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 𝑅𝑒 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 𝐻. The lower plate moves at a constant velocity 𝑢, 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 𝑞′′ and 𝑞′′ 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 𝑞 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 𝑅= and rewriting 𝑞 as 𝑞 +𝑞 , 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 = 𝜇 , U𝑢w = 𝜇 𝑢, M = 𝜇 , Da = , S = (1) H 𝑦 γH2 γH2 µ H𝐾2 √ 1 𝑌= ,𝑈 = ,𝑈 = ,𝑀 = ,𝐷𝑎= ,𝑆 = MDa (1) 𝐻 𝛾𝐻 𝛾𝐻 𝜇 𝐻 √𝑀𝐷𝑎 The Brinkman-extendedBrinkman-extended DarcyDarcy EquationEquation isis non-dimensionalizednon-dimensionalized asas follows,follows, d𝑈 2 d U −𝑆 𝑈+1=0 (2) d𝑌 S2U + 1 = 0 (2) and is subjected to the dimensionless formdY of2 −the boundary conditions, as follows: 𝑌=0, 𝑈=𝑈 and is subjected to the dimensionless form of the boundary conditions, as follows: (3) 𝑌 = 1, 𝑈 =0 Y = 0, U = Uw (3)(4)

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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) ∗ 2 Um S + ( 2 + S Uw)tan h(S/2) − The governing thermal energy equation is,

!2 ∂T ∂2T µu2 du ρ f cp f u = ke f f + + µe f f (6) , ∂x ∂y2 K dy

In Equation (6), Al-Hadhrami’s model [10] is used to represent the viscous dissipation term, where the second and third term on the right hand side of Equation (6) are termed as internal heating (I.H.) and frictional heating (F.H.) respectively. The problem is ﬁrst solved for heat ﬂux applied to moving wall only, where the thermal boundary conditions are y = 0, T = Tw , (7) ∂T y = H, = 0 . (8) ∂y In the axial direction, the temperature gradient is constant under thermally fully-developed condition, hence ∂T ∂T = m . (9) ∂x ∂x

∂TA By incorporating q00 = k at y = 0 and integrating Equation (6) throughout the height of − e f f ∂y ∂Tm the channel, ∂x is obtained as

 H H  !2  µ Z Z  ∂Tm 1  2 du  = q00 + u dy + µe f f dy . (10) ∂x ρ f cp, f Hum  K dy  0 0

Introducing the following dimensionless variables,

2 2 ke f f (T Tw) µum H Br µe f f um θ = − , Br = , BrMDa = = Br = , (11) 2 0 q00H q00K S q00H and incorporating Equation (10) in to Equation (6), the dimensionless governing thermal energy equation is obtained as 2 2 ∂ θ 2 Br du∗ = u∗A Bru∗ , (12) ∂Y2 − − S2 dY where  1 1   Z Z 2   2 Br du∗  A = 1 + Br u∗ dY + dY . (13)  S2 dY  0 0 subjected to Y = 0, θ = 0 , (14) ∂θ Y = 1, = 0 . (15) ∂Y Symmetry 2019, 11, 869 4 of 15

Solving Equation (12) alongside the boundary conditions gives rise to the exact solution,

2 θ(Y) = C1Y + C2Y + C3 cosh(SY) + C4 cosh(2SY) + C5sinh(SY) + C6sinh(2SY) + C7 (16) where C C are the coefficients defined in the AppendixA. 1 − 7 In our attempt to obtain the temperature field subjected to uniform heat flux at both plates, TB, the temperature field obtained in Equation (16) may be made use by equating

q002 TB = TA + y (17) ke f f where TA is the temperature solved subjected to uniform heat flux at moving wall only while TB denotes the temperature field subjected to uniform heat flux at both plates. Notably, TB fulfills the governing thermal energy equation, Equation (6) duly. From Equation (17), ∂T ∂T q002 B = A + . (18) ∂y ∂y ke f f

By rewriting the boundary conditions for TA in Equation (7) as

∂TA y = 0, k = q00 + q00 , (19) − e f f ∂y 1 2

∂T y = H, k A = 0 . (20) − e f f ∂y

The boundary conditions for TB may be written as

∂TB y = 0, k = q00 , (21) − e f f ∂y 1

∂TB y = H, k = q00 . (22) e f f ∂y 2 Recasting Equation (17) in dimensionless temperature gives

θ1 = θ + RY , (23)

k (T Tw) where θ = θ(Y)= e f f A− as solved in Equation (16). H(q00 1+q00 2) The ratio of heat ﬂuxes, R, is deﬁned as,

q00 R = 2 , q00 1 + q00 2

ke f f (TB Tw) θ1 = − H q00 1 + q00 2

Hence, θ1 can be written as

2 θ1 = RY + C1Y + C2Y + C3 cosh(SY) + C4 cosh(2SY) + C5sinh(SY) + C6sinh(2SY) + C7 . (24)

