Second law analysis in MHD flow and heat transfer of a rotating Casson fluid over a stretching surface

Transcription

1 Indian Journal of Pure & Applied Physics Vol. 56, January 018, pp Second la analysis in MHD flo and heat transfer of a rotating Casson fluid over a stretching surface Shalini Jain* & Rakesh Choudhary Department of Mathematics & Statistics, Manipal University Jaipur, Jaipur , India Received 15 June 017; accepted 3 November 017 Second la analysis in three-dimensional MHD boundary layer flo and heat transfer of a rotating Casson fluid over a stretching surface has been investigated. Such flo problems have relevance in extraction and manufacturing of rubber and polymer sheets. Solution of these problems is of great interest as they serve a practical purpose. Partial differential equations governing flo and heat transfer have been transformed into non-linear ordinary differential equations by using suitable similarity transformations. These non-linear differential equations have been solved numerically by using shooting techniques ith fourth order Runge-Kutta method. Effects of Casson fluid parameter β, rotation parameter λ, Prandtl number Pr and magnetic field parameter M on velocity profile, temperature profile and skin friction number have been analyzed and depicted through graphs and tables. Entropy generation number and Bejan number have also been obtained and discussed. Keyords: Rotating Casson fluid, Heat transfer, Stretching surface, Entropy generation, Runge-Kutta method 1 Introduction Flo induced by stretching surfaces has often encountered in several engineering and industrial processes and therefore its study has many promising applications for example in extraction and manufacturing of rubber and polymer sheets, ire and fiber cutting process, design of various equipment used in chemical and food stuff processing. Mechanical properties of such products depend on rate of heat transfer at the stretching sheet. Initially Crane 1 examined boundary layer flo expulsion by the continuous stretching of a surface in a fixed point. Sakiadis initiated the effects of boundary layer flo past a moving surface. Wang 3 extended Crane s problem and examined three- dimensional flo due to a stretching flat surface. Verma and Chauhan 4 studied flo beteen a torsionally oscillating impermeable disc and a stationary naturally permeable disc. Many researchers such as Jain 5, Fang et al. 6, Najar et al. 7, Mahanta and Sha 8, Jain and Choudhary 9, Raju et al. 10 and Jain and Bohra 11 investigated heat transfer phenomena on flo over a stretching sheet. Rotating fluid flos in boundary layer flos have several applications in areas like astrophysical process, biomechanics, cosmic fluid dynamics, industrial processes, in the design of turbines and rotating heat exchangers etc. *Corresponding author ( shalini.jain@jaipur.manipal.edu) Rotating fluid flos can also be observed at the migration of underater and movement of oil, petrol and gas through reservoirs. Wang 1 obtained exact results for a stretching surface in a rotating fluid. Takhar et al. 13 studied flo over a stretching surface in a rotating fluid ith a magnetic field. Zaimi et al. 14 analyzed heat transfer effects of rotating viscoelastic fluid in stretching surface. Recently, Mustafa 15 and Hayat et al. 16 also studied the heat transfer effects of nanofluids in a rotating systems ith MHD. Second la analysis is basic for many of engineering motives for formulating and solving convection problems. For example, in the development of kno ho for the heat exchanger industry. We strive for improved thermal contact and reduced pump poer loss in order to improve thermodynamic efficiency of a heat exchanger. A good heat exchanger means minimum generation of entropy. Bejan 17 played a very important role to investigate the entropy generation effects. The ideas of Bejan became a milestone for every researcher. Makinde and Osalusi 18 studied entropy analysis of laminar flo in a filled channel embedded in a saturated porous medium. Butt and Ali 19 examined investigation of entropy generation effects in MHD three-dimensional flo and heat transfer of viscous fluid over a stretching surface. Recently, Freidoonimehr et al. 0 and Qing et al. 1 investigated entropy generation analysis of a MHD Casson fluid

