Int J Thermophys (2011) 32:2079–2091DOI 10.1007/s10765-011-1069-9

Mixed Convection of Water at 4 ◦C Along a Wedgewith Variable Surface Temperature in a Porous Medium

Waqar A. Khan · Rama Subba Reddy Gorla

Received: 7 July 2009 / Accepted: 12 August 2011 / Published online: 21 September 2011© Springer Science+Business Media, LLC 2011

Abstract In this study, the mixed convection of water at 4 ◦C along a wedge ina porous medium is investigated numerically using a finite difference method. Toexplore the effect of mixed convection, both forced and free convection-dominatedregimes are considered. Non-similarity solutions are obtained for the variable wall-temperature boundary condition. Velocity and temperature profiles as well as localdimensionless skin friction and the Nusselt number are obtained and compared withavailable numerical results for various values of different parameters. The wedge anglegeometry parameter m and mixed convection parameter ξ ranged from 0 to 1 in bothregimes, whereas different values of λ are considered for the purpose of comparisonof heat transfer results.

Keywords Mixed convection · Non-similar solutions · Nusselt number ·Skin friction · Water · Wedge

List of SymbolsCf Skin frictionf Dimensionless stream functiong Acceleration due to gravity, m · s−2

h Heat transfer coefficient, W · m−2 · K−1

K Permeability of the porous medium, m2

m Wedge flow parameterNu Nusselt number

W. A. KhanDepartment of Engineering Sciences, PN Engineering College,National University of Science and Technology, Karachi 75350, Pakistan

R. S. R. Gorla (B)Department of Mechanical Engineering, Cleveland State University, Cleveland, OH, USAe-mail: r.gorla@csuohio.edu

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Pex Local Peclet numberRax Local Rayleigh numberT Temperature, ◦CU∞ Free stream velocity, m · s−1

u, v Velocity components in x- and y-directionx, y Coordinates along and normal to wedge surface

Subscriptsf Forced convection-dominated regime∞ Free stream conditionsn Free convection-dominated regimew Wall

Greek Symbols

α Thermal diffusivity of porous medium, m2 · s−1

β Thermal expansion coefficient for water at 4 ◦C, ◦C−2

η Similarity variableγ Semi-wedge angleλ Temperature variation parameterμ Absolute viscosity, Pa · sν Kinematic viscosity, m2 · s−1

ρ Fluid density, kg · m−3

ψ Stream functionθ Dimensionless temperatureξ Mixed convection parameter

1 Introduction

Fluid flow and heat transfer in porous media have been of considerable interest becauseof numerous applications of flow through porous media, such as storage of radioactivenuclear waste materials, transpiration cooling, separation processes in chemical indus-tries, filtration, transport in aquifers, and ground water pollution, etc. Fluid densitychanges as a function of temperature in a linear manner. A notable exception is liquidwater. Pure water at a pressure of 1 atm has a maximum density of 999.9720 kg · m−3

at 4 ◦C [1]. Above this temperature, the density of water decreases as the temperatureis increased in a manner similar to other fluids. For temperature below 4 ◦C, the trendis reversed, the density increases with increased temperature, giving rise to a maxi-mum density at the 4 ◦C point. Water and several other metals have their maximumdensity in the liquid phase, at a temperature Tc = 3.98 ◦C above the melting-pointtemperature. Goren [2] has shown that for a temperature sufficiently close to Tc, therelationship between the fluid temperature T and density is given by the relation,

ρ − ρc

ρc= −β(T − Tc)

2

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where ρc is the maximum density in the liquid phase and β = 8.0 × 10−6 (◦C)−2 isthe fluid thermal expansion coefficient of water at 4 ◦C. Moore and Weiss [3] state thatthe above equation is accurate within ±4 % in the range 0 ◦C ≤ T ≤ 8 ◦C. Severalstudies exist for mixed convection boundary-layer flow about vertical, inclined, andwedge surfaces immersed in a viscous fluid. The details can be found in various books[4–6] and in the literature [7–16]. However, the literature on free/mixed convectionboundary-layer flow of water at 4 ◦C past different geometries in a porous medium isvery limited. A brief review of the literature related to this topic is given below.

