Conference Proceeding

Mathematics in Space and Applied Sciences (ICMSAS-2023)
ICMSAS-2023

Subject Area: Mathematics
Pages: 331
Published On: 03-Mar-2023
Online Since: 04-Mar-2023

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Author(s): Ramesh Chand, S. K. Kango

Email(s): rameshnahan@yahoo.com

Address: Ramesh Chand1*, S. K. Kango2
1Department of Mathematics, Atal Bihari Vajpayee Government Degree College Bangana, HP, India
2Department of Mathematics, NSCBM Government College Hamirpur, HP, India
*Corresponding Author

Published In:   Conference Proceeding, Mathematics in Space and Applied Sciences (ICMSAS-2023)

Year of Publication:  March, 2023

Online since:  March 04, 2023

DOI:




Thermal Instability of Nanofluid Layer in A Porous Medium with Variable Viscosity

 

Ramesh Chand1*, S. K. Kango2

1Department of Mathematics, Atal Bihari Vajpayee Government Degree College Bangana, HP, India

2Department of Mathematics, NSCBM Government College Hamirpur, HP, India

*Corresponding Author E-mail: rameshnahan@yahoo.com

 

ABSTRACT:

Thermal instability of nanofluid with variable viscosity in a porous medium for more realistic boundary conditions is investigated theoretically. For porous medium the Darcy model is considered. The model used incorporates the effect of Brownian diffusion and thermophoresis. The eigen value problem is solved by employing the Galerkin weighted residuals method. The influence of the variable viscosity parameter, Lewis number, nanoparticle Rayleigh number, modified diffusivity ratio and porosity parameter on the stationary convection studied analytically and found that variable viscosity parameter, Lewis number, modified diffusivity ratio and nanoparticle Rayleigh number destabilizes while porosity parameter stabilize the stationary convection.

 

KEYWORDS: Nanofluid, variable viscosity parameter, Lewis number, porous medium, Galerkin method.

 

NOMENCLATURES:

a                              dimensionless resultant wave number  

d                              thickness of fluid layer

D                             differential operator

DB                                         Brownian diffusion coefficient  

DT                           thermophoretic diffusion coefficient

g                              acceleration due to gravity

km                           thermal conductivity

Le                           Lewis number

n                              growth rate of disturbances

NA                           modified diffusivity ratio

NB                                        modified particle -density increment

p                              pressure

q                                             Darcy velocity vector

Ra                                        thermal Rayleigh number

Rac                                      critical Rayleigh number

Rn                                       concentration Rayleigh number

t                               time

T                             temperature

T0                            temperature at z = 0

T1                            temperature at z = d

(u, v, w)     Darcy velocity components

(x, y, z) space co-ordinates

Greek symbols:

α                              coefficient of the thermal expansion

μ                              viscosity

ε                              porosity

ρ                             density of the nanofluid

ρ0                            density of nanofluid at z = 0

ρp                           density of nanoparticles

ρf                            density of base fluid

(ρc) m                                heat capacity  of  fluid in porous  medium                      

(ρc)p                                  heat capacity of  nanoparticles                     

φ                             volume fraction  of the nanoparticles 

φ0                            reference volume fraction  of the nanoparticles  at z = 0

                            thermal diffusivity

ω                             dimensionless frequency of oscillation

σ                              thermal capacity ratio

Γ                             viscosity parameter

                        horizontal Laplacian operator

                        Laplacian operator

 

Superscripts:

 '                              non - dimensional variables

' '                             perturbed quantities

 

Subscripts:

c                              critical

s                              stationary convection

p                              particle

b                              basic state

f                              fluid

 

