IMPACT OF HEAT TRANSFER ANALYSIS ON CARREAU FLUID-FLOW PAST A STATIC/MOVING WEDGE

Transcription

1 THERMAL SCIENCE: Year 8, Vol., No., pp IMPACT OF HEAT TRANSFER ANALYSIS ON CARREAU FLUID-FLOW PAST A STATIC/MOVING WEDGE by HASHIM * and Masood KHAN Department o Mathematics, Quaid-i-Azam University, Islamabad, Pakistan Original scientiic paper Introduction The oremost aspiration o the present endeavor is to investigate the boundary-layer low o a generalized Newtonian Carreau luid model past a static/moving wedge. In addition, the eects o heat transer on the low ield are also taken into account. The governing equations o the problem based on the boundary-layer approximation are changed into a non-dimensional structure by introducing the local similarity transormations. The subsequent system o ODE has been numerically integrated with ith-order Runge-Kutta method. Inluence o the velocity ratio parameter, the wedge angle parameter, the Weissenberg number, the power law index, and the Prandtl number on the skin riction and Nusselt number are analyzed. The variation o the skin riction as well as other low characteristics has been presented graphically to capture the inluence o these parameters. The results indicate that the increasing value o the wedge angle substantially accelerates the luid velocity while an opposite behavior is noticed in the temperature ield. Moreover, the skin riction coeicient or the growing Weissenberg number signiicantly enhances or the shear thickening luid and show the opposite behavior o shear thinning luid. However, the local Nusselt number has greater values in the case o moving wedge. An excellent comparison with previously published works in various special cases has been made. Key words: Carreau luid model, boundary-layer low, heat transer analysis, static/moving wedge, Runge-Kutta method During the last ew decades, the luid-lows across wedge shaped bodies are o undamental importance in numerous engineering applications. It has a broad occurence in the ields o aerodynamics, heat exchangers, hydrodynamics, geothermal systems, etc. Speciically, such sort o low happens as oten as possible in enhanced oil recovery, aircrat response to atmospheric gusts, packed bed reactor geothermal industries, ground water pollution and so orth. Historically, Falkner and Skan [] concentrated the low over a static wedge in early 93, which is immersed in a viscous luid and introduced the Falkner-Skan equation. In this study, they outlined the Prandtl s boundary-layer hypothesis and utilized the similarity transormations to diminish the overseeing boundary-layer equations to ODE. In the previous couple o years, analysts have demonstrated extraordinary enthusiasm or the Falkner-Skan low by considering the impacts o various parameters. The solutions and their reliance on β (the wedge angle) were likewise later inspected by Hartree []. He acquired the solutions as ar as velocity * Corresponding author, hashim_alik@yahoo.com