Steady free convection from a vertical wall/plate embedded in a porous mediumfilled with water at 4 ◦C was studied by Goren [17], and by Kumaran and Pop [18].They determined similarity solutions of the governing boundary-layer equations for aclass of problems where the variable wall temperature (VWT), the variable heat flux(VHF), or the variable heat transfer coefficient (VHTC) vary as power functions ofthe distance from the leading edge of the plate. The analytical and numerical solutionsof the similarity form of the boundary-layer equations yield velocity and temperatureprofiles as well as values of the stream function at the edge of the boundary layer, theheat transfer coefficient, and the temperature on the plate.

Takhar and Perdikis [19], Gorla and Stratman [20], and Raptis and Pop [21] alsoconsidered free and forced convection flow of water at 4 ◦C past different geometriesunder different boundary conditions. They obtained numerical solutions and providedresults for the velocity distribution, skin friction, and the dimensionless heat transferrate in graphical and tabular forms. Gorla and Stratman [20] showed that the heattransfer rate increases with increasing transverse curvature, whereas Raptis and Pop[21] discussed the effects of the Grashof number and permeability parameter on theflow characteristics.

Ling et al. [22,23] studied the steady mixed convection boundary-layer flow overa vertical impermeable surface in a porous medium saturated with water at 4 ◦C forboth variable wall temperatures and variable surface heat flux. They considered bothcases of the assisting and opposing flows and obtained numerical solutions for generalvalues of the flow parameters under both boundary conditions. They obtained anddiscussed numerical results for the dimensionless skin friction, wall heat transfer, aswell as the temperature profiles for different values of the mixed convection parameterλ and the power index m.

In this paper, we are concerned with the mixed convection of water at 4 ◦C alonga wedge with a variable surface temperature in a porous medium. To cover the entiremixed convection regime, forced and free convection-dominated regimes are stud-ied separately. The transformed boundary-layer equations are solved by means of afinite difference method. The numerical results are obtained for dimensionless veloc-ity f (ξ, η) and temperature θ(ξ, η) profiles, skin friction coefficients f ′′(ξ, 0), andlocal dimensionless heat transfer coefficients −θ ′(ξ, 0) for several values of m and λ.

2 Analysis

Consider steady mixed convection of water at 4 ◦C along an impermeable wedge(Fig. 1) with a variable wall temperature (Tw = T∞ + Axλ, where A and λ are speci-fied constants) in a porous medium. It is assumed that the physical properties of water

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2082 Int J Thermophys (2011) 32:2079–2091

Fig. 1 Flow model andcoordinate system

except the density are constant and that the Boussinesq approximation for the case ofmaximum density is valid. The pores of the porous medium and the flow velocity areassumed to be small. The free stream velocity U∞ is oriented in the upward direction,and the ambient temperature is T∞ as shown in Fig. 1, where x and y are the Carte-sian coordinates along the wedge surface and normal to it, respectively. The velocitycomponents are u along and v normal to the wedge surface. The acceleration due togravity g acts vertically downward opposite to the free stream velocity. FollowingGoren [2] and Pop and Ingham [6], the boundary-layer equations governing mixedconvection flow over a wedge embedded in a water-saturated porous medium at 4 ◦Ccan be written as

∂u

∂x+ ∂v

∂y= 0 (1)

u = U∞ + K

μρ gx β (T − T∞)2 (2)

u∂T

∂x+ v

∂T

∂y= α

∂2T

∂y2 (3)

with boundary conditions,

y = 0 : v = 0, Tw = T∞ + Axλ

y → ∞ : u → U∞, T → T∞

}(4)

where gx = g cos γ in Eq. 2 is the x-component of the acceleration due to gravity.This problem is solved for two different regimes separately.

2.1 Forced Convection-Dominated Regime

Defining a stream function ψ(x, y) such that

u = ∂ψ

∂yand v = − ∂ψ

∂x(5)

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and the following transformations,

η = y

xPe1/2

x ψ = αPe1/2x f (ξ, η)

ξf = Rax

Pexθ = T − T∞

Tw − T∞

Pex = U∞x

αRax = x

α

[ρK gxβ( Tw)

2

μ

]

U∞ = Cxm m = γ

π − γ

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(6)

Using these transformations, the above governing equations become

f ′ = 1 + ξf θ2 (7)

and

θ ′′ − λ f ′θ +(

m + 1

2

)f θ ′ = (2λ− m)ξf

[f ′ ∂θ∂ξf

− θ ′ ∂ f

∂ξf

](8)

with transformed boundary conditions

f (ξf , 0) = 0 θ(ξf , 0) = 1

f ′(ξf ,∞) = 1 θ(ξf ,∞) = 0

}(9)

where f ′ = ∂ f

∂ηand θ ′ = ∂θ

∂η. The local skin friction and dimensionless heat transfer

coefficients are given by

f ′′(0) = 2 ξf θ(0)θ′(0) (10)

