Buoyancy Driven Convection in Micropolar Fluid with Controller and Variable Boundaries

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1 Buoyancy Driven Convection in Micropolar Fluid with Controller and Variable Boundaries. F. M. MOKHTAR a,b *, I. K. KHALID a,. M. ARIFI b a Department of Mathematics Universiti Putra Malaysia UPM, Serdang Malaysia b Institute for Mathematical Research Universiti Putra Malaysia UPM, Serdang, Selangor Malaysia *norfadzillah.mokhtar@gmail.com Abstract :- The effect of controller (feedback control) is applied to the steady Rayleigh- Benard convection in a horizontal micropolar fluid layer. The bounding surfaces of the liquids are considered to be free-free, rigid-free, and rigid-rigid with combination isothermal on the spin-vanishing boundaries. A linear stability analysis is used and the Galerkin method is employed to find the critical stability parameters numerically. It is shown that the critical Rayleigh number increases as the value of feedback control increases. Key-Words: Convection, Stability, Controller, Micropolar fluid, Galerkin, Buoyancy 1. Introduction The Rayleigh-Benard situation in Eringen s [1-6] micropolar fluids has been investigated by many authors (Data and Sastry [7], Ahmadi [8], Lebon and Perez- Garcia [9], Perez-Garcia and Rubi [10], Bhattacharya and Jena [11,1], Payne and Straughan [13], Straughan [14], Franchi and Straughan [15], Lindsay and Straughan [16]). The main result from all of these studies is that for heating from below stationary convection is the preferred mode. Effect of throughflow on Marangoni convection in micropolar fluid are studied by Murty and Rao [17]. The magnetoconvection in micropolar fluid are studied by Siddheshwar and Pranesh [18]. They observed that micron sized suspended particles add to stabilizing the effect of magnetic field. The instability of convection driven by buoyancy is referred to as the Rayleigh-Benard convection. The classical Rayleigh problem on the onset of convective instabilities in a horizontal layer of fluid heated from below has its origin in the experimental observations of Benard [19] and [0]. Rayleigh s paper is the pioneering work for almost all modern theories of convection. Rayleigh [1] showed that Benard convection, which is caused by buoyancy effects, will occur when the Rayleigh number exceeds a critical value. Latest, Khalid et. al [] studied the combined effect of magnetic field and internal heat generation on Rayleigh convection in micropolar fluid. They found that the presence of magnetic field has a stability effect on Rayleigh convection in micropolar fluid. The objective of the feedback control is to delay the onset of convection while maintaining a state of no motion in the fluid layer. In proportional of feedback control of Tang and Bau [3], the thermal actuators are place at the bottom heated surface. Sensor are used to detect the departure of the surface temperature of the fluid from its conductive and they direct the actuators to take action so as to suppress unwanted disturbances. The thermal actuators modifies the bottom heated surface temperature using a proportional relationship between upper and lower thermal boundaries. Tang and Bau [4, 5] and Howle [6] have shown that the critical Rayleigh number for the onset of Rayleigh-Benard convection can be delayed. Or et al. [7] studied ISB:

