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Thermal Viscous Dissipative Couette-Poiseuille Flow in a Porous Medium Saturated Channel

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Thermal Viscous Dissipative Couette-Poiseuille Flow in a Porous Medium Saturated Channel S S symmetry Article Thermal Viscous Dissipative Couette-Poiseuille Flow in a Porous Medium Saturated Channel G. M. Chen * , M. Farrukh B. , B. K. Lim and C. P. Tso Faculty of Engineering and Technology, Multimedia University, Jalan Ayer Keroh Lama, Melaka 75450, Malaysia * Correspondence: [email protected]; Tel.: +60-06-252-3342 Received: 7 May 2019; Accepted: 24 June 2019; Published: 3 July 2019 Abstract: A Couette-Poiseuille flow between parallel plates saturated with porous medium is studied with emphasis on viscous dissipation effect on the temperature field; assuming a fully developed flow, with both plates subjected to unequal and uniform heat flux. Temperature field and Nusselt number are derived as a function of Brinkman number and porous medium shape factor. By specifying the ratio of wall to mean velocity as one, the resulting velocity and temperature fields attribute to a significant increase in Nusselt number for the moving wall as the permeability of porous medium increases. Increased permeability signifies competing effect between enhanced convection in the proximity of the moving wall and higher local viscous dissipation. When the former effect dominates, heat transfer coefficient increases. Effects of Reynolds number on the temperature field is elucidated, including a comparison between a microchannel and conventional duct to evaluate the characteristic length scale effect. As Reynolds number goes up in a microchannel, heat generation in the form of viscous dissipation intensifies and overrides the convection effect, causing an increase in the highest temperature along the duct on the contrary to the findings in conventional duct. Keywords: viscous dissipation; forced convection; Couette-Poiseuille flow; porous medium 1. Introduction The fluid flow and heat transfer in a Couette-Poiseuille flow in a channel play a vital role in a wide variety of materials processing applications which includes sheet metal forming, extrusion, wire and fiber glass drawing, continuous casting, and more, whereby in all of these applications, heat is continuously transferred to fluid from moving plane [1]. Considerably less amount of research has been directed at Couette-Poiseuille flow as compared to studies on Couette flow or Poiseuille flow. Laminar heat transfer problems in Poiseuille flow for Newtonian fluids in parallel plates have been solved [2] while Aydin and Avci [3] subsequently studied Poiseuille flow in laminar heat convection for two different thermal boundary conditions—constant heat flux and constant wall temperature respectively—accounting for the effect of viscous dissipation and highlighted the importance of viscous dissipation. Lin [4] numerically investigated the effects of viscous dissipation and pressure gradient on non-Newtonian Couette flow. On the other hand, a number of investigations have also been performed on Couette-Poiseuille flow in clear fluid between parallel plates. Aydin and Avci [5] looked into the heat transfer in a Couette-Poiseuille flow for both hydrodynamically and thermally fully developed flow between two parallel plates. Sheela-Francisca et al. [6] examined heat transfer in Couette-Poiseuille flow under asymmetric wall heat fluxes with viscous dissipation effect and obtained the closed form temperature field and Nusselt number expression. Chan et al. [7] explored the effect of viscous dissipation on the thermal aspect of a power-law fluid for a Couette-Poiseuille flow subjected to asymmetric thermal boundary conditions. Hashemabadi et al. [8] solved forced convective heat transfer problem of non-linear viscoelastic fluid flow between parallel plates for a Couette-Poiseuille flow analytically and remarked the significant effects of Brinkman number on Symmetry 2019, 11, 869; doi:10.3390/sym11070869 www.mdpi.com/journal/symmetry Symmetry 2019, 11, x FOR PEER REVIEW 2 of 14 Symmetry 2019, 11, 869 2 of 15 Couette-Poiseuille flow analytically and remarked the significant effects of Brinkman number on heat convection coefficient. Davaa et al. [1] solved heat convection problem for a non-Newtonian Couette- heatPoiseuille convection flow numerically, coefficient. stressing Davaa et on al. the [1] signific solvedance heat of convection viscous dissipation problem foreffect a non-Newtonian on temperature Couette-Poiseuilledistribution and Nusselt flow numerically,number. stressing on the significance of viscous dissipation effect on temperatureUnlike clear distribution fluid, there and Nusseltis remarkably number. less research endeavours on Couette-Poiseuille flow in porousUnlike medium. clear Aydin fluid, thereand Avci is remarkably [9] investigated less research the effect endeavours of