2 54 INDIAN J PURE & APPL PHYS, VOL. 56, JANUARY 018 over a rotating disk and porous stretching/shrinking sheet, respectively. Several fluids used in industrial and engineering process such as geological materials and poly liquid foams, exhibit flo properties that are difficult to explain by the Netonian model. One such model is the non-netonian Casson fluid model hich symbolizes a shear thinning fluid hich is assumed to have infinite viscosity at zero rate of shear. This fluid behaves like solid elastic and has yield shear stress in the consecutive equation for a Casson fluid. Analysis of the Casson and Carreau-Yasuda non- Netonian blood models in steady and oscillatory flo as discussed by Boyd et al. 3 ho used the lattice Boltzmann method for this purpose. Examples of a Casson fluid are jelly, honey, soup and human blood. Flo and heat transfer phenomena of a Casson fluid having different physical and mathematical aspects as analyzed by several researchers such as Bhargava et al. 4, Attia and Ahmed 5 and Mukhopadhyay 6. Nadeem et al. 7 discussed heat transfer and MHD boundary layer flo effects of a Casson fluid past an exponentially shrinking sheet. Bhattacharyya et al. 8 studied an analytical solution for MHD boundary layer flo of a Casson fluid over a stretching/shrinking sheet ith all mass transfer. Mukhopadhyay et al. 9 discussed Casson fluid flo and heat transfer past a symmetric edge. MHD three- dimensional Casson fluid past a porous linearly stretching sheet as analyzed by Nadeem et al. 30. Pramanik 31 investigated heat transfer effects of Casson fluid flo over an exponentially porous stretching surface ith thermal radiation. Tufail et al. 3 studied heat source/sink effects on non- Netonian magnetohydrodynamic fluid flo over a permeable stretching surface. Sumalatha and Bandari 33 analyzed heat transfer effects of Casson fluid flo over a nonlinear stretching sheet. Several authors have studied the Casson fluid model under different geometries such as Butt et al. 34, Jain and Bohra 35, Ali et al. 36 and Raju et al. 37. The forgoing discussion that deal ith a rotating Casson fluid ith stretching surface are restricted to the first la of thermodynamics and, to the best of our knoledge, none of them investigated the second la of thermodynamics. Recently Butt et al. 34 studied of flo and heat transfer on a stretching surface in a rotating Casson fluid. In this paper e extend the ork of Butt et al. 34 and aim to investigate entropy generation of flo and heat transfer over a stretching surface under transverse magnetic field, in a rotating Casson fluid. Governing equations are solved numerically and results obtained for pertinent parameters are discussed and presented through graphs and tables. Flo Analysis Consider steady, three-dimensional, laminar incompressible MHD boundary layer flo of a rotating Casson fluid over a stretching surface. Assume that the surface is stretched in the x- direction. The velocity components u, v and are in the direction of the x, y and z axes, respectively, ith an angular velocity ω in the z direction (Fig. 1). B 0 is a magnetic field hich is applied normal to the surface. The Coriolis force is the reason for flo in three dimensional MHD boundary layer flo. The fluid rotates ith angular velocity ω about the z axis. Let T be the constant temperature of the stretching surface and T be the temperature of the fluid far aay. Rheological model that describes Casson fluid is: i j (μ (μ B B p p y y / / π) e c ij π ) e, π π (1) ij c, π π here τ ij is the component of the stress tensor, p y is the yield stress of the fluid, π is the product of the component of the deformation rate ith itself, π c is a critical value of this product based on the non- Netonian model and µ B is the plastic dynamic velocity of the non-netonian fluid. The governing equations folloing Butt and Ali 19 are given as: Equation of continuity: u v 0 x y z Fig. 1 Schematic diagram of the problem. c ()

