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Thermosolutal Convection of Micropolar Rotating Fluids Saturating a Porous Medium

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Thermosolutal Convection of Micropolar Rotating Fluids Saturating a Porous Medium THERMOSOLUTAL CONVECTION OF MICROPOLAR ROTATING FLUIDS SATURATING A POROUS MEDIUM Reena* Department of Mathematics, Shri K.K. Jain (P.G.) College Khatauli, Distt. Muzaffarnagar 251001, Uttar Pradesh, India [email protected] - [email protected] U.S. Rana Department of Mathematics, D.A.V. (P.G.) College Dehradun 248001, Uttarakhand, India [email protected] *Corresponding Author (Received: March 24, 2008 – Accepted in Revised Form: February 19, 2009) Abstract Double-diffusive convection in a micropolar fluid layer heated and soluted from below in the presence of uniform rotation saturating a porous medium is theoretically investigated. An exact solution is obtained for a flat fluid layer contained between two free boundaries. To study the onset of convection, a linear stability analysis theory and normal mode analysis method have been used. For the case of stationary convection, the effect of various parameters like medium permeability, solute gradient, rotation and micropolar parameters (i.e. coupling, spin diffusion, micropolar heat conduction and micropolar solute parameters arising due to coupling between spin and solute fluxes) have been analyzed. The critical thermal Rayleigh numbers for various values of critical wave numbers (found by Newton Raphson method) for the onset of instability are determined numerically and depicted, graphically. The oscillatory modes were introduced due to the presence of the micropolar viscous effects, microinertia, rotation and stable solute gradient, which were non-existence in their absence. The principle of exchange of stabilities is found to hold true for the micropolar fluid saturating a porous medium heated from below in the absence of micropolar viscous effect, microinertia, rotation and stable solute gradient. An attempt was also made to obtain sufficient conditions for the non- existence of overstability. Keywords Thermosolutal Convection, Porous Medium, Rotation Effect, Micropolar Fluids, Medium Permeability, Stable Solute Gradient, Rayleigh Number ﭼﻜﻴﺪﻩ ﺍﻧﺘﻘﺎﻝ ﺣﺮﺍﺭﺕ ﺟﺎﺑﺠﺎﻳﻲ ﭘﺨﺶ ﺩﻭﮔﺎﻧﻪ ﺩﺭ ﻳﮏ ﻻﻳﻪ ﺳﻴﺎﻝ ﻣﻴﮑﺮﻭﻗﻄﺒﻲ ﮐﻪ ﺍﺯ ﭘﺎﻳﻴﻦ ﺗﺤﺖ ﺣﺮﺍﺭﺕ ﻭ ﺣﻼﻟﻴﺖ ﻗﺮﺍﺭ ﮔﺮﻓﺘﻪ ﻭ ﺑﺎ ﭼﺮﺧﺶ ﺛﺎﺑﺖ، ﻣﺤﻴﻄﻲ ﻣﺘﺨﻠﺨﻞ ﺭﺍ ﺍﺷﺒﺎﻉ ﻣﻲ ﮐﻨﺪ، ﺑﻪ ﻃﻮﺭ ﻧﻈﺮﻱ ﻣﻄﺎﻟﻌﻪ ﺷﺪﻩ ﺍﺳﺖ. ﺣﻞ ﺩﻗﻴﻖ ﺑﺮﺍﻱ ﻻﻳﻪ ﺗﺨﺖ ﺳﻴﺎﻝ ﻣﺤﺪﻭﺩ ﺷﺪﻩ ﺑﻴﻦ ﺩﻭ ﻣﺮﺯ ﺁﺯﺍﺩ، ﺑﻪ ﺩﺳﺖ ﺁﻣﺪﻩ ﺍﺳﺖ. ﺍﺯ ﺗﺌﻮﺭﻱ ﺗﺤﻠﻴﻞ ﭘﺎﻳﺪﺍﺭﻱ ﺧﻄﻲ ﻭ ﺭﻭﺵ ﺗﺤﻠﻴﻞ ﻭﺿﻌﻴﺖ ﻧﺮﻣﺎﻝ ﺑﺮﺍﻱ ﻣﻄﺎﻟﻌﻪ ﺷﺮﻭﻉ ﺟﺎﺑﺠﺎﻳﻲ ﺍﺳﺘﻔﺎﺩﻩ ﺷﺪﻩ ﺍﺳﺖ. ﺗﺎﺛﻴﺮ ﭘﺎﺭﺍﻣﺘﺮﻫﺎﻱ ﻣﺨﺘﻠﻔﻲ ﻣﺜﻞ ﻗﺎﺑﻠﻴﺖ ﻧﻔﻮﺫ ﻣﺤﻴﻂ ﻣﺘﺨﻠﺨﻞ، ﺗﻐﻴﻴﺮﺍﺕ ﺣﻼﻟﻴﺖ، ﭼﺮﺧﺶ ﻭ ﻋﻮﺍﻣﻞ ﻣﻴﮑﺮﻭ ﻗﻄﺒﻲ (ﮐﻮﭘﻠﻴﻨﮓ، ﻧﻔﻮﺫ spin، ﻫﺪﺍﻳﺖ ﺣﺮﺍﺭﺗﻲ ﻣﻴﮑﺮﻭﻗﻄﺒﻲ ﻭ ﺣﻼﻟﻴﺖ ﻣﻴﮑﺮﻭﻗﻄﺒﻲ ﮐﻪ ﺑﻪ ﺩﻟﻴﻞ ﮐﻮﭘﻠﻴﻨﮓ ﺑﻴﻦ ﺷﺎﺭ ﺣﻼﻟﻴﺖ ﻭ ﭼﺮﺧﺶ ﺍﻓﺰﺍﻳﺶ ﻣﻲ ﻳﺎﺑﻨﺪ) ﺑﺮ ﺣﺎﻟﺖ ﺟﺎﺑﺠﺎﻳﻲ ﺳﺎﮐﻦ، ﺗﺤﻠﻴﻞ ﺷﺪﻩ ﺍﺳﺖ. ﻋﺪﺩ ﺭﺍﻳﻠﻲ ﺣﺮﺍﺭﺗﻲ ﺑﺤﺮﺍﻧﻲ ﻭ ﻋﺪﺩ ﻣﻮﺝ ﺑﺤﺮﺍﻧﻲ ﺑﺮﺍﻱ ﺷﺮﻭﻉ ﻧﺎﭘﺎﻳﺪﺍﺭﻱ ﺑﻪ ﺩﺳﺖ ﺁﻣﺪﻩ ﻭ ﺩﺭ ﻧﻤﻮﺩﺍﺭﻫﺎ ﻧﺸﺎﻥ ﺩﺍﺩﻩ ﺷﺪﻩ ﺍﻧﺪ. ﻣﺸﺨﺺ ﺷﺪﻩ ﺍﺳﺖ ﮐﻪ ﺣﻼﻟﻴﺖ ﭘﺎﻳﺪﺍﺭ ﻣﻮﺩﻫﺎﻱ ﻧﻮﺳﺎﻧﻲ ﺑﻪ ﺩﻟﻴﻞ ﺣﻀﻮﺭ ﺍﺛﺮﺍﺕ ﻭﻳﺴﮑﻮﺯ ﻣﻴﮑﺮﻭﻗﻄﺒﻲ، ﻣﻴﮑﺮﻭﺍﻳﻨﺮﺳﻲ، ﭼﺮﺧﺶ ﻭ ﮔﺮﺍﺩﻳﺎﻥ ﺑﻪ ﻭﺟﻮﺩ ﻣﻲ ﺁﻳﻨﺪ ﮐﻪ ﺍﻳﻦ ﻣﻮﺩﻫﺎ ﺩﺭ ﺣﺎﻟﺖ ﻋﺪﻡ ﺣﻀﻮﺭ ﺍﻳﻦ ﻋﻮﺍﻣﻞ ﻭﺟﻮﺩ ﻧﺪﺍﺷﺘﻨﺪ. ﺍﺻﻞ ﺗﺒﺎﺩﻝ ﭘﺎﻳﺪﺍﺭﻱ ﻫﺎ ﺑﺮﺍﻱ ﺳﻴﺎﻝ ﻣﻴﮑﺮﻭﻗﻄﺒﻲ ﺍﺷﺒﺎﻉ ﺷﺪﻩ ﺩﺭ ﻣﺤﻴﻂ ﻣﺘﺨﻠﺨﻞ ﺑﺎ ﮔﺮﻣﺎﻳﺶ ﺍﺯ ﺯﻳﺮ ﺩﺭ ﻏﻴﺎﺏ ﺍﺛﺮﺍﺕ ﻭﻳﺴﮑﻮﺯ ﻣﻴﮑﺮﻭﻗﻄﺒﻲ، ﻣﻴﮑﺮﻭﺍﻳﻨﺮﺳﻲ، ﭼﺮﺧﺶ ﻭ ﮔﺮﺍﺩﻳﺎﻥ ﺣﻼﻟﻴﺖ ﭘﺎﻳﺪﺍﺭ ﻫﻤﭽﻨﺎﻥ ﺻﺎﺩﻕ ﺍﺳﺖ. ﺗﻼﺵ ﺷﺪ ﺷﺮﺍﻳﻂ ﻣﻨﺎﺳﺐ ﺑﺮﺍﻱ ﻋﺪﻡ ﺍﻳﺠﺎﺩ ﻓﻮﻕ ﭘﺎﻳﺪﺍﺭﻱ ﻧﻴﺰ ﺑﻪ ﻭﺟﻮﺩ ﺁﻳﺪ. 1. INTRODUCTION presented by Eringen [1,2]. According to Eringen [1], a subclass of microfluids Eringen [3] which A general theory of micropolar fluids has been exhibit the micro-rotational effects and micro- IJE Transactions A: Basics Vol. 22, No. 4, November 2009 - 379 rotational inertia is the micropolar fluid. Certain particles in the lubrication of journal and rolling anisotropic fluids, e.g., liquid crystals which are contact bearings considering cavitation. Excellent made up of dumb bell molecules are of this type. In work in micropolar simulations of human joint fact, animal blood happens to fall into this category. lubrication has been reported by Sinha, et al Other polymeric fluids and fluids containing certain [11]. He also presented a micropolar hip joint additives may be represented by the mathematical lubrication model. Chandra, et al [14] presented an model underlying micropolar fluids. Compared to analysis for blood flow through a narrow artery the classical Newtonian fluids, micropolar fluids are with mild stenosis by considering Eringen’s simple characterised by two supplementary variables, i.e., micro fluid model for blood. Power [15] was led to the spin, responsible for the micro-rotations and the use the theory of micropolar fluid for modelling micro-inertia tensor describing the distributions of the CSF (Cerebral Spinal Fluid) for low Reynolds atoms and molecules inside the fluid elements in number approximation solving linear Fredholm addition to the velocity vector. Liquid crystals, integral equations. A