# Mixed convection of a viscous dissipating fluid about a vertical flat plate

## Mixed convection of a viscous dissipating fluid about a vertical flat plate

Applied Mathematical Modelling 31 (2007) 843–853 www.elsevier.com/locate/apm Mixed convection of a viscous dissipating ﬂuid about a vertical ﬂat plat...

Applied Mathematical Modelling 31 (2007) 843–853 www.elsevier.com/locate/apm

Mixed convection of a viscous dissipating ﬂuid about a vertical ﬂat plate Orhan Aydın *, Ahmet Kaya Department of Mechanical Engineering, Karadeniz Technical University, 61080 Trabzon, Turkey Received 1 May 2005; received in revised form 1 November 2005; accepted 21 December 2005 Available online 3 March 2006

Abstract In this study, the eﬀect of the viscous dissipation in steady, laminar mixed convection heat transfer from a heated/ cooled vertical ﬂat plate is investigated in both aiding and opposing buoyancy situations. The external ﬂow ﬁeld is assumed to be uniform. The governing systems of partial diﬀerential equations are solved numerically using the ﬁnite diﬀerence method. A parametric study is performed in order to illustrate the interactive inﬂuences of the governing parameters, mainly, the Richardson number, Ri (also known as the mixed convection parameter) and the Eckert number, Ec on the velocity and temperature proﬁles as well as the friction and heat transfer coeﬃcients. Based on the facts the free stream is either in parallel or reverse to the gravity direction and the plate is heated or cooled, diﬀerent ﬂow situations are identiﬁed. The inﬂuence of the viscous dissipation on the heat transfer varied according to the situation. For some limiting cases, the obtained results are validated by comparing with those available from the existing literature. An expression correlating Nu in terms of Pr, Ri and Ec is developed.  2006 Elsevier Inc. All rights reserved. Keywords: Mixed convection; Vertical plate; Viscous dissipation; Eckert number; Richardson number

1. Introduction The mixed (combined forced and free) convection arises in many natural and technological processes (see [1,2]). Depending on the forced ﬂow direction, the buoyancy forces may aid (aiding or assisting mixed convection) or oppose (opposing mixed convection) the forced ﬂow, causing an increase or decrease in heat transfer rates [3]. The problem of mixed convection resulting from the ﬂow over a heated vertical plate is of considerable theoretical and practical interest. A detailed review of the subject, including exhaustive lists of references, can be found in the books by Gebhart et al. [2], Bejan [4], Pop and Ingham [5], Jaluria [6] and Chen and Armaly [7]. Refs. [8–13] are some examples of the recent relevant studies existing in the literature. Kafoussias et al. [8] used a modiﬁed and improved numerical solution scheme, for local non-similarity boundary layer analysis, to *

Corresponding author. Tel.: +90 462 377 2974; fax: +90 462 325 5526. E-mail address: [email protected] (O. Aydın).