2 8 THERMAL SCIENCE: Year 8, Vol., No., pp distribution or dierent estimations o pressure gradient parameters. In the meantime, an all the more ascinating problem in this regard was considered by Riley and Weidman [3] They ocused on the impacts o moving boundary on a wedge in a viscous luid and applied a similarity variable which prompted a non-linear third-order ODE and settled it numerically. Additionally, Watanabe [4] investigated the behavior o boundary-layer orced low over a wedge with uniorm suction or injection. As o late, Ishak et al. [5] examined the MHD low o a conducting luid streaming transversely with variable magnetic ield along a moving wedge. The boundary-layer theory is to a great degree prospering admiration in the historical backdrop o Newtonian luid mechanics (see Schlichting and Gersten [6]). A ew luid-low problems and their heat exchange attributes have been eectively displayed with the aid o this theory and the results agree very well with experimental observations. Some late learns about the boundary-layer low o Newtonian luids are given in [7-]. Nonetheless, numerous mechanical luids are non-newtonian in their low characteristics and are alluded to as rheological luid models. It is currently a perceived reality that non-newtonian luids have a larger number o utilization in engineering than Newtonian luids. Such examples incorporates into pharmaceuticals, iber innovation, products o everyday sustenance, crystal growth, and so on. One speciic class o such materials which is o noteworthy interest is that in which the eective viscosity relies on upon the rate o shearing. Such luids are termed as generalized Newtonian luids (GNF). The rheological equations o GNF have been cultivated rom the Newtonian constitutive relations to suspect the deormation stresses in a non-newtonian luid. The constitutive equation or the generalized Newtonian luids can be expressed: where ( ) τ = µγ ( ) A () µ γ is titled as the generalized Newtonian viscosity, τ represents the stress tensor and A is termed as the rate o strain tensor. The unction µγ ( ) is to be speciied and it has dierent expressions or various luid models, such as power-law, Sisko and Carreau models (see Bird et al. [] and Slattery []). The rheological equation or the Carreau model [3] is a particular example o the GNF in which the apparent viscosity, µ, changes with the magnitude, γ, o the deormation rate. The Carreau rheological model appropriately portrays the non-newtonian conduct o numerous lubricants [4, 5]. This rheological model has a hypothetical premise [3] and imitates the viscosity o a ew real luids, like polymer solutions, over a very extensive range or the values o γ, as appeared or occurrence in [] The governing relationship or apparent viscosity µγ ( ) that will be considered here is [3]: n ( ) ( ) µ = µ + µ µ + Γ γ () The two material parameters Γ and n appearing in the Carreau model depict the shear rate reliance o the viscosity. Here µ represents the zero shear viscosity and µ the ininite shear viscosity. The power law index n controls the slope o ( µ µ )/( µ µ ) in the power law region. The luid is characterized as shear thinning or < n < shear thickening or n > and Newtonian or n =. As the shear rate becomes larger the Carreau model behaves as power-law luid and at low shear rate it behaves as Newtonian luid. In current ormulation, we considered here a vanishing viscosity at an ininite rate o strain i. e. µ =. A very ew studies are concerned with the Carreau luid-lowed their and heat transer characteristics in the past ew years. For instance, Khella and Lauriat [6] investigated the low and heat transer in a short vertical annulus with a heated and rotating inner cylinder. Olajuwon [7] studied the con-

3 THERMAL SCIENCE: Year 8, Vol., No., pp vection heat and mass transer o MHD Carreau luid past a vertical porous plate. Also, numerical investigation o inertia and shear thinning eects on axisymmetric lows o Carreau luids was reported by Martins et al. [8]. Recently, the boundary-layer low and heat transer to a Carreau luid over a non-linear stretching sheet is discussed by Khan and Hashim [9]. This article addresses the impacts o the heat transer analysis on the low o GN Carreau luid over a static/moving wedge. New outgrowths are given in this investigation to maniest the occurrence o the locally similar solutions. Scaling group o transormations is utilized to present the local similarity representations o the governing PDE. An eicient ith-order Runge-Kutta scheme is used to solve the similarity equations. The impacts o the relevant low variables are depicted in point o interest through graphs and tables. To the best o the author s inormation, there is no endeavor highlighting the previous expressed low model or Carreau luid. Basic equations We consider the laminar, incompressible and -D low o a GN Carreau luid over a m static/ moving wedge. It is assumed that the wedge is moving with the velocity uw( x) = bx and m the velocity o the ree stream is ue ( x) = cx, where b, c, and m are constants. We use the rectangular co-ordinates ( xy, ), in which x and y are the distances measured along the wedge and normal to surace o the wedge. Further, the origin o the Cartesian co-ordinates ( xy, ) is at the tip o the wedge. The physical coniguration u e (x) and co-ordinate systems appear in ig.. Here uw ( x ) > compares to a stretching wedge surace velocity and uw( x ) < relates to a con- Ω = βπ tracting wedge surace velocity. The temperature at sheet T w is assumed to be constant and the ambient temperature is T, where T < β T w. m < Under the boundary-layer approximations, we have the ollowing governing equa- Figure. Schematic o physical model tions. Conservation o mass: Conservation o momentum [3]: u v + = x y y, v x, u (3) n n 3 ν( ) u u due u u u u u u + v = ue + ν +Γ + n Γ +Γ x y dx y y y y y (4) Conservation o energy: u + v = α x y y T T T (5)

4 8 THERMAL SCIENCE: Year 8, Vol., No., pp in which u, v are the velocity components in x- and y-directions, respectively, ν the kinematic viscosity, Γ the material parameter oten reerred as relaxation time, α the thermal diusivity o the luid, T the temperature o the luid, and n the power-law index. The relevant boundary conditions are: static wedge moving wedge To look at the low regime, we present the accompanying locally similar transormations: u =, v=, T = T w, at y = (6) e m ( ),, u = u x = cx T T as y (7) w m ( ),,, u = u x = bx v = T = T at y = (8) e m ( ),, w u = u x = cx T T as y (9) ν c m+ cm ( + ) m T T ψ( xy, ) = x ( ) = y x, θ ( ) = m+ ν T T where ψ is the stream unction, characterized in the standard way as u = ψ / yand v= ψ / x Thus, the velocity components using the previous similarity transormation are given: orm: m u = cx ( ), ( ) m b m+ ν m v= x ( ) ( ) + m + By employing eqs. () and (), the momentum and energy eqs. (4) and (5) takes the n 3 ( ) ( ) β ( ) w () () nwe We = () θ + Pr θ = (3) The changed boundary conditions are as per the ollowing: () =, () = λ, θ() = (4) ( ), θ ( ) (5) In the previous expressions, primes mean dierentiation with respect to, β is the wedge angle parameter, λ the constant velocity ratio parameter, We the local Weissenberg number, and Pr the Prandtl number. These parameters are: 3 3m m b c Γ x µ β =, λ =, We =, Pr = m+ c ν k In accordance with [], the included angle o the wedge is taken to be Ω= βπ, where β = m/( m+ ) is known as the wedge angle parameter. Physically, the wedge angle parameter, β, is related to the pressure gradient in such a way that the positive values o β representing the avorable pressure gradient and negative values o β demonstrates an adverse pressure / c p (6)

5 THERMAL SCIENCE: Year 8, Vol., No., pp gradient. Further, m =, i. e. ( β = ) means the luid-low past a lat plate and m =, i. e. ( β = ) implies the stagnation point low. Additionally, the constant velocity ratio parameter λ > and λ < identiies with a moving wedge in the same and inverse directions to the ree stream, separately, while λ = compared to a static wedge. Parameters o practical interest We are interested in the skin riction coeicient, C, and the local Nusselt number, individually. Physically, C is the shear stress at surace and Nu represents the rate o heat transer at the wall. Conventionally, these are characterized: in which C τw, Nu xqw = = ρuw k Tw T ( ) (7) τ w n, qw k u u T = µ +Γ = y y y y= y= (8) Using the similarity variables () and (), we get the corresponding expressions: n / / Re C = () + We ( ()), Re Nu = θ () β β in which the local Reynolds number is deined as Re = xu / ν. Numerical simulation The system o dierential eqs. ()-(5) is highly non-linear and partially coupled so these equations can not be solved analytically. For this purpose a numerical treatment would be much more suitable. Thus, the boundary value problems () and (3) with boundary conditions (4) and (5) have been numerically solved by applying ith-order Runge-Kutta integration technique together with shooting iteration scheme in a computer sotware MATLAB. In order to apply the said technique, we need to rewrite the boundary value problems as systems o irst-order ODE. We have placed eqs. () and (3) as irst ODE by supposing (,,, θ, θ ) = ( U, U, U, U, U ) = U: U = U, U = U3, U = The corresponding initial conditions become: UU β e ( U ) 3 3 n 3 + nwe ( U3) + We ( U3), (9) U 4 = U5, U 5 = PrUU 5 (9) (, λ,,, ) U Τ = U U () 3 5 The basic idea o shooting method is to calculate the unknown (unspeciied) boundary conditions. We noticed that previous system contains unknown values U 3 and U 5 i. e. ()