large number of references colloidal fluids, polymeric suspension, animal about modelling and applications aspect of blood, etc. are few examples of micropolar fluids micropolar fluids has been given in the book [16]. Labon, et al [4]. Kazakia, et al [5] and Eringen [6] In the review paper [17], the most comprehensive extended this theory of structure continue to account discussion of applications of fluids with for the thermal effects. microstructure, micropolar fluids in particular are Micropolar fluids abound in engineering science tabularized. Recently some work has been done on and some common examples are human blood, viscoelastic micropolar fluid by Eremeyev, et al plasma, sediments in rivers, drug suspension in [18-20]. The literature concerning applications of pharmacology, liquid crystals, etc. The past four micropolar fluids in engineering sciences is vast decades have seen an incredible interest emerge in and still quickly growing. applications of these fluid theories to numerous The theory of thermomicropolar convection began problems in engineering sciences ranging from with Datta, et al [21] and interestingly continued by biofluid mechanics of blood vessels to sediment Ahmadi [22]. Labon, et al [4], Bhattacharya, et al transport in rivers and lubrication technology. It [23], Payne, et al [24], Sharma, et al [25,26] and has wide applications in the developments of Rama Rao [27]. The above works give a good micropolar biomechanical flows and as such is understanding of thermal convection in micropolar both an engineering science one. The more fluids. The Rayleigh-Benard instability in a applications of micropolar fluid may include horizontal thin layer of fluid heated from below is lubrication theory, boundary layer theory, short an important particular stability problem. A waves for heat conducting fluids, hydrodynamics detailed account of Rayleigh-Benard instability in a of multicomponent media, magnetohydrodynamics horizontal thin layer of Newtonian fluid heated from and electrohydrodynamics, biological fluid below under varying assumptions of hydrodynamics modelling, etc. Hydrodynamics of micropolar and hydromagnetics has been given by Chandrasekhar fluids has significant applications to a variety [28]. Perez-Garcia, et al [29] have extended the of different fields of physics and engineering effects of the microstructures in the Rayleigh- (magnetohydrodynamics, tribology, etc.). We are Benard instability and