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Abdul S Gaffar - One of the best experts on this subject based on the ideXlab platform.
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modelling third grade liquid past vertical isothermal cone with variable temperature and Biot Number effects
2021Co-Authors: Abdul S Gaffar, Ramachandra V Prasad, Md Hidayathulla B Khan, K VenkatadriAbstract:The non-similar natural convection flows of an incompressible viscoelastic fluid past an isothermal cone with Biot Number effects and variable temperature are investigated. The Keller-Box technique is utilized to solve the transformed conservation equations subject to physically appropriate boundary conditions. The variations of different emerging dimensionless parameters on velocity, temperature, skin friction coefficient and heat transfer rate profiles are presented.
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magnetohydrodynamic free convection flow and heat transfer of non newtonian tangent hyperbolic fluid from horizontal circular cylinder with Biot Number effects
International Journal of Applied and Computational Mathematics, 2017Co-Authors: Abdul S Gaffar, Ramachandra V Prasad, Keshava E ReddyAbstract:This article presents the nonlinear, steady boundary layer flow and heat transfer of an incompressible Tangent Hyperbolic non-Newtonian fluid from a Horizontal Circular Cylinder in the presence of magnetic field and Biot Number effects. The transformed conservation equations are solved numerically subject to physically appropriate boundary conditions using a second-order accurate implicit finite-difference Keller Box technique. The numerical code is validated with previous studies. The influence of a Number of emerging non-dimensional parameters, namely the Weissenberg Number (We), power law index (n), Prandtl Number (Pr), Biot Number $$(\upgamma )$$ , the magnetic parameter (M) and dimensionless tangential coordinate ( $$\xi $$ ) on velocity and temperature evolution on the boundary layer regime are examined in detail. Furthermore, the effects of these parameters on surface heat transfer rate and local skin friction are also investigated. Validation with earlier Newtonian studies is presented and excellent correlation achieved. It is found that the velocity, skin friction and heat transfer rate reduce with increasing We. Whereas, there is slight increase in temperature. Increasing n is observed to increase the velocity and heat transfer rate but decreases temperature and skin friction. An increasing $$\upgamma $$ is seen to increase velocity, temperature, local skin friction and heat transfer rate. And an increasing M is found to decrease velocity, skin friction and heat transfer rate but increases the temperature. The study is relevant to chemical materials processing applications.
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computational study of non newtonian eyring powell fluid from a vertical porous plate with Biot Number effects
Journal of The Brazilian Society of Mechanical Sciences and Engineering, 2017Co-Authors: Abdul S Gaffar, Ramachandra V Prasad, Bhuvana VijayaAbstract:In this article, the nonlinear, steady-state boundary layer flow and heat transfer of an incompressible Eyring–Powell non-Newtonian fluid from a vertical porous plate is investigated. The transformed conservation equations are solved numerically subject to physically appropriate boundary conditions using a second-order versatile, implicit finite-difference Keller Box technique. The numerical code is validated with previous studies. The influence of a Number of emerging non-dimensional parameters, namely Eyring–Powell rheological fluid parameters (e), the local non-Newtonian parameter based on length scale x (δ), Prandtl Number (Pr), Biot Number (γ) and dimensionless tangential coordinate (ξ) on velocity and temperature evolution in the boundary layer regime are examined in detail. Furthermore, the effects of these parameters on surface heat transfer rate and local skin friction are also investigated. It is found that the velocity is reduced with increasing e but temperature is increased. Increasing δ enhances velocity but reduces temperature. The increasing γ is observed to enhance both velocity and temperature. And an increasing Prandtl Number Pr is found to decrease both velocity and temperature.