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study the combined free-forced convective laminar boundary layer ﬂow, past a vertical isothermal ﬂat plate, with temperature-dependent viscosity. Hossain and Munir [9] considered a two-dimensional mixed convection ﬂow of a viscous incompressible ﬂuid of temperature-dependent viscosity past a vertical impermeable ﬂuid. Mulaweh [10] conducted experiments on laminar mixed convection ﬂow adjacent to an inclined heated ﬂat plate with uniform wall heat ﬂux. Merkin and Pop [11] used a similarity transformation to analyze mixed convection boundary-layer ﬂow over a vertical semi-inﬁnite plate in which the free stream velocity is uniform and the wall temperature is inversely proportional to the distance along the plate. Steinru¨ck [12] found a new similarity solution of mixed convection ﬂow along a vertical plate. In a recent study, Pantokratoras [13] studied the steady laminar mixed convection of water with density–temperature relationship along a vertical isothermal plate. Despite its importance especially in highly viscous ﬂuids with a low thermal conductivity, the presence of viscous dissipation has been generally neglected in the existing literature. Israel-Cookey et al. [14] investigated the inﬂuence of viscous dissipation and radiation on the problem of unsteady magneto-hydrodynamic freeconvection ﬂow past an inﬁnite vertical heated plate in an optically thin environment with time-dependent suction. El-Amin [15] investigated the inﬂuence of viscous dissipation on buoyancy-induced ﬂow over a horizontal or vertical ﬂat plate embedded in a non-Newtonian ﬂuid saturated porous medium under the action of transverse magnetic ﬁeld. The objective of the present paper is to consider mixed convection from a vertical plate in the presence of viscous dissipation eﬀect. Numerical results are presented for some representative values of governing parameters, mainly, the Richardson number representing the mixed convection parameter and the Eckert number representing the eﬀect of the viscous dissipation. 2. Analysis Consider the steady, laminar, incompressible, two-dimensional, mixed convection boundary-layer ﬂow over a vertical ﬂat plate shown in Fig. 1. The viscous dissipation eﬀect is taken into consideration. The coordinate system is chosen such that x measures the distance along the plate and y measures the distance normal to it. Far away from the plate, the velocity and the temperature of the uniform main stream are U1 and T1, respectively. The entire surface of the plate is maintained at a uniform temperature of Tw. In the analysis, all the thermophysical properties are assumed to be constant except that the density in the buoyancy term. Assuming that the Boussinesq approximation is valid, the boundary-layer form of the governing equations which are based on the balance laws of mass, momentum and energy can be written as ou ov þ ¼ 0; ox oy

ð1Þ

ou ou o2 u þ v ¼ t 2  gbðT  T 1 Þ; ox oy oy   2 oT oT o2 T l ou þv ¼a 2þ . u ox oy oy qcp oy u

ð2Þ ð3Þ

Here u and v are the velocity components parallel and perpendicular to the plate, T is the temperature, b is the coeﬃcient of thermal expansion, l is the dynamic viscosity, t is the kinematic viscosity, q is the ﬂuid density, cp is the speciﬁc heat at constant pressure, g is the acceleration due to gravity, and a is the thermal diﬀusivity. The plus and minus signs of the buoyancy term denote the upward and downward ﬂows of free stream, respectively. The appropriate boundary conditions for the velocity and temperature of this problem are as follows: x ¼ 0; y > 0; x > 0; y ¼ 0; y large;

u ¼ U 1; T ¼ T 1; u ¼ 0; v ¼ 0; T ¼ T w ¼ constant; u ! U 1; T ! T 1.

Here, U1 and T1 are the free stream velocity and temperature, respectively.

ð4Þ

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845

With deﬁnition of the following dimensionless variables: U¼

u ; U1

1=2

V ¼

vReL ; U1

1=2

x X ¼ ; L

Y ¼

yReL ; L

T  T1 . Tw  T1

ð5Þ

Eqs. (1)–(3) are converted to the following dimensionless forms: oU oV þ ¼ 0; oX oY

ð6Þ

U

oU oU o2 U GrL þV ¼  ; oX oY oY 2 Re2L

ð7Þ

U

 2 oh oh 1 o2 h oU þV ¼ þ Ec ; oX oY Pr oY 2 oY

ð8Þ

where lcp U 21 ; Ec ¼ ; k cp ðT w  T 1 Þ U 1L Gr and Ri ¼ 2 . Re ¼ t Re Pr ¼

Gr ¼

gbðT  T 1 ÞL3 ; t2

ð9Þ

Pr is the Prandtl number, Ec is the Eckert number, Gr is the Grashof number, Re is the Reynolds number, Ri is the Richardson number. In the dimensionless form, the boundary conditions can be written as follows: X ¼ 0; Y > 0; X > 0; Y ¼ 0; Y large;

U ¼ 1; h ¼ 0; U ¼ V ¼ 0; h ¼ 1;

ð10Þ

U ! 1; h ! 0.