1. INTRODUCTION:

Nanofluid is a fluid colloidal mixture of nanosized particles, in base fluid. Nanoparticles are typically made of oxide ceramics (Al2O3, CuO), metal carbides (SiC), nitrides (AlN, SiN) or metals (Al, Cu) etc in base fluids are water, ethylene or tri-ethylene-glycols and other coolants, oil and other lubricants, bio-fluids, polymer solutions, other common fluids. Typical dimension of the nanoparticles is in the range of a few to about 100 nm. The term ‘nanofluid’ was first coined by Choi [1]. Nanofluids have unique properties that make them potentially useful in many applications of heat transfer and considered to be the next-generation heat transfer fluids. The characteristic feature of nanofluid is thermal conductivity enhancement, a phenomena observed by Masuda et al. [2]. Philip and Shima [3], Keblinski et al. [4], Taylor et al. [5] reported the developments in the study of heat transfer using nanofluids. Buongiorno [6] studied almost all aspects of the convective transport in nanofluids. Tzou [7] studied the thermal instability problems of nanofluid using the method of eigen function expansion and observed that nanofluids are less stable than regular fluid. The detailed study of thermal convection in a layer of nanofluid in porous medium based upon Buongiorno’s model has been given by Kuznetsov and Nield [8], Nield and Kuznetsov [9], Chand and Rana [10], Rana et al. [11], Yadav et al. [12], [13] and Chand et al. [14]. In all the above studies boundary  condition  on volume  fraction  of  nanoparticles  is  physically  not  realistic  as  it  is  difficult  to  control  the nanoparticle volume fraction on the boundaries and suggested the normal flux of volume fraction  of  nanoparticles  is  zero  on  the  boundaries  as  an  alternative  boundary  condition which  is  physically  more  realistic. Nield and Kuznetsov [15] suggested that the value of the temperature can be imposed on the boundaries, but the nanoparticle fraction adjusts so that the nanoparticle flux is zero on the boundaries. With these new boundary conditions, authors [16], [17], [18], [19] studied thermal instability in a layer of nanofluid based upon Buongiorno’s model.

 

Nanofluid properties such as viscosity and thermal conductivity have a significant affects on the onset of thermal convective instability. The effects of conductivity, viscosity variation and cross-diffusion were studied by Nield and Kuznetsov [20]. Nield and Kuznetsov [21] also studied the effect of thermal conductivity and viscosity variation with the Brownian diffusion and the thermophoresis on the onset of nanofluid convection. Yadav et al. [22] extended the problem to double diffusive convection. It is also observed that fluid viscosity varies with temperature and it significantly affects the stability characteristic of the system. Dhananjaya and Shivkumara [23] studied the effects of the variable viscosity on the thermal instability of nanofluid layer in porous medium. In this paper an attempt has been made to study the thermal instability of a horizontal layer of nanofluid in porous medium with variable viscosity for more realistic boundary conditions in porous medium.

 

2. MATHEMATICAL FORMULATIONS OF THE PROBLEM:

Consider an infinite horizontal layer of nanofluid of thickness ‘d’ bounded by horizontal boundaries z = 0 and z = d. Fluid layer is acted upon by a gravity force g(0,0,-g) and is heated from below in such a way that horizontal boundaries z = 0 and  z = d respectively maintained at a uniform temperature T0 and T1 (T0 > T1) as shown is Fig.1.

 

The normal component of the nanoparticles flux has to vanish at an impermeable boundaries and the reference scale for temperature and nanoparticles fraction is taken to be T1 and φ0 respectively.

 

Fig.1 Physical configuration of the problem

 

The equation of continuity and motion for nanofluid with variable viscosity in a porous medium under the Boussinesq approximation [15], [23] are

                (1)

                (2)

where q(u, v, w) is the Darcy velocity vector, p is the hydrostatic pressure, α is the coefficient of thermal expansion, T the temperature of the nanofluid, φ is the volume fraction  of the nanoparticles, ρp density of nanoparticles, ρf density of base fluid and  is  stands for convection derivative.

 

The viscosity μ of nanofluid is assumed to be vary with temperature in the form

,                                                                                                                                                                (3)

where  are positive constants and T1 is the reference temperature.

 The equation of energy for nanofluid in porous medium is

,                                               (4)

where (ρc)m is effective heat capacity of  fluid, (ρc)p is heat capacity of  nanoparticles and  km is effective thermal conductivity of the porous medium.

The continuity equation for the nanoparticles is

,                                                                                                                              (5)




REFERENCES:

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[2]       H. Masuda, A. Ebata, K. Teramae, N. Hishinuma. Alternation of Thermal Conductivity and Viscosity of Liquid by Dispersing Ultra-Fine Particles (Dispersion of-Al2O3, SiO2 and TiO2 Ultra-Fine Particles), Netsu Bussei (Japan), vol. 4, pp. 227-233, 1993.

[3]        J. Philip, P.D. Shima. Thermal Properties of Nanofluids. Adv., Coll. and Interface Science, vol. 15, pp. 30-45, 2012.

[4]        P. Keblinski, L.W. Hu, J. L. Alvarado. A Benchmark Study on Thermal Conductivity of Nanofluid. J. Appl. Phys., vol. 106, pp. 094312, 2009. 

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[6]        J. Buongiorno. Convective Transport in Nanofluids. ASME Journal of Heat Transfer, vol. 128, pp. 240–250, 2006.

