FINAL 03.MAY.2014

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1WEL COME TO THE DISERTATION PRESENTATION ON THE ASCOLI-ARZELA THEOREM

By Yismaw A. JAN.25,2010By,S. Narasimha Murthy, Bahir Dar University.

Mixed Convection of Couple Stress Fluid in a Vertical Channel in the Presence of Heat Generation or Heat Absorption1Prof Narasimna Murthy @ BahirDar University OUT LINE OF THE PRESENTATIONIntroduction

Mathematical Formulation Isothermal-isothermal walls Isoflux-isothermal walls Isothermal-isoflux walls Results and Discussion

2Prof Narasimna Murthy @ BahirDar University 1. INTRODUCTIONThe study of non-Newtonian fluids has received much attention due to their many practical applications in medical sciences, engineering and technology, such as liquid crystals, fluid film lubrication etc. In the category of non-Newtonian fluids, couple stress fluid has distinct features such as polar effects in addition to possessing large viscosity. The consideration of couple stress in addition to Cauchy stress has led to the recent development of several theories of fluid micro-continua.

3Prof Narasimna Murthy @ BahirDar University Introduction Continued...4The couple stress fluid model has wide applications in bio-fluids, colloidal fluids and in engineering for pumping fluids such as synthetic lubricants. Studies on natural convection in a vertical channel with non-Newtonian fluid are relatively sparse compared to the problem with Newtonian fluid. Malashetty and Umavathi (1999) analyzed the effects of couple stresses on free convective flow in a vertical channel. Umavathi (2000) analyzed the free convection flow of electrically conducting couple stress fluid in a vertical channel. Recently, Umavathi (2002) studied the free convective couple stress fluid flow between vertical parallel plates in the presence of temperature dependent heat source. Umavathiet. al. (2004) also carried out the convective flow of two immiscible viscous and couple stress fluids through a vertical channel. In spite of the various applications of non-Newtonian fluids much work has not been found in the literature on mixed convection flows. Hence the present objective is to study the problem of mixed convection couple stress fluid in a vertical channel in the presence of heat generation or heat absorption.

Prof Narasimna Murthy @ BahirDar UniversityPHYSICAL CONFIGURATIONProf Narasimna Murthy @ BahirDar University5

2. Mathematical Formulation

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Prof Narasimna Murthy @ BahirDar UniversityResults and Discussion

The theory of couple stress fluid due to Stokes is used to formulate a set of boundary layer equations for a flow of incompressible, couple stress fluid in a vertical channel for mixed convection. Analytical solutions are obtained using perturbation technique valid for small value of . Figures 1 and 2 show the effect of for .When the flow is upward, and are positive and on the other hand, the flow is downward when the and are negative. It is very interesting to note that there is a flow reversal at both the boundaries for positive, which is different from the result for viscous fluid where there is a flow reversal for positive only at cool wall. We observe that for purely viscous fluid for =-500 the flow reversal was at the hot wall whereas for couple stress fluid there is a flow reversal both at left and right walls. The effect of on the flow for couple stress fluid is dominating compare to viscous fluid both on velocity and temperature. The profile of temperature are significant for couple stress fluid for different where as the profiles were not sensible for different .

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Prof Narasimna Murthy @ BahirDar University Figures 7 illustrate the influence of couple stress parameter a with isoflux-isothermal and isothermal-isoflux wall conditions for and .

It is seen that as a increases the flow is assisted for positive and suppresses for negative at the reversal side. Also, as a increases temperature decreases for both The effect of a on velocity and temperature for isoflux-isothermal case is similar to that for isothermal-isoflux wall conditions as seen in Figure 7.