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flow and thermal convection of third grade viscoelastic fluid from a vertical porous plate with Biot Number effects
International Journal of Mathematical Archive EISSN 2229-5046, 2017Co-Authors: Bhuvana R Vijaya, Abdul S Gaffar, K Venkatadri, Md Hidayathulla B KhanAbstract:A mathematical model is developed to analyse the nonlinear, non-isothermal, steady-state, laminar boundary layer flow and heat transfer of an incompressible third grade viscoelastic non-Newtonian fluid past a vertical porous plate with Biot Number effects. The transformed conservation equations of mass, linear momentum and heat equations are solved numerically subject to physically appropriate boundary conditions using a second-order accurate implicit finite-difference Keller-Box method ( KBM ). The influence of a Number of emerging non-dimensional parameters, namely the third grade fluid parameter ( f ), material fluid parameters (e 1 , e 2 ), Prandtl Number (Pr), Biot Number ( g ) and dimensionless tangential coordinate ( x ) on velocity and temperature evolution in the boundary layer regime are examined in detail. Furthermore, the effects of these parameters on surface heat transfer rate and local skin friction are also investigated. It is observed that velocity, skin friction and heat transfer rate are reduced with increasing third-grade fluid parameter ( f ), whereas the temperature is increased. An increase in the material fluid parameter (e 1 ) reduces the velocity, skin friction and heat transfer rate but increases temperature. And increasing material fluid parameter (e 2 ) accelerates velocity, skin friction and heat transfer rate but decelerates the temperature. Detailed interpretation of the computations is included. The present simulations are of interest in chemical engineering systems and solvent and low density polymer materials processing.
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numerical study of flow and heat transfer of non newtonian tangent hyperbolic fluid from a sphere with Biot Number effects
alexandria engineering journal, 2015Co-Authors: Abdul S Gaffar, Ramachandra V PrasadAbstract:Abstract In this article, we investigate the nonlinear steady boundary layer flow and heat transfer of an incompressible Tangent Hyperbolic fluid from a sphere. The transformed conservation equations are solved numerically subject to physically appropriate boundary conditions using implicit finite-difference Keller Box technique. The numerical code is validated with previous studies. The influence of a Number of emerging non-dimensional parameters, namely Weissenberg Number ( We ), power law index ( n ), Prandtl Number ( Pr ), Biot Number ( γ ) and dimensionless tangential coordinate ( ξ ) on velocity and temperature evolution in the boundary layer regime is examined in detail. Furthermore, the effects of these parameters on heat transfer rate and skin friction are also investigated. Validation with earlier Newtonian studies is presented and excellent correlation is achieved. It is found that the velocity, Skin friction and the Nusselt Number (heat transfer rate) are decreased with increasing Weissenberg Number ( We ), whereas the temperature is increased. Increasing power law index ( n ) increases the velocity and the Nusselt Number (heat transfer rate) but decreases the temperature and the Skin friction. An increase in the Biot Number ( γ ) is observed to increase velocity, temperature, local skin friction and Nusselt Number. The study is relevant to chemical materials processing applications.
Xiao Dong Chen - One of the best experts on this subject based on the ideXlab platform.
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characteristic transport lengths ctls in porous medium evaluated with classic diffusion solutions under infinite Biot Number condition
Journal of Food Engineering, 2015Co-Authors: Xiao Dong Chen, Xin Jin, Aditya PutrantoAbstract:Abstract In process engineering practice, including those in food industry, simple mathematical solutions are more useful. Learned assumptions are necessary to support effective simplifications. Previously it has been suggested that for a conduction and convection coupled system, there is an approximately linear temperature or concentration gradient between the surface and the average temperature, which occurs at a ‘fixed location’ within the conduction domain. The local temperature at the point is also said to be similar as the average temperature. This gradient is thus approximately the same as the temperature gradient at the interface between the conduction domain and the convection medium when thermal properties are considered constants. The distance from the surface to this ‘fixed location’ is marked as the characteristic transport length (CTL), which is a fraction of the size of the conduction medium. The previous findings were based on the agreements between the numerical solutions and compartmental, and then integral solutions in different occasions The argument has been validated among moderate Biot Numbers (Bi) of 0.3. Similarly, one should find that the diffusional mass transfer process has the same property due to the same mathematical nature involved. Here, the mass diffusion process has been analyzed to yield the CTLs for the cases of infinite Biot Number, where the analytical solutions for longer times for semi-infinite slab, infinite cylinder and sphere are available which can be put to great use. When applying these classical solutions for the above purpose, there are still new discoveries, which are interesting to report here.