3 JAIN & CHOUDHARY: MHD FLOW AND HEAT TRANSFER OF A ROTATING CASSON FLUID 55 Equation of momentum: u u u u v ωv x y z 1 p 1 σb0 u 1 u x β ρ v v v u v ωu x y z 1 p 1 σb0 v 1 v ρ y β ρ 1 p u v x y z ρ z 1 σb 0 1 β ρ Equation of energy: T T T k u v T x y z ρcp 1 u v 1 Cp β z z (3) (4) (5) (6) Subject to boundary conditions: u u ( x) ax, v 0, 0 T T at z 0, and u 0, v 0, T T at z (7) here a >0 is constant, y μ B π β c p is the Casson fluid parameter, ν is the kinematic viscosity, k is thermal conductivity, C p is the specific heat of the fluid at a constant pressure, ρ is the density and T is the temperature of the fluid. We introduce the similarity transformation so that the continuity equation is identically satisfied. We define: u axf '(η), v ax g(η) and av f (η), a T T η z, θ (8) v T T Substituting Eq. (8) in Eqs (-6), Eq. () is satisfied and Eqs (3, 4, 6) become: 1 1 β ''' ff '' λ g Mf ' f ' f (9) g'' f ' g fg' λ f ' Mg 0 β 1 θ'' Pr fθ' 1 Pr Ec( f '' g' ) 0 β σb here M 0 ρa (10) (11) is the magnetic field parameter, ω λ is the ratio of rate of rotation to stretching rate, a μc p u Pr is the Prandtl number and Ec k C ( T ) is the local Eckert number. Here η is only function of z and it does not depend on x. Hence the Eckert number (Ec) does not depend only on the physical property but it increases ith the square of the velocity. The corresponding boundary conditions reduced to: f '(0) 1, f(0) 0, g(0) 0, θ(0) 1, and f '( ) 0, g( ) 0, θ( ) 0 (1) The skin friction coefficients along ith x and y directions are as follos: C τ ρu τ ρu x y fx, Cfy, (13) here x and y are the surface shear stress along the x and y directions, respectively, and are defined as: τ τ x y μ μ B B py u π z py v π z z0 z0 By using Eqs (8) and (14), e get:,. p T (14) 1 1 Rex C 1 f ''(0), fx Rex C 1 g '(0) β fy β (15) The local Nusselt number Nu is defined as follos: xq Nu x (16) k T T ) ( x

4 56 INDIAN J PURE & APPL PHYS, VOL. 56, JANUARY 018 T here q k is the heat flux. Substituting z z0 the value of q in Eq. (16), then the non-dimensional of Nusselt number is as follos: Nux Rex θ'(0) (17) 3 Numerical Solution The boundary value problems given in Eqs (9-11) are solved by the forth order Runge-Kutta Shooting method. The important steps for this method are as follos: Convert the boundary value problem into the initial value problem. Reduce the system of Eqs (9-11) into a first order system. We assume thatc 1, c and c 3 are the initial guesses for f ''(0), g '(0) and θ'(0), respectively. For this e consider a finite domain 0 η ηinstead of semiinfinite integration domain 0 η. here η has been chosen sufficiently large so that solution obtained closely approximates the boundary condition. In this study e have considered η 10. The boundary value problem is converted into a system of initial value problems, as given belo: 1 f ' z, z' p, p' ( fp λg Mz 1 1 β 1 g ' v, v' ( zg fv λz Mg) 1 1 β 1 θ' q, q' Pr fq 1 Pr Ec( p v ) β Subject to boundary conditions: z (18) f ( 0) 0, z(0) 1, g(0) 0, p( 0) c, v(0) c, θ(0) 1, q(0 c (19) 1 ) 4 Entropy Generation Bejan 17 described the local entropy generation rate per unit volume S G for a viscous fluid in the presence of magnetic field. Entropy generation equation for a Casson fluid is as follos: 3 ), G k T μ 1 S 1 T z T β u v σb0 u v z z T ( ) (0) The first term in Eq. (0) is due to heat transfer. The second term is due to fluid friction irreversibility and the third terms denote the magnetic field. To obtain anon-dimensional form of Eq. (0), e define the characteristic entropy generation rate: S G 0 kt T x (1) The non-dimensional entropy generation number is as follos: G S 1 NS Re G xθ' 1 S0 β () 1 1 BrRex( f '' g ' ) Br Re x M ( f ' g ) u x here Re x is the local Reynolds number, v μu T Br is the Brinkmann number, kt ( T ) T is the dimensionless temperature difference. Eq. () can be reritten in the form: N N N N N N (3) S H f m here N H is entropy generation due to heat transfer, Nf is entropy generation due to fluid friction and N m shos entropy effects due to magnetic field. The Bejan number is described as follos: Be N H H (4) N H NF From Eq. (4) it is clear that the range for the Bejan number is beteen 0 and 1. Also hen Be= ½ the entropy due to heat transfer fluid friction and magnetic field entropy are equal. 5 Results and Discussion Equations (9-11) subject to boundary condition (Eq. (1)) have been solved numerically using Runge- Kutta fourth order ith shooting method. F