Nu(ξf , 0)

Pe1/2x

= −θ ′(0) (11)

The dimensionless parameter ξf measures the effect of buoyancy on forced convection.For pure forced convection, ξf = 0, which reduces Eqs. 7 and 8 to a single governingequation

θ ′′ − λ θ +(

m + 1

2

)η θ ′ = 0 (12)

with boundary conditions,

θ(0) = 1, θ(∞) = 0 (13)

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It is interesting to note that for m = 0 (vertical wall), Eq. 12 has analytical solutionsfor λ = 0, 1/2, 1, and 1.5 (Table 1). For other values of m and λ, Eq. 12 is solvednumerically by using a finite difference method. The results for local dimensionlessheat transfer coefficients are compared with the results of Gorla and Kumari [9] forλ = 0.5 in Table 1. For the mixed convection regime (0 < ξf < 1), Eqs. 7 and 8with boundary conditions of Eq. 9 are solved by using the method described by Gorlaand Kumari [9]. Numerical results for local dimensionless heat transfer coefficientsare presented and compared with the results of Gorla and Kumari [9] for Newtonianfluids in Table 2. It is observed that heat transfer values increase with the increase ofthe effect of buoyancy on forced convection. The results of local heat transfer ratesare also compared with the results of Hsieh et al. [10] and Cheng [11] for m = 0. Thepresent results are found to be lower for water at 4 ◦C. It is important to note that, inall cases, the heat transfer values increase with the values of λ and ξf (Tables 3, 4).

Table 1 Analytical solutions for m = 0 and λ = 0, 1/2, and 1

λ θ(η)

0 1 − erf(η

2

)0.5

√π

2η

[erf

(η2

)− 1

]+ exp

(−η

2

4

)

11

2

[−η2erf

(η2

)+ 2 + η2 − 2√

πη exp

(−η

2

4

)− 2 erf

(η2

)]

1.5 lima→∞

⎡⎢⎢⎢⎢⎣−

hypergeom

([2], [1/2], a2

4

)(6 + η2

)η eη

2/4

a ea2/4 (6 + a2

) + hypergeom

([2], [1/2], η

2

4

)⎤⎥⎥⎥⎥⎦ e−η2/4

Table 2 Local dimensionlessheat transfer values for forcedconvection (ξf = 0) along awedge

m λ −θ ′(ξf , 0)

Present results(water at 4 ◦C)

Gorla and Kumari [9](non-Newtonian fluid)

0 0.5 0.886227 0.88602

1.0 1.128379 –

1.5 1.329340 –

1/3 0.5 0.941978 0.94151

1.0 1.170828 –

1.5 1.364482 –

1/2 0.5 0.968891 0.96833

1.0 1.191775 –

1.5 1.381976 –

1.0 0.5 1.046049 1.04522

1.0 1.253314 –

1.5 1.433966 –

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Table 3 Comparison of localheat transfer coefficients formixed convection along a wedge

ξf −θ ′(ξf , 0) for m = 1/3 and λ = 1/2

Present results(water at 4 ◦C)

Gorla and Kumari [9](Newtonian fluid)

0 0.94198 0.94151

0.1 0.96862 0.97336

0.2 0.99181 1.00412

0.3 1.01443 1.03391

0.4 1.03653 1.06281

0.5 1.05819 1.09091

0.6 1.07947 1.11826

0.7 1.10043 1.14492

0.8 1.12113 1.17095

0.9 1.14164 1.19639

1 1.16203 1.22127

Table 4 Comparison of localheat transfer coefficients formixed convection along avertical wall (m = 0)

ξf −θ ′(ξf , 0)

Present results Hsieh et al. [10] Cheng [11](Water at 4 ◦C) (Newtonian fluid)

(λ = 1/2) (λ = 1/3) (λ = 1/2) (λ = 1/3)