Eqs. (6)–(8) are coupled non-linear diﬀerential equations to be solved under the boundary conditions given in Eq. (10). However, exact or approximate solutions are not possible for this set of equations and hence we solve these equations by using the ﬁnite-diﬀerence method. The equivalent ﬁnite diﬀerence schemes corresponding Eqs. (6)–(8) are given by U i1;j

U i;j  U i1;j U i;jþ1  U i;j1 U i;jþ1 þ U i;j1  2U i;j þ V i1;j ¼  Rihi;j ; 2 DX 2DY ðDY Þ

 2 hi;j  hi1;j hi;jþ1  hi;j1 1 hi;jþ1 þ hi;j1  2hi;j U i;jþ1  U i;j1 þ V i1;j ¼ þ Ec . U i1;j 2 Pr DX 2DY 2DY ðDY Þ After some derivations V values can be determined from the continuity equation as    DY V i;j ¼ V i;j1  U i;j  U i1;j þ U i;j1  U i1;j1 . 2DX

ð11Þ

ð12Þ

ð13Þ

Here the index i refers to x and j to y. The above equations are explicit in x-direction, while they are implicit in y-direction. After specifying the conditions along some initial i = 1, U values and, in the following, h values can be obtained on the i = 2 line. Then, V values are obtained on the i = 2 line. Having in this way determined the values of all the variables on the i = 2 line, the same procedure can then be used to ﬁnd the values on the i = 3 line and so on. More detail on the numerical procedure can be found in the textbook by Oosthuizen and Naylor [16]. A mesh system with 100 · 100 nodes is proven to suggest mesh-independent results. The Nusselt number can be deﬁned as follows:   Nux X oh ¼ . ð14Þ 1=2 hw oY Y ¼0 ReL

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3. Results and discussion The schematic of the problem examined in this study has been already shown in Fig. 1. The governing equations (6) and (8) subject to the boundary conditions given by Eq. (10) have been solved numerically for the following ranges of the main parameters: Ri = 10, 1, 0.1, 0.01, 0.01, 0.1, 1, 10; Pr > 1; Ec = 0.5– 0.5. The combined eﬀects of the Richardson number and the Eckert number on the momentum and heat transfer are analyzed and discussed. The Richardson number, Ri, represents a measure of the eﬀect of the buoyancy in comparison with that of the inertia of the external forced or free stream ﬂow on the heat and ﬂuid ﬂow. Outside the mixed convection region, either the pure forced convection or the free convection analysis can be used to describe accurately the ﬂow or the temperature ﬁeld. Forced convection is the dominant mode of transport when Ri ! 0, whereas free convection is the dominant mode when Ri ! 1. Buoyancy forces can enhance the surface heat transfer rate when they assist the forced convection, and vice versa. Viscous dissipation, as an energy source, severely distorts the temperature proﬁle. Remember positive values of Ec correspond to wall heating (heat is being supplied across the walls into the ﬂuid) case (Tw > Tc), while the opposite is true for negative values of Ec. First of all, in order to assess the accuracy and validity of our method, we have compared our results for the case without the eﬀect of the viscous dissipation (Ec = 0) with those given by Saeid [17]. As seen from Table 1, 1=2 there exists a good correspondence for the results of Nu=ReL , which gives a credit to the validity of the approach followed here. As it is clear from this table, increasing the Prandtl number results in the increase in the heat transfer due to the decreasing thermal boundary layer thickness. Interestingly, according to the direction of the free stream ﬂow and thermal boundary condition applied at the wall, four diﬀerent ﬂow situations arise, which are

Tw

g x, u U∞, T∞

y, v Fig. 1. The schematic representation of the problem.

