origin in the lower subjected to the temperatures boundary and z-axis vertically upwards. Let T be the 1 temperature difference between the lower and upper T  T0  T[1 cos(t)], (7) boundaries. (See Fig. 1.) 2 and H0 Z 1 T  T  T[1 cos(t  )]. (8) 1 0 T  T0  T[1 cos(t  )] 2 Z =d 2 respectively, where  is the small amplitude,  is the

frequency and  is the phase angle.

III. BASIC STATE Y The basic state of the fluid being quiescent is described by:

 Z = 0 1 qb  0, P  Pb (z,t),   b (z,t), O T  T0  T[1  cos(t)] 2 X   Q  0,0,Qb (z), T Tb (z,t), , (9)   Figure 1. Schematic diagram of the Rayleigh – Bénard situation. ˆ  Hb  H0k  The basic governing equations are:  The temperature Tb , pressure Pb , Qb heat flux and density Continuity equation:  b satisfy .q  0, (1) T 2T Conservation of linear momentum: b   b , (10)  t z2 q   ˆ  o   (q.)q  P  gk P  t  , (2)  b   g, (11)   z b 2    q  m H.H   T Q   b , (12) Conservation of energy: b z T    qT  .Q, (3) b  0[1 (Tb  T0)]. (13) t

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The solution of “(10)” that satisfies the thermal boundary   0T. (25) conditions of “(7)” and “(8)” is Operating divergence on the “(23)” and substituting Tb  Ts (z)  T1(z,t), (14) in “(22)”, on using “(20)”, we get

The steady temperature field Ts and an oscillating field T1    T 1  Tb  2  are given by 1    1  W where,  t  t 2  z   , (26) T T  T  (d  2z), (15) 2    Tb  s 0   T  1  W 2d  t  z  z z     where,     d d   it  1 T1  Re a()e  a()e e . (16) The perturbation “(21)”, “(24)” and “(26)” are non-       dimensionalised using the following definitions:

In the “(16)”   (x, y,z)  q  1 (x*,y*,z*) , q*  ,    d2  2 d      (1 i)  , (17) d    2 * W * t    W  , t  , , (27)   2  T  ei  e   d d   a()   . (18)     2  e  e      T  H T*  , H*   and Re stands for the real part.  T Ho  We now superpose infinitesimal perturbation on this basic state and study the stability of the system. Using “(25)” in “(21)”, operating curl twice on the resulting equation and non-dimensionalising the resulting IV. LINEAR STABILITY ANALYSIS equation and also “(24)” and “(26)”, using “(27)”, we get: Let the basic state be disturbed by an infinitesimal thermal perturbation. We now have 1  2 2 4     ( W)  R1 T  W    Pr t  q  qb  q , P  Pb  P ,  , (28)     Pr 2 Hz    b  , Q  Qb  Q, , (19) Q      Pm  z  ˆ   T Tb T, Hb  H0k  H  Hz W Pr 2 The primes indicate that the quantities are    H , (29) z z Pm z infinitesimal perturbations and subscript ‘b’ indicates basic state value.    T T0 2 2  1 2C   C W   T Substituting “(19)” in to “(1)” to “(6)” and using the  t  t z  basic state solutions “(10)” to “(13)”. We get the linearised . (30) T0     equations governing the infinitesimal perturbations in the  1 2C W form z  t    .q  0, (20) where, the asterisks have been dropped for  simplicity and the non-dimensional parameters q 2 ˆ  0    P   q  gk R, Q, Pr, Pm and C are as defined as  t   , (21) 3   ogTd H  R  (Rayleigh Number),  mH0  z  2 2 2 T T  m H0 d  W b  .Q, (22) Q  (Chandrasekhar Number) t z     1 T q      b   Pr (), 1  Q     W  t  2  z  z , (23) o   T    Pm () and H W  m  H kˆ   2H, (24) t 0 z m

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 2 C  (Cattaneo Number). R,W  R0,W0  R1,W1  R2,W2 .... 2d2 (37)