6 84 THERMAL SCIENCE: Year 8, Vol., No., pp and θ () that needs to be determined by guessing in order to solve previous system o eq. (9) subject to initial conditions (). To apply this method, the most important step is to pick some pertinent inite value o namely = max. In this way, the solutions were acquired by various initial speculations or the missing estimations o () and θ (). The technique is to regard these terms as certain values that ought to be resolved ahead o time, and aterward an additional iterative loop in the program has been connected to locate these speciic values in a way that ulills the ar ield conditions, i. e. ( max ) = and θ( max ) =. Ater that integration is carried out and compare with the boundary conditions at =. In the present analysis, we have chosen a suitable inite value o, namely =, max between 5 and. A step size o =. was chosen to be acceptable or a convergence basis o 6 or all cases. Testing o code In order to validate the current numerical scheme, we have computed the values o / skin riction coeicient Re C or distinct values o β with those results obtained by Rajagopal et al. [], Kuo [], and Ishak et al. [] or Newtonian case. It is conspicuous rom tab., that the inormation delivered by the present code is in radiant concurrence with prior published results. Thus we are very sure that the present results are exact. / Table. Numerical values o the skin riction coeicient Re C or various β when n =. and We =. are ixed β [] [] [] Present study Furthermore, to evaluate the exactness o the numerical technique, estimations o the / local Nusselt number Re Nu are contrasted with already published work. Table looks at the estimations o θ () or various Prandtl numbers in the limiting cases. We watched that the outcomes acquired in the present study are observed to be in brilliant concurrence with those got by beore analysts. / Table. Numerical values o the Nusselt number Re Nu or various values o Pr and β when n =. and We =. are ixed Pr β = β =.3 [3] Present study [3] Present study

7 THERMAL SCIENCE: Year 8, Vol., No., pp Results and discussion Numerical simulation was perormed to investigate the momentum and also the thermal boundary-layer analysis or a -D low o GN Carreau luid generated by a static/moving wedge. For this reason, the impact o non-dimensional overseeing parameters n, We, β, λ, and Pr on dimensionless luid velocity and temperature proiles, alongside with the riction actor coeicient and local Nusselt numbers, are concentrated on and displayed through graphs. Henceorth, the numerical results are exhibited in two segments-luid-low and heat transer, respectively. Computational results or luid-low Velocity proiles portrayed in igs. (a) and (b) demonstrates the impact o velocity ratio parameter, λ, or both shear thinning and shear thickening regimes. All the model curves ulill the ar ield boundary conditions (4) asymptotically, along these lines supporting the numerical calculations. It regards see that the velocity proiles are bestowed or two distinctive instances o β, speciically, β =. that relates to the wedge edge o zero degree, i. e. low past a lat plate and β =. which compares to the wedge point o 9 i. e. a stagnation-point low. From these trajectories, we perceived that the luid velocity is accelerated by increasing values o the velocity ratio parameter or both cases. We notice rom ig. that momentum boundary-layer thickness is thinner in case o stagnation-point low when contrasted with low over a lat plate. From physical perspective, it is because o the way that the pressure orce together with inertia orce has a tendency to decrease the impact o viscous orces in the boundary-layer. It can be urther seen that in case o low near the stagnation-point, the velocity curves are closer to each other. Additionally, these igures present that the thickness o the momentum boundary-layer or shear thickening luid is higher in comparison with shear thinning luid. '().8.6 '() β =. β =..8 β =. β = λ =.3,.,.,.3 λ =.3,.,., n =.5 n = (a) (b) Figure. Velocity proiles ( ) or dierent λ with We = 3. Figures 3(a) and 3(b) illustrates the dimensionless luid proiles inside the boundary-layer or various estimations o the wedge angle parameter β i there should arise an occurrence o static or moving wedge. For each situation, a considerable increment in the velocity proile is sighted or more grounded estimations o β. From a physical perspective, the wedge angle parameter, β, represent the pressure gradient, so the positive values o β mean a avorable pressure gradient which accelerates the low. It is important to mark that i there should arise an occurrence o accelerated lows, i. e. positive estimations o β, the velocity