Table 1 Comparing results for Nu/Re1/2 at Ec = 0 Ri

Pr = 0.72

Pr = 7.0

Saeid [17]

Present

Saeid [17]

Present

0.0 0.2 0.4 0.6 0.8 1.0

0.309 0.332 0.361 0.382 0.402 0.416

0.298 0.333 0.359 0.381 0.398 0.413

0.628 0.698 0.752 0.791 0.822 0.851

0.625 0.694 0.749 0.788 0.819 0.848

O. Aydın, A. Kaya / Applied Mathematical Modelling 31 (2007) 843–853 1.0

847

1.0 Ec=0.0 Pr=10

Ec=0.1 Pr=10

0.8

0.8

0.6

U

U 0.4

0.4

0.2

0.2

0.0 0.00

Ri=2, 1, 0

0.6 Ri=2, 1, 0

0.02

0.04

0.06

0.08

0.10 X

0.12

0.14

0.16

0.18

0.0 0.00

0.20

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

X 1.0

1.0

Ec=0.1 Pr=10

Ec=0.0 Pr=10

0.6

0.6

θ

0.8

θ

0.8

0.4

0.4

Ri=2, 1, 0 Ri=2, 1, 0

0.2

0.2

0.01

0.02

0.03

0.04

0.05

0.0 0.00

0.06

0.01

0.02

0.03

(b)

X

0.04

0.05

X

Fig. 2. Velocity and temperature proﬁles for diﬀerent values of Ri at (a) Ec = 0 and (b) Ec = 0.1 (Case A).

1.20 1.15 1.10

Pr=10

1.05 1.00

0.90 0.85 0.80 Ec=0.1, 0.0 0.75 0.70

Ri 1=2

Fig. 3. Variation of Nu=ReL

for Case A.

9.00

6.00

3.00

0.90

0.60

0.30

0.60

0.09

0.65

0.06

Nu/Re1/2

0.95

0.03

0.0 0.00

(a)

0.06

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O. Aydın, A. Kaya / Applied Mathematical Modelling 31 (2007) 843–853

1.0

1.0

0.8

0.8

Ec= - 0.1 Pr=10

Ec=0.0 Pr=10

0.6

0.6

U

Ri=0, - 1, - 2

U

Ri=0, - 1, - 2 0.4

0.4

0.2

0.2

0.0 0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.0 0.00

0.20

0.02

0.04

0.06

0.08

X

0.10

0.12

0.14

0.16

0.18

0.20

X

1.0

1.0

Ec=0.0 Pr=10

Ec= - 0.1 Pr=10 0.8

0.6

0.6

θ

θ

0.8

0.4

0.4

Ri=0, - 1, - 2 Ri=0, - 1, - 2

0.2

0.01

0.02

0.03

(a)

0.04

0.05

0.0 0.00

0.06

0.01

0.02

0.03

(b)

X

0.04

0.05

X

Fig. 4. Velocity and temperature proﬁles for diﬀerent values of Ri at (a) Ec = 0 and (b) Ec = 0.1 (Case B).

0.70 Pr=10

0.68 0.66 0.64 0.62

Ec=0.0, - 0.1

0.58 0.56 0.54 0.52 0.50

Ri 1=2

Fig. 5. Variation of Nu=ReL

for Case B.

9.00

6.00

3.00

0.90

0.60

0.30

0.46

0.09

0.48

0.06

Nu/Re1/2

0.60

0.03

0.0 0.00

0.2

0.06

O. Aydın, A. Kaya / Applied Mathematical Modelling 31 (2007) 843–853

849

1. Case A. Aiding mixed ﬂow with Ec+: The buoyancy force for the wall heating case, Tw > T1 accelerates the upward external forced or free stream ﬂow. 2. Case B. Opposing mixed ﬂow with Ec: The buoyancy force for the wall cooling case, Tw < T1 retards the upward external forced or free stream ﬂow. 3. Case C. Opposing mixed ﬂow with Ec+: The buoyancy force for the wall heating case, Tw > T1 retards the downward external forced or free stream ﬂow. 4. Case D. Aiding mixed ﬂow with Ec: The buoyancy force for the wall cooling case, Tw < T1 accelerates the downward external forced or free stream ﬂow. 3.1. Case A: Aiding mixed ﬂow with Ec+ For the heated wall case (Tw > T1), the upward free ﬂow caused by the buoyancy is in the same direction with the external forced ﬂow. This case is called aiding mixed ﬂow. For the case without the viscous dissipation eﬀect (Ec = 0), Fig. 2(a) shows velocity and temperature proﬁles for diﬀerent values of the Richardson