 T   T  The expansion “(37)” is substituted into “(35)” and In “(30)”,  0  is the non – dimensional from of  b  ,  z   z  the coefficients of various powers of  are equated on either side of the equation. The resulting system of equation is where, T LW  0, (38) 0  1 f (z), (31) 0 z   Pr     LW   2 C2  1 2C f (z)  Re A()ez  A()ez eit , (32) 1         t Pm   t , (39)  ei  e  2 2  and A()   . (33) f R 01 W0  R11 W0   2  e  e      Pr 2  2    LW      C  1 2C   Equations “(28)” to “(30)” are solved subject to the following 2 t Pm  t  conditions:     . 2  2 2    W H f R 01 W1  R11 W1  W   T  z  0 at z  0,1 (34)   2 z  2 2   z  f R11 W0  R 21 W0   Eliminating T and Hz from “(28)” to “(30)”, we get (40) a differential equation of order 8 for W in the form where, 2 2   2  L  M1 M2 M3  R 01 M4 M3 [K4 ] K2 .K1    2 T0 2 , (41)  2 W  R1 K3 W, (35) Pr 2  Pr 2  z  Q  M5   Q  K2  Pm z2   Pm z2   1  2    Pr 2  M1    , where, K1     ,  Pr t   t Pm      2      2  M2  1 2C    , K2  1 2C    ,  t  t   t  t    Pr 2   2    M3     , K3  K1 C  1 2C ,  t Pm    t   2     1  2  M4  C  1 2C , K4     .   t   Pr t  In dimensionless form, the velocity boundary     2  M5  1 2C    . conditions for solving “(35)” are obtained from “(28)” to  t  t  “(30)” and “(34)” in the form: 2W 4W 6W VI. SOLUTION TO THE ZEROTH ORDER PROBLEM W     0 at z  0,1 (36) z2 z4 z6 The zeroth order problem is equivalent to the Rayleigh-Bénard problem with magnetic field using

Maxwell-Cattaneo law in the absence of temperature

modulation. The stability of the system in the absence of V. METHOD OF SOLUTION thermal modulation is investigated by introducing vertical velocity perturbation W0 lowest mode of convection as We seek the eigen-function W and eigenvalues R of the “(35)” for the basic temperature distribution “(31)” that W0  Sin(z)expilx  my, (42) T where, l and m are wave numbers in xy-plane with departs from the linear profile 0  1by quantities of z a2  l2  m2. Substituting “(42)” into “(38)” we obtain the order  . Thus, the eigenvalues of the present problem differ expression for Rayleigh number in the form from those of the ordinary Bénard convection by quantities of K6  Q2K2 order  . We seek the solution of “(35)” in the form R  1 1 , (43) 0 2 2 a CK1 1 2 2 2 2 2 2 where, K1    a and a  l  m .

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Setting Q  0 and C  0, the “(43)”reduces the classical 2 2 2 2 and Kn  n   a (see Siddheshwar and Pranesh [26]). Rayleigh-Bénard result. The equation for W2 , then becomes 6 2 2 K1 LW2  A1R2a W0  A2a f R0W1, (48) R 0  . (44) a 2 where, VII. SOLUTION TO THE FIRST ORDER PROBLEM  2 Pr 2 2  A2  CK n 1 Kn  2C   Pm  Equation “(39)” for W1 now takes the form (49) Pr LW  A fR a2 sin(z)  R a2 sin(z) , (45)  2 2  1 1 0 1   i CK n 1 2 CK n   Pm  2  Pr 2  where, A1  CK1 1 K1 . We shall not solve “(48)”, but will use this to determine. The  Pm  solvability condition requires that the time-independent part If the above equation is to have a solution, the right of the right hand side of “(48)” must be orthogonal to hand side must be orthogonal to the null-space of the operator sin(nz) , and this results in the following equation, L . This implies that the time independent part of the RHS of 2 2 2 the “(45)” must be orthogonal to sin(z) . Since f varies R0a Bn () A2 Y1  R 2  Re  , (50) sinusoidal with time, the only steady term on the RHS of 2 2  2  L1(,n)   “(45)” is A R a2 sin(z),so that R  0 . It follows that 1 1 1 Y  L (,n)  L* (,n), 1 1 1 all the odd coefficients i.e. R1  R3  ......  0 in “(37)”. L (,n)  L(,n)A* , 1 2 To solve “(45)”, we expand the right-hand side using A* and L* (,n) are the conjugates of A and Fourier series expansion and obtain W1 by inverting the 2 1 2 operator L term by term as L1(,n) respectively.