8 86 THERMAL SCIENCE: Year 8, Vol., No., pp '().8 λ =. λ =.3 '().8 λ =. λ = β =.,.4,.8,.6 β =.,.4,.8, n =.5 n = (a) (b) Figure 3. Velocity proiles ( ) or dierent β with We = 3. curves crush closer and nearer to the surace o the wedge, and reverse low does not happen. Further, the momentum boundary-layer thickness diminishes by expanding, β, and then again it is higher i there should be an occurrence o shear thickening luid. The impacts o the velocity ratio parameter, λ, the wedge angle parameter, β, and power / law index, n, on the local skin riction coeicient, Re C, are portrayed in igs. 4 and 5. It is shown in ig. 4 that the velocity ratio parameter has a great eect on the skin riction coeicient in both shear thinning and thickening cases. Besides, a watchul inspection o ig. 4 reveals that the local skin riction o Carreau luid can be reduced by enlarging the values o λ. On the other hand, we noticed that with rising values o the Weissenberg number, the skin riction increases or shear thickening luids while an opposite is true in case o shear thinning luids. In addition, ig. 5 indicates the dependency o local skin riction on the wedge angle parameter. Concerning the riction coeicient, a monotonic growth is ound with an expansion in the values o β. Also, the local skin riction is lessor in case o a moving wedge in comparison with in the case o static wedge. Computational results or heat transer Some physical bits o knowledge into the way o heat exchange can be picked up by analyzing the non-dimensional temperature proiles and local Nusselt number variation rom the surace o the wedge.. β =.5 Re / C.8 n =.5 n =.5 Re / C.4 β =.,.4,.8,.6 λ =. λ = λ =.3,.,., We Figure 4. Variation o skin riction coeicient / Re C with We or dierent λ.4 n = We Figure 5. Variation o skin riction coeicient / Re C with We or dierent β

9 THERMAL SCIENCE: Year 8, Vol., No., pp Figure 6 is a plot o the variation in the non-dimensional temperature, θ ( ), within the boundary-layer or distinct values o the velocity ratio parameter, λ. These igs. 6(a) and 6(b) reveal that the luid temperature shows a decreasing behavior with the lourishing values o the velocity ratio parameter in both cases, i. e. or shear thinning as well as shear thickening luids. The thermal boundary-layer thickness decreases or both the low over lat plate and near the stagnation-point. Anyhow, the temperature proiles merely nearer to each other i there should be an occurrence o low near to the stagnation-point and the thickness o the thermal boundary-layer is higher or shear thickening luid. θ().8 n =.5 β =. n =.5 β =. θ().8 β =. β = λ =.3,.,.,.3 λ =.3,.,., (a) (b) Figure 6. Temperature proiles θ ( ) or dierent λ with We = 3. and Pr =. Concerning the impact o wedge angle parameter, β, on thermal boundary-layer, igs. 7(a) and 7(b) maniests the temperature proiles or dierent values o β = (,.4,.8,.6). We observed that the luid temperature is irmly depressed with increasing the wedge angle parameter. Moreover, the maximum temperature o the luid can be accomplished or the low over lat plate β =. Physically, this happens on the grounds that or β =, the driving orce to the luid motion (pressure gradient) is zero and as a result o that the luid temperature at the surace o the wedge enhances. In the event o static wedge thickness o the thermal boundary-layer is higher. The variance in non-dimensional temperature or various estimations o the Weissenberg number is lit up in igs. 8(a) and 8(b). These igures delineate that the dimensionless temperature proiles are contracted or shear thinning luid with an expansion in the Weissen- θ().8 n =.5 λ =. λ =.3 θ() n =.5.8 λ =. λ = β =.,.4,.8,.6 β =.,.4,.8, (a) (b) Figure 7. Temperature proiles θ ( ) or dierent β with We = 3. and Pr =.