1.0 Ec=0.1 Pr=10

0.8

0.6 U

Ri=0, - 1, - 2

0.4

0.2

0.0 0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

X 1.0 Ec=0.1 Pr=10 0.8

θ

0.6

0.4 Ri=0, - 1, - 2

0.2

0.0 0.00

0.01

0.02

0.03 X

0.04

0.05

0.06

Fig. 6. Velocity and temperature proﬁles for diﬀerent values of Ri at Ec = 0.1 (Case C).

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number, Ri and for diﬀerent thermal cases at wall. An increase at Ri results in increasing velocities due to addition of buoyancy-induced ﬂow onto the external forced ﬂow. Ec assumes positive values for the heated wall case. Fig. 2(b) shows the eﬀect of Ri on the velocity and temperature distributions at Ec = 0.1. For the case of Ri = 0, the Eckert number does not have any inﬂuence on velocity proﬁle since the momentum and energy equations are not coupled; however, it does on the temperature proﬁle. The viscous dissipation, as a heat generation inside the ﬂuid, increases the bulk ﬂuid temperature. The eﬀect of Ri on the heat transfer from the wall into the ﬂuid for various Ec is illustrated in Fig. 3. As expected, for the aiding or assisting mixed convection, increasing Ri increases Nu. However, magnitude of this increase decreases with an increase at Ec as a result of decreased temperature gradient at the wall, as explained above. 3.2. Case B: Opposing mixed ﬂow with Ec For the wall cooling case (Tw < T1), the buoyancy causes a downward free ﬂow which is in the opposite direction to that of upward external forced ﬂow, which is called the opposing mixed ﬂow. For the case without the eﬀect of the viscous dissipation, Fig. 4(a) shows velocity and temperature proﬁles for diﬀerent values of the Richardson number, Ri. An increase at Ri in negative direction results in decreasing velocities due to retarding eﬀect of downward buoyancy-induced ﬂow onto the upward external forced ﬂow. For the cooled wall case, Ec receives negative values. For Ec = 0.1, Fig. 4(b) displays the eﬀect of Ri on the velocity and temperature distributions. In this case, for the negative value of Ec, since Tw < T1, the viscous dissipation will again increase the temperature distribution in the ﬂow region. Finally, this leads to an increased temperature gradient, as will be shown later, which will result in increased heat transfer values. For this opposing mixed convection case, the eﬀect of Ri on the heat transfer from the wall into the ﬂuid for various Ec is illustrated in Fig. 5. As expected, increasing Ri in the negative direction decreases Nu. However, an increase at Ec in the negative direction as a result of increased temperature gradient at the wall increases Nu. 3.3. Case C: Opposing mixed ﬂow with Ec+ In Cases A and B, an upward forced ﬂow is assumed, while Cases C and D assuming a downward one. The upward ﬂow caused by the buoyancy for the heated wall case (Tw > T1) has a retarding eﬀect on the external forced ﬂow, which is called as opposing mixed ﬂow. For the case without the viscous dissipation eﬀect (Ec = 0), the velocity and temperature proﬁles for diﬀerent values of the Richardson number, Ri, are symmetrically identical to those given above for the Case B. An increase at Ri results in decreasing velocities because

0.68 Pr=10

0.66 0.64 Ec=0.1, 0.0

0.62

0.58 0.56 0.54 0.52 0.50

Ri 1=2

Fig. 7. Variation of Nu=ReL

for Case C.