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a linear relationship between dimensionless crossing point temperature and frank kamenetskii reactivity parameter in self heating test at infinite Biot Number for slab geometry
Fire Safety Journal, 2013Co-Authors: Xiao Dong Chen, H S Sidhu, Mark I NelsonAbstract:Abstract Self-heating/ignition is one of the well-known practical causes for fires and explosions in industry and in nature. The Transient Method (or Chen Method) is a cost-effective approach for determining the thermal ignition parameters of packed particulate or loose materials (activation energy E, the product of the heat of reaction and the pre-exponential constant QA). The crossing-point-temperature (CPT) method to establish the ignition kinetics was initiated by the first author in 1994. A finite difference solution obtained in 1998 showed that for Biot Number approaching infinity the dimensionless CPT, θcpt (when the conduction term becomes zero at symmetry), is proportional to the Frank–Kamenetskii reactivity parameter δ, i.e. θ c p t = 0.1 δ . In this study, this relationship has been re-confirmed firstly by new Matlab simulations, and secondly, derived analytically with the characteristic transport dimension concept and a new simple idea of a three-region approximation. The dimensionless thickness of the third region (next to the solid-gas boundary), defined as (1−β2)self-heat, is remarkably similar to that for the heat conduction (1−β2)cond=0.333 which leads to θ c p t = 0.093 δ . A small adjustment of (1−β2)self-heat to 0.339 leads to the exact relationship. This work shows a general applicability of the approximate linear relationship, making the method more useful.
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surface center temperature differences within milk droplets during convective drying and drying based Biot Number analysis
Aiche Journal, 2008Co-Authors: Kamleshkumar Chhanabhai Patel, Xiao Dong ChenAbstract:An assumption of uniform temperature is frequently used when evaluating average temperature-time and average moisture content-time profiles for the convective drying of small droplets or thin-layer materials. In most studies, the assumption of uniform temperature was justified by estimating the heat-transfer Biot Number at the beginning and end of the drying process. However, the conventional Biot Number analysis performed in the literature does not reflect the evaporative effect. In this article, we have examined the temperature uniformity and the heat-transfer Biot Number during the drying of skim milk droplets under laboratory drying conditions following the entire drying process. Surface-centre temperature differences and conventional and drying-based Biot Numbers are calculated during the drying of skim milk droplets. A simple procedure is outlined to estimate the extent of temperature nonuniformity within the droplet. Results demonstrated that temperature nonuniformity within the skim milk droplets under drying conditions examined is very small, thus the uniform temperature assumption is more likely to be a reliable approach to model heat and mass transfer processes in industrial spray dryers. The analyses provided in this study help in understanding a few assumptions used in literature, and offer a framework that may be used in the future. © 2008 American Institute of Chemical Engineers AIChE J, 2008
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modified Biot Number in the context of air drying of small moist porous objects
Drying Technology, 2005Co-Authors: Xiao Dong Chen, Xiaofeng PengAbstract:Abstract: In this study, a modified Biot Number analysis of hot air drying of small objects has been described. The derivations outlined in this article provide a platform for testing a Number of assumptions made for the simple models. In particular, the uniform temperature assumption is focused upon. A Number of calculation examples are given based on previous experimental results. It is expected that the new calculation procedures can provide a better justification for the assumptions made in modeling air drying.
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microwave heating of an infinite solid slab and its thermal stability analysis using steady state bifurcation theory
Journal of Food Engineering, 1998Co-Authors: Xiao Dong ChenAbstract:This paper presents an analytical solution of the steady state temperature profiles in a microwave heated slab. The analytical solution has been obtained by taking advantages of the well known Frank-Kamenetskii thermal ignition theory. Both the finite Biot Number and infinite Biot Number boundary conditions have been considered and their effects upon the critical parameters for the onset of thermal instability evaluated.
Ramachandra V Prasad - One of the best experts on this subject based on the ideXlab platform.
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modelling third grade liquid past vertical isothermal cone with variable temperature and Biot Number effects
2021Co-Authors: Abdul S Gaffar, Ramachandra V Prasad, Md Hidayathulla B Khan, K VenkatadriAbstract:The non-similar natural convection flows of an incompressible viscoelastic fluid past an isothermal cone with Biot Number effects and variable temperature are investigated. The Keller-Box technique is utilized to solve the transformed conservation equations subject to physically appropriate boundary conditions. The variations of different emerging dimensionless parameters on velocity, temperature, skin friction coefficient and heat transfer rate profiles are presented.
