ISTP-6, 5, PAGUE 6 TH INTENATIONAL SYMPOSIUM ON TANSPOT PHENOMENA COUETTE FLOW IN A PATIALLY POOUS CUVED CHANNEL WHICH IS SLIGHTLY ECCENTIC Leong, J.C., Tsai, C.H., Tai, C.H. Department of Vehicle Engineering, National Pingtung University of Science and Technology, Pingtung 9, Taiwan.O.C. Corresponding author: Email: jcleong@mail.npust.edu.tw, Fax: +886-8-774398 Keywords: porous media, polar Couette flow, Brinkman-extended Darcy law, heat transfer, eccentricity Abstract This paper presents analytical solutions of the flow and temperature distributions in a configuration similar to the polar Couette flow except that a porous layer is inserted on top of the lower curved surface and the upper sliding surface is allowed to slightly displace. The solution is obtained through the use of the perturbation method with the eccentricity ratio being the perturbation parameter that is no greater than.. With the end effects close to both ends of the curved surface being neglected, a slight displacement of the upper sliding surface either compresses or decompressed the flow. With the presence of the porous layer, the velocity profile is further compressed within the fluid layer leading to a greater shear stress experienced on the upper sliding surface. If viscous dissipation is more important than the heat conduction in the fluid, the relative motion of fluid particles close to the sliding surface heats up the fluid. Introduction The classical Couette flow describes the flow profile within two infinite parallel plates. The flow parallel to these plates is induced by the relative motion between these plates. Later, the classical Couette flow was extended for flows in the polar coordinate. This flow is then referred to as the polar Couette flow [, ]. This work considers a similar configuration except that a porous layer is inserted on top of the lower curved surface and the upper sliding surface is allowed to slightly displace. The lower curved surfaces and porous layer considered have a span of. Different from the isothermal classical Couette flow, the temperature on the upper sliding surface is assumed higher than that on the lower stationary surface. This paper presents analytical solutions of the flow and temperature distributions in it obtained through the use of the perturbation method with the eccentricity ratio being the perturbation parameter. When the eccentricity ratio is zero, both the upper and lower surfaces are concentric. Furthermore, if the thickness of the porous layer is zero, the velocity profile obtained will be identical to the solution of the polar Couette flow. Although the solutions of the flow and temperature profiles were obtained up to the second order, it was found that only the first two leading terms (the zero and first orders) produce significant modifications on the solution for the eccentricity ratio considered is no greater than. in this study. The end effects close to both ends of the curved surface will definitely affect the flow patterns between the curved surfaces. At the entrance, recirculation cells will develop right after the flow makes a sharp turn into the channel [3]. Also, right before the exit, the flow in the channel tends to diverge and leaves the channel where flow resistance is higher [4]. The consideration of these phenomena is not included in this paper.
Jik-Chang LEONG, Chien-Hsiung TSAI, Chang-Hsien TAI Even though Couette flow is a classical fluid problem solved for a long time, theoretical investigation of heat transfer related to Couette flow through a porous medium are surprisingly rare. Nakayama [5] successfully obtained analytical solutions of isothermal Couette flow through an inelastic porous medium under various situations. In his problem, the channel is fully filled with a porous medium. A plate moving at a given velocity is placed on top of the medium without any gap. This problem Nakayama tackled is rather unrealistic because the moving plate may cause the porous medium to deform considerably. ecognizing the unrealistic boundary condition, Kuznetsov [6] considered a composite channel in which a porous medium was filled. In between the porous medium and the moving plate, a gap was inserted. The flow in the porous medium was modeled using the Brinkman-Forchheimer-extended Darcy model. The velocity and temperature profiles he obtained were under the boundary layer approximation. A configuration somewhat similar to the present study has been considered by Meena and Kandaswamy [7]. They considered a full solid journal bearing with a thin porous lining. Because of its negligible thickness, the flow field in the porous lining was not solved for. Only the velocity profile in the fluid region was