9.00

6.00

3.00

0.90

0.60

0.30

0.09

0.46

0.06

0.48 0.03

Nu/Re1/2

0.60

O. Aydın, A. Kaya / Applied Mathematical Modelling 31 (2007) 843–853

851

of opposing eﬀect of the upward buoyancy-induced ﬂow on the downward external forced ﬂow. Including the viscous dissipation eﬀect (Ec = 0.1), the eﬀect of Ri on the velocity and temperature distributions is shown in Fig. 6. The eﬀect of Ri on the heat transfer from the wall to the ﬂuid for various Ec is illustrated in Fig. 7. The increasing viscous dissipation decreases the temperature gradient near the wall by increasing ﬂuid temperature for the wall heating case (Ec > 0). Then, in addition to opposing eﬀect of the buoyancy, viscous dissipation will have an opposing eﬀect on the heat transfer, too. 3.4. Case D: Aiding mixed ﬂow with Ec The downward ﬂow caused by the buoyancy for the cooled wall case (Tw < T1) will aid the downward external forced ﬂow, which is called as aiding or assisting mixed ﬂow. The case without the viscous dissipation eﬀect (Ec = 0) represents velocity and temperature proﬁles symmetrically identical to those given above for the Case A. An increase at Ri increases velocities because of aiding eﬀect of the downward buoyancy-induced ﬂow on the downward external forced ﬂow. Including the viscous dissipation eﬀect (Ec = 0.1), the eﬀect of Ri on the velocity and temperature distributions is shown in Fig. 8. The eﬀect of Ri on the heat transfer from the 1.0 Ec= - 0.1 Pr=10 0.8

0.6 U

Ri=2, 1, 0

0.4

0.2

0.0 0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

X 1.0 Ec= - 0.1 Pr=10 0.8

θ

0.6

0.4

Ri= 2, 1, 0

0.2

0.0 0.00

0.01

0.02

0.03

0.04

0.05

0.06

X

Fig. 8. Velocity and temperature proﬁles for diﬀerent values of Ri at Ec = 0.1 (Case D).

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O. Aydın, A. Kaya / Applied Mathematical Modelling 31 (2007) 843–853 1.3 1.2

Pr=10

1.1

Nu/Re1/2

1.0 0.9 Ec=0.0, - 0.1

0.8

9.00

6.00

3.00

0.90

0.60

0.30

0.09

0.06

0.6

0.03

0.7

Ri

Fig. 9. Variation of Nu=Re1=2 for Case D. L

Table 2 Values of the coeﬃcients seen in Eq. (15) Coeﬃcient

Case A

Case B

Case C

Case D

a b c R

0.398 0.111 0.173 0.997

0.294 0.016 0.074 0.988

0.294 0.0217 0.067 0.987

0.397 0.128 0.171 0.995

ﬂuid to the wall (the wall cooling case) for various Ec is illustrated in Fig. 9. The increasing viscous dissipation increases the temperature gradient near the wall by increasing ﬂuid temperature for the wall cooling case (Ec < 0). In the following, in addition to aiding eﬀect of the buoyancy, viscous dissipation will have an aiding eﬀect on the heat transfer, too. Finally, for the practical use, an expression correlation Nu in terms of Pr, Ri and Ec is developed. Note that the viscous dissipation becomes considerable for highly viscous ﬂuids, for which Pr > > 1. Therefore, in addition to the values of Pr studied above, Pr = 1, the results are also obtained for Pr = 10. Above, we only cover the range of Ec, 0.1–0.1. For deriving such a Nu correlation, this range of Ec is extended to 0.5–0.5. After obtaining Nu values for the above values of Pr, Ri and Ec, the following correlation is obtained: Nu ¼ aðRiÞb ðPrÞ1=4 ð1  cEcÞ; 1=2 Re

ð15Þ

where R is the correlation coeﬃcient representing the degree of the harmony. The values of the coeﬃcients are given in Table 2. 4. Conclusions Mixed convection ﬂow about a vertical ﬂat plate considering the eﬀect of viscous dissipation is analyzed. Uniform suction/injection was allowed at the wall. Depending on the thermal boundary conditions applied at the wall (heated/cooled wall) and the direction of forced or free stream ﬂow (upward/downward), four different mixed convection ﬂow situations have been identiﬁed: (i) aiding buoyancy with opposing viscous dissipation, (ii) opposing buoyancy with aiding viscous dissipation,