2  Bn () it  VIII. MINIMUM RAYLEIGH NUMBER FOR W1  A1R0a Re e sin(z), (46)  L(,n)  CONVECTION where, The value of Rayleigh number R obtained by this Bn ()  A()gn1()  A()gn1(), procedure is the eigenvalue corresponding to the eigen- 2n22 e  e  1n ei  ei function W which, though oscillating, remains bounded in B ()     , n   2 2 2 2 2 2 time. Since, R is a function of the horizontal wave number ‘ e  e   n 1    n 1   a ’and the amplitude of modulation  , we have (47) 2 Ra,  R0a  R2a...... (51)  2 Pr 2 2  L, n  X1  R 0a X2  Q n  X3  It was shown by Venezian [21] that the critical value  Pm  of thermal Rayleigh number is computed up to O(2 ) , by  2 Pr 2 2   iX4  R 0a X5  Q n  , evaluating R 0 and R 2 at a  a0 . It is only when one wishes  Pm  to evaluate R 4 that a 2 must be taken into account where  2 Pr 6 2 4 Pr 6  2C K   K  K a  a minimizes R . Evaluate the critical value of R  Pm n n Pm n  2 2 2 X   , 1 2 2 (denoted by R 2c ) one has to substitute a  a0 in R 2 , where  4 1 2  4  4   2C K n  K n  K n   Pm Pm Pr  a 0 is the value at which R 0 given by “(44)” is minimum. We now evaluate R for three cases:  2 Pr 2 2  2c X2  CK n  1 K n  2C , (a) When the oscillating field is symmetric, so that the  Pm  wall temperatures are modulated in phase with   0 . In this X  K 6  2C 2K 2 , 3  n n  case, Bn ()  bn or 0, accordingly as n is even or 2 odd.  1 4  2 2 1 4   K  K  2C K  (b) When the wall temperature field is antisymmetric, Pm n Pr n Pm n X4   . corresponding to out-of phase modulation with   2 4 Pr 6 6 . In this case, or , accordingly  2C K n  K n  K n     Bn ()  0 b n  Pm  as n is even or odd.  2 Pr 2  (c) When only the temperature of the bottom wall is X5   CK n 1 2 CK n  modulated, the upper plate being held at constant  Pm 

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temperature, with   i . In this case, classical Fourier law of heat conduction by non-classical Maxwell-Cattaneo law. b B ()  n for integers values of n. n 2 The analysis presented in this paper is based on the assumption that the amplitude of the modulating temperature The are given by bn is small compared with the imposed steady temperature 4n22 difference. The validity of the results obtained here depends bn  on the value of the modulating frequency  . When  1, 2  n 12 2 2  n 12 2    the period of modulation is large and hence the disturbance The variable  defined in “(17)”, in terms of the grows to such an extent as to make finite amplitude effects 1 important. When    , R2c  0 , thus the effect of    2 dimensionless frequency, reduces to   (1 i)  and thus modulation becomes small. In view of this, we choose only  2  moderate values of  in our present study. 2 4 2 2 16n   b  . The results have been presented in Fig. 2 to Fig. 13, n 2 4 4 2 4 4   n 1    n 1   from these figures we observe that the value of R 2c may be Hence from “(50)” and using the above expression positive or negative. The sign of the correction Rayleigh of Bn () , we can obtain the following expression for R 2c number R 2c characterizes the stabilizing or destabilizing in the form effect of modulation on R 2c . A positive R 2c means the 2 2 2 R0a Bn () A2 Y1  modulation effect is stabilizing while a negative R 2c means R 2c  Re   . (52) 2 2  2  the modulation effect is destabilizing compared to the system L1(,n) in which the modulation is absent. where, Y  L (,n)  L* (,n) 1 1 1 Fig. 2 is the plot of R 2c versus frequency of In “(52)” the summation extends over even values of modulation  for different values of Cattaneo number C , in n for case (a), odd values of n for case (b) and for all values the case of in-phase modulation. In the figure, we observe of n for case (c).The infinite series “(52)” converges rapidly that as C increase R 2c become more and more negative, in all cases. The variation of R 2c with  for different values which represents Cattaneo number has a destabilizing of C, Pr,Pm and Q are depicted in figures. influence. Increase in Cattaneo number leads to narrowing of