10 88 THERMAL SCIENCE: Year 8, Vol., No., pp θ().8 n =.5 Pr =. n =.5 Pr = 3. θ().8 Pr =. Pr = We =.,., 3., 5. We =.,., 3., (a) (b) Figure 8. Temperature proiles θ ( ) or dierent We with β =.4 and λ =.3 berg number. Be that as it may, a remarkable inverse conduct is caught in the shear thickening luid. The impacts o Prandtl number on the luid temperature are to decelerate the temperature as appeared in ig. 8. The Prandtl number is the proportion o momentum diusivity to thermal diusivity. At the point when Pr =., both momentum and thermal diusivities are equivalent, however when Pr >., the momentum diusivity is more prominent than thermal diusivity and the thermal boundary-layer thickness diminishes with the large Prandtl number. Likewise, the thermal boundary-layer thickness is higher in shear thickening luids. / At long last, the physical quantity, or example, the local Nusselt number, Re Nu, is plotted in igs. 9 and against the Weissenberg number or dierent parameters like n, β, and λ. The results or variation o the local Nusselt number with ew estimations o the velocity ratio parameter, λ, are presented in ig. 9. A substantial increase in the local Nusselt number is seen or the larger values o λ. However, an enhancement in the wedge angle parameter, β, accelerates the magnitude o the local Nusselt number as depicted in the ig. as expected the values o local Nusselt number show an increasing behavior by upliting the Weissenberge number..4 Re / Nu λ =.3,.,.,.3 n =.5 n =.5 Re / Nu β =.,.4,.8,.6.4 λ =. λ = β =.4, Pr = We Figure 9. Variation o the Nusselt number / Re Nu with We or dierent λ n =.5, Pr = We Figure. Variation o the local Nusselt number / Re Nu with We or dierent β

11 THERMAL SCIENCE: Year 8, Vol., No., pp Conclusions This review manages to ocus on the momentum and thermal boundary-layer characteristics o the Carreau constitutive model past a static/moving wedge. The local similarity variables have been exhibited which change the basic governing dierential equations into a set o ODE. We acilitate inding the solutions o these conventional dierential equations with a numerical strategy known as shooting technique. Results or the skin riction coeicient, local Nusselt number, velocity, and temperature proiles were accounted or. From the outcomes it was noticed that the indings o this study were in great concurrence with already distributed works or some speciic cases. The guideline consequences o the paper can be compacted as takes ater: y Reduction in the momentum and additionally the thermal boundary-layer thicknesses were seen with the development o the wedge angle parameter. y As the velocity ratio parameter turned out to be huge the velocity o the luid was expanded, however, an inverse conduct was seen in the luid temperature. y The local skin riction was larger or the moving wedge with increasing value o wedge angle parameter. y We got the greatest estimation o skin riction on account o low close to the stagnation point ( β =.) or the static wedge. y The local Nusselt number and local skin riction have displayed opposite behaviors with increasing values o the velocity ratio parameter. y It was seen that the impact o expanding the Weissenberg number was to upgrade the local Nusselt number. Nomenclature b, m, c constants C skin riction c p speciic heat, dimensionless stream unction k thermal conductivity Nu Nusselt number n power law index Pr Prandtl number q w surace heat lux Re local Reynold number T luid temperature T w surace temperature T ree stream temperature u, v velocity components u w stretching velocity u e ree stream velocity We local Weissenberg number x, y space co-ordinates Greek symbols α β Γ γ θ λ μ μ μ ν ρ τ τ w ψ Ω thermal diusivity wedge angle parameter relaxation time magnitude o deormation rate similarity variable dimensionless temperature velocity ratio parameter generalized Newtonian viscosity zero shear viscosity ininite shear viscosity kinematic viscosity thickness stress tensor surace shear stress stream unction wedge angle Reerences [] Falkner, V. M., Skan, S. W., Some Approximate Solutions o the Boundary-Layer Equations, Philos. Mag., (93), pp [] Hartree, D. H., On an Equation Occurring in Falkner and Skan s Approximate Treatment o the Equations o the Boundary Layer, Proc. Camb. Philos. Soc., 33 (937), pp [3] Riley, N., Weidman, P. D., Multiple Solutions o the Falkner-Skan Equation or Flow Past a Stretching Boundary, SIAM J. Apple. Math., 49 (989), 5, pp