[7]        D.Y. Tzou. Thermal Instability of Nanofluids in Natural Convection, Int. Journal of Heat and Mass Transfer, vol. 51, pp. 2967-2979, 2008.

[8]        D.A. Nield, A.V. Kuznetsov. Thermal Instability in a Porous Medium Layer Saturated by a Nanofluid. Int. J. Heat Mass Trans. vol. 52, pp. 5796-5801, 2009.

[9]           A.V. Kuznetsov, D. A. Nield. Thermal Instability in a Porous Medium Layer Saturated by a Nanofluid: Brinkman Model. Trans. Porous Medium, vol. 81, pp. 409-422, 2010.

[10]     R. Chand, G. C. Rana. On The Onset of Thermal Convection in Rotating Nanofluid Layer Saturating a Darcy-Brinkman Porous Medium. Int. J. of Heat and Mass Transfer, vol. 55, pp. 5417-5424, 2012.

[11]     G.C. Rana, R. C. Thakur, S. K. Kango. On the Onset of Double-Diffusive Convection in a Layer of Nanofluid under Rotation Saturating a Porous Medium. Journal of Porous Media, vol. 17, no. 8, pp.657-667, 2014.

[12]     D. Yadav, G.S. Agrawal, R. Bhargava. Numerical solution of a thermal instability problem in a rotating nanofluid layer. Int. J Heat Mass Transf., vol. 63, pp. 313–322, 2013.

 [13]    D. Yadav, G.S. Agrawal, R. Bhargava.Thermal instability in a nanofluid layer with vertical magnetic field. J. Eng. Math. vol. 80, pp.147–164, 2013.

[14]     R. Chand, G.C. Rana, S. Kumar. Variable Gravity Effects on Thermal Instability of Nanofluid in Anisotropic Porous Medium. Int.  J. of Appl. Mech. and Engg., vol. 18, no. 3, pp. 631-642, 2013.

 [15]    D.A. Nield, A. V.  Kuznetsov. Thermal Instability in a Porous Medium Layer Saturated by a Nanofluid: A Revised Model, Int. J. of Heat and Mass Transfer, vol. 68, no. 4, pp.  211-214, 2014.

[16]     R. Chand, G.C. Rana. Magneto Convection in a Layer of Nanofluid in Porous Medium-A More Realistic Approach. Journal of Nanofluids, vol. 4, pp.196-202, 2015.

[17]     R. Chand, G.C. Rana. Thermal Instability in a Brinkman Porous Medium Saturated by Nanofluid with No Nanoparticle Flux on Boundaries, Special Topics & Reviews in Porous Media: An International Journal, vol. 5, no. 4, pp. 277-286, 2014.

[18]     R. Chand, S.K. Kango, G.C. Rana. Thermal Instability in Anisotropic Porous Medium Saturated by a Nanofluid-A Realistic Approach, NSNTAIJ, vol. 8, no. 12, pp. 445-453, 2014.

[19]     G.C. Rana, R. Chand.  On the Thermal Convection in a Rotating Nanofluid Layer Saturating a Darcy-Brinkman Porous Medium: A More Realistic Model, Journal of Porous Media, vol. 18, no. 6, pp. 629-635, 2015.

[20]     D. A. Nield, A.V. Kuznetsov. Effects of Temperature-Dependent Viscosity On Forced Convection in a Porous Medium: Layered Medium Analysis, Journal of Porous Media, vol. 6, pp. 213-222, 2003.

[21]     D. A. Nield, A.V. Kuznetsov. The Onset of Convection in a Layer of Porous Medium saturated by nanofluid: effects of conductivity and viscosity variation and cross-diffusion, Trans. Porous Media, vol. 92, pp. 837-846, 2012.

[22]     D. Yadav, G.S. Agrawal, R. Bhargava. The onset of double diffusive nanofluid convection in a layer of a saturated porous medium with thermal conductivity and viscosity variation. J. Porous media, vol. 16, pp.105-121, 2013.

[23]     M. Dhananjaya and I.S. Shivkumara. Effect of Variable Viscosity on Thermal Convective Instability in a Nanofluid Saturated Porous Layer, Int. Journal of Mathematical Archive, vol. 5, no. 3, pp. 38-46, 2014. 

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Author/Editor Information

Dr. Sanjay Kango

Department of Mathematics, Neta Ji Subhash Chander Bose Memorial, Government Post Graduate College, Hamirpur Himachal Pradesh-177 005, INDIA