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FLOW AND HEAT TRANSFER OVER AMOVING VERTICAL PLATE IN A PARALLELFREE STREAM: ROLE OF INTERNAL HEATGENERATION OR ABSORPTIONP. M. Patil a , S. Roy b & I. Pop ca J.S.S.'s Banashankari Arts, Commerce and Shanti Kumar GubbiScience College, Dharwad, Karnataka, Indiab Indian Institute of Technology, Madras, Chennai, Indiac Faculty of Mathematics, University of Cluj, Cluj, Romania

Available online: 06 Mar 2012

To cite this article: P. M. Patil, S. Roy & I. Pop (2012): FLOW AND HEAT TRANSFER OVER A MOVINGVERTICAL PLATE IN A PARALLEL FREE STREAM: ROLE OF INTERNAL HEAT GENERATION OR ABSORPTION,Chemical Engineering Communications, 199:5, 658-672

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Flow and Heat Transfer over a Moving VerticalPlate in a Parallel Free Stream: Role of Internal

Heat Generation or Absorption

P. M. PATIL,1 S. ROY,2 AND I. POP3

1J.S.S.’s Banashankari Arts, Commerce and Shanti Kumar GubbiScience College, Dharwad, Karnataka, India2Indian Institute of Technology, Madras, Chennai, India3Faculty of Mathematics, University of Cluj, Cluj, Romania

In this article, a numerical investigation on a steady two-dimensional mixedconvection boundary layer flow along a moving semi-infinite vertical plate is pre-sented. The plate is assumed to move with a constant velocity in the direction ofthe flow. The influence of internal heat generation or absorption is also includedin the analysis. The nonlinear partial differential equations governing the flow andthermal fields are written in nondimensional form using a suitable group of transfor-mations. The final nondimensional set of coupled nonlinear partial differential equa-tions is solved using an implicit finite difference scheme in combination with thequasi-linearization technique. The effects of various parameters on the velocityand temperature profiles are reported. In addition, numerical results are presentedfor the skin friction coefficient and the Nusselt number. The present results are com-pared with previously published work and are found to be in excellent agreement.

Keywords Finite difference scheme; Heat generation; Mixed convection;Nonsimilar solution; Parallel free stream; Quasi-linearization

Introduction

Boundary layer flow on a continuous moving surface has many importantapplications in manufacturing processes such as hot rolling, wire drawing, metalextrusion, spinning of artificial fiber, continuous casting, glass fiber production,and paper production. Further, the boundary layer along a liquid film in conden-sation processes and a polymer sheet or filament extruded continuously from adie, or along thread traveling between a feed roll and a wind-up roll are examplesof practical applications of continuous moving surfaces.

There are several practical applications in which a significant temperature differ-ence between the surface of the body and the ambient fluid exists. This temperaturedifference causes a density gradient in the fluid medium, and free convection effectsbecome important in the presence of gravitational body force. A situation where

Address correspondence to P. M. Patil, Department of Mathematics, J.S.S.’s Banashan-kari Arts, Commerce and Shanti Kumar Gubbi Science College, Vidyagiri, Dharwad – 580 004,Karnataka State, India. E-mail: pmpmath@gmail.com

Chem. Eng. Comm., 199:658–672, 2012Copyright # Taylor & Francis Group, LLCISSN: 0098-6445 print=1563-5201 onlineDOI: 10.1080/00986445.2011.614978

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both the forced and free convection are of comparable order is called mixedconvection. The heat transfer and skin friction rates can be quite different undermixed convection conditions relative to the forced convection case or the freeconvection case. Karwe and Jaluria (1988, 1991) presented mixed convection flowover a continuous plate moving at a uniform speed in material processes such ashot rolling, extrusion, and drawing. Ingham (1986) investigated the existence ofsingular and nonunique solutions for free convection boundary layer flow near acontinuously moving vertical plate with the temperature inversely proportional tothe distance along the plate. Ali and Al-Yousef (1998) discussed the problem oflaminar mixed convection flow adjacent to a uniformly moving vertical plate withsuction or injection.