the convection cells and thus lowering of the critical Rayleigh IX. SUBCRITICAL INSTABILITY number. It is also observed from the figures that influence of Cattaneo number is dominant for small values because the The critical value of Rayleigh number R c is convection cells have fixed aspect ratio. It is interesting to determine to be of order 2 , by evaluating R and R , 0c 2c note that for a given value of C, R 2c decreases for small and is of the form values of  and increases for the moderate values of  . Thus 2 small values of  destabilize the system and moderate values Rc  R0c   R2c (53) of  stabilize the system. This is due to the fact that when the where, R and R can be obtained from “(44)”, “(52)” 0c 2c frequency of modulation is low, the effect of modulation on respectively. the temperature field is felt throughout the fluid layer. If the If R 2c is positive, sub critical instability exists and plates are modulated in phase, the temperature profile R c has a minimum at   0 . When R 2c is negative, sub consists of the steady straight line section plus a parabolic critical instabilities are possible. In this case from “(47)” we profile which oscillates in time. As the amplitude of have modulation increases, the parabolic part of the profile becomes more and more significant. It is known that a R 2  0c (54) parabolic profile is subject to finite amplitude instabilities so R 2c that convection occurs at lower Rayleigh number than those Now, we can calculate the maximum range of  , by predicted by the linear theory. assigning values to the physical parameters involved in the Fig. 3 is the plot of R versus frequency of above condition. Thus, the range of the amplitude of 2c modulation, which causes sub critical instabilities in different modulation  for different values of Chandrasekhar number physical situations, can be explained. Q , in the case of in-phase modulation. In the figure, we

X. RESULTS AND DISCUSSION observe that as Q increases R 2c becomes more and more In this paper we make an analytical study of the negative, for small values of  and becomes more and more effects of temperature modulation and magnetic field at the positive for moderate values of  . In making the conclusions onset of convection in Newtonian fluid by replacing the from the figure we should recollect that Q influences R 0c .

