International Journal of Current Research and Review
ISSN: 2231-2196 (Print)ISSN: 0975-5241 (Online)
Bootstrap Slider

Indexed and Abstracted in: SCOPUS, Crossref, CAS Abstracts, Publons, CiteFactor, Open J-Gate, ROAD, Indian Citation Index (ICI), Indian Journals Index (IJINDEX), Internet Archive, IP Indexing, Google Scholar, Scientific Indexing Services, Index Copernicus, Science Central, Revistas Medicas Portuguesas, EBSCO, BOAI, SOROS, NEWJOUR, ResearchGATE, Ulrich's Periodicals Directory, DocStoc, PdfCast, getCITED, SkyDrive, Citebase, e-Print, WorldCat (World's largest network of library content and services), Electronic Journals Library by University Library of Regensburg, SciPeople.

Search Articles

Track manuscript

Readers around the world

Full Html

IJCRR - Vol 05 Issue 15 , August, 2013

Pages: 102-110

Date of Publication: 17-Aug-2013


Print Article   Download XML  Download PDF

THERMOSOLUTAL INSTABILITY OF AN INCOMPRESSIBLE, VISCOUS FLUID CONFINED IN A POROUS MEDIUM IN THE PRESENCE OF MAGNETIC FIELD

Author: Arti Bansal

Category: Technology

Abstract:The paper critically examines, within the framework of linear analysis, thermosolutal instability of an incompressible, viscous fluid confined in a porous medium in the presence of magnetic field, analytically and numerically both. The analytical discussion provides the sufficient conditions of stability and instability and the characterization of modes. By actually calculating the root of eigenvalue equation (of degree 4) neutral stability curves are drawn. The numerical results show the effect of various physical parameters on the critical wave number ac. It is concluded that RD-1 , S and R4 have stabilizing character and Richardson number J has destabilizing character. The non-dimensional parameter R2 shows a dual character, which depends upon thermal diffusivity ?

Keywords: Thermosolutal instability, porous medium, magnetic field

Full Text:

INTRODUCTION
The problems on thermal instability (Bénard convection) in a fluid layer under varying assumptions of hydrodynamics and hydromagnetic have been discussed in detail by Chandrasekhar[1] in his celebrated monograph. The problem of thermosolutal instability in fluids through a porous medium is of importance in geophysics, soil sciences, ground-water hydrology and astrophysics. The development of geothermal power resources holds increased general interest in the study of the properties of convection in a porous medium.The instability of fluid flows in a porous medium under varying assumptions has been well summarized by Scheidegger [2] and Yih [3]. While investigating the flows or flow instabilities through porous media, the liquid flow has been assumed to be governed by Darcy’s Law [4] by most of the research workers, which neglects the inertial forces on the flow. Beavers et al. [5] demonstrated experimentally the existence of shear within the porous medium near surface, where the porous medium is exposed to a freely flowing fluid, thus forming a zone of shearinduced flow field. Darcy’s equation however, cannot predict the existence of such a boundary zone, since no macroscopic shear term is included in this equation (Joseph and Tao [6]). To be mathematically compatible with the Navier-Stokes equations and physically consistent with the experimentally observed boundary shear zone mentioned above, Brinkman [7] proposed the introduction of the term 1 k μ V in addition to μ 2  V in the equations of fluid motion. The thermosolutal convection in a porous medium was studied by Nield and Bejan [8]. Instability of compressible or incompressible flow has been studied extensively by a number of research workers in past few decades. In almost all such investigations, the Boussines’q approximation is used to simplify the equations of motion. Goel et.al. [9] examined the shear flow instability of an incompressible viscoelastic second order fluid in a porous medium in which the modified Darcy’s law is replaced by the celebrated Brinkman model so that both the inertia and viscous terms are included in their usual forms. The behaviour of conducting fluid is very much different in the absence and in the presence of a magnetic field. The interesting properties associated with a magnetic field, have attracted a number of research workers to work in this direction. Bansal and Agrawal [10] have studied the thermal instability of a compressible shear flow in the presence of a weak applied magnetic field. The problem for compressible shear layer in the presence of a weak applied magnetic field through porous medium has been studied by Bansal et.al. [11]. In the present paper, an attempt has been made to examine the thermosolutal instability of an incompressible, viscous fluid in the presence of magnetic field and confined in a porous medium following Brinkman model. The Boussinesq approximation is used throughout the paper. It states that variations of density in the equations of motion can safely be ignored everywhere except its association with the external force. The approximation is well justified in the case of incompressible fluids.