The steady laminar flow and heat transfer characteristics of a continuously mov-ing vertical sheet of extruded material were studied close to and far downstreamfrom the extrusion slot by Al-Sanea (2004). The velocity and temperature variationswere calculated by the finite volume method. Abdelhafez (1985) and Chappidi andGunnerson (1989) studied the laminar boundary layer in two cases, namely Uw>U1and U1>Uw, separately and formulated two sets of boundary value problems.Afzal et al. (1993) framed a single set of equations by employing the compositereference velocity U irrespective of whether Uw>U1 or U1>Uw. Lin and Huang(1994) analyzed a horizontal isothermal plate moving in parallel or reversibly to afree stream where similarity and nonsimilarity equations are used to obtain thevelocity and temperature fields. Tsou et al. (1967) showed experimentally that theflow and heat transfer problem from a continuously moving surface is a physicallyrealizable one and studied its basic characteristics. Later, various aspects of the prob-lem have been treated by many authors. Soundalgekar and Murty (1980) discussedthe effects of power law surface temperature variation on the heat transfer from acontinuous moving surface with constant surface velocity. The effects of variablesurface temperature and linear stretching surface were investigated by Grubka andBobba (1985). Similarity solutions were reported by Ali (1994) for the case of power-law surface velocity and three different thermal boundary conditions. Ali (1995)extended his work for a stretching surface subject to suction or injection. Moutso-glou and Chen (1980) considered buoyancy effects on flow and heat transfer froman inclined continuous sheet with either uniform wall temperature or uniform sur-face heat flux. More recently, Afzal (2003) obtained the similarity solution of themomentum equation for power law stretching plate, and Chamkha et al. (2006)investigated the effect of heat generation or absorption on a free convection flowfrom a vertical plate in a fluid saturated porous medium. An analysis of the steadylaminar flow of an incompressible viscous fluid impinging on a permeable stretchingsurface with heat generation and uniform suction or injection was made by Hazemand Seddeek (2007). Very recently, Sawant and Rao (2010) studied the combinedconduction-mixed convection-surface radiation from a uniformly heated verticalplate. Magnetohydrodynamic (MHD) free convection flow of a nanofluid past a ver-tical plate in the presence of heat generation or absorption effects was investigatedby Chamkha and Aly (2011). Makinde (2011) discussed the effects of MHD mixedconvection interaction with thermal radiation and nth order chemical reaction pasta vertical porous plate embedded in a porous medium. Kumar and Rao (2011) exam-ined the effects of interaction of surface radiation with conjugate mixed convectionfrom a vertical plate with multiple nonidentical discrete heat sources. Effects ofunsteady mixed convection with Soret and Dufour effects past a porous plate

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moving through a binary mixture of chemically reacting fluid were studied byMakinde and Olanrewaju (2011).

The present numerical study investigates the simultaneous effects of internalheat generation=absorption and thermal diffusion on a steady mixed convection flowalong a semi-infinite vertical plate. The plate is supposed to move with a constantvelocity parallel to the free stream velocity. The coupled nonlinear partial differentialequations governing the flow have been solved numerically using an implicit finitedifference method in combination with the quasi-linearization technique (see Inouyeand Tate (1974) and Patil et al. (2010)). Results are compared with some resultsreported by Tsou et al. (1967), Soundalgekar and Murty (1980), Ali (1995), andMoutsoglou and Chen (1980) and are found to be in excellent agreement.

Problem Formulation and Governing Equations

Consider a steady incompressible viscous heat-generating or -absorbing mixed con-vection boundary layer flow along a semi-infinite vertical plate moving with velocityUw in the x-direction. The x-axis is taken along the plate in the vertically upwarddirection and the y-axis is taken normal to it. Figure 1 shows the coordinate systemand physical model for the flow configuration. The free stream velocity U1 and theplate velocity Uw are constants in the same direction. The surface is maintained at auniform temperature Tw. The buoyancy force arises due to the temperature differ-ence in the fluid. All the thermophysical properties of the fluid in the flow modelare assumed to be constant except the density variations causing a body force inthe momentum equation. The Boussinesq approximation is invoked for the fluid

Figure 1. Physical model and coordinate system.

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properties to relate the density changes and to couple in this way the temperaturefield to the flow field (Schlichting, 2000). Under these assumptions, the equationsof the conservation of mass, momentum, and energy governing the mixed convectionboundary layer flow over a moving vertical plate are given by