0 0.88623 0.79110 0.8862 0.8540

0.5 0.98743 0.88092 1.0428 0.9816

1.0 1.07853 0.96173 1.1780 1.0930

10 2.11814 1.88355 2.5960 2.3110

20 2.85459 2.53689 3.5602 3.1520

30 3.43577 3.05263 4.3140 –

40 3.93165 3.49276 4.9545 –

50 4.37151 3.88319 5.5212 –

100 6.11266 5.42892 7.7570 –

500 13.5411 12.0247 17.265 –

1000 19.1272 16.9850 24.419 –

2.2 Free Convection-Dominated Regime

Using the following transformations for the free convection-dominated regime,

η = y

xRa1/2

x ψ = αRa1/2x f (ξ, η)

ξn = Pex

Raxθ = T − T∞

Tw − T∞

⎫⎪⎬⎪⎭ (14)

Equations 1–3 reduce to

f ′ = ξn + θ2 (15)

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and

θ ′′ +(λ+ 1

2

)f θ ′ − λ f ′θ = (m − 2λ)ξn

[f ′ ∂θ∂ξn

− θ ′ ∂ f

∂ξn

](16)

with transformed boundary conditions,

f (ξn, 0) = 0 θ(ξn, 0) = 1

f ′(ξn,∞) = ξ θ(ξn,∞) = 0

}(17)

In this regime, the local dimensionless skin friction and heat transfer coefficients aregiven by

f ′′(0) = 2θ(ξn, 0)θ ′(ξn, 0) (18)

Nux (0)

Ra1/2 = −θ ′(ξn, 0) (19)

The controlling parameter ξn determines the forced flow effect on free convection. Forpure free convection, ξn = 0, whereas in the limiting case, ξn → ∞; it correspondsto pure forced convection. For pure free convection, Eqs. 15 and 16 reduce to

d f (η)

dη− θ2 (η) = 0 (20)

θ ′′(η)+(λ+ 1

2

)f(η) θ ′(η)− λ θ(η) f ′(η) = 0 (21)

with boundary conditions,

f (0) = 0 θ(0) = 1

f ′(∞) = 0 θ(∞) = 0

}(22)

In this case, the local dimensionless skin friction and heat transfer coefficients aregiven by

f ′′(0) = 2θ(0)θ ′(0) (23)

Nux

Ra1/2x

= −θ ′(0) (24)

The results for the local dimensionless skin friction and heat transfer coefficients forthe case of free convection are presented and compared with Kumaran and Pop [19]in Table 5 for water at 4 ◦C along a vertical wall (m = 0). Table 6 shows the effect offorced flow on free convection heat transfer rates. The results are compared with Gorlaand Kumari [9] for Newtonian fluids. The present results of the local dimensionlessskin friction and heat transfer coefficients for for forced convection-dominated regime

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Table 5 Comparison ofdimensionless heat transfervalues for free convection(ξn = 0) of water at 4 ◦C alonga vertical wall (m = 0)

λ −θ ′(0) (present) −θ ′(0) [18]

0 0.4702 0.3765

0.25 0.5723 0.5651

0.50 0.7992 0.7071

0.75 0.9050 0.8255

1.00 1.0000 0.9291

1.50 1.1675 1.1080

2.00 1.3141 1.2618

5.00 1.9783 1.9445

10.0 2.7500 2.7259

20.0 3.8550 3.8380

50.0 6.0722 6.0253

100 8.5608 8.5514

200 12.0936 12.0881

Table 6 Comparison ofdimensionless heat transfervalues for mixed convectionalong a wedge(m = 1/3, λ = 1/2)

ξn −θ ′(0) (present) −θ ′(0) [9](Water at 4 ◦C) (Newtonian fluid)

0 0.69139 0.77584

0.1 0.76404 0.83043

0.2 0.83451 0.88220

0.3 0.90212 0.93134

0.4 0.96622 0.97810

0.5 1.02611 1.02270

0.6 1.08114 1.06528

0.7 1.13062 1.10557

0.8 1.17389 1.14247

0.9 1.21027 1.17753

1 1.23909 1.22119

are presented in Table 7. It shows the effects of temperature variation and wedge flowparameters on both local dimensionless skin friction and heat transfer coefficients. It isclear that both local dimensionless skin friction and heat transfer coefficients increasewith the increase in both temperature variation and wedge flow parameters.