Defining θ2 as ke f f (TB Tw) θ2 = − , (25) Hq00 1 Symmetry 2019, 11, 869 5 of 15 relates 1 θ = θ , (26) 2 1 × 1 R − and solves the temperature field subjected to uniform heat fluxes at both boundaries as

1 h i θ = RY + C Y + C Y2 + C cosh(SY) + C cosh(2SY) + C sinh(SY) + C sinh(2SY) + C . (27) 2 1 R 1 2 3 4 5 6 7 − 2.2. Nusselt Number Nusselt number deﬁned based on the moving wall temperature is,

q00 H Nu = 1 . (28) k (Tw Tm) e f f − Computing the bulk mean temperature, Nu can be expressed as:

n 2 2 o Nu = 1/6(R 1)S [S + S Uw 2 tanh(S/2)] S 3S[ C + C + (C C + 2C + R)S] − − − − 5 6 1 − 3 7 2 o +2C2 6 + S + (3C5 4C6)Scos h(S) + C6Scos h(2S) + S[3C3 4C4 + 2C4cos h(S)] sin h(S) n − − +S3 6(C + R) + (3C + 4C )S 3C S2 4C Scos h(S) + S(2C 6C 3C S)cot h(S) (29) − 1 5 6 − 3 − 6 4 − 7 − 5 2 +2S[3(C + C + C + R) C cos h(2S)]cos ech(S) Uw + [S (6C 4C + 12C + 6R + 3C S) 1 2 7 − 4 1 − 4 7 5 2 2 o +6C 4 + S 12C S Uw]tanh(S/2) 2 − 2 2.3. Nusselt Number Verification Reducing and comparing the solutions to cases for Poiseuille flow and Couette-Poiseuille flow in clear fluid, as presented in Table1, shows, that Nu in this study is in excellent agreement with the literature, as presented in Table1.

Table 1. Comparison of Nusselt number with the literature.

Nu, Nu, Nu, Nu, Tan Nu, Chen Nu*, Tso Br R S U /U Present Aydin Davaa and w m et al. [7] et al. [12] Study et al. [9] et al. [11] Chen [13] 0 0 1/ √10 0 5.385 5.385 5.385 5.385 5.385 — 0 0 1/ √10 1 7.238 — 7.241 — — — 0.2 0 1/ √10 0 3.805 3.804 3.804 3.804 — — 0.2 0 1/ √10 1 9.992 10 — 10 — — 0 0.5 1/ √10 0 8.237 8.235 — — 8.235 8.235 0.5 0.5 1/ √10 0 3.183 3.182 — — — 3.182

3. Results and Discussion

3.1. Velocity Profile In order to facilitate the discussion of Couette-Poiseuille flow in a saturated porous medium, the velocity field in the channel for different porous medium shape factors S is depicted in Figure2 for Uw/Um = 1, that results in profile with a maximum velocity. Velocity gradient in the vicinity of a moving wall is higher for low S porous medium while larger S porous medium gives rise to steeper gradient in the vicinity of the fixed wall. Flow velocity is also higher for lower S porous medium in the vicinity of the moving wall. Figure3 illustrates the viscous dissipation profile at Br = 0.1, for S = 1, and S = 10, respectively, in order to elucidate the effects of local viscous dissipation on the temperature field in Section 3.2. It is noteworthy to point out that overall viscous dissipation corresponding to S = 1 drops to a minimum and picks up tremendously towards the fixed wall, revealing a much lower viscous dissipation at the Symmetry 2019, 11, x FOR PEER REVIEW 5 of 14

Nu, Nu, Nu, Nu, Nu*, Nu, Tan 𝑼 𝑩𝒓 𝑹 𝑺 𝒘 present Chen et Aydin et Davaa et Tso et and Chen /𝑼𝒎 study al. [7] al. [9] al. [11] al. [12] [13] 1 0 0 0 5.385 5.385 5.385 5.385 5.385 --- /√10 1 0 0 1 7.238 --- 7.241 ------/√10 1 0.2 0 0 3.805 3.804 3.804 3.804 ------/√10 1 0.2 0 1 9.992 10 --- 10 ------/√10 1 0 0.5 0 8.237 8.235 ------8.235 8.235 /√10 1 0.5 0.5 0 3.183 3.182 ------3.182 /√10

