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): Gian. C. Rana, Sanjay K. Kango, Poonam K. Gautam

Email(s): Email ID Not Available

Address: Gian. C. Rana1, Sanjay K. Kango1, Poonam K. Gautam2
1Department of Mathematics, NSCBM Government College, Hamirpur-177 005, Himachal Pradesh, India
2School of Basic Sciences (Mathematics), Bahra University, Waknaghat -173 234, Himachal Pradesh, 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:




Effect of suspended particles in a porous medium layer heated from below saturating a Jeffrey fluid: A Mathematical Theorem

 

Gian. C. Rana1, Sanjay K. Kango1, Poonam K. Gautam2

1Department of Mathematics, NSCBM Government College, Hamirpur-177 005, Himachal Pradesh, India

2School of Basic Sciences (Mathematics), Bahra University, Waknaghat -173 234, Himachal Pradesh, India

*Corresponding Author E-mail:

 

ABSTRACT:

In this study, the influence of suspended particles on thermal convection in a porous layer saturating a Jeffrey fluid is examined. Linear stability theory based on normal modes is employed to derive a mathematical theorem on thermal convection in a porous layer saturating a Jeffrey fluid which states that the viscoelastic thermal convection at marginal state cannot manifest as stationary convection if the thermal Rayleigh number R, the medium permeability parameter Pl, the Jeffrey parameter  and suspended particles parameter B, satisfy the inequality

.

 

KEYWORDS: Jeffrey fluid, porous medium, suspended dust particles, thermal convection.

 

1. INTRODUCTION:

The problem of thermal convection in porous media has received considerable attention over the last few decades due to its various uses in geophysics, food processing, soil science, groundwater hydrology, nuclear reactors, and more. A detailed description of the thermal instability of Newtonian fluids under various assumptions of fluid dynamics and magnetohydrodynamics was given by Chandrasekhar [1]. Lapwood [2] used a linearized stability theory to study convection in a porous medium. Rayleigh instability of the thermal boundary layer flowing through a porous medium was investigated by Wooding [3]. Scanlon and Segel [4] studied the effect of airborne particles on the initiation of Be′nard convection and found that the critical Rayleigh number was reduced simply because the heat capacity of the pure gas was supplemented by the particles. Suspended particles were found to destabilize the layer.

 

In all the above studies the fluid is considered to be Newtonian. The problem of thermal convection has been extensively studied for Newtonian liquids, but little attention has been paid to it for non-Newtonian liquids. Due to the increasing importance of non-Newtonian fluids containing suspended particles in modern technology and industry, the study of such fluids is desired. One such type of liquid is Jeffrey liquid. Jeffrey [5] studied the stability of a liquid layer heated from below. He has numerically solved several problems related to the stability of layers of incompressible liquids with increasing temperature. Various researchers have studied the Jeffrey fluid model and found it to be the best fluid model for representing the behavior of physiological and industrial fluids [6-10].

 

 

The study of porous media started with the simple Darcy model and gradually extended to the Darcy-Brinkman model. The problem of convection in porous media is well described by Vafai and Hadim [11], Ingham and Pop [12], Nield and Bejan [13]. Rana and  Kumar [14] studied the thermal instability of Rivlin-Ericksen elastico-viscous rotating fluid permeated with suspended particles and variable gravity field in porous medium whereas Rana and Thakur [15] derived a mathematical theorem for the initiation of couple stress fluids infiltrated with airborne dust particles that saturate the porous medium and found that airborne particles have an unstable effect on the physical system. The flow through the porous layer has a variety of practical uses. That is, the Earth's molten cores, oil reservoirs, tires, ropes, cushions, seats, sand floor flows, etc. Sandstone, limestone, human lungs, bile ducts, and gallbladder vessels with stones are some examples of natural porous media.

 

Interest in the study of non-Newtonian fluids is also motivated by a wide range of engineering applications, such as soil contamination with non-Newtonian chemicals such as lubricants and polymers, and treatment of sewage sludge on dry beds. Recently, polymers have been used in agricultural, telecommunications equipment, and biomedical applications. Examples of these applications include filtration processes, fixed bed reactors, adiabatic systems, ceramic treatment, enhanced oil recovery, and chromatography.

 

Given the importance of the various applications mentioned above, this study derives a mathematical theorem to study the effects of suspended particles on the thermal instability of incompressible Jeffrey fluids in porous media.

 

2              Mathematical model and perturbation equations:

Here, we consider an infinite, horizontal, incompressible Jeffrey fluid of depth d, bounded by the planes z = 0 and z = d in an isotropic and homogeneous medium of porosity  and permeability k1, which is acted upon by gravity g(0, 0, -g) as shown in figure 1.  This layer is heated from below to maintain a constant negative temperature gradient   The equilibrium characteristics of this initial static state are determined by assuming that the system is slightly perturbed and tracking its further evolution.




REFERENCES:

[1]          Chandrasekhar, S.,  Hydrodynamic and Hydromagnetic Stability, Dover Publication, New York, (1981).

[2]          Lapwood, E.R., Convection of a fluid in porous medium, Proc. Camb. Phil. Soc. 44 (1948) 508-519. 

[3]          Wooding, R.A., Rayleigh instability of a thermal boundary layer in flow through a porous medium, J. Fluid Mech. 9 (1960) 183-192.

[4]          Scanlon, J.W., Segel, L.A., Effect of suspended particles on the onset of Be′nard convection,  Physics Fluids 16 (1973) 1573-78.

[5]          Jeffrey, H.,  The stability of a layer of fluid heated below,  The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 2 (1926) 833-844.

[6]          Nallapu, S.  Radhakrishnamacharya, G., A.J. Chamkha, Flow of a Jeffrey fluid through a porous medium in narrow tubes, J. of Porous Media 18 (2015) 71–78.

[7]          Sushma, K., Sreenadh, S., Dhanalakshmi, P., Mixed convection flow of a Jeffrey nanofluid in a vertical channel, Middle-East Journal of Scientific Research 25 (2017) 950-959.

[8]          Imran, M.A.,  Miraj, F.,  Khana, I., Tlili, I., MHD fractional Jeffrey’s fluid flow in the presence of thermo diffusion, thermal radiation effects with first order chemical reaction and uniform heat flux, Results in Physics 10 (2018) 10–17.

[9]          Rana, G.C., Gautam, P.K., On the onset of thermal instability of a porous medium layer saturating a Jeffrey nanofluid, Engineering Transactions 70 (2022) 123-139.

[10]      G.C. Rana, P.K. Gautam, V. Sharma, Stability analysis of a non-newtonian nanofluid layer: Jeffrey model,  J. Theoretical and Applied Mechanics (2022) Accepted

[11]      Ingham, D.,  Pop, I.,  Transport Phenomena in Porous Media- Elsevier, New York, (1981).

[12]      Nield, D.A. , Bejan, A., Convection in Porous Medium- Springer, New York, 2006.

[13]      Vafai, K., Hadim, H.A., Hand Book of Porous Media- M. Decker, New York, 2000.

[14]      Rana, G.C., Kumar, S., Thermal instability of Rivlin-Ericksen elastico-viscous rotating fluid permeated with suspended particles and variable gravity field in porous medium, Studia Geotechnica et  Mechanica  XXXII (2010) 39-54.

[15]      Rana, G.C., Thakur, R.C.,  A mathematical theorem on the onset of Couple-Stress fluid permeated with suspended particles saturating a porous medium, Int. J. of Multiphysics 6 (2012) 61-72.

 



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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