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magnetohydrodynamic free convection flow and heat transfer of non newtonian tangent hyperbolic fluid from horizontal circular cylinder with Biot Number effects
International Journal of Applied and Computational Mathematics, 2017Co-Authors: Abdul S Gaffar, Ramachandra V Prasad, Keshava E ReddyAbstract:This article presents the nonlinear, steady boundary layer flow and heat transfer of an incompressible Tangent Hyperbolic non-Newtonian fluid from a Horizontal Circular Cylinder in the presence of magnetic field and Biot Number effects. The transformed conservation equations are solved numerically subject to physically appropriate boundary conditions using a second-order accurate implicit finite-difference Keller Box technique. The numerical code is validated with previous studies. The influence of a Number of emerging non-dimensional parameters, namely the Weissenberg Number (We), power law index (n), Prandtl Number (Pr), Biot Number $$(\upgamma )$$ , the magnetic parameter (M) and dimensionless tangential coordinate ( $$\xi $$ ) on velocity and temperature evolution on the boundary layer regime are examined in detail. Furthermore, the effects of these parameters on surface heat transfer rate and local skin friction are also investigated. Validation with earlier Newtonian studies is presented and excellent correlation achieved. It is found that the velocity, skin friction and heat transfer rate reduce with increasing We. Whereas, there is slight increase in temperature. Increasing n is observed to increase the velocity and heat transfer rate but decreases temperature and skin friction. An increasing $$\upgamma $$ is seen to increase velocity, temperature, local skin friction and heat transfer rate. And an increasing M is found to decrease velocity, skin friction and heat transfer rate but increases the temperature. The study is relevant to chemical materials processing applications.
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computational study of non newtonian eyring powell fluid from a vertical porous plate with Biot Number effects
Journal of The Brazilian Society of Mechanical Sciences and Engineering, 2017Co-Authors: Abdul S Gaffar, Ramachandra V Prasad, Bhuvana VijayaAbstract:In this article, the nonlinear, steady-state boundary layer flow and heat transfer of an incompressible Eyring–Powell non-Newtonian fluid from a vertical porous plate is investigated. The transformed conservation equations are solved numerically subject to physically appropriate boundary conditions using a second-order versatile, implicit finite-difference Keller Box technique. The numerical code is validated with previous studies. The influence of a Number of emerging non-dimensional parameters, namely Eyring–Powell rheological fluid parameters (e), the local non-Newtonian parameter based on length scale x (δ), Prandtl Number (Pr), Biot Number (γ) and dimensionless tangential coordinate (ξ) on velocity and temperature evolution in the boundary layer regime are examined in detail. Furthermore, the effects of these parameters on surface heat transfer rate and local skin friction are also investigated. It is found that the velocity is reduced with increasing e but temperature is increased. Increasing δ enhances velocity but reduces temperature. The increasing γ is observed to enhance both velocity and temperature. And an increasing Prandtl Number Pr is found to decrease both velocity and temperature.
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non newtonian thermal convection of eyring powell fluid from an isothermal sphere with Biot Number effects
International Journal of Industrial Mathematics, 2016Co-Authors: S Abdelghaffar, Ramachandra V Prasad, Keshava E ReddyAbstract:This article investigates the nonlinear, steady boundary layer ow and heat transfer of an incom- pressible Eyring-Powell non-Newtonian uid from an isothermal sphere with Biot Number eects. The transformed conservation equations are solved numerically subject to physically appropriate boundary conditions using a second-order accurate implicitnite-dierence Keller Box technique. The inuence of a Number of emerging dimensionless parameters, namely the Eyring-Powell rheological uid pa- rameter ( "), the local non-Newtonian parameter based on length scale ( ), Prandtl Number (Pr), Biot Number ( ) and dimensionless tangential coordinate ( ) on velocity and temperature evolution in the boundary layer regime are examined in detail. Furthermore, the eects of these parameters on surface heat transfer rate and local skin friction are also investigated. It is found that the velocity and heat transfer rate (Nusselt Number) decrease with increasing ( "), whereas temperature and skin friction increase. An increasing ( ) is observed to enhance velocity, local skin friction and heat transfer rate but reduces the temperature. An increase ( ) is seen to increase velocity, temperature, local skin friction and Nusselt Number. The study is relevant to chemical materials processing applications.
