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Thermodynamical Analysis of the Flow and Heat Transfer Over a Static and a Moving Wedge Hindawi Publishing Corporation ISRN Thermodynamics Volume 2013, Article ID 264095, 6 pages http://dx.doi.org/10.1155/2013/264095 Research Article Thermodynamical Analysis of the Flow and Heat Transfer over a Static and a Moving Wedge Adnan Saeed Butt and Asif Ali Department of Mathematics, Quaid-I-Azam University, Islamabad 45320, Pakistan Correspondence should be addressed to Adnan Saeed Butt; [email protected] Received 12 August 2013; Accepted 24 September 2013 Academic Editors: G. L. Aranovich, A. Chagovetz, and M. Sanati Copyright © 2013 A. S. Butt and A. Ali. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The first and second law characteristics of fluid flow and heat transfer over a static and a moving wedge are investigated. With the help of suitable similarity transformations, the governing boundary layer equations for the velocity and temperature fields are transformed into ordinary differential equations and are solved numerically. The velocity and the temperature profiles are obtained for various parameters and are utilized to compute the entropy generation number Ns and the Bejan number Be. The effects of various physical parameters on the entropy generation number and the Bejan number are depicted through graphs and are discussed qualitatively. It is observed that the entropy production rate is less in case of wedge moving in the opposite direction to flow as compared to static wedge. 1. Introduction forcedflowpastahorizontalflatplatewasexaminedby Al-Odat et al. [6]. Selamet and Arpaci [7]investigatedthe Newton’s second law of motion and laws of thermodynam- entropy effects in boundary layer flows. Makinde and Osalusi ics are the fundamental principles on which all the flow [8] studied entropy effects in a liquid film falling along and heat transfer systems are built today. All other laws an inclined heated porous flat plate. Makinde9 [ ] extended play a supportive role. First law of thermodynamics gives his work and analyzed the irreversibility effects in non- information about the energy of the system quantitatively. Newtonian liquid film falling under the influence of gravity On the other hand, second law of thermodynamics states force. The entropy generation effects in the slip MHDflow thatallreallifeprocessesareirreversible.Theirreversibility in a single rotating disk were studied by Arikoglu et al. [10]. of the processes is measured by entropy generation. The pioneer work on entropy generation in flow systems was Makinde [11] discussed the irreversibility effects in variable done by Bejan [1, 2]. He showed that the engineering design viscosity channel flow with convective cooling at the walls. of a thermal system can be improved by minimizing the Second law analysis for a gravity driven variable viscosity entropy generation. Later on, a lot of research has been done liquid film along an inclined heated plate with convective by many other investigators by taking different geometrical cooling was made by Makinde [12]. Rev´ eillere` and Baytas¸ configurations related to the thermally designed systems and [13] studied entropy generation in laminar boundary layer fluid flow processes. Sahin3 [ ]analyzedtheentropyeffects flow over an isothermal permeable flat surface. Makinde and in viscous fluid flow in a circular duct with the isothermal Beg´ [14] used perturbation technique together with Hermite- boundary condition. The first and second law characteristics Pade approximation to examine the entropy generation in a of fluid flow and heat transfer in a channel having two parallel reactive hydromagnetic channel flow. The first and second plates with a finite gap between them were analytically law characteristics of fluid flow in a porous medium over a investigated by Mahmud and Fraser [4]. Yilbas et al. [5] permeable stretching surface were investigated by Tamayol discussed the entropy production effects in a semiblocked et al. [15]. The thermodynamical analysis of MHD boundary pipe by taking into consideration the swirling effects. The layer flow over a flat plate with variable viscosity was done influence of magnetic field on entropy generation in laminar by Makinde [16]. He considered that the thermal radiation 2 ISRN Thermodynamics y The corresponding boundary conditions for the flow are x (i) static wedge: u =0, V =0 at =0, (4) u (x) = () →∞, T∞ e at Ω= (ii) moving wedge: = () , V =0 at =0, (5) U = () →∞. U∞ w at Figure 1: Schematic diagram of the problem. The thermal boundary conditions are = at =0, =∞ at →∞. (6) is also present there, and the surface is convectively heated. It was found that the variable viscosity results in more Here and V are the velocity components along the and - entropy production. Makinde and Aziz [17]discussedthe axes, respectively, ] is the kinematic viscosity, is the thermal entropy generation effects in a long hollow cylinder with diffusivity, is the density of the fluid, is the specific heat time dependent internal heating and convective heating at of the fluid at a constant pressure, is the temperature of the both inner and outer surfaces. Analytical solution of the fluid, and is the wall temperature, ∞ is the temperature problem was obtained, and a thermal stability criterion was far away from the surface. also examined. Chen et al. [18] made use of differential Introducing the similarity transformation, we have transform method to study the entropy effects in the mixed convective flow in a vertical channel. Butt et al.