Entropy 2011, 13, ; doi: /e OPEN ACCESS

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1 Entropy 011, 13, ; doi: /e OPEN ACCESS entropy ISSN Article Second Law Analysis or Variable Viscosity Hydromagnetic Boundary Layer Flow with Thermal Radiation and Newtonian Heating Oluwole Daniel Makinde Institute or Advance Research in Mathematical Modelling and Computations, Cape Peninsula University o Technology, P. O. Box 1906, Bellville 7535, South Arica; makinded@cput.ac.za; Tel.: ; Fax: Received: 6 May 011; in revised orm: 3 July 011 / Accepted: 16 July 011 / Published: 5 August 011 Abstract: The present paper is concerned with the analysis o inherent irreversibility in hydromagnetic boundary layer low o variable viscosity luid over a semi-ininite lat plate under the inluence o thermal radiation and Newtonian heating. Using local similarity solution technique and shooting quadrature, the velocity and temperature proiles are obtained numerically and utilized to compute the entropy generation number. The eects o magnetic ield parameter, Brinkmann number, the Prandtl number, variable viscosity parameter, radiation parameter and local Biot number on the luid velocity proiles, temperature proiles, local skin riction and local Nusselt number are presented. The inluences o the same parameters and the dimensionless group parameter on the entropy generation rate in the low regime and Bejan number are calculated, depicted graphically and discussed quantitatively. It is observed that the peak o entropy generation rate is attained within the boundary layer region and plate surace act as a strong source o entropy generation and heat transer irreversibility. Keywords: lat plate; variable viscosity; Newtonian heating; Thermal radiation; magnetic ield; entropy generation rate PACS Codes: a; j; gm

2 Entropy 011, Introduction Theoretical study o thermodynamic irreversibility in hydromagnetic boundary layer low in the presence o thermal radiation appears to be increasingly important due to its various applications in engineering and industries [1,] such as in the design o cooling systems or electronic devices, in the ield o solar energy collection, geothermal reservoirs, heat exchangers, thermal insulation, enhanced oil recovery, packed-bed catalytic reactors, cooling o nuclear reactors, etc. Moreover, many engineering processes occur at high temperature and acknowledge radiation heat transer become very important or the design o pertinent equipment [3 5]. Nuclear power plants, gas turbines and the various propulsion devices or air crat, missiles, satellites and space vehicles are other example o such applications. Several excellent studies on hydromagnetic boundary layer lows with thermal radiation have been communicated [6 8]. These orgoing research works have covered a wide range o problems involving the hydromagnetic boundary layer low and heat transer phenomenon; they have been restricted, rom thermodynamic point o view, to only the irst law analysis. The contemporary trend in the ield o luid low, heat transer and thermal design is the second law o thermodynamics analysis and its related concept o entropy generation minimization [9]. Entropy generation is closely associated with thermodynamic irreversibility, which is encountered in all heat transer processes [10]. Dierent sources are responsible or entropy generation like heat transer along a temperature gradient, luid riction, magnetic ield eect, thermal radiation eect, etc. The analysis o thermodynamic irreversibility enables us to identiy the irreversibility associated with various components and to avoid loss o available energy. Such inormation can be employed to design thermal systems, guide eorts to reduce sources o irreversibility, estimate the cost o engineering systems and optimize complex systems. In a pioneering work, Bejan [11,1] introduced the theoretical concept o entropy generation minimization based on the second-law analysis into heat transer and thermal design problems. Thereater, several works on entropy generation minimisation have appeared in the literature [13 15]. Makinde and Aziz [16] gave an analytical and numerical analysis o the second law or a variable viscosity plane Poiseuille low with asymmetric convective heat transer. The study o entropy generation in a alling variable viscosity liquid ilm along an inclined heated plate with convective cooling was carried out by Makinde [17]. Mahmud et al. [18] studied the eect o a magnetic ield on the entropy generation in a mixed convection channel low. Makinde and Beg [19] applied the second law analysis to the problem o inherent irreversibility in a reactive hydromagnetic channel low. The double-diusive (natural) convection in vertical annuluses with opposing temperature and concentration gradients was report by Chen et al. [0]. They proposed that lattice Boltzmann-based numerical method is an eective alternative to traditional CFD or evaluating the irreversibility due to viscosity with a more straightorward way. Chen and his co-workers [1 4] made use o this important numerical approach to investigate the entropy generation rate in several low related problems, such as impinging low conined by planar opposing jets, natural convection in a rectangle cavity, premixed hydrogen-air combustion, etc. In this paper, our main ocus is on the entropy generation characteristics or hydromagnetic boundary layer low under the inluence o thermal radiation and Newtonian heating at the plate surace. This study extends the work o Aziz [5] to include the combined eects o magnetic ield, viscous dissipation, radiative heat transer and entropy generation analysis. Pertinent results on the