@u

@xþ @v

@y¼ 0 ð1Þ

u@u

@xþ v

@u

@y¼ n

@2u

@y2þ g b T � T1ð Þ ð2Þ

u@T

@xþ v

@T

@y¼ a

@2T

@y2þ Q0 T � T1ð Þ

qCpð3Þ

subject to the following physical boundary conditions:

u ¼ Uw; v ¼ 0; T ¼ Tw at y ¼ 0;

u ! U1; T ! T1 as y ! 1:ð4Þ

We introduce the following transformations:

n ¼ U x

n

� �1=2

; g ¼ U

n x

� �1=2

y; wðx; yÞ ¼ ðnU xÞ1=2 f ðn; gÞ

Tðx; yÞ ¼ TwðxÞ � T1ð ÞHðn; gÞ; TwðxÞ � T1 ¼ ðTw 0 � T1Þ x� 1

ð5Þ

where U ¼ UwþU12 and w is the stream function, which is defined as u¼ @ w=@ y and

v¼� @ w=@ x. Thus, we have

u ¼ U fg; v ¼ � 1

2

nUx

� �1=2

f ðn; gÞ þ xU

n

� �1=2

fn � g fg

" #: ð6Þ

On substituting Equations (5) and (6) into Equations (1)–(3), we find that Equation(1) is satisfied identically, and Equations (2) and (3) reduce to

Fg g þ 1

2f Fg þ 2 k H ¼ n

2F Fn � fn Fg� �

; ð7Þ

Hg g þ Pr

2f Hg þ Pr Q n2 H ¼ Pr n

2F Hn � fn Hg� �

; ð8Þ

where f ¼R g0 F dg þ fw; fw ¼ 0. Further, k is the mixed convection parameter, Pr

is the Prandtl number, and Q is the dimensionless heat generation or absorptionparameter, which are defined as

k ¼ Grx

Re2x; Pr ¼ n

a; Q ¼ Q0

qCp U2; ð9Þ

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with Grx¼ g b (Tw0�T1) x2=n2 being the local Grashof number and Rex¼Ux=n isthe local Reynolds number. It is worth mentioning that k in Equation (6) representsthe buoyancy force effect on the flow field and it has� signs; the plus sign (k> 0)indicates the buoyancy-upward (or buoyancy-assisted) flow, while the negative sign(k< 0) stands for the buoyancy-downward (or buoyancy-opposed) flow and k¼ 0corresponds to the forced convection flow.

The boundary conditions (4) reduce to

F ¼ 1� e; H ¼ 1 at g ¼ 0;

F ! e; H ! 0 as g ! 1;ð10Þ

where e¼U1=(U1þUw) corresponds to the ratio of the free stream velocity to thecomposite reference velocity and is called the moving parameter.

The physical quantities of interest are the local skin friction coefficient Cf andthe local Nusselt number Nux, which are defined as

Cf ¼l

qU2

@ u

@ y

� �y¼0

; Nux ¼ x

TwðxÞ � T1ð Þ � @ T

@ y

� �y¼0

: ð11Þ

Using variables (5), we get

ffiffiffi2

pRe1=2x Cf ¼ 2Fgðn; 0Þ; 2Rexð Þ�1=2 Nux ¼ �Hgðn; 0Þ: ð12Þ

Solution Procedure

Nonlinear coupled partial differential equations (7) and (8) under the boundaryconditions (10) have been solved numerically using an implicit finite differencescheme in combination with the quasi-linearization technique (see Inouye and Tate(1974) and Patil et al. (2010)). An iterative sequence of linear equations is carefullyconstructed to approximate the nonlinear equations (7) and (8) for achieving thequadratic convergence and monotonicity. Using the quasi-linearization technique,the nonlinear coupled partial differential equations (7) and (8) with boundary con-ditions (10) are replaced by the following sequence of linear partial differentialequations:

Fiþ1gg þ Ai

1 Fiþ1g þ Ai

2 Fiþ1 þ Ai

3 Fiþ1n þ Ai

4 Hiþ1 ¼ Ai

5 ð13Þ

Hiþ1gg þ Bi

1 Hiþ1g þ Bi

2 Hiþ1 þ Bi

3 Hiþ1n þ Bi

4 Fiþ1 ¼ Bi

5 ð14Þ

The coefficient functions with iterative index i are known and the functions withiterative index (iþ 1) are to be determined. The corresponding boundary conditionsare given by

Fiþ1 ¼ 1� e; Hiþ1 ¼ 1 at g ¼ 0

Fiþ1 ¼ e; Hiþ1 ¼ 0; at g ¼ g1 ð15Þ

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The coefficients in Equations (13) and (14) are given by

Ai1 ¼

1

2f þ n fnð Þ; Ai

2 ¼ � n2Fn; Ai

3 ¼ � n2F ; Ai

4 ¼ 2 k; Ai5 ¼ � n

2F Fn;