obtained. On the rotating shaft, the fluid velocity was the same as the linear velocity of the shaft. At the interface between the fluid region and the porous lining, a slip velocity boundary condition was applied. The purposes of this study are to demonstrate the feasibility of solving for the velocity and temperature fields in an eccentric curved channel partially filled with a porous layer, to investigate the main effect the eccentricity and the presence of the porous layer have on the flow and temperature distribution within the channel, and to consider secondary factors such as the porous layer thickness and the relative importance of viscous dissipation to heat conduction. e a T L Formulation T H Figure A partially porous eccentric channel subject to rotation and differential heating (T H > T L ). ϖ. Governing Equations The physical configuration of the present study consists of two infinitely long curved solid surfaces maintained at different temperatures, T H (on the upper sliding solid surface) and T L (on the lower stationary solid surface), with which T H > T L (Fig. ). The radii of the inner and outer surfaces are a and c, respectively. A homogeneous, isotropic porous layer, whose inner and outer curvatures are b and c, is securely attached to the lower solid surface. The centers of these two surfaces are offset by a distance e. This parameter is generally referred to as the eccentricity. The configuration has a span of. Both the porous layer and the absolutely hollow space above it are filled and saturated with a fluid of constant properties. Furthermore, it is assumed that the buoyancy effect within the solid surfaces is negligible in comparison with the shearing effect the upper sliding solid surface induces. Under these idealizations, the governing equations for the two-dimensional, steady, laminar flow within the solid surfaces can be greatly simplified. The governing equations obtained are presented along with their boundary and interface θ r b c
COUETTE FLOW IN A PATIALLY POOUS CUVED CHANNEL WHICH IS SLIGHTLY ECCENTIC conditions in this section. With the presence of the porous layer, two sets of momentum equation must be solved simultaneously. In the fluid region, the Navier- Stokes equation is simplified to fit the current case. In the porous layer, the momentum equation takes the form based on the Brinkmanextended Darcy model. Since the eccentricity is assumed very small in this study, it is reasonable to conclude that the azimuthal velocity component is much greater than the radial velocity. Thus, the momentum equation in the radial direction is eliminated from the sets of governing equations. With proper means of normalization, the governing equations for this study become θ-momentum equations: µ µ d dv V = d d % d dv b d d K V = Energy transport equation d dθ dv V Br d d + = d d dθ dv V Br d d + = d At the upper solid surface, At the lower solid surface, At the interface, () () (3) (4) V = Θ = (5) V = Θ = (6) dv dv V = V = d d (7) where the dimensionless and normalized parameters are Br ν = α U C(T T ) p H L (9) = r / b V = u θ / ϖa () Θ = (T T L ) / (T H T L ) () The radius b is used instead of the gap between the two solid surfaces (c - a) because this definition of simplifies the process of obtaining the solution. The product of ϖa in Eq. () represents the linear velocity of the upper curved surface.. Perturbation Method To make use of the perturbation method, a possible form of solution must be assumed before making an attempt on solving the ordinary differential equations [8, 9]. In this study, it is assumed that the general solutions take the form of a power series of the product of the eccentricity ratio and a trigonometric function (ε cosθ). The velocity and temperature profiles look like V = V + εcosθ V + ε cos θ V +K () i i i i Θ = Θ + εcosθ Θ + ε cos θ Θ +K (3) i i i i where ε is the eccentricity ratio defined as e / b and the subscript i = refers to the fluid layer and i = the porous layer. Accordingly, the boundary and interface conditions are expanded in terms of the same quantity. Applying the above assumed solution forms, the boundary condition at the upper sliding solid surface can be expressed as ( ' ) V + ε cos θ V + V + ε cos θ V + V ' + V" + = K (4) dθ k dθ Θ = Θ d k d = p f (8) 3