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We find that R 2c increases with increase in Q . When the The effect of 1 on R with respect to out-of-phase magnetic field strength permeating the medium is modulation and only lower wall modulation is illustrated in considerably strong, it induces viscosity into the fluid, and the Fig. 15 and Fig. 16. We observe that as 1 increases R magnetic lines are distorted by convection. Then these magnetic lines hinder the growth of disturbances, leading to increases thus increase in 1 stabilizes the system. It is also the delay in the onset of instability. observed from these figures that the frequency of modulation  destabilizes the system. Fig. 4 is the plot of R 2c versus frequency of The results of this study are useful in controlling the modulation  for different values of Prandtl number Pr,in convection by thermal modulation with Maxwell-Cattaneo the case of in-phase modulation. In the figure, we observe law. that as Pr increases R 2c becomes more and more negative. We can infer from this is that the effect of increasing the viscosity of the fluid is to destabilize the system. 100 In Phase Temperature Fig. 5 is the plot of R versus frequency of Modulation 2c C = 0.001 80 modulation  for different values of magnetic Prandtl C = 0.01 C = 0.03 number Pm, in the case of in-phase modulation. In the figure, C = 0.05 60 C = 0.07  we observe that as Pm increase R 2c becomes more and Pr = 0.9, Q = 10, Pm = 5 more negative. 40 We now discuss the results pertaining to out-of- phase modulation. Comparing Fig. 2 to Fig. 5 and Fig. 6 to 20 Fig. 9 respectively we find that R is positive for the out- 2c 0 of-phase modulation where as it is negative for in-phase -12 -10 -8 -6 -4 -2 0 2 4 6 8 R modulation. Thus, Q, Pr and Pmhave opposing influence 2c in in-phase and out-of-phase modulations. However, C has Figure 2: identical influence on R 2c in both in-phase and out-of-phase Plot of R2c versus frequency of modulation  for different values of and these can be seen in Fig. 2 and Fig. 6. The above results Cattaneo number C. are due to the fact that the temperature field has essentially a linear gradient varying in time, so that the instantaneous 100 Rayleigh number is super critical for half cycle and In Phase Temperature subcritical during the other half cycle. Modulation Q = 0 80 Q = 10 The above results on the effect of various parameters Q = 25 on for out-of-phase modulation do not qualitatively Pr = 0.9, C = 0.01, Pm = 5 60 change in the case of temperature modulation of just the  lower boundary. This is illustrated with the help of Fig. 10 to 40 Fig. 13. 20 From the above result, we can conclude that the system is more stable when boundary temperature is 0 modulated in out-of-phase in compare to only lower wall -15 -10 -5 0 5 R modulation and in-phase modulation. In-phase temperature 2C modulation leads to sub critical motions. Sub critical motions Figure 3: are ruled out in the case of out-of-phase modulation and Plot of R2c versus frequency of modulation  for different values of lower wall temperature modulation. Chandrasekhar number Q. Fig. 14 is the plot of Rayleigh number R versus amplitude of modulation 1 for different values of  , in the case of in-phase modulation. From the figure we observe that has amplitude of modulation 1 increases the Rayleigh number R also increases thus amplitude of modulation stabilizes the system. It can be clearly seen that as  increases R increases for smaller values of  and decreases for moderate values of  which reconforms our earlier discussion on this.

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100 100 Out of Phase Temperature In Phase Temperature Modulation Modulation 80 Q = 0 Pr = 0.5 80 Pr = 0.7 Q = 10 Pr = 0.9 Q = 25 60 Q = 10, C = 0.01, Pm = 5 60 Pr = 0.9, C = 0.01, Pm = 5   40 40

20 20

0 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 0 R 0 50 100 150 200 250 2c R 2c Figure 4: Plot of R2c versus frequency of modulation  for different values Figure 7: of Prandtl number Pr. Plot of R2c versus frequency of modulation  for different values of Chandrasekhar number Q.

100

In Phase Temperature 100 Modulation 80 Out of Phase Temperature Pm = 05 Modulation Pm = 15 80 Pr = 0.5 Pm = 25 60 Pr = 0.7 Pr = 0.9 Pr = 0.9, Q = 10, C = 0.01  60 Q = 10, C = 0.01, Pm = 5 40 

40 20

20 0 -6 -5 -4 -3 -2 -1 0 1 2 3 0 R 2c 0 50 100 150 200 250 R Figure 5: 2c Plot of R2c versus frequency of modulation  for different values of Magnetic Prandtl number Pm. Figure 8: Plot of R2c versus frequency of modulation  different values of Prandtl number Pr.

100 Out of Phase Temperature Modulation 100 C = 0.001 Out of Phase Temperature 80 C = 0.01 Modulation 80 C = 0.03 Pm = 05 C = 0.05 Pm = 15 60 C = 0.07 Pm = 25 60  Pr = 0.9, Q = 10, Pm = 5  Pr = 0.9, Q = 10, C = 0.01 40 40

20 20

0 0 0 50 100 150 200 250 300 0 50 100 150 200 250 R R 2c 2c

Figure 6: Plot of R2c versus frequency of modulation  for different values Figure 9: Plot of R2c versus frequency of modulation  for different values of Cattaneo number C. of Magnetic Prandtl number Pm.