5 JAIN & CHOUDHARY: MHD FLOW AND HEAT TRANSFER OF A ROTATING CASSON FLUID 57 Tables (1-3) sho a comparison beteen present results and existing results of Wang 1, Najar 7 and Butt et al. 34. It is clearly seen that our results are very ell in agreement ith existing results. It is observed that our results are ell in agreement ith Butt et al. 34 for particular conditions of our problem. Figures and 3 sho a comparison beteen our ork and Butt et al. 34 for velocity and temperature profiles. Figures 4 and 5 sho velocity profiles f '(η) and g (η) in the x and y directions for magnetic parameter M. An increase in magnetic field reduces the velocity component in both the x and y direction of stretching surface, due to an increase in Lorentz force, hich offers greater resistance to the flo, and hence flo decelerates. Hoever, in Fig. 6, ith an increase in magnetic field M, temperature increases. The Casson parameter also reduces velocity boundary layer. This is due to the introduction of tensile stress. Elasticity creates resistance in the fluid flo, and as a result velocity decreases. For higher values of Casson fluid parameter β, there is a decrease in thickness of the momentum boundary layer. Hoever, ith increase in β, the thermal boundary layer thickness increases. Figures 7-9 indicate the effect of the Casson fluid parameter β on velocity profile and temperature profile. The velocity profiles f '(η) and g(η) decrease ith the increasing values of the Casson fluid parameter β. For higher values of β the boundary layer thickness decreases. An increase in the Casson fluid parameter β the temperature profile (η) increases. In Fig. 9 at a particular set of parameters, hen magnetic field parameter M=0 and Prandtl Number Pr = 0, the results obtained are exactly the same as those mentioned by Butt et al. 34 Figures 10 and 11 depict the effect of rotation parameter λ on velocity profiles f '(η) and g (η). With an Table 1 Comparison of the values of β. f ''(0) hen M = 0 and Wang 1 Najar et al. 7 Butt et al. 34 Present Study f ''(0) f ''(0) f ''(0) f ''(0) Table Comparison of the values of g '(0) hen M= 0 and β. Fig. Comparison of velocity profile. Wang 1 Najar et al. 7 Butt et al. 34 Present Study g '(0) g '(0) g '(0) g '(0) Fig. 3 Comparison of temperature profile. Table 3 Comparison of the values of θ'(0) hen M = 0, Ec= 0 and β. Pr = 0.7 Pr =.0 Pr = 7.0 Wang 1 Butt et al. 34 Present study Wang 1 Butt et al. 34 Present study Wang 1 Butt et al. 34 Present study

6 58 INDIAN J PURE & APPL PHYS, VOL. 56, JANUARY 018 increase in the value of λ the velocity profile f '( ) decreases and the boundary layer thickness decreases. Similarly for velocity profile g(η) hen e increase the rotation parameter λ the velocity profile g (η) decreases. The results obtained in our investigation are ell in agreement ith Butt et al. 34 at a particular condition. In these figures for larger values of λ the Fig. 4 Velocity profile of f '(η) for variation in M hen β =. Fig. 7 Velocity profile of f '(η) for variation in β hen M =. Fig. 5 Velocity profile of g (η) for variation in M hen β =. Fig. 8 Velocity profile of g(η) for variation in β hen M =. Fig. 6 Temperature profile for variation in M hen β =. Fig. 9 Temperature profile for variation in β hen M =.