3. Results and Discussion

3.1. Velocity Profile In order to facilitate the discussion of Couette-Poiseuille flow in a saturated porous medium, the velocity field in the channel for different porous medium shape factors 𝑆 is depicted in Figure 2

for 𝑈⁄𝑈 =1, that results in profile with a maximum velocity. Velocity gradient in the vicinity of a moving wall is higher for low 𝑆 porous medium while larger 𝑆 porous medium gives rise to steeper gradient in the vicinity of the fixed wall. Flow velocity is also higher for lower 𝑆 porous medium in the vicinity of the moving wall. Symmetry 2019 11 Figure, , 8693 illustrates the viscous dissipation profile at 𝐵𝑟 = 0.1 , for 𝑆=1, and 𝑆=10,6 of 15 respectively, in order to elucidate the effects of local viscous dissipation on the temperature field in Section 3.2. It is noteworthy to point out that overall viscous dissipation corresponding to 𝑆=1 moving wall than the fixed wall. On the other hand, for S = 10, viscous dissipation is dominated drops to a minimum and picks up tremendously towards the fixed wall, revealing a much lower by internal heating, which varies in a similar pattern to the velocity profile. The average viscous viscous dissipation at the moving wall than the fixed wall. On the other hand, for 𝑆=10, viscous dissipationdissipation is computed is dominated and by tabulated internal heating, in Table which2 at a fixedvaries Brin afor similar the velocitypattern to field the velocity in Figure profile.2. Table 2 indicatesThe average that smaller viscousS porous dissipation medium is computed has a much and tabulated higher overall in Table viscous 2 at a fixed dissipation 𝐵𝑟 for duethe velocity to the more significantfield in frictional Figure 2. heatingTable 2 indicates towards that fixed smaller wall at𝑆 a porous fixed Brmedium, defined has based a much on higher the relative overall magnitudesviscous of internaldissipation heating due to to heat the more applied significant at wall. frictional heating towards fixed wall at a fixed 𝐵𝑟, defined based on the relative magnitudes of internal heating to heat applied at wall.

FigureFigure 2. Velocity 2. Velocity distribution distribution in in thethe channelchannel for different diﬀerent porous porous medium medium shape shape factor, factor, 𝑆 andS and Symmetry 2019, 11, x FOR PEER REVIEW 6 of 14 Uw/U𝑈m⁄=𝑈1.= 1.

FigureFigure 3. Viscous 3. Viscous Dissipation Dissipation (I.H (I.H= =Internal Internal Heating,Heating, F.HF.H = =FrictionalFrictional Heating) Heating) in the in channel the channel for for diﬀerentdifferent porous porous medium medium shape shape factor, factor,S and𝑆 andBr 𝐵𝑟= 0.1. = 0.1.

Table 2. Eﬀects of viscous dissipation, Br = 0.1 (I.H = Internal Heating, F.H = Frictional Heating). Table 2. Effects of viscous dissipation, 𝐵𝑟 = 0.1 (I.H = Internal Heating, F.H = Frictional Heating) Porous Medium Shape Factor, S I.H/q”.HF.H/q”.H Porous Medium Shape Factor, 𝑺 𝑰. 𝑯/𝒒.𝑯 𝑭. 𝑯/𝒒.𝑯 1 0.1129 0.4001 √101 0.11290.1106 0.4001 0.0412 10 0.1048 0.0064 √10 0.1106 0.0412 10 √10 0.1015 0.0016 10010 0.1048 0.1005 0.0064 0.0005

10√10 0.1015 0.0016 3.2. Temperature Distribution 100 0.1005 0.0005 The transverse dimensionless temperature profile in the channel is depicted in Figure4a,b for various3.2. shapeTemperature factor DistributionS and heat flux ratio. Both figures show increasing transverse dimensionless temperature with increasing S in the vicinity of the heated wall. In the presence of viscous dissipation, The transverse dimensionless temperature profile in the channel is depicted in Figure 4a and 4b S Figurefor4 bvarious indicates shape a factor more 𝑆 significant and heat flux increase ratio. Both in figures temperature show increasing field for transverse lower dimensionlessdue to gradually moretemperature dominant convectionwith increasing effect 𝑆 overin the heat vicinity generation of the inheated the adjoiningwall. In the region presence of the ofmoving viscous wall. dissipation, Figure 4b indicates a more significant increase in temperature field for lower 𝑆 due to gradually more dominant convection effect over heat generation in the adjoining region of the moving wall. Hence, viscous dissipation is more significant to lower 𝑆 porous medium shape factor

at a fixed 𝐵𝑟 and specified 𝑈⁄𝑈 =1. Expectedly, at a fixed 𝐵𝑟, temperatures increase in both figures as 𝑅 increases, for an increase in fluid temperature adjoining the fixed wall. In Figure 4b, the temperature at fixed wall rises to 0.3 when 𝑅=0.5, reflecting that increasing 𝑅 causes a higher surface temperature at the fixed wall than moving wall, due to a higher heat transfer coefficient at the moving wall than the fixed wall, as depicted by the velocity distribution in Figure 2, when heat flux is applied at both boundaries.