Jik-Chang LEONG, Chien-Hsiung TSAI, Chang-Hsien TAI ( ' ) Θ + εcos θ Θ + Θ + cos + ' + " + = K ε θ Θ Θ Θ.3 Solution (5) By collecting the terms of identical power of ε cosθ, one obtains the governing equations and their corresponding boundary and interface conditions. The method of solution for these equations has been elaborated presented (Nayfeh, 9xx). For brevity, these steps are excluded in this paper. Instead, the complete solutions for these equations are presented. ab V = f4 + f5 f3 f + f (f f ) + + ab V = f4 + f5 f3 Θ K c K β I( β) b + K c I β K( β) b b = Br f + a a Fln + F + F b 6 3 5 ( 6 7 8) ( 4) Br f + f + f + F + F + F ln() +K (6) (7) (8) b Θ = + Br( f6 + f7 + f8) a b Br( f6 + f7) a 4 b b 6 + 3Br f + F a a b ( F+ F) a b F+ F F 4ln a ( 6 7 8) ( 4) + Br f + f + f ln() + F + F F ln() ab 4Br + f +K 3 ( ) 4 + ab + f3 f3 4 f F() (9) where for simplicity, we write = ε cos(θ) and β = K / b. The coefficients appeared in (6)- (9) are c c f = I β K( β ) K β I( β ) b b γ c β β c f = I K( ) + K β I( β ) b b 3 () () f = (a + b )f (a b )f () ab f = (a b)f (a+ b)f (3) [ ] 4 f3 c b f5 = ln+ (4) b a f ab 6 = f3 f + f (5) 4
COUETTE FLOW IN A PATIALLY POOUS CUVED CHANNEL WHICH IS SLIGHTLY ECCENTIC f 7 = ab ( + ) f f f 4 ab c + Br f5f6 4Br F f 3 b F = c a ln ln b b F 4 F 3 b Br f6 + f7f9 a b f4 c F + 8Brab F a f3 b = c a ln ln b b 3 b b F3 = Br f6 + f7 a a b a F Fln a b 4 3 b b 3 f6 f7 a a Br c b + f8 + ln b a = c ln b b b F F a a f 5 c 4Br ab + f 4 F( ) f3 b c ln a f 3 (6) (7) (8) (9) (3) Normalized Thickness.5.75.5.5 ε=..5..5.. -6-4 - 4 6 Angle Figure. Variation of channel thickness for different eccentricities. 4 3 b b b F5 = Br 3 f6 f7 + f8 a a a b b c + F 4 F -Fln a a b F() = β β ln() c K β I ( β) b c + I β K( β) b d c K β I ( β) b ln() d c + I β K( β) b c c + K β I( β ) I β K( β ) ln() b b [ ln() ] + c K β I( β) b d c I β K( β) b c K β I( β) d I K( β) b ln() b c β (3) (3) Note that F(c/b) in the expressions of F, F, and F 4 is a constant which is evaluated as F(x) with x = c/b. 5
Jik-Chang LEONG, Chien-Hsiung TSAI, Chang-Hsien TAI 3 esult and Discussions Since the regular perturbation method is used in this study to obtain the flow and temperature fields, these solutions are valid only for small eccentricity ratios. For the context that follows, the configuration of the curved surfaces are limited to a =, b =.5, and c =., unless otherwise specified. Figure shows the height of the curved channel for -6 θ 6 with -. ε.. The normalized thickness shown is defined as (c r) / (c - a). The broken line in the middle of the plot represents the location of the interface between the fluid region and the porous layer. The upper solid lines represent the locations of the upper rotating solid surfaces. Notice that the height of curved channel remains unchanged when ε =. If ε >, the channel height reduces gradually and becomes minimum at θ =. After that, the height of the channel increases. For ε <, the geometry of the channel reverses. Figures 3-5 show the flow and temperature fields between two curved solid surfaces with a porous layer inserted within. The upper and lower solid curves represent the rotating upper and stationary lower curved solid surfaces. The curved broken line in between indicates the location of the interface between the fluid region and the porous layer. In Figs. (a), the magnitude and direction of the flow are visualized using vectors. The magnitude of these vectors is normalized by the linear velocity on the upper surface. Therefore, the magnitude of the normalized velocity field varies from at the lower surface to at the upper surface. Due to the shearing effect the upper surface induces, the velocity profile in the fluid region is similar to those Meena and Kandaswamy presented [7]. The magnitude of the velocity reduces linearly in the radial direction from the upper surface to the interface. In the porous layer, a small but finite velocity field is found. This can be examined more clearly in the velocity profiles shown in Fig. 6. In Figs. (b), the contours of temperature (isotherms) are presented. Although the temperature field is normalized using the temperature difference between the upper and lower surfaces, the normalized temperature at the vicinity of the upper surface is greater than unity due to viscous dissipation. As shown in the figures, the uppermost isotherm and the one next to it correspond to θ =. This implies that the temperature between these isotherms is greater than T H. In this case, both the upper and lower solid surfaces act as heat sinks through which excessive heat energy is removed in additional to the channel outlet. In Figs. 3, the eccentricity ratios of the curved channel vary from -. to.. At ε = -. when the channel height is the greatest at θ =, the velocity is more widely distributed across the fluid region. As the eccentricity ratio increases, the channel height at θ = reduces and eventually leads to the compression of the flow at that angle. For ε >, the flow in the fluid region is gradually compressed (when θ < ) and then gradually decompressed (when θ > ) as it enters the channel. Apparently, the compression