3 Results and Discussion

Figures 2 and 3 show the dimensionless velocity f ′(ξ, η) and temperature θ(ξ, η) pro-files for both forced and free convection-dominated regimes for the selected values ofthe wedge parameter m, the temperature variation parameter λ, and mixed convectionparameter ξ . It can be seen from Fig. 2 that as the effect of buoyancy on forced con-vection (Fig. 2a) or the effect of forced flow on free convection (Fig. 2b) increases, the

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Table 7 Local dimensionless skin friction and heat transfer coefficients for forced convection-dominatedregime

m ξf λ = 0.5 λ = 1.0 λ = 1.5

−Cf (ξ, 0) − Nu(ξ, 0)

(Pex )1/2−Cf (ξ, 0) − Nu(ξ, 0)

(Pex )1/2−Cf (ξ, 0) − Nu(ξ, 0)

(Pex )1/2

0 0 0 0.88623 0 1.12838 0 1.32934

0.1 0.18149 0.90745 0.23113 1.15566 0.27233 1.36165

0.2 0.37126 0.92814 0.47291 1.18228 0.55727 1.39317

0.3 0.56901 0.94835 0.72496 1.20827 0.85438 1.42397

0.4 0.77448 0.9681 0.98695 1.23369 1.16327 1.45408

0.5 0.98743 0.98743 1.25857 1.25857 1.48356 1.48356

0.6 1.20763 1.00636 1.53953 1.28294 1.81494 1.51245

0.7 1.43487 1.02491 1.82957 1.30683 2.15708 1.54077

0.8 1.66898 1.04311 2.12845 1.33028 2.5097 1.56856

0.9 1.90976 1.06098 2.43594 1.3533 2.87253 1.59585

1 2.15707 1.07853 2.75183 1.37591 3.24533 1.62266

1/3 0 0 0.94198 0 1.17083 0 1.36448

0.1 0.19289 0.96445 0.23981 1.19903 0.2795 1.39753

0.2 0.39454 0.98636 0.49061 1.22653 0.5719 1.42976

0.3 0.60465 1.00774 0.75203 1.25339 0.87675 1.46125

0.4 0.82292 1.02865 1.02372 1.27965 1.19363 1.49204

0.5 1.0491 1.0491 1.30534 1.30534 1.52218 1.52218

0.6 1.28296 1.06913 1.59662 1.33052 1.86204 1.5517

0.7 1.52427 1.08877 1.89727 1.35519 2.21291 1.58065

0.8 1.77284 1.10802 2.20705 1.3794 2.57448 1.60905

0.9 2.02847 1.12693 2.52571 1.40317 2.94649 1.63694

1 2.291 1.1455 2.85305 1.42653 3.32868 1.66434

1/2 0 0 0.96889 0 1.19177 1.38198

0.1 0.19839 0.99197 0.24409 1.22043 0.28308 1.41539

0.2 0.40579 1.01447 0.49935 1.24838 0.5792 1.44799

0.3 0.62186 1.03643 0.7654 1.27567 0.8879 1.47983

0.4 0.84632 1.0579 1.04188 1.30235 1.20877 1.51096

0.5 1.0789 1.0789 1.32845 1.32845 1.54143 1.54143

0.6 1.31936 1.09947 1.62483 1.35403 1.88553 1.57128

0.7 1.56748 1.11963 1.93074 1.3791 2.24075 1.60054

0.8 1.82304 1.1394 2.24591 1.40369 2.6068 1.62925

0.9 2.08586 1.15881 2.57011 1.42784 2.9834 1.65744

1 2.35576 1.17788 2.90312 1.45156 3.37029 1.68514

1 0 0 1.04605 0 1.25331 0 1.43397

0.1 0.21418 1.07088 0.25667 1.28333 0.2937 1.4685

0.2 0.43803 1.09508 0.52504 1.31259 0.60087 1.50219

0.3 0.67123 1.11871 0.8047 1.34117 0.92105 1.53508

0.4 0.91344 1.1418 1.09528 1.3691 1.25379 1.56724

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Table 7 continued

m ξf λ = 0.5 λ = 1.0 λ = 1.5

−Cf (ξ, 0) − Nu(ξ, 0)

(Pex )1/2−Cf (ξ, 0) − Nu(ξ, 0)

(Pex )1/2−Cf (ξ, 0) − Nu(ξ, 0)