3 Entropy 011, eects o various embedded parameters controlling the system are presented in tabular and graphical orms and discussed quantitatively.. Mathematical Model The steady two-dimensional hydromagnetic boundary layer low with heat transer over a lat plate in a stream o variable viscosity electrically conducting luid at temperature T in the presence o magnetic ield and thermal radiation is considered. It is assumed that the lower surace o the plate is heated by convection rom a hot luid at temperature T which provides a heat transer coeicient h. The luid on the upper side o the plate is then subjected to Newtonian heating and its property variations due to temperature are limited to viscosity. A uniorm transverse magnetic ield B 0 is imposed along the y-axis as shown in Figure 1. Both the induced magnetic ield due to the motion o the electrically-conducting luid and the electric ield due to the polarization o charges are assumed to be negligible. Figure 1. Flow coniguration and coordinate system. y U T (ree stream) v B 0 (Magnetic ield) u (Cold Fluid) u = 0, v = 0, T = T (Hot Fluid) x Under the usual boundary layer approximations, the low is governed by the ollowing Equations [6 8,19]: u v + = 0 x y (1) u u 1 u σ B u + v = μ O ( u U) x y ρ y y ρ () T T T 1 qr μ u σ B0 α x y y ρcp y ρcp y ρcp u + v = + + ( u U ) (3) where U is the ree stream velocity, c p is the speciic heat at constant pressure, is the thermal diusivity, is the luid density, is the luid electrical conductivity. The luid dynamical viscosity μ is assumed to be an inverse linear unction o temperature, Lai and Kulacki [6], as given by: μ μ ( T ) = (4) 1+ γ ( T T )

4 Entropy 011, where μ is cold luid viscosity and is a constant. Using the Rosseland approximation [3] or radiation, the radiative heat lux is simpliied as: q r * 4σ = * 3k * where σ is the Stephan-Boltzmann constant and k* is the mass absorption coeicient. The temperature dierences within the low are assumed to be suiciently small so that T 4 may be expressed as a linear unction o temperature T using a truncated Taylor series about the ree stream temperature T i.e., T y T 4T T 3T. (6) The boundary conditions at the plate surace and ar into the ree stream may be written as: T u ( x,0) = 0, v(x,0)=0, k ( x,0) = h [ T T ( x,0)] y u(x, ) = U, T(x, ) = T (7) where k is the thermal conductivity coeicient. The stream unction, satisies the continuity Equation (1) automatically with: u = ψ y and ψ v = (8) x A similarity solution o Equations (1) (8) is obtained by deining an independent variable and a dependent variable in terms o the stream unction ψ as: (5) U μ η = y, ψ = υxu ( η), υ =, υx ρ T T T T θ (η) = (9) Ater introducing Equation (9) into Equations (1) (8), we obtain: 1 aθ "' + (1 + aθ ) " Ha(1 + aθ )( ' 1) = 0 (10) (1 + aθ ) ( ) 1 Brβ θ " + Pr βθ ' + + βbrha( 1) = 0 (11) (1 + aθ ) (0) = 0, (0) = 0, θ (0) = Bi[ θ(0) 1] (1) ( ) = 1, θ ( ) = 0 (13) where β = 3 Ra /(3Ra + 4) and = 1 corresponds to the absence o thermal radiation inluence. The prime symbol represents the derivative with respect to and: h Bi = k σ 0 B x Ha = (local magnetic ield parameter), ρ U υx U (local convective heat exchange parameter),