and decompression of the fluid elements are exactly reverse for ε <. Although the flow pattern in the fluid region depends strongly on the eccentricity ratio, the flow pattern in the porous layer appears to be almost independent of this parameter. In other words, the flow resistance the porous layer introduces is much greater than that by the eccentricity. In Fig. 3(b), the overall heat transfer mechanism in the fluid region is mainly by conduction. The comparison of these three cases reveals the fact that the warmer fluid layer close to the sliding upper surface becomes thicker as the eccentricity ratio increases. Not only so, the region whose temperature is greater than T H also shrinks with increasing eccentricity ratio. Since the velocity in the porous layer is almost negligible, heat flux from the fluid region through the porous layer is conducted out to the lower surface. However, it is found that the amount of heat energy conducted through the 6
COUETTE FLOW IN A PATIALLY POOUS CUVED CHANNEL WHICH IS SLIGHTLY ECCENTIC porous layer increases with the eccentricity ratio. Also, this amount of heat energy conducted is only weakly dependent on the azimuthal direction. Figures 4 and 5 show the flow and temperature fields between the eccentric surfaces for different porous layer thicknesses (b =.5, b =.5, and b =.75). Among these three cases, the first case (b =.5) corresponds to the thickest porous layer while the last case (b =.75) is associated to the thinnest porous layer. (a) ε=. ε=. ε=. (b) ε=. ε=. ε=...8...8...8. Figure 3. (a) Velocity and (b) temperature distributions for b =.5, Br =, and β =. (a) b =.5 b =.5 b =.75 (b). b =.5.8. b =.5.8... b =.75.8. Figure 4. (a) Velocity and (b) temperature distributions for ε = -., Br =, and β =. (a) b =.5 b =.5 b =.75 (b) b =.5 b =.5 b =.75..8...8...8. Figure 5. (a) Velocity and (b) temperature distributions for ε =., Br =, and β =. 7
Jik-Chang LEONG, Chien-Hsiung TSAI, Chang-Hsien TAI These figures show the influence the porous layer thickness has on the flow and temperature fields. Although a slight displacement of the upper sliding solid surface either compresses or decompressed the flow, the presence of the porous layer further compresses the velocity profile within the fluid layer. This eventually leads to a greater velocity gradient across the fluid region. As a result, a greater shear stress is expected on the upper sliding surface. It is interesting to observe from Figs. 4(b) and 5(b) that fluid temperature tends to increase with porous layer thickness. From Figs. 4(a) and 5(a), it is obvious that the velocity gradient in the fluid region increases with porous layer thickness. Since heat generation by viscous dissipation is directly proportional to the velocity gradient, heat generation is therefore also proportional to the porous layer thickness. For a given porous layer thickness, it is observed that the fluid temperature in the fluid region is higher for greater eccentricity ratios. The reasoning for this trend is exactly the same as that discussed above. Careful examination on the figures suggests that the temperature profiles depend on the azimuthal direction. This angular dependence is only relatively noticeable for thick porous layers. Even so, the dependence is still very insignificant for the Normalized velocity.9.8.7.5.3.. b* =..3.5.7.9...3.5.7.8.9 Normalized distance Figure 6. Velocity profiles for various porous layer thicknesses with Br =., ε =.5, and β =. span of channel considered in this study. For the ease of comparison, the radial distance from the upper to lower solid surface is normalized as * = (r a) / (c a) (33) In accordance with the above definition, * =. corresponds to the upper surface whereas * = corresponds to the lower surface. Normalized temperature.5.4.3...9.8.7.5.3.. b* =..3.5.7.9...3.5.7.8.9 Normalized radial distance Figure 7. Temperature profiles for various porous layer thicknesses with various Br =., ε =.5, and β =. If the thicknesses of the fluid region and the porous layer are the same, the location of the interface will be given as * =.5 if ε =. Figure 6 presents the effect of porous layer thickness on the overall velocity profiles. In this figure, b* indicates the location of the interface. For fixed configurations of the upper and lower surfaces, the greater the value of b* is, the thinner the porous layer and therefore the thicker the fluid region. As proven through the figure, the flow velocity significantly decreases as the porous layer thickness increases. This is no doubt accompanied with a greater velocity gradient. This also implies that a higher friction loss is anticipated with a thicker porous layer. Among the five cases shown, the angular velocity component associated with b* =. is the smallest due to the greatest flow resistance the porous layer introduces. If b* =.9, the 8