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Prof Narasimna Murthy @ BahirDar UniversityREFERENCES [1] Lavine, A.S., Analysis of fully developed opposing mixed convection between inclined parallel plates. Warme-und Stoffubetragung, vol. 23, pp. 249-257, 1988.[2] Malashetty M. S., Umavathi J. C. Oberbeck convection flow of couple stress fluid through a vertical porous stratum. Int. J. Non-Linear Mech. vol. 34, pp. 1037-1045, 1999.[3] Umavathi J.C., Free convection of couple stress fluid for radiating medium in a vertical channel. ASME, vol. 69, pp. 1-20, 2000.[4] Stokes V. K. Couple stresses in fluids. The physics of fluids. vol. 9 pp.1709, 1966.[5] Srivastava L. M. Flow of couple stress fluid through stenotic blood vessles. J. Bio. Mech. vol. 18, pp. 479-485, 1986.[1] Shehawey E. F. and Mekheimer K.S. Couple stresses in peristaltic transport of fluids. J. Phy. vol. 27, pp. 1163-1172, 1994. [2] Barletta, A., Analysis of combined forced and free flow in a vertical channel with viscous dissipation and isothermal-isoflux boundary conditions. Journal Heat Transfer, vol. 121, pp. 349356, 1999.[3] Aung, W., and Worku, G., Theory of fully developed, combined convection including flow reversal. Journal Heat Transfer, vol. 108, pp. 485488, 1986.[4] Barletta, A., Fully developed mixed convection and flow reversal in a vertical rectangular duct with uniform wall heat flux. International Journal Heat Mass Transfer, vol. 45, pp. 641654, 2002.[5] Cheng, C.H., Kou, H.S., and Huang, W.H., Flow reversal and heat transfer of fully developed mixed convection in vertical channels. Journal Thermo Physics Heat Transfer, vol. 4, pp. 375383, 1990.[6] Hamadah, T.T., and Wirtz, R.A., Analysis of laminar fully developed mixed convection in a vertical channel with opposing buoyancy. ASME Journal Heat Transfer, vol. 113, pp. 507510, 1991[7] Barletta, A., Heat transfer by fully developed flow and viscous heating in a vertical channel with prescribed wall heat fluxes, International Journal Heat Mass Transfer, vol. 41, pp. 35013513, 1986.[8] Barletta, A., Laminar mixed convection with viscous dissipation in a vertical channel, International Journal Heat Mass Transfer, vol. 42, pp. 38733885, 1999.

Prof Narasimna Murthy @ BahirDar University37[1] Barletta, A., Fully developed mixed convection and flow reversal in a vertical rectangular duct with uniform wall heat flux. International Journal Heat Mass Transfer, vol. 45, pp. 641654, 2002.[2] Cheng, C.H., Kou, H.S., and Huang, W.H., Flow reversal and heat transfer of fully developed mixed convection in vertical channels. Journal Thermo Physics Heat Transfer, vol. 4, pp. 375383, 1990.[3] Hamadah, T.T., and Wirtz, R.A., Analysis of laminar fully developed mixed convection in a vertical channel with opposing buoyancy. ASME Journal Heat Transfer, vol. 113, pp. 507510, 1991[4] Barletta, A., Heat transfer by fully developed flow and viscous heating in a vertical channel with prescribed wall heat fluxes, International Journal Heat Mass Transfer, vol. 41, pp. 35013513, 1986.[5] Barletta, A., Laminar mixed convection with viscous dissipation in a vertical channel, International Journal Heat Mass Transfer, vol. 42, pp. 38733885, 1999. [6] Aung, W., and Worku, G., Developing flow and flow reversal in a vertical channel with asymmetric wall temperature. ASME Journal Heat Transfer, vol. 108, 299304, 1986[7] Barletta, A., Laminar mixed convection with viscous dissipation in a vertical channel, International Journal of Heat and Mass Transfer, vol. 41, pp. 3501-3513, 1998.[8] Batchelor, G.K., Heat transfer by free convection across a closed cavity between vertical boundaries at different temperatures. Quarterly of Applied Mathematicsvol. 12, pp. 209-233, 1954.[9] Cheng, K.C. and Wu, R.S., Viscous dissipation effects on convective instability and heat transfer in plane Poiseuille flow heated from below. Applied Scientific Research, vol. 32, pp. 327-346, 1976.


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