have found that in the absence highlighted some key areas of applications in of coupling between thermal and micropolar effects, Figures 1 and 2. the Principle of Exchange of Stabilities (PES) Rosenberg [7] has developed the variational holds good. Perez-Garcia, et al [30] have shown formulation of the boundary value problems of the that when coupling between thermal and micropolar continuum and hence applied to the micropolar effect is present, the Principle of modelling of the bone. Micropolar fluid theory has Exchange of Stabilities (PES) may not be fulfilled been applied successfully by many authors, and hence oscillatory motions are present in Shukla, et al [8], Sinha, et al [9,10] and Sinha, et al micropolar fluids. The effect of rotation on thermal [11,12] ,to study various problems in lubrication. convection in micropolar fluids is important in Prakash et al [13] applied micropolar fluid certain chemical engineering and biochemical theory to study theoretically the effects of solid situations. Qin, et al [31] has considered a thermal 380 - Vol. 22, No. 4, November 2009 IJE Transactions A: Basics Paint Rheology Sedimentary Contaminated (Mud, Sand) and Clean Geomorpho- Engine Logical Fluids Lubricants Micropolar Liquid Fluid Crystal Applications in Colloids and Systems Engineering Polymeric Suspensions Radial Blood Flows Diffusion and Thrust Bearing Technologies Figure 1. Micropolar fluid applied in engineering. Arterial Blood Flows and Cardiovascular Flows Peristaltic Synovial Transport by Lubrication Micropolar (Kneecap Fluid Pumping Mechanics) Micropolar Hydrodynamics Applications in Bioengineering Pharmaco- Cervical Flows Dynamics of (Spermatozoa Drug Delivery Propulsion Dynamics) Figure 2. Micropolar hydrodynamics applied in bioengineering. IJE Transactions A: Basics Vol. 22, No. 4, November 2009 - 381 instability problem in a rotating micropolar fluid. solid and pores, porous rollers, etc. In addition, They found that the rotation has a stabilizing these flows are applicable to bio-mathematics effect. The effect of rotation on thermal convection particularly in the study of blood flow in lungs, in micropolar fluids has also been studied by arteries, cartilage and so on. The study of a layer of Sharma, et al [32], whereas the numerical solution a fluid heated from below in porous media is of thermal instability of a rotating micropolar fluid motivated both theoretically as
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