[19] examined 2] () (,)=√ (), the viscoelastic effects on the entropy generation in fluid flow +1 andheattransferthroughaporousmediumoverapermeable (7) stretching surface. The study of entropy production in MHD (+1) () (,)=√ , slip flow over a vertical surface with convective boundary was 2] done by Butt et al. [20].MunawarandAli[21] examined the entropy generation in fluid flow and heat transfer over an where is the stream function defined such that = oscillating stretching cylinder. The effects of magnetic field /, V = −/ satisfy (1) identically. The similarity on entropy generation in flow over a radially stretching sheet transformation for temperature is were investigated by Butt and Ali [22]. −∞ = . (8) 2. Mathematical Formulation of the Problem −∞ Consider a steady state two-dimensional boundary layer flow Substituting (7)and(8)into(1)–(6), we have of a laminar incompressible fluid past a static and a moving 2 wedge as shown in the Figure 1.Thecoordinatesystemis + + (1 − 2)=0, +1 chosen such that the -coordinate is parallel to the surface of (9) thewedgeandthe-coordinate is normal to it. The properties 2 + Pr + PrEc =0. of the fluid are assumed to be constant. It is assumed that the velocity of the free stream (inviscid flow) is () =∞ The boundary conditions for flow are andthatofthemovingwedgeis() = ,where∞, ,and are constants with 0≤≤1.Usingtheboundary (i) static wedge: layer approximations, the governing equations for the steady flow and heat transfer over a wedge are given by (0) =0, (0) =0, (∞) =1, (10) V + =0, (ii) moving wedge: (1) (0) =0, (0) =, (∞) =1. (11) 2 + V = ] + , 2 (2) The boundary conditions for temperature in nondimen- 2 ] 2 sional form are + V = + ( ) . 2 (3) (0) =1, (∞) =0, (12) ISRN Thermodynamics 3 where prime denotes the differentiation with respect to , 3.0 2 Pr =/ is the Prandtl number, Ec =/( −∞) m = 0.0909 2.5 Pr = 1.0 istheEckertnumber,andtheparameters and are defined Ec = 0.1 2.0 = 3.0, 2.5, 2.0, 1.5, 1.0 as 2 1.5 () = ,= . (13) f 1.0 ∞ +1 0.5 Here is the moving wedge parameter where >0 0.0 corresponds to the motion of wedge in the direction of free = −0.55, −0.35, 0.0, 0.5 stream and <0corresponds to a moving wedge in the −0.5 =0 opposite direction of the free stream velocity. The case 0123456 corresponds to a static wedge. () 3. Entropy Generation Figure 2: Effects of moving wedge parameter on . According to Bejan [2, 23], the entropy generation per unit volume is defined as 1.0 m = 0.0909 2 2 2 Pr = 1.0 0.8 = [( ) +( ) ]+ ( ) . Ec = 0.1 2 (14) 0.6 The first term in(14) is the entropy generation due to heat = −0.55, −0.35, 0.0, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0 transfer, and the second term is the entropy generation due () 0.4 to viscous dissipation. The dimensionless form of the entropy generation is given by 0.2 Ns = 0.0 0123456 −1 2 +1 =(( ) 2 +( ) )2 (15) 2 2 Re Figure 3: Effects of moving wedge parameter on (). +1 + Br ( ) 2, Ω 2 Re the numerical solution, and the accuracy goal is kept equal to 2 2 2 −7 where =( −∞) / is the characteristic entropy 10 . The effects of different physical parameters on velocity generation rate, Ω=( −∞)/ is the dimensionless and temperature profiles are analyzed by plotting graphs. temperature difference, Re =/] is the local Reynolds Moreover, the graphs of the entropy generation number and number, and Br = PrEc is the Brinkman number. the Bejan number are presented to examine the entropy An alternative irreversibility distribution parameter is the production in the considered problem. Bejan number, which gives an idea whether the fluid friction Figure 2 illustrates that for nonpositive values of ;thatis, irreversibility dominates over heat transfer irreversibility or whenthewedgeismovingintheoppositedirectionofstream the heat transfer irreversibility dominates over fluid friction velocity, the momentum boundary layer thickness is more rreversibility. It is simply the ratio of entropy generation due than the case when both wedge and stream are moving in the to heat transfer to the total entropy generation: same direction ≥0.Moreover=1corresponds to the case whenthefluidandthewedgemovewithsamevelocities.In Entropy generation due to heat transfer Be = . (16) this situation, there is no shear stress at the surface. Figure 3 Total entropy generation depicts the effects of moving wedge parameter on temper- ature profile. It is observed that the thermal boundary layer When Be ≫ 0.5, the irreversibility due to heat transfer thickness decreases all values of .
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