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Hence, viscous dissipation is more significant to lower S porous medium shape factor at a fixed Br and specified Uw/Um = 1. Expectedly, at a fixed Br, temperatures increase in both figures as R increases, for an increase in fluid temperature adjoining the fixed wall. In Figure4b, the temperature at fixed wall rises to 0.3 when R = 0.5, reflecting that increasing R causes a higher surface temperature at the fixedSymmetry wall 2019 than, 11, movingx FOR PEER wall, REVIEW due to a higher heat transfer coefficient at the moving wall than the7 fixed of 14 wall, as depicted by the velocity distribution in Figure2, when heat flux is applied at both boundaries. Symmetry 2019, 11, x FOR PEER REVIEW 7 of 14

(a) (a) (b)(b)

FigureFigure Figure4. Dimensionless 4. 4.Dimensionless Dimensionless temperature temperature temperature distribution distribution distribution subject subject subject to to co co tonstantnstant constant heatheat heatflux at fluxat both both at boundaries both boundaries boundaries for fordifferent different for heat fluxdiheatfferent ratio, flux heat R. ratio, (a flux) R.𝐵𝑟 ( ratio,a) 𝐵𝑟 =0 and R. ( =0a and()bBr) 𝐵𝑟(=b) 0𝐵𝑟 and = .0.1 ( =b ).0.1 Br = 0.1.

3.3. Nusselt3.3. Nusselt Number Number Variation Variation 3.3. Nusselt Number Variation FigureFigure5 illustrates 5 illustrates the variation the variation of Nusselt of Nusselt number number defined defined at at thethe moving wall wall versus versus Brinkman Brinkman numberFigurenumber with 5 shapeillustrates with factorshape the Sfactor beingvariation 𝑆 thebeing parameter,of Nusseltthe parameter, number indicating indicating defined its heavy its at theheavy reliance moving reliance on Swall. Thereon versus𝑆. There is an Brinkman expectedis an increasenumberexpected inwithNu shape forincreaseS =factor 1in, particularly𝑁𝑢 𝑆 forbeing 𝑆=1 the with, particularly parameter, a large Rwith, indicating attributed a large 𝑅 , byits attributed aheavy reduced relianceby a heat reduced fluxon 𝑆. atheat theThere flux moving atis an wall,expected andthe increase therefore,moving wall,in a 𝑁𝑢 decrease and for therefore, 𝑆=1 in the, particularly temperaturea decrease inwith ditheff erenceatemperature large between𝑅, attributed difference moving by between a wall reduced temperature moving heat wall flux and at bulkthe moving meantemperature temperature wall, andand bulk whentherefore, mean both temperature platesa decrease are when heated, in boththe as temperaturepl reflectedates are heated, in Figure difference as 4reflecb. Whileted between in higher Figure moving 4b.Br Whileindicates wall higher 𝐵𝑟 indicates more intense viscous dissipation, competing effect between the more dominant moretemperature intense and viscous bulk dissipation,mean temperature competing when eff bothect between plates are the heated, more dominantas reflected convection in Figure 4b. and While heat convection and heat generation leads to a higher 𝑁𝑢. Such effect is caused by enhanced heat generationhigher 𝐵𝑟 leads indicates to a more higher intenseNu. Such viscous effect dissipation, is caused by competing enhanced heateffect convection between the eff ectmore at adominant specified convectionconvection and heateffect generationat a specified leads 𝐵𝑟, which to a ishigher more significant𝑁𝑢. Such for effect low 𝑆is. As caused 𝑆 approaches by enhanced infinitely heat Br, which is more𝐵𝑟 significant for low S. As S approaches𝑁𝑢 infinitely large value, Br has no noticeable convectionlarge effectvalue, at a hasspecified no noticeable 𝐵𝑟, which effect ison more , indicating significant the for heat low generation 𝑆. As 𝑆 effectapproaches being offset infinitely by effect onconvectionNu, indicating due to the the velocity heat generation and viscous e ffdissipatioect beingn distribution offset by convection depicted in dueFigure to 2 the and velocity Figure 3 and large value, 𝐵𝑟 has no noticeable effect on 𝑁𝑢, indicating the heat generation effect being offset by viscousat dissipation a fixed 𝐵𝑟, distributiona higher 𝐵𝑟 is depicted accompanied in Figures by heat2 and convection3 at a fixed enhancementBr, a higher at theBr movingis accompanied wall. by heatconvection convection due to enhancement the velocity atand the viscous moving dissipatio wall. n distribution depicted in Figure 2 and Figure 3 at a fixed 𝐵𝑟, a higher 𝐵𝑟 is accompanied by heat convection enhancement at the moving wall.

Figure 5. Nusselt number versus Brinkman number for 𝑅 = 0, 𝑅 = 0.25, 𝑅=0.5.

3.4. Temperature Contour Plots

Figure 5. Nusselt number versus Brinkman number for R𝑅= 0, = R0,= 0.25, 𝑅 = 0.25,R = 0.5. 𝑅=0.5 3.4.1. 𝑅=0Figure (Heat 5. Nusselt flux applied number to versusthe moving Brinkman plate only)number for .

