Magneto-Nanofluid Dynamics in Convergent ...

Defect and Diffusion Forum ISSN: 1662-9507, Vol. 377, pp 95-110 doi:10.4028/www.scientific.net/DDF.377.95 © 2017 Trans Tech Publications, Switzerland

Submitted: 2017-07-04 Revised: 2017-07-26 Accepted: 2017-07-27 Online: 2017-09-06

Magneto-Nanofluid Dynamics in Convergent-Divergent Channel and its Inherent Irreversibility Md. S. Alam1,a*, M.A.H. Khan2,b, O.D. Makinde3,c 1

Department of Mathematics, Jagannath University, Dhaka-1100, Bangladesh

2

Department of Mathematics, Bangladesh University of Engineering and Technology, Dhaka-1000, Bangladesh 3

Faculty of Military Science, Stellenbosch University, South Africa

a

[email protected], [email protected], [email protected]

Keywords: Convergent-divergent channel, Magnetic field, Nanofluid, Irreversibility analysis, Hermite- Padé approximation.

Abstract. The effects of Cu-nanoparticles on the entropy generation of steady magnetohydrodynamic incompressible flow with viscous dissipation and Joule heating through convergent-divergent channel are analysed in this paper. The basic nonlinear partial differential equations are transformed into a system of coupled ordinary differential equations using suitable transformations which are then solved using power series with Hermite- Padé approximation technique. The velocity profiles, temperature distributions, entropy generation rates, Bejan number as well as the rate of heat transfer at the wall are presented in convergent-divergent channels for various values of nanoparticles solid volume fraction, Eckert number, Reynolds number and channel angle. A stability analysis has been performed for the shear stress which signifies that the lower solution branch is stable and physically realizable, whereas the upper solution branch is unstable. It is interesting to remark that the entropy generation of the system increases at the two walls as well as the heat transfer irreversibility is dominant there whereas the fluid friction irreversibility is dominant along the centreline of the channel. 1. Introduction The two dimensional flow of a viscous, incompressible fluid between converging-diverging channels separated by a fixed angle and driven by a source or sink at the apex is known as the classical Jeffery-Hamel flow which was studied first by Jeffery [1] and Hamel [2]. It has important applications particularly in fluid mechanics, chemical, mechanical and bio-mechanical engineering. Many available studies [3-6] have considered different properties related to this problem and have tried to investigate the flow characteristics by varying the angle between the walls. These flows have discovered similarity solutions of the Navier-Stokes equations depending on two nondimensional parameters, the flow Reynolds number and channel angular width. Fraenkel [7] then investigated the laminar flow in symmetrical channels with slightly curved walls. In his analysis the velocity field of the flow was obtained as a power series in small curvature parameter where the leading term is the Jeffery-Hamel solution. Sobey and Drazin [8] studied some instabilities and bifurcations of two-dimensional Jeffery-Hamel flows using analytical, numerical and experimental methods. Moreover, the steady flow of a viscous incompressible fluid in a slightly asymmetrical channel was studied by Makinde [9]. He expanded the solution into a Taylor series with respect to the Reynolds number and performed a bifurcation study. Meanwhile, the study of electrically conducting viscous fluid flowing through convergent-divergent channels under the influence of an external magnetic field is not only fascinating theoretically, but also finds applications in mathematical modelling of several industrial and biological systems. Clearly, the motion in the region with intersecting walls may represent a local transition between two parallel channels with different cross-sections, an expansion or a contraction of the flow. Makinde and Mhone [10] investigated the MHD flows in convergent- divergent channels which was an extension of the classical Jeffery-Hamel flows to MHD. He interpreted that the effect of All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of Trans Tech Publications, www.ttp.net. (#100470121-06/09/17,13:22:34)