5 Entropy 011, μu Br = k( T T ) (the Brinkmann number), υ Pr = (the Prandtl number), α a = γ ( T T ) (viscosity variation parameter), Ra = kk * * 3 4σ T (thermal radiation parameter). It is noteworthy that the local parameters Ha and Bi in Equations (10) (13) are unctions o x. In order to have a similarity solution, all the parameters must be constant and we thereore assume [6,7,18]: where c, and l are constants..1. Numerical Procedure 1 1 h = cx, σ = lx (14) The set o Equations (10) and (11) under the boundary conditions (1) have been solved numerically by applying the Nachtsheim and Swigert [7] shooting iteration technique together with Runge-Kutta sixth-order integration scheme. Let = x1, ' = x, '' = x3, θ = x4, θ = x5, Equations (10) and (11) are transormed into a system o irst order dierential equations as ollows: x = x x 5 = 3 5, x = x, x = Ha(1 + ax x = x, 4 1 Pr βx1x subject to the ollowing initial conditions: )( x 5 ax5x3 1) + (1 + ax Brβx 1+ ax ) BrβHa 1 (1 ( x 1) + ax 4 ) x x x 1 (0) = 0, x (0) = 0, x 3 ( 0) = s1, x 4 ( 0) = s, x 5 ( 0) = Bi( s 1) (16) The unspeciied initial conditions s 1 and s are assumed and Equation (15) is then integrated numerically as an initial valued problem to a given terminal point. Improvement is made on the values o assumed missing initial conditions by iteratively comparing the calculated value o the dependent variable at the terminal point with its given value there. The computations were done by a written program which uses a symbolic and computational computer language MAPLE. A step size o Δ η = was selected to be satisactory or a convergence criterion o 10 7 in nearly all cases. From the process o numerical computation, the plate surace temperature, the local skin-riction coeicient and the local Nusselt number which are respectively proportional to (0), (0 ) and θ (0), are also worked out and their numerical values are presented in a tabular orm. 1 3, (15)

6 Entropy 011, Entropy Generation Analysis The hydromagnetic boundary layer low past a lat surace under the inluence o thermal radiation and Newtonian heating is inherently irreversible. This is due to the exchange o energy and momentum, within the luid and solid boundaries leading to continuous entropy generation [9 1]. Two major o entropy production can be identiied. One part is due to the combined eects o heat transer and thermal radiation in the direction o inite temperature gradients while the other arises due to the combined eects o luid riction and Joule heating. Following Woods [1], the local volumetric rate o entropy generation in the presence o a magnetic ield is given by: S m k T T μ u σb0 = + u + + (17) T x y T y T The irst term in Equation (17) is the irreversibility due to heat transer, the second term is the entropy generation due to viscous dissipation and the third term is the local entropy generation due to the eect o the magnetic ield. Using Equation (9), we express the entropy generation number in dimensionless orm as: Ns = m x T S η dθ Re Br 1 d d = Re+ + + Ha k( T T ) 4 dη Ω (1 + aθ ) dη dη (18) where Ω = ( T T ) / T is the temperature dierence parameter and Re = U x/ is the local Reynolds number. We assigned N 1 to the irst term in Equation (18) due to heat transer and the second term due to the combined eects o viscous dissipation and magnetic ield as N, i.e., N 1 η dθ = Re+ 4 dη, N Re Br 1 d = Ω (1 + aθ ) dη d + Ha dη The irreversibility distribution ratio is then given as = N /N 1. Thereore, whenever 0 < 1, the heat transer irreversibility dominates. The luid riction together with magnetic ield dominates when > 1. The contribution o both heat transer and luid riction with magnetic ield to entropy generation are equal when = 1. However, in many engineering designs and energy optimisation problems, the contribution o heat transer entropy (N 1 ) and luid riction with magnetic ield eect (N ) to overall local entropy generation rate Ns x is needed. In order to achieve this, we deine the ollowing Bejan numbers (Be) mathematically as: N1 1 Be = = (0) Ns 1 +Φ From Equation (0), it is very obvious that the Bejan number ranges rom 0 to 1. The zero value o the Bejan number corresponds to the limit where the irreversibility is dominated by the combined eects o luid riction and magnetic ield while Be = 1 is the limit where the irreversibility due to heat transer by virtue o inite temperature dierences dominates the low system. The contributions o both the actors to entropy generation are equal when Bejan number is equal to hal. (19)