3.4. TemperatureTwo dimensional Contour Plots temperature field is plotted to compare the effects of viscous dissipation on a conventional channel and a microchannel. A microchannel of height, 𝐻 = 50 μm and length, 𝐿= 3.4.1. 𝑅=030 mm (Heatis filled flux with applied water to saturated the moving porous plate medium only) made up of silicon. Table 3 shows the thermophysical properties of fluid and porous medium for computation. The properties of porous Two dimensional temperature field is plotted to compare the effects of viscous dissipation on a conventional channel and a microchannel. A microchannel of height, 𝐻 = 50 μm and length, 𝐿= 30 mm is filled with water saturated porous medium made up of silicon. Table 3 shows the thermophysical properties of fluid and porous medium for computation. The properties of porous

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3.4. Temperature Contour Plots

3.4.1. R = 0 (Heat Flux Applied to the Moving Plate Only) Two dimensional temperature field is plotted to compare the effects of viscous dissipation on a conventional channel and a microchannel. A microchannel of height, H = 50 µm and length, L = 30 mm is filled with water saturated porous medium made up of silicon. Table3 shows the thermophysical properties of fluid and porous medium for computation. The properties of porous medium and geometry size are derived from Ting et. al. [14] who applied porous medium modeling to solve a forced convection problem in microchannel. The resulting porous medium shape factor is calculated to be S = 30. The temperature contour plots are based on a constant Re, from which the corresponding flow velocity and Br are computed. Inlet temperature is specified as 300 K and heat flux applied is taken as 1 104 W/m2 for all the computation hereinafter. Figure6a–d depict the temperature × contour plots in a microchannel with flow velocity, u = 1.90 m/s and u = 0.952 m/s, respectively and Re fixed at 100 and 50, respectively. Viscous dissipation causes an increase in the axial temperature as reflected by Figure6b,d, corresponding to Br = 6.20 and Br = 1.55. The transverse temperature profile is flatter at a higher Re indicating an improved heat transfer coefficient. Notwithstanding the larger heat convection coefficient with increasing Re, a higher Br elevates the axial temperature along the axial direction slightly as reflected by Figure6b,d, when viscous dissipation is accounted for, reflecting the significance of viscous dissipation to a microchannel.

Table 3. Thermophysical properties of ﬂuid and porous medium in a microchannel [11].

Fluid Water Solid Silicon Porosity, 0.9 3 Density of fluid, ρ f (kg/m ) 997 Specific heat of fluid, c (J/kg K) 4179 p, f · Viscosity of fluid, µ (N s/m2) 8.55 10-4 f · × Heat flux, q (W/m2) 1 104 00 × Moving wall temperature, Tw (K) 300 Thermal conductivity of fluid, k (W/m K) 0.613 f · Thermal conductivity of solid, ks (W/m K) 148 · Effective thermal conductivity of porous material, k (W/m K) 15.3 e f f ·

In order to illustrate the effects of channel size on the temperature distribution, two-dimensional temperature distribution is also computed for a conventional size channel of height, H = 10 mm and length, L = 50 mm based on another set of thermophysical properties in Table4. The properties of porous medium are obtained from Hwang and Chao [15] who performed experimental investigation on the heat transfer measurement of sintered porous channel. Figure7a–d show the temperature profile for S = 296 at Re = 250 and Re = 150, with velocities, u = 1.06 m/s and u = 0.636 m/s respectively. For a conventional channel duct, Figure7a,c indicate marked change in temperature di fference along the axial direction where the highest temperature declines with larger Re. Compared to the effects of Br on a microchannel at a fixed Re, the effects of viscous dissipation on the temperature difference diminishes in a conventional channel by Figure7b,d, corresponding to Br = 0.061 and Br = 0.022 As Reynolds number goes up, the effect of heat generation is withered by a more marked convection, hence causing the decrease in the highest temperature along axial direction. Symmetry 2019, 11, x FOR PEER REVIEW 9 of 14

2 -5 Viscosity of fluid, 𝜇 (N·s/m ) 1.846 × 10 Heat flux, 𝑞"(W/m 2) 0.8 x 104

Moving wall temperature,𝑇 (K) 300 -3 Thermal conductivity of fluid, 𝑘 (W/m·K) 26.14 × 10

SymmetryThermal2019 ,conductivity11, 869 of solid, 𝑘 (W/m·K) 10.287 9 of 15

Effective thermal conductivity of porous material, 𝑘 (W/m·K) 6.5

(a) (b)

Figure 6. Temperature ﬁeld in a microchannel, for R = 0, S = 30 and k = 15.3 W/m K based on e f f · the thermal physical properties of Table3.( a) Re = 100 without viscous dissipation, (b) Re = 100 with viscous dissipation, (c) Re = 50 without viscous dissipation, (d) Re = 50 with viscous dissipation.

Table 4. Thermophysical properties of ﬂuid and porous medium in a conventional channel [12].