96

New Development for Heat Transfer in Solids and Fluid Flow

external magnetic field works as a parameter in solution of the MHD flows in convergent divergent channels. Makinde and Mhone [11] investigated the temporal development of small disturbances in MHD Jeffery-Hamel flows to understand the stability of hydromagnetic steady flows in convergent-divergent channels at very small magnetic Reynolds number Re using Chebyshev spectral collocation method. However, a numerical investigation of the effect of arbitrary magnetic Reynolds number on steady flow of an incompressible conducting viscous liquid in convergent-divergent channels under MHD was presented by Makinde [12]. He solved the nonlinear 2D Navier-Stokes equations modeling the flow field using a perturbation technique applying the special type of Hermite-Pade´ approximation method and a bifurcation study was also performed. The increasing values of magnetic Reynolds number cause a general decrease in the fluid velocity around the central region of the channel whereas the flow reversal control is also observed by increasing magnetic field intensity. Presently there is an increasing attention of the researchers in the analysis of nanofluids. The word nanofluid was introduced by Choi [13]. In fact, a nanofluid is a dilute suspension of solid nanoparticles with the average size below 100 nm in a base fluid, such as: water, oil and ethylene glycol. These nanoparticles are good conductors of heat and enable the base fluids to enhance their thermal properties. The effects of magnetic field and nanoparticles on the Jeffery-Hamel flow using a powerful analytical method called the Adomian decomposition method were studied by Sheikholeslami et al. [14]. Moreover, the effects of heat transfer and viscous dissipation on the Jeffery-Hamel flow of nanofluids were investigated by Moradi et al. [15]. Seth and Singh [16] studied the combined free and forced convection MHD flow in a rotating channel with perfectly conducting wall. Finally, a study of velocity and temperature slip effects on flow of water based nanofluids in converging and diverging channels was done by Syed et al. [17]. For any thermal system, as the entropy generation increases, the energy decreases. Thus, to enhance the efficiency of the system, the rate of entropy generation must be effectively controlled. The idea of thermodynamic irreversibility is central to the understanding of entropy. Everyone has an intuitive knowledge of irreversibility. The second law of thermodynamics states that all real processes are irreversible. Entropy generation provides a measure of the amount of irreversibility associated with real process. Bejan [18] studied the entropy-generation for forced convective heat transfer due to temperature gradient and viscosity effects in a fluid. Bejan [19] also presented various reasons for entropy-generation in applied thermal engineering where the generation of entropy destroys the available work of a system. The purpose of this study is to investigate the entropy generation analysis on magnetohydrodynamic nanofluid flow through convergent - divergent channel considering viscous dissipation and Joule heating effect. The resulting problem is solved applying the power series along with Hermite–Padé approximation (HPA). The effects of various thermophysical parameters namely nanoparticles solid volume fraction  , Eckert number Ec, Hartmann number Ha, Reynolds number and channel angle on fluid velocity, temperature distributions, entropy generation rates with Bejan profiles, shear stress and Nusselt number are discussed graphically in details. A comparative study between the previously published results and the present results in a limiting sense reveals excellent agreement between them. 2. Derivation of Mathematical Equation Consider a steady two-dimensional laminar incompressible viscous Cu-water nanofluid flow from a source or sink between two channel walls that intersect at an angle 2  shown in Fig. 1. A cylindrical coordinate system (r ,  , z ) is used and assumed that the velocity is purely radial and depends on r and  so that there is no change in the flow parameter along the z-direction. Further it is presumed that there is an external magnetic field acting vertically downward to the top wall. Let  be the semi-angle and the domain of the flow be       . The continuity equation, the Navier-Stokes equations and energy equation considering viscous dissipation and Joule heating in reduced polar coordinates are

Defect and Diffusion Forum Vol. 377

nf 

(ru(r , ))  0,

r r u (r ,  )

97

(1)

  2u (r ,  ) 1 u (r ,  ) 1  2u (r ,  ) u (r ,  )  u (r ,  ) 1 p     nf    2  2 2 2  r  nf r r  r  r r   r   

 nf B0 2  nf r 2

u (r ,  ),

p 2 nf u (r ,  )  2  0,  nf r   r 1

u (r ,  )

 nf T (r ,  )  r  cp





nf

 nf

(2)

(3)

  2 T (r ,  ) 1 T (r ,  ) 1  2 T (r ,  )      2 2  r 2  r  r r    

 c p nf

  u (r ,  )  2 1  u (r ,  )  2   nf B0 2  4   u (r ,  ) 2     2  2   r  r      c p r nf  





(4)

The flow is assumed to be symmetrically radial, so that v  0 . Then the volumetric flow rate through the channel is

Q





ur d

(5)

The boundary conditions are u(r,  )  0, T  Ta at   

(6)

Where B0 is the electromagnetic induction, u is the velocity along radial direction and p is the fluid pressure. The effective density  nf , the effective dynamic viscosity  nf ,the electrical conductivity

 nf and the kinematic viscosity  nf of the nanofluid are given as Das et al. [20].

 nf   f (1   )   s , nf 

f (1   )

,  nf  2.5

nf ,  nf

          nf  1  3 s  1 /  s  2    s  1     f   f   f    f      

(7)

The corresponding effective thermal conductivity and heat capacity of nanofluid are

k nf  k f

k s  2k f  2 (k f  k s ) k s  2k f   ( k f  k s )

 nf

, c p

  f   c p s

 (1   ) c p

(8)

Here,  is the solid volume fraction of the nanoparticles. The thermophysical properties of the nanofluid are given in Table 1.