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100 Lower Wall Temperature 100 Modulation Lower Wall Temperature C = 0.001 Modulation 80 C = 0.01 80 Pm = 05 C = 0.03 C = 0.05 Pm = 15 60 C = 0.07 60 Pm = 25  Pr = 0.9, Q = 10, Pm = 5 Pr = 0.9, Q = 10, C = 0.01  40 40

20 20

0 0 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 R 2c R 2c

Figure Figure 10: Plot of R2c versus frequency of modulation  for different 13: Plot of R2c versus frequency of modulation  for different values of values of Cattaneo number C. Magnetic Prandtl number Pm.

100 791.4 In Phase Temperature Modulation Lower Wall Temperature 791.2  80 Modulation  Q = 0 791.0  Q = 10 790.8 Pr = 0.5, Pm = 5, Q = 10, C = 0.01 60 Q = 25 Pr = 0.9, C = 0.01, Pm = 5  R 790.6 c 40 790.4

790.2 20 790.0

789.8 0 0 10 20 30 40 50 60 70 0.0 0.2 0.4 0.6 0.8 1.0 R 2c   Figure 11: Plot of R2c versus frequency of modulation  for different values of 2 Figure 14: Plot of critical Rayleigh number Rc  R0c   R2c versus Chandrasekhar number Q. 1 amplitude of modulation 1 for in phase temperature modulation for different values of frequency of modulation  .

100 815 Lower wall Temperature Out of Phase Temperature 80 Modulation Modulation Pr = 0.5 810  Pr = 0.7   60 Pr = 0.9 805 Pr = 0.5, Pm = 5, Q = 10, C = 0.01  Q = 10, C = 0.01, Pm = 5

40 R c 800

20 795

0 790 0 10 20 30 40 50 60 0.0 0.2 0.4 0.6 0.8 1.0 R 2c  

Figure 12: Plot of R2c versus frequency of modulation  different values of Figure 15: Plot of Rc versus  for out of phase temperature modulation for Prandtl number Pr. 1 different values of  .

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[3] G. Lebon and A. Cloot, “A nonlinear stability analysis of the Bénard–Marangoni problem,” J. Fluid 797 Lower Wall Temperature Modulation Mech., vol. 145, 1984, pp. 447-469. 796  [4] P.C. Dauby, M. Nelis and G. Lebon, “Generalized Fourier  equations and thermo-convective instabilities,” Revista 795  Mexicana de Fisica., vol. 48, 2001, pp. 57-62. 794 Pr = 0.5, Pm = 5, Q = 10, C = 0.01 [5] B. Straughan, “Oscillatory convection and the Cattaneo law of heat conduction,” Ricerche mat., vol. 58, 2009, pp. R 793 c 157-162. [6] P.G. Siddheshwar, “Rayleigh-Bénard 792 Convection in a second order ferromagnetic fluid with 791 second sounds,” Proceedings of 8th Asian Congress of Fluid Mechanics, Shenzen, December 6-10, 1999, pp. 631. 790 [7] S. Pranesh, “Effect of Second sound on the onset of 789 Rayleigh-Bénard convection in a Coleman – Noll Fluid,” 0.0 0.2 0.4 0.6 0.8 1.0 Mapana Journal of sciences, vol. 13, 2008, pp. 1-9.   [8] S. Pranesh and R.V. Kiran, “Study of Rayleigh-Bénard magneto convection in a Micropolar fluid with Maxwell- Cattaneo law,” Applied Mathematics, 1, 2010, pp. 470-