FORMULATION OF THE PROBLEM
Here we consider an infinite, horizontal, incompressible, viscous fluid saturating an isotropic porous medium and which is subjected to uniform magnetic field in the horizontal direction. Uniform temperature and concentration gradients are maintained along z-axis. Equations expressing the conservation of momentum, mass, magnetic field, temperature, solute mass concentration and equation of






NUMERICAL RESULTS AND DISCUSSION
Eq.(10) is a fourth degree equation in ? with real coefficients, which depends upon RD -1 , S, J, J’, R2, R3, R4, l and a. Our aim has been to examine the effect of various parameters such as RD -1 , R2, R4, magnetic force number S and the Richardson number J on the stability of the system. This has been achieved by actually calculating the roots of the equation (3.2) correct upto three decimal places. The calculation of critical wave numbers has lead to the neutral stability curves. Fig.2. shows the critical wave number ac for different value of RD -1 (curve-I). When RD -1 =0, for a < 1.321 unstable modes are non-oscillatory i.e., the eigenvalue equation has all real roots with at least one positive root. For 1.321 < a < 2.153 unstable modes are oscillatory and system becomes stable for a ≥ 2.153 . As RD -1 increases, ac decreases so that the range of stable wave numbers increases. It is concluded that RD -1 has a stabilizing character and a large value of RD -1 is required to stabilize the system for all wave numbers. Curve-II separates the unstable modes into unstable oscillatory and unstable nonoscillatory modes. As RD -1 increases, the range of both the unstable oscillatory and the unstable nonoscillatory modes decreases. Fig.3. shows the stabilizing character of magnetic field (curve-I). Short wave length perturbations are more stable. The unstable modes are divided into oscillatory and non-oscillatory modes. The large wave length perturbations are unstable and nonoscillatory and the modes which are intermediatory between stable and unstable nonoscillatory modes are unstable and oscillatory, i.e., a > ac are stable, ac* < a < ac are unstable and oscillatory and a < ac* are unstable and nonoscillatory. It is important to observe that whereas the unstable modes are non-oscillatory in the absence of magnetic field, some oscillatory unstable modes are introduced in its presence . Fig.4. shows the destabilizing character of J. As J increases, ac increases which decreases the range of stable wave numbers. As is clear from this figure, the range of both the non-oscillatory unstable and the oscillatory unstable modes increases with J. Fig.5. shows the dual character of the nondimensional parameter R2, which in fact, depends upon the thermal diffusivity ?. As R2 increases from 0 to 0.3, the range of stable wave increases. As R2 increases from 0.3 to 3.5 approximately, the range of unstable wave numbers decreases rapidly, however, this decrease in the range of unstable wave numbers is slow as R2 further increases beyond 3.5. The neutral stability curve also shows that the critical wave number for maximum instability is given by ac = 2.081 and it occurs for R2 = 0.3. Similar character is exhibited for R3. Stabilizing character of fluid viscosity is exhibited in Fig.6. As R4 increases, the range of stable wave numbers increases sharply upto R4 = 4 (approx.) and then increases slowly as R4 further increases.

CONCLUSION
The analytical discussion provides the sufficient conditions of stability and instability and the characterization of modes. Theorem 2 states that non-oscillatory modes are stable if ? ? 0 and 0 ' ? ? . This situation is known as potentially stable arrangement. The numerical results show the effect of various physical parameters on the critical wave number ac . On the basis of numerical discussion & neutral stability curves obtained in paper, it is concluded that medium porosity parameter RD -1 , magnetic force number S and fluid viscosity have stabilizing character and Richardson number J has destabilizing character. The non-dimensional parameter R2, which depends upon thermal diffusivity ?, shows a dual character,

ACKNOWLEDGEMENT
The author is grateful to Professor S.C. Agrawal, Retd. H.O.D., Department of Mathematics, C.C.S. University, Meerut, for providing valuable support.

References:

1. Chandrasekhar S., Hydrodynamic and Hydromagnetic Stability. – Oxford, UK: Clarendon Press, 1961.

2. Scheidegger A. E. Growth of instabilities on displacement fronts in porous media. Phys Fluids 1960; 3 94.

3. Yih C. S. Flow of a non-homogeneous fluid in a porous medium. J. Fluid Mech. 1961; 10 133.

4. H. Darcy Les fontaines publiques de la ville de Dijon Paris : Victor Dalmont (1856).

5. Beavers G.S., Sparrow E.M. and Magnuson R.A. , Experiments on coupled parallel flows in a channels and a bounding porous medium. – J. Basic Engng. Trans. 1970, ASME, vol.D92, pp.843-845.