Fluid Air Solid Sintered Bronze Beads Porosity, 0.37 9 Permeability, K 0.422 10− 3 × Density of fluid, ρ f (kg/m ) 1.177 Specific heat of fluid, c (J/kg K) 1005 p, f · Viscosity of fluid, µ (N s/m2) 1.846 10 5 f · × − Heat flux, q (W/m2) 0.8 104 00 × Moving wall temperature,Tw (K) 300 3 Thermal conductivity of fluid, k f (W/m K) 26.14 10 · × − Thermal conductivity of solid, ks (W/m K) 10.287 · Effective thermal conductivity of porous material, k (W/m K) 6.5 e f f · (a) (b) Symmetry 2019, 11, x FOR PEER REVIEW 9 of 14

2 -5 Viscosity of fluid, 𝜇 (N·s/m ) 1.846 × 10 Heat flux, 𝑞"(W/m 2) 0.8 x 104

Moving wall temperature,𝑇 (K) 300 -3 Thermal conductivity of fluid, 𝑘 (W/m·K) 26.14 × 10

Thermal conductivity of solid, 𝑘 (W/m·K) 10.287

Effective thermal conductivity of porous material, 𝑘 (W/m·K) 6.5

(a) (b)

Symmetry 2019, 11, 869 10 of 15

Symmetry 2019, 11, x FOR PEER REVIEW 10 of 14

(a) (b)

(c) (d) Figure 7. Temperature field in a conventional duct for R = 0, S = 296 and k = 6.5 W/m K based on Figure 7. Temperature field in a conventional duct for 𝑅 = 0, 𝑆 = 296 ande f𝑘 f =6.5 W/m·K· based theon the thermal thermal physical physical properties properties of Table of Table4.( a )4.Re (a=) Re250 = without250 without viscous viscous dissipation, dissipation, (b) Re (b=) 250Re = with 250 viscouswith viscous dissipation, dissipation, (c) Re (c=) Re150 = without150 without viscous viscou dissipation,s dissipation, (d) Re (d)= Re150 = 150 with with viscous viscous dissipation. dissipation 3.4.2. R = 0.5 (Equal Heat Flux Applied to Both Plates) 3.4.2. 𝑅=0.5 (Equal heat flux applied to both plates) Figures8 and9 represent two-dimensional temperature field for heat flux applied at both Figures 8 and 9 represent two-dimensional temperature field for heat flux applied at both boundaries for microchannel and conventional channel ducts, respectively. The same thermophysical boundaries for microchannel and conventional channel ducts, respectively. The same thermophysical properties of fluid and porous material are applied, as given in Tables3 and4, respectively. Figure8a,c properties of fluid and porous material are applied, as given in Table 3 and Table 4, respectively. depict the same range of temperature achieved in the channel as R = 0 in Figure6a,c but shows much Figure 8a and 8c depict the same range of temperature achieved in the channel as 𝑅=0 in Figure 6a higher temperature in transverse direction due to an increase in heat transfer coefficient, similar to and 6c but shows much higher temperature in transverse direction due to an increase in heat transfer Figure9a,c as R increases from 0 to 0.5. Likewise, Figure8b,d reflect significant viscous dissipation coefficient, similar to Figure 9a and 9c as 𝑅 increases from 0 to 0.5. Likewise, Figure 8b and 8d reflect effect on the axial temperature variation whereas for conventional channel in Figure9b,d, the change significant viscous dissipation effect on the axial temperature variation whereas for conventional in transverse temperature is more noticeable than the microchannel. In a microchannel, the surface channel in Figures 9b to 9d, the change in transverse temperature is more noticeable than the temperature approaches the mean temperature due to its apparently much higher heat transfer microchannel. In a microchannel, the surface temperature approaches the mean temperature due to coefficient than a conventional channel. Expectedly, a more gradual temperature change is noted in the its apparently much higher heat transfer coefficient than a conventional channel. Expectedly, a more proximity of the moving wall due to its higher heat transfer coefficient than the fixed wall. A decreasing gradual temperature change is noted in the proximity of the moving wall due to its higher heat temperature field with increasing Reynolds number indicates the enhancement in convection. transfer coefficient than the fixed wall. A decreasing temperature field with increasing Reynolds number indicates the enhancement in convection.

(a) (b)

Symmetry 2019, 11, x FOR PEER REVIEW 10 of 14

Figure 7. Temperature field in a conventional duct for 𝑅 = 0, 𝑆 = 296 and 𝑘 =6.5 W/m·K based on the thermal physical properties of Table 4. (a) Re = 250 without viscous dissipation, (b) Re = 250 with viscous dissipation, (c) Re = 150 without viscous dissipation, (d) Re = 150 with viscous dissipation