Bi1 ¼

Pr

2f þ n fnð Þ; Bi

2 ¼ Pr Q n2; Bi3 ¼ � Pr

2nF ; Bi

4 ¼ � Pr

2nHn;

Bi5 ¼ � Pr

2nF Hn: ð16Þ

Since the method is presented for ordinary differential equations by Inouye andTate (1974) and also presented for partial differential equations in a recent studyby Patil et al. (2010), its detailed description is not provided here. At each iterationstep, the sequence of linear partial differential equations (13) and (14) are expressedin difference form using the central difference scheme in the g-direction and thebackward difference scheme in the n-direction. Thus, in each step, the resultingequations are then reduced to a system of linear algebraic equations with a blocktri-diagonal matrix, which is solved by Varga’s algorithm (2000). To ensure the con-vergence of the numerical solution to the exact solution, the step sizes Dg andDnareoptimized and taken as 0.01 and 0.005, respectively. The results presented here areindependent of the step sizes at least up to the fourth decimal place. A convergencecriterion based on the relative difference between the current and previous iterationvalues is employed. When the difference reaches 0.0001, the solution is assumed tohave converged and the iteration process is terminated.

Results and Discussion

Computations have been carried out for various values of Pr (0.7�Pr� 7.0), k(�0.11� k� 2.5), e (0.1� e� 0.9), and Q (�2.0�Q� 2.0). The edge of the boundarylayer (g1) has been taken to be between 5.0 and 10.0 depending on the values of theparameters. To verify the accuracy of the presented approach, we have verifiedsteady state results of heat transfer rate by direct comparison with the results pre-viously reported by Tsou et al. (1967), Soundalgekar and Murty (1980), Ali(1995), and Moutsoglou and Chen (1980). The results of this comparison are pre-sented in Table I and are found to be in excellent agreement. Also, we have presentedsome of the results for skin friction coefficient and heat transfer coefficient in Table IIfor nonsimilar solution. The effects of the buoyancy or mixed convection parameter

Table I. Comparison of �Hg(0) for k¼ 0, n¼ 0, and e¼ 0 and selected values of Prto previously published work

Pr 0.7 1.0 2.0 7.0 10.0 100.0

Tsou et al. (1967) 0.3492 0.4438 — — 1.6804 5.545Soundalgekar andMurty (1980)

0.3508 — 0.6831 — 1.6808 —

Ali (1995) 0.3476 0.4416 — — 1.6713 —Moutsoglou andChen (1980)

Present work 0.35004 0.44401 0.68314 1.38625 1.68011 5.54610

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(k) and the Prandtl number (Pr) on the velocity and temperature profiles (F(n, g),H(n, g)) are presented in Figures 2–4. The velocity profiles are displayed inFigures 2 and 3 for different values of e (the ratio of the free stream velocity tothe composite reference velocity). In buoyancy aiding or assisting flow (k> 0), thebuoyancy force shows significant overshoot in the velocity profiles near the surfacefor lower Pr number fluid, whereas the velocity overshoot is not present for higher Prnumber fluid. Although the magnitude of the overshoot increases with the buoyancyparameter k (k> 0), it decreases as the Pr number increases. The physical reason isthat the buoyancy force (k) has more effects in smaller Pr number fluid (air, Pr¼ 0.7)due to the lower viscosity of the fluid. Hence, the velocity increases within the mov-ing boundary layer as the assisting buoyancy force acts like a favorable pressuregradient and the velocity overshoot occurs. For higher Pr number fluid the over-shoot is not present because higher Pr number (water, Pr¼ 7.0) with more viscousfluid makes it less sensitive to the buoyancy force. Comparative studies inFigures 2 and 3 indicate that the increase in e enhances the magnitude of the velocity

Figure 2. Effects of k and Pr on velocity profile for e¼ 0.5, n¼ 0.5, and Q¼ 0.5.

Table II. Skin friction coefficient ðð2RexÞ1=2 Cf Þ and heat transfercoefficient ðð2RexÞ�1=2 NuxÞ when n¼ 1.0, Pr¼ 0.7, and k¼ 1.0

Q e ðð2RexÞ1=2 Cf Þ ðð2RexÞ�1=2 NuxÞ

0.5 0.1 2.6732 0.26990.5 0.5 3.5803 0.25210.5 0.9 4.4099 0.2298�0.5 0.1 2.3424 0.6992�0.5 0.5 3.2139 0.6641�0.5 0.9 4.0102 0.6641

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overshoot within the boundary layer. It is interesting to note from Figures 2 and 3that for the opposing buoyancy flow, i.e., k< 0, the buoyancy opposing forcereduces the magnitude of the velocity significantly within the boundary layer forthe low Pr number fluid (Pr¼ 0.7, air) as well as for the high Pr number fluid(Pr¼ 7.0, water).