(Pex )1/2

0.5 1.16439 1.16439 1.39643 1.39643 1.59871 1.59871

0.6 1.42381 1.18651 1.70784 1.4232 1.95545 1.62954

0.7 1.69146 1.20819 2.02922 1.44944 2.32367 1.65977

0.8 1.96712 1.22945 2.36029 1.47518 2.70307 1.68942

0.9 2.25059 1.25033 2.70082 1.50045 3.09335 1.71853

1 2.54166 1.27083 3.05056 1.52528 3.49426 1.74713

η

f' (ξf,η

)

0.5 1 1.5 20.5

1

1.5

2

2.5

0.30.61.0

ξf

m = 1/3

λ= 1/2

ξf

(a)

η

f' ( ξn

,η)

0 0.5 1 1.5 2 2.5 30.5

1

1.5

2

2.5

0.30.61.0

ξn

m = 1/3

λ= 1/2

ξn

(b)

Fig. 2 Velocity profiles for (a) forced convection-dominated region and (b) free convection-dominatedregion for various values of ξ with m = 1/3 and λ = 1/2

η

θ(ξ f,η)

0 1 2 3

0.2

0.4

0.6

0.8

1

00.61.0

m = 1/3

λ = 1/2

ξf

ξf(a)

η

θ(ξ n,η)

0 1 2 30

0.2

0.4

0.6

0.8

1

00.61.0

m = 1/3

λ = 1/2ξn

ξn(b)

Fig. 3 Temperature profiles for (a) forced convection-dominated region and (b) free convection-dominatedregion for various values of ξ with m = 1/3 and λ = 1/2

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ξn

f'' (ξn,

0)

0 5 10 15 20 250

100

200

300

4000.51.01.5

m = 1/3

λ

(b)λ

ζf

f'' (ξf,

0 )

0 5 10 15 20 25

50

100

150

200

250

0.51.01.5

m = 1/3λ

(a)λ

Fig. 4 Local shear stress for (a) forced convection-dominated region and (b) free convection-dominatedregion for various values of λ with m = 1/3

flow velocity increases at the wedge surface but approaches to the free stream velocityas η increases. It is also observed that the flow velocity approaches the free streamvelocity quickly in the forced convection-dominated regime. Figure 3 shows that thedimensionless temperature depends on the mixed convection parameter ξ along withm and λ. As ξ increases, the dimensionless temperature decreases in both forced andfree convection-dominated regions for all values of m and λ and approaches to ambientconditions as the streamwise distance along the wedge increases.

Skin friction coefficients at the wedge surface are shown in Fig. 4 for both forcedand free convection-dominated regimes for different values of λ. It can be seen in Fig.4a and b that as the value of λ increases, the skin friction increases with the parameterξ in both regimes. This is due to the fact that the skin friction depends strongly onthe parameter ξ . It is also observed that the skin friction increases with the wedgeparameter m and the temperature parameter λ.

Local dimensionless heat transfer rates at the wedge surface are shown in Fig. 5 forboth regimes for different values of λ. Forced and free convection asymptotes are alsoshown to compare the heat transfer rates in the mixed flow regime. It can be seen thatthe heat transfer rates increase with an increase in the temperature parameter λ. Asthe value of ξ increases/decreases, the heat transfer rates approach to the asymptotes.These heat transfer rates are also compared with the available numerical results forspecified values of m and λ.

4 Conclusions

Mixed convection of water at 4 ◦C along a wedge in a porous medium is investigatednumerically using a finite difference method. The boundary-layer equations are solvednumerically for both forced and free convection-dominated regimes, and non-simi-lar solutions are obtained for specified values of ξ,m, and λ and are compared withavailable numerical results. The dimensionless velocity and temperature profiles are

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Int J Thermophys (2011) 32:2079–2091 2091

ξn

Nu

x/R

a x1/2

10 100 101-1

-1

10210

100

101

Mixed convection

Forced convection asymptote

Free convection asymptote

m = 1/3

λ 1.51.00.5

(b)

λ

ξf

Nu x

/Pe x1/

2

10 100 101-1

-1

10210

100

101

Mixed convection

Forced convection asymptote

Free convection asymptote

m = 1/3

λ 1.51.00.5

(a) λ

Fig. 5 Local dimensionless heat transfer for (a) forced convection-dominated region and (b) free convec-tion-dominated region for various values of λ with m = 1/3

presented for both regimes. The results of local skin friction and dimensionless heattransfer rates are tabulated (Table 7) and presented graphically as well. It is shownthat both skin friction and dimensionless heat transfer rates depend strongly on allparameters.

References

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