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numerical study of flow and heat transfer of non newtonian tangent hyperbolic fluid from a sphere with Biot Number effects
alexandria engineering journal, 2015Co-Authors: Abdul S Gaffar, Ramachandra V PrasadAbstract:Abstract In this article, we investigate the nonlinear steady boundary layer flow and heat transfer of an incompressible Tangent Hyperbolic fluid from a sphere. The transformed conservation equations are solved numerically subject to physically appropriate boundary conditions using implicit finite-difference Keller Box technique. The numerical code is validated with previous studies. The influence of a Number of emerging non-dimensional parameters, namely Weissenberg Number ( We ), power law index ( n ), Prandtl Number ( Pr ), Biot Number ( γ ) and dimensionless tangential coordinate ( ξ ) on velocity and temperature evolution in the boundary layer regime is examined in detail. Furthermore, the effects of these parameters on heat transfer rate and skin friction are also investigated. Validation with earlier Newtonian studies is presented and excellent correlation is achieved. It is found that the velocity, Skin friction and the Nusselt Number (heat transfer rate) are decreased with increasing Weissenberg Number ( We ), whereas the temperature is increased. Increasing power law index ( n ) increases the velocity and the Nusselt Number (heat transfer rate) but decreases the temperature and the Skin friction. An increase in the Biot Number ( γ ) is observed to increase velocity, temperature, local skin friction and Nusselt Number. The study is relevant to chemical materials processing applications.
Aditya Putranto - One of the best experts on this subject based on the ideXlab platform.
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characteristic transport lengths ctls in porous medium evaluated with classic diffusion solutions under infinite Biot Number condition
Journal of Food Engineering, 2015Co-Authors: Xiao Dong Chen, Xin Jin, Aditya PutrantoAbstract:Abstract In process engineering practice, including those in food industry, simple mathematical solutions are more useful. Learned assumptions are necessary to support effective simplifications. Previously it has been suggested that for a conduction and convection coupled system, there is an approximately linear temperature or concentration gradient between the surface and the average temperature, which occurs at a ‘fixed location’ within the conduction domain. The local temperature at the point is also said to be similar as the average temperature. This gradient is thus approximately the same as the temperature gradient at the interface between the conduction domain and the convection medium when thermal properties are considered constants. The distance from the surface to this ‘fixed location’ is marked as the characteristic transport length (CTL), which is a fraction of the size of the conduction medium. The previous findings were based on the agreements between the numerical solutions and compartmental, and then integral solutions in different occasions The argument has been validated among moderate Biot Numbers (Bi) of 0.3. Similarly, one should find that the diffusional mass transfer process has the same property due to the same mathematical nature involved. Here, the mass diffusion process has been analyzed to yield the CTLs for the cases of infinite Biot Number, where the analytical solutions for longer times for semi-infinite slab, infinite cylinder and sphere are available which can be put to great use. When applying these classical solutions for the above purpose, there are still new discoveries, which are interesting to report here.
Li Zhang - One of the best experts on this subject based on the ideXlab platform.
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three dimensional numerical simulation on thermocapillary convection of moderate prandtl Number fluid in an annular shallow pool with surface heat dissipation
Microgravity Science and Technology, 2019Co-Authors: Li ZhangAbstract:In order to understand the effect of surface heat dissipation on thermocapillary convection of moderate Prandtl Number fluid in an annular shallow pool, we performed a series of three-dimensional numerical simulations by using the finite volume method. An annular shallow pool is full of 0.65cSt silicone oil with Prandtl Number of 6.7. The radius ratio and the aspect ratio of the pool are respectively fixed at 0.5 and 0.1. Surface heat dissipation Biot Number is varied from 0 to 50. Results indicate that the critical Marangoni Number of flow destabilization mainly depends on the coupling effect of the thermocapillary force and surface heat dissipation on the free surface, which decreases first, and then increases gradually. When Biot Number is small, the steady axisymmetric flow after the flow destabilization will bifurcate to the hydrothermal wave. With the increase of Marangoni Number, the flow pattern evolution at a small Biot Number is similar to that of an adiabatic free surface, and the fundamental oscillatory frequency increases gradually. With the increase of Biot Number, the radial roll cells near the outer cylindrical wall gradually extend to the inner wall, and eventually occupy the whole liquid pool. When Biot Number is large, the flow pattern after the flow destabilization is a radial rolling cell pattern with alternating azimuthal direction. Then it gradually evolves into azimuthal waves, and the fundamental oscillatory frequency has a slight decrease.