Figure 4. Effects of k and Pr on temperature profile for e¼ 0.5, n¼ 0.5, and Q¼ 0.5.

Figure 3. Effects of k and Pr on velocity profile for e¼ 0.9, n¼ 0.5, and Q¼ 0.5.

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The effect of the buoyancy parameter (k) on the temperature profile H(n, g) iscomparatively less, as shown in Figure 4. Further, it is observed from Figure 4 thatan increase in the (higher) Pr number (water, Pr¼ 7.0) clearly induces a strongreduction in the temperature of the fluid and thus results in thinner thermal bound-ary layer. The Pr number is inversely proportional to the thermal conductivity and assuch lower Pr fluid will possess higher thermal conductivities and therefore diffusesthe heat energy more than momentum.

Figure 5 presents the effect of e on the velocity profile F(n, g) at k¼ 1.0, Q¼ 0.5,and n¼ 0.5 for Pr¼ 0.7 and 7.0. It has been observed that the magnitude of the velo-city within the boundary layer increases with the increase ofe. The physical reasonis that the combined effects of the assisting buoyancy force due to the thermal gra-dients and heat generation act like a favorable pressure gradient, which acceleratesthe fluid for lower Pr number (air, Pr¼ 0.7), exerting the velocity overshoot nearthe surface within the moving boundary layer. The velocity overshoot reducesas the Pr number (water, Pr¼ 7.0) increases.

Figures 6 and 7 illustrate the influence of the heat generation or absorption para-meter Q on the velocity and temperature profiles, respectively, for k¼ 0.5, n¼ 1.0,and e¼ 0.8. It is noted that owing to the presence of a heat generation or a heatsource effect (Q> 0), the thermal state of the fluid increases. Hence, the velocityand temperature of the fluid increase within the boundary layers. In the event thatthe strength of the heat source is relatively large, the overshoot is observed in thevelocity and temperature profiles within the momentum and thermal boundarylayers, as can be seen in Figures 6 and 7.

Further, the effect of heat generation is more pronounced on temperature pro-files for higher Pr number fluid (Pr¼ 7.0, water). Figure 7 shows that for Pr¼ 7.0,the temperature profile has approximately 90% overshoot with a thinner thermalboundary layer due to the higher thermal conductivity. Conversely, the presence

Figure 5. Effects of e and Pr on velocity profile for k¼ 1, n¼ 0.5, and Q¼ 0.5.

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of heat absorption or a heat sink effect (Q< 0) has the tendency to reduce the fluidtemperature. This causes the thermal buoyancy effects to decrease, resulting in a netreduction in the fluid velocity. These behaviors are clearly observed in Figures 6 and7, in which the magnitude of both the velocity and temperature fields decrease forQ< 0. Moreover, it is also observed that the thicknesses of both the hydrodynamic

Figure 6. Effects of Q and Pr on velocity profile for e¼ 0.8, n¼ 1, and k¼ 0.5.

Figure 7. Effects of Q and Pr on temperature profile for e¼ 0.8, n¼ 1, and k¼ 0.5.

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(velocity) and thermal (temperature) boundary layers decrease as the heat absorptionor heat sink effect increases.

The effects of e and Pr number on the skin friction coefficient ðð2RexÞ1=2 Cf Þ fork¼ 1.0 and Q¼ 0.5 are shown in Figure 8. Results indicate that the skin friction coef-ficient ðð2RexÞ1=2 Cf Þ increases with e when Pr increases from Pr¼ 0.7 to Pr¼ 7.0 fora fixed value of e. This is due to the fact that the increase of e enhances the fluidacceleration, and hence the skin friction coefficient increases. In particular, forPr¼ 0.7 at n¼ 1.0, the skin friction coefficient ðð2RexÞ1=2 Cf Þ increases approxi-mately 60%, as e increases from 0.10 to 0.90. Figure 9 displays the effect of skinfriction coefficient ðð2RexÞ1=2 Cf Þ on the buoyancy parameter k and the heat gener-ation=absorption parameter Q. The skin friction coefficient increases with the buoy-ancy parameter k and the heat generation parameter Q. The physical reason is thatthe combined effect of positive buoyancy force (k> 0) and heat generation (Q> 0)implies favorable pressure gradient, and thus the fluid is accelerated, which resultsin thinner hydrodynamic and thermal boundary layers. However, the skin frictioncoefficient ðð2RexÞ1=2 Cf Þ decreases with the increase of heat absorption Q (<0).In particular, for Q¼ 2.0 at n¼ 1.0, the skin friction coefficient ðð2RexÞ1=2 Cf Þincreases approximately 130% as k increases from 0.5 to 1.5.