12 8 THERMAL SCIENCE: Year 8, Vol., No., pp [4] Watanabe, T., Thermal Boundary Layer over a Wedge with Uniorm Suction or Injection in Force Flow, Acta Mech., 83 (99), -4, pp. 9-6 [5] Ishak, A., et al., MHD Boundary Layer Flow Past a Moving Wedge, Magnetohydrodynamics, 45 (9),, pp. 3- [6] Schlichting, H., Gersten, K., Boundary Layer Theory, Springer-Verlag, Berlin, [7] Hayat, T., et al., Impact o Cattaneo-Christov Heat Flux Model in Flow o Variable Thermal Conductivity Fluid-low over a Variable Thicked Surace, Int. J. Heat Mass trans., 99 (6), Aug., pp. 7-7 [8] Hayat, T., et al., Stagnation Point Flow with Cattaneo-Christov Heat Flux and Homogeneous-Heterogeneous Reactions, J. Mol. Liq., (6), Aug., pp [9] Hayat, T., et al., Unsteady Flow o Nanoluid with Double Stratiication and Magnetohydrodynamics, Int. J. Heat Mass trans., 9 (6), Jan., pp. -9 [] Hayat, T., et al., Impact o Melting Phenomenon in the Falkner-Skan Wedge Flow o Second Grade Nanoluid: A Revised Model, J. Mol. Liq., 5 (6), Mar., pp [] Bird, R. B., et al., Dynamics o Polymeric Liquids, John Wiley and Sons Inc., New York, USA, 987 [] Slattery, J. C., Advanced Transport Phenomena, Cambridge University Press, Cambridge, USA, 999 [3] Carreau, P. J., Rheological Equations rom Molecular Network Theories, Trans. Soc. Rheol., 6 (97),, pp [4] Chapkove, A. D., et al., Film Thickness in Point Contacts under Generalized Newtonian EHL Conditions: Numerical and Experimental Analysis, Tribol. Int., 4 (7), -, pp [5] Jang, J. Y., et al., On the Elastohydrodynamic Analysis o Shear-Thinning Fluids, Proc. R. Soc. A., 463 (7), Dec., pp [6] Khella, K., Lauriat, G., Numerical Study o Heat Transer in a Non-Newtonian Carreau-Fluid between Rotating Concentric Vertical Cylinders, J. Non-Newtonian Fluid Mech., 89 (), -, pp [7] Olajuwon, B. K., Convection Heat and Mass Transer in a Hydromagnetic Carreau Fluid Past a Vertical Porous Plate in Presence o Thermal Radiation and Thermal Diusion, Thermal Science, 5 (), Suppl., pp. S4-S5 [8] Martins, R. R., et al., Numerical Investigation o Inertia and Shear-Thinning Eects on Axisymmetric Flows o Carreau Fluids by a Galerkin Least-Squares Method, Latin American Appl. Research, 38 (8), 4, pp [9] Khan. M., Hashim, A., Boundary Layer Flow and Heat Transer to Carreau Fluid over a Nonlinear Stretching Sheet, AIP Advances 5, (5),, 73 [] Rajagopal, K. R., et al., A Note on the Falkner-Skan Flows o a Non-Newtonian Fluid, Int. J. Nonlinear Mech., 8 (983), 4, pp [] Kuo, B. L., Application o the Dierential Transormation Method to the Solutions o Falkner-Skan Wedge Flow, Acta Mech., 64 (3), 3-4, pp [] Ishak, A., et al., Moving Wedge and Flat Plate in a Micropolar Fluid, Int. J. Eng. Sci., 44 (6), 8-9, pp [3] White, F. M., Viscous Fluid Flow, McGraw-Hill, New York, USA, 99 Paper submitted: January 5, 6 Paper revised: June 7, 6 Paper accepted: June 3, 6 8 Society o Thermal Engineers o Serbia Published by the Vinča Institute o Nuclear Sciences, Belgrade, Serbia. This is an open access article distributed under the CC BY-NC-ND 4. terms and conditions