7 Entropy 011, Results and Discussion The boundary layer low and heat transer in a variable viscosity luid under the inluence o a transverse uniorm magnetic ield and thermal radiation has been solved numerically using shooting iteration technique together with Runge-Kutta integration scheme and numerical expressions o the velocity and temperature have been used to compute the entropy generation. This type o problem arises in many industrial applications such as geothermal reservoirs, cooling o nuclear reactors, thermal insulation, petroleum reservoirs, electronic packages and microelectronic devices during their operation. In order to study the inluence o all parameters involved on the low and thermal ields along with their entropy generation characteristics, a selected set o graphical results is presented in Figures 1. I emphasize here that an increase in the parameter a > 0 indicates a decrease in the luid viscosity. Extensive calculations have been perormed to obtain the local skin riction and local Nusselt number or a wide range o parameters. Also, the results have been compared with the one reported by Aziz [18] as shown in Table 1 or constant viscosity luid without the magnetic ield, thermal radiation and viscous dissipation eects and it is ound that they are in good agreement. This serves as a benchmark or the accuracy o our numerical procedure. Table shows the eect o various thermophysical parameters on the local skin riction ( (0 ) ), Nusselt number ( θ (0) ) and plate surace temperature variation ( θ (0) ). I notice that both the local skin riction and local Nusselt number increase with an increase in the parameter values o Bi, Br, a. This can be attributed to the act that as the convective heat transer rom the hot luid on the lower side o the plate to the upper side increases due to Newtonian heating, the luid viscosity on the upper side o the plate decreases leading to an increase in velocity gradient and viscous dissipation. It is interesting to note that the local skin riction decreases while the surace heat transer rate increases with an increase in thermal radiation (Ra) and Prandtl number (Pr). The physical reason or this trend is that at higher Prandtl number and radiation, the luid thermal boundary layer becomes thinner leading to an increase in the temperature gradient. Moreover, an increase in magnetic ield intensity (Ha) causes a decrease in the local Nusselt number and an increase in local skin riction. Table 1. Computations showing comparison with Aziz [18] results or Ha x = Br = a = 0, = 1, Pr = 0.7. Bi θ (0) Aziz [18] θ (0) Aziz [18] θ (0) Present θ (0) Present

8 Entropy 011, Table. Computation showing (0 ), θ (0), and θ (0) or various values o key parameters. Bi a Br Ra Pr Ha (0 ) θ (0) θ (0) Eect o Parameter Variation on Velocity Proiles Figure shows the variation o longitudinal velocity as a unction o at dierent values o magnetic ield parameter. The luid velocity is lowest at the plate surace and increases to the ree stream value satisying the ar ield boundary condition. Application o the magnetic ield creates a resistive orce similar to the drag orce that acts in the opposite direction o the luid motion, thus causing the velocity o the luid to overshoot towards the plate surace. Similar trend is observed in Figure 3 with a decrease in the luid viscosity (i.e., as parameter a increases). Since as luid viscosity decreases, the momentum boundary layer becomes thinner, leading to an increase in the luid velocity gradient. Figure. Velocity proile or Pr = 0.7, Br = 0.1, Ra = 0.7, a = 0.1, Bi = 0.1.

9 Entropy 011, Figure 3. Velocity proile or Pr = 0.7, Br = 0.1, Ra = 0.7, Ha = 0.1, Bi = Eects o Parameter Variation on Temperature Proiles Dimensionless temperatures are plotted as a unction o transverse distance in Figures 4 9. Generally, the luid temperature is highest at the plate surace and decreases exponentially to the ree stream zero value away rom the plate satisying the boundary condition. As it can be seen in Figure 4, the luid temperature increases with an increase in Ha accordingly leading to an increase in thermal boundary layer. As explained above, the transverse magnetic ield gives rise to a resistive orce known as the Lorentz orce o an electrically conducting luid. This orce makes the luid experience a resistance by increasing the riction between its layers and thus increases its temperature. Similar trend is observed in Figures 5 and 6 with increasing values o Br and Bi. The thermal boundary layer generates energy due to combined eects o viscous heating and Newtonian heating. This causes the temperature o the luid to increase. In Figure 7, it is noteworthy that the luid temperature decreases with an increase in radiation parameter (Ra) leading to a decrease in the thermal boundary layer thickness. Figure 8 represents graphs o temperature proiles taking dierent values o the variable viscosity parameter (a). It can be seen that thermal boundary layer decreases due to a decrease in the luid viscosity and this causes the temperature o the luid to decrease. Figure 9 depicts the temperature proiles or dierent values o the Prandtl number. The luid temperature decreases with an increase in Prandtl number, consequently the thermal boundary layer decreases. Figure 4. Temperature proiles or Pr = 0.7, Br = 0.1, Ra = 0.7, a = 0.1, Bi = 0.1.