COUETTE FLOW IN A PATIALLY POOUS CUVED CHANNEL WHICH IS SLIGHTLY ECCENTIC velocity profile within the curved channel bears absolute resemblance with that of the classical polar Couette flow. ecall that Brinkman number (Br) signifies the relative importance of viscous dissipation to heat conduction. If Br =, no viscous dissipation is considered. If Br <, the rate of heat conduction is greater than the rate of heat generation due to viscous dissipation. If Br >, heat is generated due to viscous dissipation at a rate greater than the rate required to be removed through conduction. To amplify the relative importance of viscous dissipation, Br = is used to plot Fig. 7. This implies that heat energy is generated by viscous dissipation twice as fast as the heat energy capable of being conducted out of the channel. In this case, the relative motion of fluid particles close to the sliding upper surface heats up the fluid. This trend shown suggests that the effect of viscous dissipation is more apparent when the porous layer is thick. Once again, the great heat generation due to viscous dissipation is because of the large velocity gradient. Since the thermal properties of the fluid are assumed the same as those of the porous layer, no sudden change in the temperature profile is expected across the interface. porous layer is very thick, a high velocity gradient takes place within the fluid region causing a high shear stress, leading to viscous dissipation, and creating a region of overheated fluid. As the thickness of the porous layer is reduced, the shear flow in it becomes finite and its velocity profile approaches that of the polar Couette flow. Eventually, the velocity gradient, the shear stress, and the heat generation by viscous dissipation decrease. If viscous dissipation is more important than the heat conduction in the fluid, the relative motion of fluid particles close to the sliding surface heats up the fluid. The amount of heat energy generated is proportional to the shear stress on the upper sliding surface and both the curved surfaces serve as heat sinks. If heat conduction in the fluid is greater than viscous dissipation, heat generated on the upper surface is dumped through the lower surface. While the present study has provided useful insights on the flow and temperature distribution in a porous bearing, the solutions however are limited to the locations far enough from the edges of the channel. For a complete understanding of the problem, the solutions of flow and temperature fields must be obtained via computational approach. 4 Conclusions The present study has shown that a closed form solution is possible for the flow and temperature fields in a partially porous slightly eccentric curved channel whose upper surface slides. It was found that eccentricity either compresses (when the eccentricity ratio is positive) or decompresses (when the eccentricity ratio is negative) the velocity and temperature fields at θ =. When ε >, the velocity and temperature gradients at θ = are larger than those at any other angular position. With the presence of a porous layer, the velocity profile is further compressed within the fluid layer leading to an even greater shear stress experienced on the upper sliding surface. If the eferences [] White FM. Viscous Fluid Flow. second edition, McGraw-Hill Company, New York, 99. [] Panton L. Incompressible Flow. second edition, John Wiley & Sons, New York, 996. [3] Syuhada A, Hirota M, Fujita H, Araki S, Yanagida M and Tanaka T. Heat (mass) transfer in serpentine flow passage with rectangular cross-section. Energy Conversion & Management, Vol. 4, pp 867-885,. [4] Stokes SD, Glauser MN and Gatski TB. An examination of a 3D corner-step experiment. Experimental Thermal and Fluid Science, Vol. 7, pp 3-38, 998. [5] Nakayama A. Non-Darcy Couette flow in a porous medium filled with an inelastic non-newtonian fluid. Journal of Fluid Engineering, Vol. 4, pp 64-647, 99. [6] Kuznetsov AV. Analytical investigation of Couette flow in a composite channel partially filled with a 9
Jik-Chang LEONG, Chien-Hsiung TSAI, Chang-Hsien TAI porous medium and partially with a clear fluid. International Journal of Heat and Mass Transfer, Vol. 4, No. 6, pp 556-56, 998. [7] Meena S and Kandaswamy P. Effect of slip on the flow between eccentric cylinders. Journal of Porous Media, Vol. 4, No., pp 79-88,. [8] Nayfeh AH. Introduction to Perturbation Technique. John Wiley & Sons, New York, 98. [9] Aziz A and Na TY. Perturbation Methods in Heat Transfer. Hemisphere Publishing corporation, New York, 984.