Figure 16: Plot of Rc versus  for lower wall temperature modulation for 480. 1 different values of  . [9] S. Pranesh and R.V. Kiran, “Effect of non-uniform temperature gradient on the onset of Rayleigh–Bénard– XI. CONCLUSIONS Magnetoconvection in a Micropolar fluid with Maxwell– Cattaneo law,” Mapana Journal of sciences, vol. 23, 2012, Following conclusions are drawn from the problem: pp.195-207. [10] S. Pranesh and R.V. Kiran, “The study of effect of 1. The system is more stable when boundary temperature is suction-injection-combination (SIC) on the modulated in out-of-phase. onset of Rayleigh-Bénard- Magnetoconvection in a micropolar fluid with Maxwell-Cattaneo law,” 2. In-phase temperature modulation leads to sub critical American Journal of Pure Applied Mathematics, motions. vol. 2, No.1, 2013, pp. 21-36 3. The results of the study throw light on an external means [11] S. Pranesh and S.N. Smita, “Rayleigh-Bénard convection in a second-order fluid with Maxwell-Cattaneo law,” of controlling Rayleigh-Bénard convection either Bulletin of Society for mathematical services & standard, advancing or delaying convection by thermal vol. 1, No. 2, 2012, pp. 33-48. modulation. [12] P. Puri and P.M. Jordan, “Stokes’s first problem for a 4. It is observed that for large frequencies, the effect of dipolar fluid with non-classical heat conduction,” Journal of Engineering Mathematics, vol. 36, 1999, pp. 219-240. modulation disappears. [13] P. Puri and P.M. Jordan, “Wave structure in stokes second 5. The non – classical Maxwell – Cattaneo heat flux law problem for dipolar fluid with non classical heat involves a wave type heat transport and does not suffer conduction,” Acta Mech., Vol. 133, 1999, pp. 145-160. from the physically unacceptable drawback of infinite [14] P. Puri and P.K. Kythe, “Non classical thermal effects in stokes second problem,” Acta Mech., vol. 112, 1995, pp. heat propagation speed. The classical Fourier flux law 1-9. overpredicts the critical Rayleigh number compared to [15] P. Puri and P.K. Kythe, “Discontinuities in velocity that predicted by the non-classical law. Overstability is gradients and temperature in the stokes first problem with the preferred mode of convection. non-classical heat conduction,” Quart. Appl. Math., vol. 55, 1997, pp. 167-176. ACKNOWLEDGMENT [16] W.B. Thomson, “Thermal convection in a magnetic field,” Authors would like to acknowledge management of Christ Philosophical Magazine, 42, 1951, pp. 1417. University and Christ Junior College for their support in [17] S. Chandrasekhar, “Hydrodynamic and hydromagnetic stability,” Oxford: Clarendon Press., 1961. completing the work and also to Prof. Pradeep. G. [18] Y. Nakagawa, “An experiment in the inhibition of thermal Siddheshwar, Professor, Department of Mathematics, convection by a magnetic field,” Nature, London, 175, Bangalore University, Bangalore for his valuable suggesting 1955, pp. 417. during the completion of their work which increased the [19] Y. Nakagawa, “Experiments on the inhibition of thermal quality of the paper. convection by a magnetic field,” Proceedings of Royal Society of London, A240, 1957, pp.108-113. REFERENCES [20] K. Jirlow, “Experimental investigation of the inhibition of convection by a magnetic field,” Tellus, vol. 8, 1956, pp. [1] J.C. Maxwell, “On the dynamical theory of gases,” 252. Philosophical Transactions of the Royal Society of [21] G. Venezian, “Effect of modulation on the onset of London, vol. 157, 1867, pp. 49-88. thermal convection,” Journal of Fluid Mechanics, vol. 35, [2] C. Cattaneo, “Sulla condizione del Calore, Atti 1969, pp. 243. del Semin. Matem. eFis.,” Della Univ. Modena, vol. 3, [22] R.J. Donnelly, “Experiments on the stability of viscous 1948, pp. 83-101. flow between rotating cylinders. III Enhancement of

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