10 Entropy 011, Figure 5. Temperature proiles or Pr = 0.7, Ha = 0.1, Bi = 0.1, Ra = 0.1, a = 0.1. Figure 6. Temperature proiles or Pr = 0.7, Ha = 0.1, a = 0.1, Ra = 0.7, Br = 0.1. Figure 7. Temperature proiles or Pr = 0.7, Ha = 0.1, a = 0.1, Ra = 0.7, Br = 0.1.

11 Entropy 011, Figure 8. Temperature proiles or Pr = 0.7, Br = 0.1, Ra = 0.7, Ha = 0.1, Bi = 0.1. Figure 9. Temperature proiles or Bi = 0.1, Ha = 0.1, Br = 0.1, Ra = 0.7, a = Eects o Parameter Variation on Entropy Generation Rate The inluence o various parameters on entropy generation rate is displayed in Figures In Figure 10, we notice that the entropy generation number is higher or higher magnetic parameter (Ha). The presence o the magnetic ield creates entropy in the luid. The eects o the local Biot number Bi and the group parameter Br 1 on the entropy generation number is represented in Figures 11 and 1. It seems that these parameters have similar eect on the entropy generation number by creating more entropy in the luid. However, it is interesting to note that the entropy generated in the luid attained it maximum value at a distance near the plate surace within the boundary layer region. The inluence o the Prandtl number on the entropy generation number is shown in Figure 13. As the Prandtl number increases, the entropy generation number increases gradually rom the plate surace, then to its highest value within the boundary layer and decreases to its lowest zero value at the ree stream. It is noteworthy that the entropy generation rate peak value or Pr = 0.7 (Air) is higher than that o Pr = 7.1 (Water). Similar trend is observed in Figures 14 and 15 with increasing value o parameters a and Ra. As the luid viscosity decreases and thermal radiation increases, the entropy generation rate

12 Entropy 011, increases rom the plate surace, reaching its peak values and then decreases to zero value at ree stream. The peak value or Ra = 0.7 is higher to that o Ra = 3. Figure 10. Entropy generation rate or Pr = 0.7, Br 1 = 0.1, Re= 0.1, Bi = 0.1, a = 0.1, Ra = 0.7. Figure 11. Entropy generation rate or Pr = 0.7, Br 1 = 0.1, Re = 0.1, Ha = 0.1, Ra = 0.7, a = 0.1.

13 Entropy 011, Figure 1. Entropy generation rate or Pr = 0.7, Bi = 0.1, Re = 0.1, Ha = 0.1, a = 0.1, Ra = 0.7. Figure 13. Entropy generation rate or Bi = 0.1, Br 1 = 0.1, Ra = 0.7, Re = 0.1, Ha = 0.1, a = 0.1.

14 Entropy 011, Figure 14. Entropy generation rate or Bi = 0.1, Br 1 = 0.1, Ra = 0.7, Re = 0.1, Ha = 0.1. Figure 15. Entropy generation rate or Bi = 0.1, Br 1 = 0.1, a = 0.1, Re = 0.1, Ha = Eects o Parameter Variation on Bejan Number In Figures 16 1, the Bejan number is displayed as a unction o the transverse distance. It is interesting to note that the heat transer irreversibility dominates within the boundary layer while the viscous dissipation and magnetic ield irreversibility dominate at the plate surace and in the ree stream region. Meanwhile, the dominant eect o heat transer irreversibility within the boundary layer region decreases with an increase in magnetic ield intensity (Ha), group parameter Br 1, variable viscosity parameter (a) and Prandtl number (Pr) as illustrated in Figures In Figure 1, we notice that the eect o heat transer irreversibility increases as the local Biot number (Bi) parameter

15 Entropy 011, increase due to Newtonian heating o the plate surace. Hence, the plate surace acts as a strong source o heat transer irreversibility. Figure 16. Bejan number or Bi = 0.1, Br 1 = 0.1, a = 0.1, Ra = 0.7, Re = 0.1. Figure 17. Bejan number or Pr = 0.7, Bi = 0.1, Re = 0.1, Ha = 0.1, a = 0.1, Ra = 0.7.

16 Entropy 011, Figure 18. Bejan number or Bi = 0.1, Br 1 = 0.1, Ra = 0.7, Re = 0.1, Ha = 0.1. Figure 19. Bejan number or Bi = 0.1, Br 1 = 0.1, a = 0.1, Re = 0.1, Ha = 0.1. Figure 0. Bejan number or Bi = 0.1, Br 1 = 0.1, Ra = 0.7, Re = 0.1, Ha = 0.1, a = 0.1.