6. Joseph D.D. and Tao L.N., The effect of permeability on the slow motion of a porous sphere in a viscous liquid, 1964.

7. Brinkman H. C. A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles 1947; Appl. Sci. Res., A1 27.

8. Nield D. A. and Bejan A. Convection in porous medium 1992; Springer.

9. Goel A. K., Agrawal S. C. and Jaimala shear flow instability of an incompressible viscoelastic second order fluid in a porous medium. Indian J. pure appl. Math. 1997 28(4) 563.

10. Bansal N. and Agrawal S. C. thermal instability of a compressible shear flow in the presence of a weak applied magnetic field. Proc. Nat. Acad. Sci. India 1999 ;69(A) III .

11. Bansal A., Bansal N. and Agrawal S. C. Thermal instability of compressible shear layer in a porous medium in the presence of a weak applied magnetic field. Ganita 2004; vol.55, no.2,187-197.

12. Goel A. K., Agrawal S. C. and Agarawal G.S. Hydromagnetic stability of an unbounded couple stress binary fluid mixture having vertical temperature and concentration gradients with rotation.Indian J. pure appl. Math. 1997; 30(10) 991.




Research Incentive Schemes

Awards, Research and Publication incentive Schemes by IJCRR

Best Article Award: 

One article from every issue is selected for the ‘Best Article Award’. Authors of selected ‘Best Article’ are rewarded with a certificate. IJCRR Editorial Board members select one ‘Best Article’ from the published issue based on originality, novelty, social usefulness of the work. The corresponding author of selected ‘Best Article Award’ is communicated and information of award is displayed on IJCRR’s website. Drop a mail to editor@ijcrr.com for more details.

Women Researcher Award:

This award is instituted to encourage women researchers to publish her work in IJCRR. Women researcher, who intends to publish her research work in IJCRR as the first author is eligible to apply for this award. Editorial Board members decide on the selection of women researchers based on the originality, novelty, and social contribution of the research work. The corresponding author of the selected manuscript is communicated and information is displayed on IJCRR’s website. Under this award selected women, the author is eligible for publication incentives. Drop a mail to editor@ijcrr.com for more details.

Emerging Researcher Award:

‘Emerging Researcher Award’ is instituted to encourage student researchers to publish their work in IJCRR. Student researchers, who intend to publish their research or review work in IJCRR as the first author are eligible to apply for this award. Editorial Board members decide on the selection of student researchers for the said award based on originality, novelty, and social applicability of the research work. Under this award selected student researcher is eligible for publication incentives. Drop a mail to editor@ijcrr.com for more details.