2 theoretically the use of feedback control strategies to stabilize long wavelength instabilities in the Marangoni-Benard convection. Bau [8] has shown independently how such a feedback control can delay the onset of Marangoni- Benard convection on a linear basis with no-slip boundary conditions at the bottom. Arifin et al.[9] have shown that a feedback control also can delay the onset of Marangoni-Benard convection with free-slip boundary conditions at the bottom. The effect of feedback control has been investigated by Mokhtar and Arifin in ferrofluid [30]. They studied the coupled convection with the effect of controller. Marangoni convection in micropolar fluid with feedback control have been studied by Abidin et. al [31]. They claimed that the onset of instability can be delayed through the use of feedback control. Mokhtar et. al [3] investigated the effect of internal heat generation on Benard-Marangoni convection in micropolar fluid with feedback control. The present study deals with feedback control on the onset of Rayleigh-Benard convection in micropolar fluid. The resulting eigenvalue is solved numerically using the Galerkin technique with lower and upper boundary conditions that are rigid-free, rigid-rigid, and free-free.. Mathematical Formulation Consider an infinite horizontal layer of quiescent micropolar fluid of depth d, where the fluid is heated from below. The stability of a horizontal layer of micropolar fluid in the presence of thermal feedback control is examined. The no-spin boundary condition is assumed for micro rotation at the boundaries. Let T be the temperature difference between the lower and upper surfaces with the lower boundary at a higher temperature than the upper boundary and these boundaries maintained at constant temperature. The upper free surface is assumed to be non-deformable and the governing equations for the Rayleigh-Benard situation in Boussinesquian micropolar fluid are v = 0, (1) v ρ ˆ 0 + ( v ) v = p ρgk t + ( ζ + η ) v + ζ( ω), () ω ρ0i + ( v ) ω = ( λ + η ) ( ω) t + ( η ω) + ζ( v ω), T β + ( v ) T = ( ω) T (3) t ρ0cv + κ T, (4) where is the velocity, ρ 0 is the density at T a, t is the time, p is the pressure term, g is the acceleration due to the gravity, k is the unit of vector in the z direction, ζ is the coupling viscosity coefficient for vortex viscosity, λ and η is the bulk and shear kinematic viscosity coefficients, ω is the micro rotation, I is the moment of inertia, λ and η is the bulk and shear spin-viscosity coefficients, T is the temperature, β is the micropolar heat conduction coefficient, C ν is the specific heat, κ is the thermal conductivity, h g is the overall uniformly distributed volumetric internal heat generation within the micropolar fluid layer and ρ is the density. The basic state of the fluid is quiescent and is described by q = ( 0,0,0 ), ( ) b ωb = 0,0,0, p = pb ( z), T = Tb ( z), (5) with ρ = ρ0 1 α( T T a ). Subject to the boundary conditions Tb = T0 at z = 0 and T = T T at z = d. b 0 Let the basic state be disturbed by an infinitesimal thermal perturbation and we now have ISB:

3 q = qb + q, ω = ω b + ω, p = pb + p, and T = Tb + T, (6) where the primes indicate that the quantities are infinitesimal perturbations. Substituting (6) into Eqs. (1) (4), we obtained the linearised equations in the form q = 0, (7) p ρ gk ζ η q 0 = + ( + ) + ζ( ω ), (8) 0 = ( + ) ( ) + + ζ( q ω ), (9) λ η ω η ω T β T W = ω k T d ρ0c ν d zh g hgd T + κ T + + W. κ κ d (10) The perturbations Eqs.(7) (10) are nondimensionalised using the following definitions * * * x y z * W ( x, y, z ) =,,, W =, d d d xd * ( ω ) z * T Ω =, T =. (11) 3 xd T Substituting Eq.(11) into Eq.(8) (10), and apply the normal mode analysis, we will have (1) (13) 1 ( D a ) W + G 3( D a ) G = 0 (14) where 1 = ζ η + ζ is the coupling parameter, η 3 = ( η+ ζ ) d is the couple stress, β = is the heat conduction, 5 ρ0cd ν 3 αg Tρ0d Ra = is the Rayleigh number. η ζ χ ( + ) Following the proportional feedback control in [4], the continuously distributed actuators and sensors are arranged in a way that for every sensor, there is an actuator positioned directly beneath it. The determination of a control; qt ( ), can be accomplished using the proportionalintegral-differential (PID) controller of the form, [ ] qt () = r+ K et (), et () = mt () + mt (), (15) where r is the calibration of the control, et () an error or deviation from the measurement, mt ( ), from some desired or reference value, mt ( ), p D l 0 K = K + K d dt + K dt with K p is the proportional gain, K D differential gain and K integral gain. Based on [4], for l one sensor plane and proportional feedback control, the actuator modifies the heated surface temperature using a proportional relationship between the upper, z = 1 and the lower, z = 0 thermal boundaries for perturbation field, T (, xy,0,) t = KT (, xy,1,), t (16) where T denotes the deviation of the fluid s temperature from its conductive state and K is the scalar controller gain in which it will be used to control our system. Eqs.(1) (14) are solved subject to appropriate boundary conditions at z = 0, W = DW = G =Θ 0 + KΘ 1 = 0 and For upper free boundary ( ) ( ) t (17) W= DΘ= G= DW= 0 at z = 1. (18) For upper rigid boundary W = DΘ= G = DW = 0 at z = 1. (19) 3. Method of solution We now employ a Galerkin-type weighted residuals method to obtain an apptoximate ISB:

4 solution to the system of Eqs.(1) (14). We choose as trial functions (satisfying the boundary conditions) W p, θ p, ξ p, where p = 1,, 3, and write W= AW p p, θ = Bpθp, ξ = C p= 1 p= 1 p= 1 (0) Substitute (0) into Eqs. (1) (14) and make the expressions on the left-hand sides of those equations orthogonal to the trial functions, thereby obtaining a system of 3 linear algebraic equations with A p, B p, C p are three unknowns. The vanishing of coefficients produces the eigenvalue equation for the system. 4. Results and Discussion The criterion for the onset of Rayleigh- Benard convection in micropolar fluid in the presence of feedback control is investigated theoretically. The sensitiveness of critical Rayleigh number, Ra c to the changes in the micropolar fluid parameters; 1, 3 and 5 are also studied. Three cases are involved in the system which are rigid-free, rigid-rigid and freefree surfaces. Table 1 shows the comparison of the critical Rayleigh number; Ra c for different values of coupling parameter; 1 and controller effect; K when 3 = and 5 = 1. Our results are compared with Siddheshwar and Pranesh [18] in the absence of feedback control and the results are in a good agreement. Table 1. Comparison of Ra c for different values of K Ra c 1 [18] Present study K = 0 K = 0 K= 1 K = p and yield stability in the system. Besides, this shows that the controller parameter has a significant impact on the system. Table shows the comparison of the critical Rayleigh number; Ra c with controller; K for three different types of surfaces. Table. Critical Rayleigh number for various boundary conditions Ra c Rigid-free Rigid-Rigid Free-Free K It can be clearly seen that the critical Rayleigh number; Ra c values increases as the value of controller; K increase for three cases considered. This shows that the controller; K can delay the onset of convection and thus stabilize the system. If we look the differences in the critical Rayleigh number; Ra c values for rigidfree, rigid-rigid and free-free surfaces, the critical Rayleigh number; Ra c values in rigid-rigid surfaces are the highest. It is interesting to take note that the rigid-rigid surface is the most stable in micropolar fluid system. Figure 1 indicates the variation values of Rayleigh number; Ra for K = 0,, 4 in three different cases. The parameters chosen are 1 = 0.5, 3 = and 5 = 1. It is found that as the number of Rayleigh number; Ra increases, the value of gain controller; K also increases and thus stabilize the system. From this table we found that the critical Rayleigh number; Ra c increases as the value of coupling parameter; 1 and controller; K increases. This is because the microelement concentration is increase ISB:

5 Fig.1 The variation of Rayleigh number with wavenumber Figure show the plot of the critical Rayleigh number; Ra c versus the coupling parameter; 1 respectively for various values of controller; K when 3 = and 5 = 1. In each of these plots, the critical number increases with increasing of 1 for all values of K in three different cases considered. 1 indicates the concentration of microelements, and increasing of 1 is to elevate the concentration of microelements number. When this happened, a greater part of the energy of the system is consumed by these elements in developing gyrational velocities of the fluid and thus delayed the onset of convection. Fig. Variation of critical Rayleigh number with 1 The illustration of the couple stress parameter; 3 can be seen in Figure 3 when 1 = 0.1 and 5 = 10. From the graph, it can be clearly seen that in the presence of controller, increased of 3 decrease the values of Ra c for all K in three cases considered. This situation revealed that the system become more unstable much faster when the couple stress parameter increasing. However, the presence of controller K in the system, helps to delay the onset of convection in micropolar fluid. Fig. 3. Variation of critical Rayleigh number with 3 Figure 4 show the plot of Ra c versus micropolar heat conduction parameter; 5 when 1 = 0.1 and 3 = in the presence of feedback control. From the graph, it can be seen that the increasing of the micropolar heat conduction; 5, increased the critical Rayleigh number as well. The reason behind this is, when 5 increases, the heat induced into the fluid due to the microelements is also increased and thus reducing the heat transfers from the bottom to the top of the system. The decrease in heat transfer is responsible for delaying the onset of convection. Thus, increasing 5, promotes stability in the micropolar system. As for the controller effect, the values of K are increases in all cases considered when increase 5. Fig. 4. Variation of critical Rayleigh number with Conclusions ISB:

6 The stability analysis of the Rayleigh- Benard convection in micropolar fluid with feedback control is investigated theoretically. It is found that the effect of controller; K in the micropolar fluid is clearly has a stabilizing effect to make the system more stable. For three cases considered, rigid-free, rigid-rigid and freefree surfaces, it is found that the critical values of the Rayleigh number in rigidrigid surfaces are the highest. This shows that the use of rigid-rigid surfaces can delay the onset of convection. The coupling parameter; 1, couple stress parameter 3, and micropolar heat conduction; 5 has a significant effect on the onset of Rayleigh-Benard convection. The increase of the microelement concentration; 1 and 5 helps to slow down the process of destabilizing. Acknowledgements Authors gratefully acknowledge the financial support received from Government of Malaysia under FRGS scheme. References [1]A. C. Eringen, Simple microfluids, Int. J. Engng Sci.,Vol., 1964, pp []A. C. Eringen, Micropolar fluids with stretch, Int. J. Engng Sci., Vol.7, 1969, pp [3]A. C. Eringen, Theory of thermomicrofluids, J. Math. Anal Appl., Vol.38, o., 197, pp [4]A. C. Eringen, Theory of anistropic micropolar fluids, Int. J. Engng Sci., Vol.18, 1980, pp [5]A. C. Eringen, Theory of thermomicrostrecth fluids and bubbly liquids, Int. J. Engng Sci., Vol.9, 1990, pp [6]A. C. Eringen, Memory-dependent orientable nonlocal micropolar fluids, Int. J. Engng Sci., Vol.7, 1991, pp [7]A. B. Datta, V. U. K. Sastry,Thermal instability of a horizontal layer of micropolar fluid heated from below, Int. J. Engng Sci., Vol.14, o.7, 1976, pp [8]G. Ahmadi, Stability of a micropolar fluid layer heated from below, Int. J. Engng Sci., Vol.14, o.1, 1976, pp [9]G. Lebon, C. Perez-Garcia, Convective instability of a micropolar fluid layer by the method of energy, Int. J. Engng Sci., Vol.19, 1981, pp [10]C. Perez-Garcia, J. M. Rubi, On the possibility of overstable motions of micropolar fluids heated from below, Int. J. Engng. Sci., Vol.0, o.7, 198, pp [11]S. P. Bhattacharyya, S. K. Jena, On the stability of a hot layer micropolar fluid, Int. J. Engng. Sci., Vol.1, 1983, pp [1]S. P. Bhattacharyya, S. K. Jena, Thermal instability of a horizontal layer of micropolar fluid with heat source, Proc. Indian Acad. Sci. (Math. Sci.), Vol.93, 1984, pp [13]L. E. Payne, B. Straughan, Critical Rayleigh numbers for oscillatory and nonlinear convection in an isotropic thermomicropolar fluid, Int. J. Engng. Sci., Vol.7, 1989, pp [14]B. Straughan, The energy method, stability and non-linear convection, Springer Ser in Appl. Math. Sci., 199. [15]F. F. Franchi, B. Straughan, A nonlinear energy stability analysis of a model for deep convection, Int. J. Engng. Sci.,Vol.30, 199, pp [16]K. A. Lindsay, B. Straughan, Penetrative convection in a micropolar fluid, Int. J. Engng. Sci., Vol.30,199, pp [17] Y.. Murty, V. V. Ramana Rao, Effect of throughflow on Marangoni convection in micropolar fluids, Acta Mech., Vol. 138, 1999, pp [18] P. G. Siddheshwar, S. Pranesh, Magnetoconvection in fluids with ISB:

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