3.4.2. 𝑅=0.5 (Equal heat flux applied to both plates) Figures 8 and 9 represent two-dimensional temperature field for heat flux applied at both boundaries for microchannel and conventional channel ducts, respectively. The same thermophysical properties of fluid and porous material are applied, as given in Table 3 and Table 4, respectively. Figure 8a and 8c depict the same range of temperature achieved in the channel as 𝑅=0 in Figure 6a and 6c but shows much higher temperature in transverse direction due to an increase in heat transfer coefficient, similar to Figure 9a and 9c as 𝑅 increases from 0 to 0.5. Likewise, Figure 8b and 8d reflect significant viscous dissipation effect on the axial temperature variation whereas for conventional channel in Figures 9b to 9d, the change in transverse temperature is more noticeable than the microchannel. In a microchannel, the surface temperature approaches the mean temperature due to its apparently much higher heat transfer coefficient than a conventional channel. Expectedly, a more gradual temperature change is noted in the proximity of the moving wall due to its higher heat Symmetry 2019, 11, 869 11 of 15 transfer coefficient than the fixed wall. A decreasing temperature field with increasing Reynolds number indicates the enhancement in convection.

Symmetry 2019, 11, x FOR PEER REVIEW 11 of 14 Symmetry 2019, 11, x FOR PEER REVIEW 11 of 14 (a) (b)

(c) (d) (c) (d) Figure 8. Temperature ﬁeld in a microchannel, for R = 0.5, S = 30 and k = 15.3 W/m K based on e f f · the thermal physical properties of Table3.( a) Re = 100 without viscous dissipation, (b) Re = 100 with viscous dissipation, (c) Re = 50 without viscous dissipation, (d) Re = 50 with viscous dissipation.

(a) (b) (a) (b)

Figure 9. Cont.

(c) (d) (c) (d) Figure 9. Temperature field in a conventional duct for 𝑅 = 0.5, 𝑆 =296 and 𝑘 =6.5 W/m·K based Figure 9. Temperature field in a conventional duct for 𝑅 = 0.5, 𝑆 =296 and 𝑘 =6.5 W/m·K based on the thermal physical properties of Table 4. (a) Re = 250 without viscous dissipation, (b) Re = 250 withon viscous the thermal dissipation, physical (c )properties Re = 150 without of Table viscou 4. (as) dissipation,Re = 250 without (d) Re viscous= 150 with dissipation, viscous dissipation (b) Re = 250 with viscous dissipation, (c) Re = 150 without viscous dissipation, (d) Re = 150 with viscous dissipation 4. Conclusions 4. Conclusions The temperature field for a Couette-Poiseuille flow characterized by a maximum velocity betweenThe two temperature parallel plates field in afo saturatedr a Couette-Poiseuille porous medium flow is solvedcharacterized with emphasis by a maximum on the effects velocity of viscousbetween dissipation. two parallel The platestemperature in a saturated distribution porous is subjectmedium to is heat solved flux with appl emphasisied at both on boundaries, the effects of viscous dissipation. The temperature distribution is subject to heat flux applied at both boundaries, Symmetry 2019, 11, x FOR PEER REVIEW 11 of 14

Symmetry 2019, 11, 869 12 of 15

(a) (b)

(c) (d) Figure 9. Temperature field in a conventional duct for 𝑅 = 0.5, 𝑆 =296 and 𝑘 =6.5 W/m·K based Figure 9. Temperature ﬁeld in a conventional duct for R = 0.5, S = 296 and k = 6.5 W/m K based on on the thermal physical properties of Table 4. (a) Re = 250 without viscouse dissipation, f f (b· ) Re = 250 the thermal physical properties of Table4.( a) Re = 250 without viscous dissipation, (b) Re = 250 with with viscous dissipation, (c) Re = 150 without viscous dissipation, (d) Re = 150 with viscous dissipation viscous dissipation, (c) Re = 150 without viscous dissipation, (d) Re = 150 with viscous dissipation.

4.4. Con Conclusionsclusions TheThe temperaturetemperature field field for fo ar Couette-Poiseuillea Couette-Poiseuille flow flow characterized characterized by a maximum by a maximum velocity betweenvelocity betweentwo parallel two platesparallel in plates a saturated in a saturated porous mediumporous me is solveddium is with solved emphasis with emphasis on the e onffects the of effects viscous of viscousdissipation. dissipation. The temperature The temperature distribution distribution is subject is to subject heat flux to appliedheat flux at appl bothied boundaries, at both boundaries, whereby a heat flux ratio is defined as the heat flux applied to the fixed boundary to the total heat flux applied to both boundaries. The velocity field at lower porous medium shape factor S in particular, envisages the competing effect between convection and heat generation in the form of viscous dissipation. Due to the more significant viscous dissipation effect particularly in the vicinity of the fixed wall, Nusselt number depends and varies significantly with Br for smaller porous medium shape factor S. The two-dimensional temperature contour plots based on the properties derived from the literature show that viscous dissipation causes appreciable temperature hike in a microchannel along the axial direction at fixed Reynolds numbers. The transverse temperature change is however more apparent in a conventional channel due to a comparatively lower heat transfer coefficient at the wall for its correspondingly larger hydraulic diameter. Therefore, while the effect of viscous dissipation is significant to the enthalpy change for flows in a microchannel, its effect on the transverse temperature profile becomes more obvious as the characteristic size of the channel increases. As Reynolds number goes up, the attendant increase in viscous dissipation is more significant than the inertial effect in a microchannel, while the contrary is true in a conventional duct.