viscous dissipation on Couette-Poiseuille on the heat transfer flow in porousrate for medium.Couette-Poiseuille Aydin and flow Avci in [ 9a] saturated investigated porous the emediumffect of viscous between dissipation two plane onparallel the heat plates. transfer The ratestudy for concurred Couette-Poiseuille on the significance flow in a saturatedof viscous porousdissipation medium on Couette-Poiseuille between two plane flow parallel but did plates. not Thefurther study anyconcurred explorations on the on significancethe thermalof boundary viscous dissipation condition implementation on Couette-Poiseuille at the flowfixed but boundary did not furthernor the anylength explorations scale effect on of the the thermal parallel boundary plate channel condition on dimensional implementation temperature at the fixed field. boundary nor the lengthHence, scale this e ffstudyect of would the parallel like to plate fill in channel the gap on by dimensional looking into temperature the thermal field. viscous dissipative effectsHence, on a Couette-Poiseuille this study would flow like toin filla saturate in the gapd porous by looking medium, into subjected the thermal to unequal viscous and dissipative uniform eheatffects flux on aapplied Couette-Poiseuille at both plates, flow assuming in a saturated a steady, porous laminar medium, and subjected fully developed to unequal flow and with uniform local heatthermal flux equilibrium applied at both inside plates, the assuming porous medium. a steady, laminarThe study and would fully developedalso compare flow the with significance local thermal of equilibriumviscous dissipation inside the to porousthe temperature medium. Thefield study in microchannel would also compareand conventional the significance size channel of viscous for dissipationdifferent to. the temperature field in microchannel and conventional size channel for different Re. 2. Problem Formulation and Analytical Solution Figure1 is1 ais schematic a schematic diagram diagram of the problem of the where problem there iswhere a steady, there laminar, is a hydrodynamically,steady, laminar, andhydrodynamically, thermally fully-developed and thermally flow fully-developed through porous fl mediumow through between porous two medium plates separated between bytwo a gapplates of height H. The lower plate moves at a constant velocity uw, while the upper plate is stationary. Uniform separated by a gap of height . The lower plate moves at a constant velocity , while the upper heat flux is applied to both plates whereby q001 and q002 are applied to the moving plate and stationary plate is stationary. Uniform heat flux is applied to both plates whereby ′′ and ′′ are applied to plate,the moving respectively. plate Inand solving stationary the governing plate, respecti thermalvely. energy In equation,solving temperaturethe governing field thermal is first derivedenergy q00 2 subjected to a uniform heat flux q00 at a moving wall. By defining a heat flux ratio R = and equation, temperature field is first derived subjected to a uniform heat flux at a movingq00 1+ wall.q00 2 By rewriting q00 as q00 1 + q00 2, the temperature field for the prior solved single heated wall only solution defining a heat flux ratio = and rewriting as + , the temperature field for the can then be transformed to temperature field having both boundaries subjected to uniform heat flux. Theprior details solved is single provided heated in Section wall only 2.1. solution can then be transformed to temperature field having both boundaries subjected to uniform heat flux. The details is provided in Subsection 2.1. Figure 1.1.Schematic Schematic Diagram Diagram of theof Problemthe Problem subject subject to unequal to unequal and uniform and heatuniform fluxes heat at both fluxes boundaries. at both 2.1. Governingboundaries. Equation 2.1. GoverningThis study Equation adopts the Brinkman-extended Darcy equation solved by [9] as the governing momentum equation. By defining the following dimensionless variables, This study adopts the Brinkman-extended Darcy equation solved by [9] as the governing momentum equation.y By definingµe f f u the followingµe f f u wdimensionlessµe f f variables,K 1 Y = , U = , Uw = , M = , Da = , S = (1) H γH2 γH2 µ H2 p 1 = , = , = , = ,= , = MDa (1) √ The Brinkman-extendedBrinkman-extended DarcyDarcy EquationEquation isis non-dimensionalizednon-dimensionalized asas follows,follows, d 2 d U − +1=0 (2) d S2U + 1 = 0 (2) and is subjected to the dimensionless formdY of2 −the boundary conditions, as follows: =0, = and is subjected to the dimensionless form of the boundary conditions, as follows: (3) = 1, =0 Y = 0, U = Uw (3)(4) Symmetry 2019, 11, 869 3 of 15 Y = 1, U = 0 (4) Solving Equation (2) alongside boundary conditions (3) and (4) yields the dimensionless velocity, u∗ defined and expressed as [9] 2 U S 1 cos h(SY) + S cos ech(S)sin h(S SY)Uw + sin h(SY)tan h(S/2) u = = − − (5)
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