Best Article Award

A Study by Jabbar Desai et al. entitled "Prevalence of Obstructive Airway Disease in Patients with Ischemic Heart Disease and Hypertension" is awarded Best Article for Vol 12 issue 17
A Study by Juna Byun et al. entitled "Study on Difference in Coronavirus-19 Related Anxiety between Face-to-face and Non-face-to-face Classes among University Students in South Korea" is awarded Best Article for Vol 12 issue 16
A Study by Sudha Ramachandra & Vinay Chavan entitled "Enhanced-Hybrid-Age Layered Population Structure (E-Hybrid-ALPS): A Genetic Algorithm with Adaptive Crossover for Molecular Docking Studies of Drug Discovery Process" is awarded Best article for Vol 12 issue 15
A Study by Varsha M. Shindhe et al. entitled "A Study on Effect of Smokeless Tobacco on Pulmonary Function Tests in Class IV Workers of USM-KLE (Universiti Sains Malaysia-Karnataka Lingayat Education Society) International Medical Programme, Belagavi" is awarded Best article of Vol 12 issue 14, July 2020
A study by Amruta Choudhary et al. entitled "Family Planning Knowledge, Attitude and Practice Among Women of Reproductive Age from Rural Area of Central India" is awarded Best Article for special issue "Modern Therapeutics Applications"
A study by Raunak Das entitled "Study of Cardiovascular Dysfunctions in Interstitial Lung Diseas epatients by Correlating the Levels of Serum NT PRO BNP and Microalbuminuria (Biomarkers of Cardiovascular Dysfunction) with Echocardiographic, Bronchoscopic and HighResolution Computed Tomography Findings of These ILD Patients" is awarded Best Article of Vol 12 issue 13 
A Study by Kannamani Ramasamy et al. entitled "COVID-19 Situation at Chennai City – Forecasting for the Better Pandemic Management" is awarded best article for  Vol 12 issue 12
A Study by Muhammet Lutfi SELCUK and Fatma COLAKOGLU entitled "Distinction of Gray and White Matter for Some Histological Staining Methods in New Zealand Rabbit's Brain" is awarded best article for  Vol 12 issue 11
A Study by Anamul Haq et al. entitled "Etiology of Abnormal Uterine Bleeding in Adolescents – Emphasis Upon Polycystic Ovarian Syndrome" is awarded best article for  Vol 12 issue 10
A Study by Arpita M. et al entitled "Estimation of Reference Interval of Serum Progesterone During Three Trimesters of Normal Pregnancy in a Tertiary Care Hospital of Kolkata" is awarded best article for  Vol 12 issue 09
A Study by Ilona Gracie De Souza & Pavan Kumar G. entitled "Effect of Releasing Myofascial Chain in Patients with Patellofemoral Pain Syndrome - A Randomized Clinical Trial" is awarded best article for  Vol 12 issue 08
A Study by Virendra Atam et. al. entitled "Clinical Profile and Short - Term Mortality Predictors in Acute Stroke with Emphasis on Stress Hyperglycemia and THRIVE Score : An Observational Study" is awarded best article for  Vol 12 issue 07
A Study by K. Krupashree et. al. entitled "Protective Effects of Picrorhizakurroa Against Fumonisin B1 Induced Hepatotoxicity in Mice" is awarded best article for issue Vol 10 issue 20
A study by Mithun K.P. et al "Larvicidal Activity of Crude Solanum Nigrum Leaf and Berries Extract Against Dengue Vector-Aedesaegypti" is awarded Best Article for Vol 10 issue 14 of IJCRR
A study by Asha Menon "Women in Child Care and Early Education: Truly Nontraditional Work" is awarded Best Article for Vol 10 issue 13
A study by Deep J. M. "Prevalence of Molar-Incisor Hypomineralization in 7-13 Years Old Children of Biratnagar, Nepal: A Cross Sectional Study" is awarded Best Article for Vol 10 issue 11 of IJCRR
A review by Chitra et al to analyse relation between Obesity and Type 2 diabetes is awarded 'Best Article' for Vol 10 issue 10 by IJCRR. 
A study by Karanpreet et al "Pregnancy Induced Hypertension: A Study on Its Multisystem Involvement" is given Best Paper Award for Vol 10 issue 09
Late to bed everyday? You may die early, get depression
Egg a day tied to lower risk of heart disease
88 Percent Of Delhi Population Has Vitamin D Deficiency: ASSOCHAM Report

List of Awardees

Awardees of COVID-19 Research

Woman Researcher Award

A Study by Neha Garg et al. entitled "Optimization of the Response to nCOVID-19 Pandemic in Pregnant Women – An Urgent Appeal in Indian Scenario" published in Vol 12 issue 09

A Study by Sana Parveen and Shraddha Jain entitled "Pathophysiologic Enigma of COVID-19 Pandemic with Clinical Correlates" published in Vol 12 issue 13

A Study by Rashmi Jain et al. entitled "Current Consensus Review Article on Drugs and Biologics against nCOVID-19 – A Systematic Review" published in Vol 12 issue 09

Emerging Researcher Award

A Study by Madhan Jeyaraman et al. entitled "Vitamin-D: An Immune Shield Against nCOVID-19" published in Vol 12 issue 09

Study by Dheeraj Kumar Chopra et al. entitled "Lipid-Based Solid Dispersions of Azilsartan Medoxomil with Improved Oral Bioavailability: In Vitro and In Vivo Evaluation" published in Vol 12 issue 19


RSS feed

Indexed and Abstracted in


Antiplagiarism Policy: IJCRR strongly condemn and discourage practice of plagiarism. All received manuscripts have to pass through "Plagiarism Detection Software" test before forwarding for peer review. We consider "Plagiarism is a crime"

IJCRR Code of Conduct: We at IJCRR voluntarily adopt policies on Code of Conduct, and Code of Ethics given by OASPA and COPE. To know about IJCRRs Code of Conduct, Code of Ethics, Artical Retraction policy, Digital Preservation Policy, and Journals Licence policy click here

Disclaimer: International Journal of Current Research and Review (JICRR) provides platform for researchers to publish and discuss their original research and review work. IJCRR can not be held responsible for views, opinions and written statements of researchers published in this journal.



Company name

International Journal of Current Research and Review (JICRR) provides platform for researchers to publish and discuss their original research and review work. IJCRR can not be held responsible for views, opinions and written statements of researchers published in this journal

Contact

148, IMSR Building, Ayurvedic Layout,
        Near NIT Complex, Sakkardara,
        Nagpur-24, Maharashtra State, India

editor@ijcrr.com

editor.ijcrr@gmail.com


Copyright © 2021 IJCRR. Specialized online journals by ubijournal .Website by Ubitech solutions