Rayleigh-Benard-Taylor Convection in Temperature-sensitive Newtonian Liquid with Heat Source/Sink

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Rayleigh-Benard-Taylor Convection in Temperature-sensitive Newtonian Liquid with Heat Source/Sink

Rayleigh-Bénard-Taylor Convection in Temperature-sensitive Newtonian Liquid with Heat Source/Sink

  1. Ramachandramurthy1

    1Department of Mathematics, Ramaiah Institute of Technology, Bengaluru-560054, India

    A.S. Aruna2

    2Department of Mathematics, Ramaiah Institute of Technology, Bengaluru-560054, India

    Abstract The present paper deals with a linear stability analysis of Rayleigh-Bénard convection in a rotating Newtonian fluid with heat source/sink confined between two parallel, infinitely extended horizontal surfaces. It is proved that internal Rayleigh number; Thermo rheological parameter and the Taylor number influence the onset of convection. It is found that the effect of increasing the strength of rotation is to stabilize the system whereas the increase of internal heat source and variable viscosity parameter is to destabilize the system. The result has possible applications in the astrophysical and geophysical context.

    Keywords Rayleigh-Benard convection; Taylor number; variable heat source; temperature-dependent viscosity.

    1. INTRODUCTION

      The Rayleigh-Benard instability problem with internal heat generation, thermo rheological effect and rotation have been received great attention due to their implications in heat and mass transfer. The rotation is one of the important external mechanisms in controlling the onset of convection. The effect of non-inertial acceleration on the stability analysis of buoyancy driven convection is discussed in several books and monographs such as Chandrasekhar [1961], Platten and Legros [1984] and Drazin and Reid [2004].

      Thermal convection occurs when a fluid is heated from below. Rayleigh-Bénard convection as it is called is discussed in two excellent books Chandrasekhar [1] and Drazin and Reid [2]. Over several decades, many researchers are trying to control the convective phenomena in Rayleigh-Bénard convection by imposing external constraints such as temperature, Magnetic field, rotation etc. Rotation plays a very important role in controlling the convection. Motivated by the experiments of Donnelly [3] on the effect of rotation on the onset of instability in a fluid flow between two concentric cylinders, Venezian [4] performed a linear analysis under temperature modulation with free-free surfaces. He obtained an expression for critical Rayleigh number and found that by suitably tuning the frequency of rotation one can regulate the heat transport effectively. Later on, RayleighBénard problem under many constraints were studied by various researchers, considering different physical models. Kloosterziel and Carnevale [5] investigated the effect of rotation on the stability of a thermally modulated system, and determined analytically the critical points on the marginal stability boundary above which an increment either in viscosity or in diffusivity is

      destabilizing the system. Finally, they showed that when the fluid has zero viscosity the system is always unstable, in contradiction to Chandrasekhars [1] conclusion. Siddheshwar and Vanishree [7] studied the effect of rotation on thermal convection in an anisotropic porous medium with temperature dependent viscosity. The Galerkin variant of the weighted residual technique is used to obtain the Eigen value of the problem. They presented some new result on the parameters influence on convection in the presence of rotation, for both high and low rates. Bhadauria [8] investigated rotational influence on Darcy convection and found that both rotation and permeability suppress the onset of thermal instability. The effect of non-uniform temperature gradient on Rayleigh- Bénard convection in micro polar fluid has been studied by Siddheshwar and Pranesh [9]. The Eigen value is obtained for free-free, rigid-free and rigid-rigid boundary conditions on the spin vanishing boundaries. They presented some important results on the micro polar fluid parameters and the internal heat generation on the onset of convection. Bhadauria et al. [10] investigated the non-linear thermal instability in a rotating viscous fluid layer under temperature/gravity Modulation. They found that by suitably adjusting the frequency or amplitude of modulation one can control the convective flow. Recently Siddheshwar and Chan [11] investigated the thermo rheological effect on Benard and Marangoni convection in an anisotropic porous media. They showed that the effect of increasing thermo rheological parameter is to destabilize the system. A brief study of the combined effect of thermal modulation and rotation on the onset of convection in a rotating fluid layer was made by Rauscher and Kelly [12]. It is clear most of the available studies considered are not involving all the three effects such as internal heat generation, rotation and the temperature-sensitivity of the liquid on the onset of convection. However there are only few studies available, This paper deals with the study of convective non-linear Rayleigh- B\'{e}nard-Taylor instability problem in an electrically conducting temperature-sensitive Newtonian liquid with heat source; we use the truncated representation for half range Fourier cosine series expansion to represent the basic non- uniform temperature gradient and basic viscosity. The Galerkin procedure is utilized to discover the systematic articulation for the thermal Rayleigh number as the function of Internal heat source parameter, thermo rheological parameter and Taylor number.