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853

(iii) opposing buoyancy with opposing viscous dissipation, (iv) aiding buoyancy with aiding viscous dissipation. Finally, a Nu = f(Pr, Ri, Ec) correlation which is valid for all the cases described above is suggested. References [1] Y. Jaluria, Natural Convection Heat and Mass Transfer, Pergamon Press, Oxford, 1980. [2] B. Gebhart, Y. Jaluria, R.L. Mahajan, B. Sammakia, Buoyancy-induced Flows and Transport, Hemisphere, 1988. [3] O. Aydın, Aiding and opposing mechanisms of mixed convection in a shear- and buoyancy-driven cavity, Int. Commun. Heat Mass Transfer 26 (7) (1999) 1019–1028. [4] A. Bejan, Convection Heat Transfer, Wiley, New York, 1995. [5] I. Pop, D.B. Ingham, Convective Heat Transfer, Pergamon, Amsterdam, 2001. [6] Y. Jaluria, Basic of natural convection, in: S. Kakac, R.K. Shah, W. Aung (Eds.), Handbook of Single-Phase Convective Heat Transfer, Wiley, New York, 1987. [7] T.S. Chen, B.F. Armaly, Mixed convection in external ﬂow, in: S. Kakac, R.K. Shah, W. Aung (Eds.), Handbook of Single-Phase Convective Heat Transfer, Wiley, New York, 1987. [8] N.G. Kafoussias, D.A.S. Rees, J.E. Daskalakis, Numerical study of the combined free-forced convective laminar boundary layer ﬂow past a vertical isothermal ﬂat plate with temperature-dependent viscosity, Acta Mech. 127 (1–4) (1998) 39–50. [9] M.A. Hossain, M.S. Munir, Mixed convection ﬂow from a vertical ﬂat plate with temperature dependent viscosity, Int. J. Therm. Sci. 39 (2) (2000) 173–183. [10] H.I. Abu-Mulaweh, Measurements of laminar mixed convection ﬂow adjacent to an inclined surface with uniform wall heat ﬂux, Int. J. Therm. Sci. 42 (1) (2003) 57–62. [11] J.H. Merkin, I. Pop, Mixed convection along a vertical surface: similarity solutions for uniform ﬂow, Fluid Dyn. Res. 30 (2002) 233– 250. [12] H. Steinruck, About the physical relevance of similarity solutions of the boundary-layer ﬂow equations describing mixed convection ﬂow along a vertical plate, Fluid Dyn. Res. 32 (1–2) (2003) 1–13. [13] A. Pantokratoras, Laminar assisting and mixed convection heat transfer from a vertical isothermal plate to water with variable physical properties, Heat Mass Transfer 40 (2004) 581–585. [14] C. Israel-Cookey, A. Ogulu, V.B. Omubo-Pepple, Inﬂuence of viscous dissipation and radiation on unsteady MHD free-convection ﬂow past an inﬁnite heated vertical plate in a porous medium with time-dependent suction, Int. J. Heat Mass Transfer 46 (2003) 2305– 2311. [15] M.F. El-Amin, Combined eﬀect of magnetic ﬁeld and viscous dissipation on a power-law ﬂuid over plate with variable surface heat ﬂux embedded in a porous medium, J. Magn. Magn. Mater. 261 (1–2) (2003) 228–237. [16] P.H. Oosthuizen, D. Naylor, Introduction to Convective Heat Transfer Analysis, McGraw-Hill, New York, 1999. [17] N.W. Saeid, Mixed convection ﬂow along a vertical plate subjected to time-periodic surface temperature oscillations, Int. J. Therm. Sci. 44 (2005) 531–539.