The effects of Pr and Q on the heat transfer rate ((2Rex)�1=2Nux) are shown in

Figure 10. It may be noted that negative heat transfer rates are obtained for Q> 0.Negative values of �H0 n; 0ð Þ indicate that heat is transferred from the fluid to themoving surface in spite of the excess of surface temperature over that of the freestream fluid. The physical reason is that for heat generation (Q> 0), the fluidparticles are heated within the boundary layers and the heat transfer rate at the wallbecomes negative when the fluid particles near the wall reach higher temperaturethan the surface temperature.

Figure 8. Effects of e and Pr on skin friction coefficient for k¼ 1 and Q¼ 0.5.

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Further, it is observed that positive heat transfer rates ((2Rex)�1=2Nux) are

obtained for heat absorption Q (< 0) and increase monotonously with increasingn. This is evident from the fact that negative values of Q (<0) would cause anenlargement of the temperature difference between the moving surface and the

Figure 9. Effects of k and Q on skin friction coefficient for e¼ 0.4 and Pr¼ 0.7.

Figure 10. Effects of Pr and Q on heat transfer rate for e¼ 0.4 and k¼ 0.5.

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convective fluid, and thus enhance the heat transfer rate. In particular, for Q¼ 0.25at n¼ 1.0, the heat transfer rate ((2Rex)

�1=2Nux) decreases approximately 65% as Prdecreases from 7.0 to 0.7, while for Q¼�0.25 at n¼1.0, the heat transfer rate((2Rex)

�1=2Nux) decreases approximately 70% when the Pr number decreases from7.0 to 0.7. Furthermore, the heat transfer rates are steady when there is no effectof heat generation or absorption (Q> 0 or Q< 0).

Conclusions

A detailed numerical study has been carried out for the effects of internal heatgeneration or absorption on the mixed convection flow over a moving vertical plate.The conclusions of the study are as follows:

1. The buoyancy force causes overshoot in the velocity profile for the lower Prnumber fluid (air, Pr¼ 0.7) but for the higher Pr number fluid (water,Pr¼ 7.0) the velocity overshoot is not present.

2. The effect of e is significant on the velocity profile.3. The higher Pr number Pr¼ 7.0 (water) causes thinner thermal boundary layer in

the presence of heat generation or absorption (Q> 0 or Q< 0).4. The heat generation effect (Q> 0) causes thicker velocity and thermal boundary

layers, while the heat absorption effect (Q< 0) has the tendency to reduce thethickness of the velocity and thermal boundary layers.

5. The negative heat transfer rates are obtained for Q> 0 (heat generation).

Acknowledgments

One of the authors, Dr. P. M. Patil, expresses his sincere thanks to Dr. Ajith Prasad,Principal, J.S.S.’s Banashankari Arts, Commerce and Shanti Kumar Gubbi ScienceCollege, Vidyagiri, Dharwad, India for his constant encouragement in researchactivities. Also, Dr. Patil dedicates this article to one of his close friends, Shri M.G. Hiremath, Deputy Commissioner of Commercial Taxes, Government ofKarnataka, India, for his valuable service to the people of Karnataka. The authorswish to express their thanks to the anonymous reviewer for the valuable commentsand suggestions.

Nomenclature

Cf local skin friction coefficientCp specific heat at constant pressuref dimensionless stream functiong acceleration due to gravityGrx local Grashof numberk thermal conductivityNux local Nusselt numberPr Prandtl numberQ0 heat generation coefficientQ dimensionless heat generation or absorption parameterRex local Reynolds numberT temperature

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U composite reference velocityUw moving plate velocityU1 free stream velocityu velocity component in the x-directionv velocity component in the y-direction

Greek lettersa thermal diffusivityn, g transformed variablesk constant dimensionless mixed convection or buoyancy parameterl dynamic viscosityn kinematic viscosityq densityw stream function

Subscriptsw, 1 conditions at the wall and infinity, respectivelyn, g denote the partial derivatives w.r.t. these variables, respectively

References

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