    2. MATHEMATICAL FORMULATION

      = , =

      (8)

      Eliminating the pressure in the equation (2) and using the equations (8) in the resulting equation, we get the following equations

      (2) =

      (2) +

      4

      0

      + (2) + 4

      +

      (2) 0

      2

      + (,2), (9)

      0 0

      (,)

      = + 2 + + (,), (10)

      Consider a Newtonian liquid confined between two infinite, parallel horizontal planes of depth between = 0 and =

      1 (,)

      . The lower and the upper plates are maintained at different

      Further from the momentum equation we can write

      temperatures, the lower plate is hotter than the upper plate (see

      =

      4 + 2

      + (,). (11)

      Fig. 1). The system is rotated about axis. We assume the

      0

      0 0

      (,)

      Oberbeck-Boussinesq approximation is valid and consider only small-scale convective motions (Lorenz) and the

      We non-dimensionalize the equations (12), (13) and (14) using the following definitions:

      boundaries are assumed to be to be stress-free and isothermal.

      Density, dynamic viscosity and heat source are assumed to be temperature-dependent. The governing equations for the study

      (, , ) = ( ,

      ,

      ) , =

      , =

      2

      ,

      }, (12)

      of Rayleigh-Bénard convection in variable viscosity Newtonian liquids with the Coriolis force are given by

      = T

      T

      and obtain

      1 (2) =

      (2) +

      4

      . = 0 (1)

      + (2) + 4

      0 [ + (. ) + 2()] = + () +

      +

      (2) +

      . [ ()( + )], (2)

      + 1 (,2), (13)

      + (. ) = 2T + Q (T T ), (3)

      2

      (,)

      (,)

      1 0

      =

      () + + +

      (,)

      , (14)

      () = 0[1 ( 0)], (4)

      () = 0(0). (5)

      1 =

      2 + + (,) , (15)

      (,)

      To make a finite amplitude analysis, we consider the following perturbations:

      where, = is Prandtl number,

      = 3

      is the

      = + , = + , = + ,

      thermal Rayleigh number,

      = (

      202

      2

      ) is Taylor number

      }, (6)

      and () = 1 + 2 cos() ,

      = + , = +

      =1 22

      where the primes indicates perturbed quantities. The basic state quantities (), (), () have the forms

      0 0

      = + () , = (1 ())

      . (7)

      The boundaries are assumed to be stress free and isothermal, hence the boundary conditions for solving the equations (13),

      1. and (15) are

        = 2 = = = 0 at = 0,1 (16)

        = ( ) ( ) + , =

        0 0

        ( )

        }

        In the next section, the linear stability analysis is performed using fourier series and the Galerkin technique is used to find

        sin[(1 )]

        Where, ( ) = , 0 is the constant of integration

        out the analytical expression for the critical Rayleigh number

        sin[]

        and

        = 12 is the internal Rayleigh number. Since the flow

        which is of great utility in performing the linear stability analysis.

        considered is two dimensional, we introduce the stream function as follows:

    3. LINEAR STABILITY ANALYSIS

      In order to study the linear theory, the linearized version of equations (13),(14) and (15) is considered along with the boundary conditions (16). This means that the Jacobeans

      (,) and (,) in equations. (13), (14) and (15) is neglected.

    4. RESULTS AND DISCUSSIONS

      The effect of a variable heat source (sink) and rotation on Rayleigh-Bénard convection is studied. The effect of heat source (sink) appears in the equation in the form of an internal Rayleigh number and rotation in the form of Taylor number

      (,)

      (,)

      . The external Rayleigh number is the Eigen value of the

      The solution of the linearized system is assumed to be periodic waves of the form:

      (, ) = 0 sin(X) sin(Y)

      (, ) = 0 cos(X) sin(Y)} , (17)

      (, ) = 0 sin(X) cos(Y)

      Where, = , 2= 2(1 + 2)

      1

      1

      1

      2(2 )(4 2 )

      problem.

      The highlights of the linear study are

        1. Derivation of a useful analytical expression for the stationary critical Rayleigh number by using the half- range Fourier cosine series expansion for the basic non uniform temperature gradient and for the basic viscosity.

        2. Discounting the possibility of oscillatory motions

      The focus in the paper is on Rayleigh-Benard convection

      =

      422

      influenced by a heat source (sink) and for this reason only those values of that do not allow dominance of heat source

      1 + 2

      ( +

      2

      1 2

      2 2

      2

      2 2

      ), (18)

      (sink) over buoyancy in effecting convection are considered.