7 JAIN & CHOUDHARY: MHD FLOW AND HEAT TRANSFER OF A ROTATING CASSON FLUID 59 velocity profile oscillates increasingly, ith no oscillation for small values of λ. On the other hand Fig. 1 illustrates that ith an increase in the rotation parameter λ the temperature profile θ(η)increases. Hence hen e increase the rotation parameter λ, the fluid temperature rises. Figure 13 exhibits the effect of Prandtl number on temperature profileθ(η). As value of Prandtl number Pr increases the temperature profile decreases. Figure 14 depicts the effects of magnetic field parameter M ON Ns, local entropy generation. Figure 14 shos that the local entropy generation Ns increases ith increase in magnetic field parameter M. This is due to an increase in the Lorentz force. An increment in M causes enhancement of the Lorentz force and results in more friction hich causes an increase in the entropy generation. Figures sho variation of rotation parameter λ and group parameter BrΩ -1 on Ns and Be. It is observed that increment in both the parameters, increases the entropy generation number hereas the Bejan number decreases. Fig. 1 Temperature profile for variation in λ hen β =. Fig. 10 Velocity profile of f '(η) for variation in λ hen β =. Fig. 13 Temperature profile for variation in Pr hen β =. Fig. 11 Velocity profile of g(η) for variation in λ hen β =. Fig. 14 Entropy effects of Ns for variation in M hen β =.

8 60 INDIAN J PURE & APPL PHYS, VOL. 56, JANUARY 018 It is observed that the magnetic field effect and fluid friction are prominent near the stretching surface. In the region aay from stretching surface, entropy due to heat surface is prominent. Table 4 shos variation of magnetic field parameter M, rotation parameter λ and Casson fluid parameter β on the skin friction coefficient Re 1/ x C fx along the x 1/ direction and Rex C along the y direction. From Table fy 4 it is noted that ith an increase in magnetic field parameter M, the coefficient of skin friction along the x direction decreases hereas along the y direction it increases. Hoever ith an increment in the Casson fluid parameter β the skin friction coefficient increases. 1/ For an increment in the rotation parameter λ, Re x Cfx decreases. Table 5 shos variation of Nusselt number for different values of parameters. Nusselt number increases for increasing values of Prandtl number Pr but decreases as e increase the Casson fluid parameter β, Eckert number Ec, magnetic field parameter M and rotation parameter λ. Fig. 15 Entropy effects of Ns for variation in λ hen β =. Fig. 18 Entropy effects of Be for variation in BrΩ -1. Fig. 16 Entropy effects of Ns for variation in BrΩ -1. 1/ Table 4 Numerical values of Re x C fx and Re 1/ x C fy for different values of rotation parameter λ, magnetic field parameter M and Casson fluid parameter β. M β λ Re 1/ x C fx Re 1/ x C fy Fig. 17 Entropy effects of Be for variation in λ hen β =