17 Entropy 011, Figure 1. Bejan number or Pr = 0.7, Br 1 = 0.1, Re = 0.1, Ha = 0.1, Ra = 0.7, a = Conclusions Analysis o entropy generation rate in hydromagnetic boundary layer low o a variable viscosity luid in the presence o thermal radiation, viscous dissipation and Newtonian heating is carried out. The velocity and temperature proiles are obtained numerically and used to compute the entropy generation number. The eects o the magnetic parameter, the Prandtl number, local Biot number, variable viscosity parameter and radiation parameter on velocity and temperature proiles are presented. The inluences o the same parameters and the dimensionless group parameter on the entropy generation rate and Bejan number are also discussed. From the results the ollowing conclusions could be drawn: I. The skin riction and the Nusselt number increase with increasing values o Bi, Br, a. An increase Ha causes a decrease in the Nusselt number and an increase in the skin riction. As Ra and Pr increase, the skin riction decreases while the Nusselt number increases. II. The velocity boundary layer thickness decreases with Ha, a. III. The thermal boundary layer thickness increases with Ha, Br, Bi and decreases with Ra, Pr, a. IV. The plate surace act as a strong source o entropy generation and heat transer irreversibility. However, the peak o entropy generation rate is attained within the boundary layer region. V. The entropy generation number increases as Ha, Bi and Br 1 increase while it decreases as the Pr, Ra and a increase. VI. The optimum design and eicient perormance o the low system can be enhanced by the ability to clearly identiy the source and location o entropy generation. The present results show that minimum entropy generation in the low system can be achieved with appropriate choice and combination o the various thermophysical parameters.

18 Entropy 011, Acknowledgements The author would like to thank the National Research Foundation o South Arica Thuthuka programme or inancial support. Reerences 1. Woods, L.C. Thermodynamics o Fluid Systems; Oxord University Press: Oxord, UK, Aboud, S.; Saouli, S. Entropy analysis or viscoelastic magnetohydrodynamic low over a stretching surace. Int. J. Non-Linear Mech. 010, 45, Sparrow, E.M.; Cess, R.D. Radiation Heat Transer; Harpercollins College Div: New York, NY, USA, 1978; pp Raptis, A.; Perdikis, C.; Takhar, H.S. Eect o thermal radiation on MHD low. Appl. Math. Comput. 004, 153, Mbeledogu, I.U.; Amakiri, A.R.C.; Ogulu, A. Unsteady MHD ree convection low o a compressible luid past a moving vertical plate in the presence o radiative heat transer. Int. J. Heat Mass Trans. 007, 50, Bataller, R.C. Radiation eects or the Blasius and Sakiadis lows with a convective surace boundary condition. Appl. Math. Comput. 008, 06, Makinde, O.D.; Aziz, A. MHD mixed convection rom a vertical plate embedded in a porous medium with a convective boundary condition. Int. J. Therm. Sci. 010, 49, Makinde O.D. On MHD heat and mass transer over a moving vertical plate with a convective surace boundary condition. Can. J. Chem. Eng. 010, 88, Narusawa, U. The second-law analysis o mixed convection in rectangular ducts. Heat Mass Trans. 1998, 37, Sahin, A.Z. Second law analysis o laminar viscous low through a duct subjected to constant wall temperature. J. Heat Trans. 1998, 10, Bejan, A. Second-law analysis in heat transer and thermal design. Adv. Heat Trans. 198, 15, Bejan, A. Entropy Generation Minimization; CRC: Boca Raton, FL, USA, Kobo, N.S.; Makinde, O.D. Second law analysis or a variable viscosity reactive Couette low under Arrhenius kinetics. Math. Probl. Eng. 010, 010, Jalaal, M.; Ganji, D.D.; Ahmadi, G. Analytical investigation on acceleration motion o a vertically alling spherical particle in incompressible Newtonian media. Adv. Powder Technol. 010, 1, Jalaal, M.; Ganji, D.D. An analytical study on motion o a sphere rolling down an inclined plane submerged in a Newtonian luid. Powder Technol. 010, 198, Makinde, O.D.; Aziz, A. Second law analysis or a variable viscosity plane Poiseuille low with asymmetric convective cooling. Comput. Math. Appl. 010, 60, Makinde, O.D. Thermodynamic second law analysis or a gravity driven variable viscosity liquid ilm along an inclined heated plate with convective cooling. J. Mech. Sci. Technol. 010, 4,

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