      In order to understand better the results arrived at in the

      1

      4 (

      32 +

      4 )

      0

      15

      problem we analyze the nonlinear basic state temperature distribution, which throws light on the observed effect of heat

      0

      = 20

      1 V

      sin[(1)]

      esin[] ,

      0

      source (sink) on the stability. A scaled, dimensionless temperature distribution is considered in the following form:

      2

      = 20

      1 V

      sin[(1)]

      esin[] cos(2Y) ,

      () =

      0

      sin[( 1)]

      =

      sin[]

      4

      = 20

      0

      1 V

      sin[(1)]

      esin[] cos(4Y) ,

      0

      Fig.2 is the plot of the non uniform basic temperature gradient

      () versus for different values of . It is evident that the plots are not symmetric about the line = 1 , which is the basic temperature distribution when there is no heat source

      In equation (18), is the scaled horizontal wave number. The

      quantities 0 , 0 and 0 are respectively, amplitudes of the stream function, temperature, and the vorticity function. Substituting equation (17) into the linear version of equations

      (13) to (15) and integrating the above equation with respect to

      ]

      X in [0, 2 and also with respect to in [0, 1], a set

      of homogeneous equations in 0 , 0 and 0 is obtained. In obtaining the non-trivial solution of the linear system, the above expression of the critical Rayleigh number is obtained.

      It is now clear that defined by equation (20) is the critical Rayleigh number of the marginal stationary state. The scaled critical wave number for the preferred mode satisfies the following equation:

      3 4 4

      (sink).

      1.0

      2

      0.8 1

      0

      0.6

      b Y

      -1

      0.4 -2

      0.2

      0.0

      0.0 0.2 0.4 0.6 0.8 1.0

      Y

      ) 4 6

      0

      (

      1

      + (

      2 ) 4

      Fig.2: Temperature profile of quiescent basic state for different values of

      0 2

      2 02 + 22

      ( + )4 2 2

      , (22 )

      This asymmetry due to temperature-dependent heat

      1 0 2

      0

      2

      source/sink helps in making the inference that the results on

      +

      2

      ( 30(2)

      1 2 4 0

      2(10 +2 15 )

      ) = 0

      }

      the problem with a heat sink cannot be obtained from that of a heat source through a suitable transformation as can be done in the case of a constant heat source.

      3000

      2500

      RI 1, V 0

      3000

      2500

      RI 1, V 0

      2000 2000

      R

      1500

      1000

      500

      Ta 0

      Ta 100

      Ta 1000

      1500

      R

      1000

      500

      Ta 0

      Ta 100

      Ta 1000

      0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 2.0 2.5

      3000

      2500

      Fig.3 (a)

      RI 1, V 0.5

      3000

      2500

      Fig.3 (e)

      RI 1, V 0.5

      2000

      R

      R

      1500

      2000

      1500

      1000

      500

      Ta 0

      Ta 100

      Ta 1000

      1000

      500

      Ta 0

      Ta 100

      Ta 1000

      0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 2.0 2.5

      3000

      2500

      2000

      R

      1500

      1000

      Fig.3 (b)

      RI 0, V 0

      Ta 0

      Ta 100

      Fig.3 (f)

      Figs. 3(a),(b),(c),(d),(e) and (f) illustrate the variation of Rayleigh number with respect to the wave number for different parameters combinations. It is clear that as an increase in Taylor number, , results in an increase in the value of critical Rayleigh number, , and the critical wave number, , and also it is clear that if increases the critical Rayleigh number and the critical wave number

      500

      Ta 1000

      decreases. The negative value of represents the heat sink

      and the positive value corresponds to the heat source. Clearly,

      0.0 0.5 1.0 1.5 2.0 2.5

      Fig.3 (c)

      3000

      RI 0, V 0.5

      2500

      2000

      R

      1500

      the wave number decreases with increase in , for both the heat source and heat sink.

    5. CONCLUSION

The paper presents an analytical study of Rayleigh-Benard convection with variable heat source. Closed form expressions for and are otained as functions of the parameters of the problem. It is found that one can regulate the flow by suitably tuning the parameters involved in the problem. The effect of increasing the strength of rotation is to stabilize the

1000

500

Ta 0

Ta 100

Ta 1000

system where as the increasing the Thermo rheological parameter and the internal Rayleigh number lead to the destabilization of the system.

0.0 0.5 1.0 1.5 2.0 2.5

Fig.3 (d)

ACKNOWLEDGMENT

The authors are very much thankful to the Management and Principal of Ramaiah Institute of Technology, Bengaluru-54 and Prof. P. G. Siddheshwar, Department of Mathematics, Bangalore University, Bengaluru-56 for his valuable suggestions towards the preparation of the article.

NOMENCLATURE LATIN SYMBOLS

depth of the fluid layer

acceleration due to gravity

unit normal in direction

unit normal in direction

Prandtl number

pressure

velocity of the fluid, (, )

heat source / sink

internal Rayleigh number

external Rayleigh number

Ta Taylor number

GREEK SYMBOLS

wave number

thermal expansion coefficient

constant thermal diffusivity

constant of dynamic viscosity

two dimensional Laplacian

kinematic viscosity

phase angle

stream function

perturbed stream function

density

perturbed temperature

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