9 JAIN & CHOUDHARY: MHD FLOW AND HEAT TRANSFER OF A ROTATING CASSON FLUID 61 1/ Table 5 Numerical values of Rex Nux for different values of rotation parameter λ, magnetic field parameter M, Casson fluid parameter β, Eckert number Ec and Prandtl number Pr. Ec M Pr β λ 1/ Rex Nux Conclusions The current study focused on investigation of entropy generation for three dimensional MHD boundary layer flo and heat transfer due to a stretching surface in a rotating Casson fluid. Some important results are as follos: The parallel velocity profiles f '(η) and g (η) in the x and y directions decrease as the Casson fluid parameter β and magnetic parameter M increase. The velocity profiles f '(η) and g(η) decrease as the rotation parameter λ increases. An increment in the Casson fluid parameter β, magnetic field parameter M and rotation parameter λ increases the thermal boundary layer thickness. On the other hand an increment in Prandtl number Pr, decreases the noted in temperature profile. The local entropy generation number Ns increases ith increase in magnetic field parameter M, rotation parameter λ and group parameter BrΩ -1.On the other hand Bejan number Be decreases as the rotation parameter λ and group parameter BrΩ -1 increase. For an increase in magnetic field parameter M, the coefficient of skin friction along the x direction decreases hereas along they direction it increases. 1/ 1/ The skin friction coefficients Rex C and fx Rex C in fy the x and y directions increase ith the Casson fluid parameter β hereas Re x C 1/ 1/ Rex C decreases and fx fy increases ith magnetic field parameter M. Net heat transfer through the flo decreases ith the increasing value of M. Acknoledgement The support provided by Manipal University Jaipur through Endoment Fund Felloship to one of the author RC is gratefully acknoledged. References 1 Crane L J, Zeit Ange Math und Phys, 1 (1970) 645. Sakiadis B C, AIChe J, 7 (1961) 6. 3 Wang C Y, Phys Fluids, 7 (1984) Verma P D & Chauhan D S, Indian J Pure Appl Math, 10 (1979) Jain S, J Rajasthan Acad Phys Sci, 5 (006) Fang T, Zhang J & Yao S, Commun Nonlinear Sci Numer Simul, 14 (009) Najar R, Amin N & Pop I, Mech Res Commun, 31 (004) Mahanta G & Sha S, Alexandria Eng J, 54 (015) Jain S & Choudhary R, Proc Eng, 17 (015) Raju C S K, Naramgari S, Babu M J & Sugunamma V, Alexandria Eng J, 55 (016) Jain S & Bohra S, Indian J Pure Appl Phys, 55 (017) Wang C Y, Z Ange Math Phys, 39 (1988) Takhar H S, Chamkha A J & Nath G, Int J Therm Sci, 4 (003) Zaimi K, Ishak A & Pop I, Appl Math Mech Engl Ed, 34 (013) Mustafa M, Int J Heat Mass Trans, 108 (017) Hayat T, Muhammad T, Shehzad S A & Alsaedi A, Comput Meth Appl Mech Eng, 315 (017) Bejan A, CRC Press, Boca Raton, (1996). 18 Makinde O D & Osalusi E, Entropy, 7 (005) Butt A S & Ali A, J Braz Soc Mech Sci Eng, 37 (015) Freidoonimehr N, Rashidi M M, Abelman S & Lorenzini G, Entropy, 18 (016) Qing J, Bhatti M M, Abbas M A, Rashidi M M & Ali M El- S, Entropy, 18 (016) 13. Casson N A, Rheology of disperse systems, (Pergamon Press: Oxford), Boyd J, Buick J M & Green S, Phys Fluids, 19 (007) Bhargava R, Takhar H S, Raat S, Beg T A & Beg O A, Nonlinear Anal Model, 1 (007) Attia H A & Ahmed M E S, Italian J Pure Appl Math, 7 (010) Mukhopadhyay S, Chin Phys B, 7 (013) Nadeem S, Haq R U & Lee C, Scientia Iranica B, 19 (01) Bhattacharyya K, T Hayat & Alsaedi A, Chin Phys B, (013) Mukhopadhyay S, Mondal I C & Chamkha A J, Heat Trans Asian Res, 4 (013) Nadeem S, Haq R U, Akbar N S & Khan Z H, Alexandria Eng J, 5 (013) 577.

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