Author Contributions: Conceptualization, G.M.C., B.K.L., C.P.T.; methodology, G.M.C., M.F.B. software, M.F.B.; validation, G.M.C., M.F.B.; formal analysis, G.M.C., M.F.B.; investigation, G.M.C., M.F.B.; resources, G.M.C.; writing—original draft preparation, G.M.C., M.F.B.; writing—review and editing, G.M.C., M.F.B., B.K.L., C.P.T.; supervision, G.M.C., B.K.L., C.P.T.; funding acquisition, G.M.C., B.K.L., C.P.T. Funding: Ministry of Education Malaysia under the research grant FRGS/1/2013/TK01/MMU/02/01. Acknowledgments: This research is funded by the Ministry of Education Malaysia under the research grant FRGS/1/2013/TK01/MMU/02/01. Conﬂicts of Interest: The authors declare no conﬂict of interest. Symmetry 2019, 11, 869 13 of 15

Nomenclature

A Constant, defined in Equation (13) Br Brinkman number C C Coefficients in Equation (16), listed in AppendixA 1 − 7 c Specific heat of fluid, J/kg K p, f · Da Darcy number, K/H2 H Height of the channel, m K Permeability of the porous medium, m2 k Thermal conductivity, W/m K · k Effective thermal conductivity of porous medium, W/m K e f f · L Length of the channel, m µe f f M Ratio of effective viscosity to viscosity, defined as, µ q H Nu Nusselt number, defined as, 00 ke f f (Tw Tm) 2 − q00 Heat flux, W/m ρ u H Re Reynolds number, defined as f m µe f f q R The fraction of heat flux applied to fixed wall, defined as, 00 2 q00 1+q00 2 S Porous medium shape factor, defined as, 1 √MDa T Fluid temperature, K Tm Mean temperature, K Tw Wall temperature at lower plate, K u Fluid velocity, m/s uw Moving wall velocity, m/s um Mean velocity, m/s Um Dimensionless mean velocity µe f f u U Dimensionless velocity, defined as, γH2 µe f f uw Uw Dimensionless wall velocity, defined as, γH2 u Dimensionless velocity, defined as, u = U ∗ um Um x Axial coordinate of the channel, m x X Dimensionless length, L y Y Dimensionless transverse distance, defined as, H y Vertical coordinate, m γ Pressure gradient, N/m3 3 ρ f Density of the fluid, kg/m µ Viscosity of the fluid, N s/m2 · µ Effective viscosity of the porous medium, N s/m2 e f f · k (TA Tw) e f f − θ Dimensionless temperature, ( + ) q100 q200 H k (TB Tw) e f f − θ1 Dimensionless temperature, ( + ) q100 q200 H ke f f (TB Tw) θ2 Dimensionless temperature, − q100H Symmetry 2019, 11, 869 14 of 15

Appendix A : List of coeﬃcients

n h i = 1 ( )2 + ( ) ( ) + 2 [ C1 2 2 cos ech S/2 1 cos h S Ssin h S S Uw 2A 2[ 2+Scot h(S/2)+S Uw] − − − − 2 o +( Br + A)Scot h(S/2) + AS Uw] − 2 Scot h(S/2)[ 2A+( Br+A)Scot h(S/2)+AS Uw] = − − C2 2 2 2[ 2+Scot h(S/2)+S Uw] − 2 2 cot h(S/2)( 1+S Uw)[ 2A+(A 2Br)Scot h(S/2)+AS Uw] = − − − C3 2 2 S[ 2+Scot h(S/2)+S Uw] − 4 2 2 2 Brcos ech(S/2) 1+2 cosh(S)( 1+S Uw)+cosh(2S)( 1+S Uw) − − C4 = 2 16[ 2+Scot h(S/2)+S2] − − 2 2 2 cos ech(S/2) [ 2A+(A 2Br)Scot h(S/2)+AS Uw][1+cos h(S)( 1+S Uw)] = − − − C5 2 2 2S[ 2+Scot h(S/2)+S Uw] − − 2 2 2 Brcot h(S/2)cos ech(S/2) ( 1+S Uw)[ 1+cos h(S)( 1+S Uw)] = − − − C6 2 2 4[ 2+Scot h(S/2)+S Uw] − n = 1 ( )2 [ + ( ) ( )+ C7 2 2 cos ech S/2 4 4BrS 2AS 3Br 2A Scos h S 16S[ 2+Scot h(S/2)+S Uw] − − − − 2 4Asin h(S)] + S Uw 4S[3Br 2A + 2(Br A)cos h(S)] + 24Asin h(S)+ h − − io S2 BrScos h(2S)cos ech(S/2)2 8Asin h(S) − }}

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