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thermocapillary convection of moderate prandtl Number fluid in a shallow annular pool heated from inner cylinder with surface heat dissipation
International Journal of Thermal Sciences, 2019Co-Authors: Sen Zhang, Li ZhangAbstract:Abstract This paper presents a series of three-dimensional numerical simulations on thermocapillary convection of moderate Prandtl Number (Pr = 6.7) fluid in a shallow annular pool heated from the inner cylinder with surface heat dissipation. The aspect ratio and radius ratio of the annular pool are respectively fixed at e = 0.05 and η = 0.5. Biot Number varies from 0 to 40 and Marangoni Number varies from 0.1 × 105 to 6.0 × 105. Results show that there is a large temperature drop near the heated cylinder and the flow becomes more intense than those when the outer cylinder is heated at adiabatic free surface. With the increase of Marangoni Number, the flow will transit from the axisymmetric steady flow to the three-dimensional oscillatory flow. The critical Marangoni Number decreases first, and then increases with the increase of Biot Number. After the flow destabilization, thermocapillary convection with adiabatic free surface exhibits the multicellular wave pattern when the inner cylinder is heated, which is different from that when the outer cylinder is heated. The temperature fluctuation pattern depends on the coupling effect of thermocapillary convection and heat exchange on the free surface. The wave Number increases first, and then decreases with the increase of Marangoni Number at a fixed Biot Number. Furthermore, as Marangoni Number increases, the temperature fluctuation on the free surface will radially extend from the inner cylinder to the outer cylinder gradually, and the amplitude of temperature fluctuation increases obviously.
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surface heat dissipation dependence of thermocapillary convection of moderate prandtl Number fluid in an annular pool
Microgravity Science and Technology, 2019Co-Authors: Li ZhangAbstract:In order to understand surface heat dissipation dependence of thermocapillary convection for moderate Prandtl Number fluid in a deep annular pool, a series of three-dimensional numerical simulations have been carried out by using the finite volume method. The radius ratio and the aspect ratio of an annular pool are fixed at 0.5 and 1.0, respectively. The working fluid is 0.65cSt silicone oil with Prandtl Number of 6.7. Surface heat dissipation Biot (Bi) Number is varied from 0 to 50. Results indicate that with the increase of Biot Number, the radial temperature gradient near the inner cylindrical wall decreases, and near the outer cylindrical wall it increases, so the flow is enhanced. When 0 < Bi < 10, with the increase of Marangoni Number, the axisymmetric steady flow first transits to the standing wave, and then to the azimuthal waves. The standing wave should be attributed to Marangoni-Benard instability. However, the azimuthal waves should be corresponded to hydraulic instability, which is mainly driven by the azimuthal motion of temperature fluctuation from the sudden change of flow direction near the bottom and the inner cylindrical wall. When Bi ≥ 10, when the flow destabilizes, the axisymmetric steady flow transits directly to the azimuthal waves. With the increase of Biot Number, the critical Marangoni Number of the flow destabilization increases. Furthermore, the fundamental frequency and the wave Number of three-dimensional oscillatory flow increase gradually with the increase of Biot Number.
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effect of surface heat dissipation on thermocapillary convection of moderate prandtl Number fluid in a shallow annular pool
Journal of Crystal Growth, 2019Co-Authors: Li Zhang, Lu ZhangAbstract:Abstract This paper presents a series of three-dimensional numerical simulations on the effect of surface heat dissipation on thermocapillary convection of moderate Prandtl Number fluid in a shallow annular pool. The annular pool with a fixed aspect ratio of 0.05 and radius ratio of 0.5 is filled with 0.65 cSt silicone oil. Its Prandtl Number is 6.7. Biot Number of surface heat dissipation ranges from 0 to 50. The results show that when Marangoni Number is small, thermocapillary convection is the axisymmetric steady flow. When Marangoni Number exceeds the critical value, the basic flow will destabilize and transit to the hydrothermal waves that propagate along the clockwise or anticlockwise direction. The critical Marangoni Number of the flow destabilization increases with the increase of Biot Number. Furthermore, when Biot Number increases, the radial temperature gradient on the free surface decreases near the inner cylindrical wall, but increases near the outer cylindrical wall, which results in the thermocapillary convective cell moving gradually from the inner to the outer cylindrical walls. After the flow destabilizes, the temperature fluctuation on the free surface mainly appears near the outer cylinder at a large Biot Number.