98


B0

T  Ta , u  0

 

Source or Sink



U max

r

T  Ta , u  0

  

Fig. 1 Geometry of the problem. Table 1 Thermo physical properties of water and Cu-nanoparticles Das et al. [20]. Physical properties Water Cu (Copper) 3 997.1 8933  (kg m ) 4179 385 c p ( J kg K )

 (W m K ) 

0.613

401

5.5 106

59.6 106

If it requires Q  0, then for   0 the flow is diverging from a source at r  0 . Let    (r,  ) be the stream function, then

   ur , 0  r Introducing the dimensionless variables



 ( ) T  , F ( )  ,  ( )  Ta Q 

(9)

The governing Eqs. (2)-(4) reduce to the following ordinary differential equations F (iv )  2 Re A1 (1   ) 2.5 F F   (4  (1   ) 2.5 A4 Ha2 ) 2 F   0, A Ec Pr 2    2 4 2 F 2  F   (1   ) 2.5 A4 Ha2 2 F 2  0, 2.5 A3 (1   )



(10)



(11)

The boundary conditions are reduced as follows:

F  1, F   0,   1 at   1 Where Re 

Q

f

is Reynolds number, Pr 

(12)

( c p ) f

f

is Prandtl number, Ec 

U max2 (c p ) f Ta

 f B0 2 number, Ha  is square of Hartmann number and channel angle  . Also  f f k s  2k f  2 (k f  k s ) ( c p )s  , A1  (1   )  s  , A2  (1   )   , A3  k s  2k f   ( k f  k s ) f ( c p ) f 2

         A4  1  3 s  1 /  s  2    s  1  are the constants.    f   f      f      

is the Eckert


99

3. Series Analysis The following power series expansions are considered in terms of the parameter  as Eqs. (10) (11) are non-linear for stream function and temperature profile 



i 0

i 0

F ( )   Fi ( ) i ,  ( )   i ( ) i , as   1

(13)

The non-dimensional governing equations are then solved into series solution by substituting the Eq. (13) into Eq. (10) and Eq. (11) with boundary conditions Eq.(12) and equating the coefficients of powers of  . Order zero ( 0 ) : F

(iv )

0

 0,

 0 

A2 Ec Pr F02  0, 2.5 A3 (1   )

(14)

 F0 (1)  1, F0 (1)  0, 0 (1)  1

(15)

Order one ( 1) : (iv )

F1

   2 Re A1 (1   ) 2.5 F0 F0  0,

1 

A2 Ec Pr 2F0F1  0, A3 (1   ) 2.5

 F1 (1)  0, F1 (1)  0, 1 (1)  0

(16) (17)

With the help of algebraic programming language MAPLE, we have computed the first 18 coefficients for the series of the stream function F ( ) and temperature  ( ) in terms of  , Ha , Re, Pr, Ec,  , A1 , A2 , A3 , A4 . The first few coefficients of the series for F ( ) and  ( ) are as follows: 3 1 3 1 52 F ( ; ,Re, Ha, , A1, A4 )     3  Re A1 (1   )  ( 2  5)(  1) 2 (  1) 2    (  1) 2 2 2 280 431200 (  1)2 (43120  10780  9590 4 Re 2 A12 2  9590 4 Re 2 A12 3  4795 4 Re 2 A12 4  959 4 Re 2 A12 5 4795 4 Re2 A12  24720 2 Re 2 A12 2  24720 2 Re 2 A12 3  12360 2 Re2 A12 4  2472 2 Re2 A12 5 12360 2 Re2 A12  980 6 Re2 A12 2  980 6 Re2 A12 3  490 6 Re2 A12 4  98 6 Re2 A12 5 490 6 Re2 A12  2875Re2 A12  10780 Ha 2 1   A4  28750Re 2 A12 2  28750Re 2 A12 3  98 6 Re 2 A12 2472 2 Re2 A12  959 4 Re2 A12  10780 Ha 2 1   A4 2  21560 Ha 2 1   A4  2875Re 2 A12 5 14375Re2 A12  14375Re 2 A12 4 ) 2  O( 3 )

 ( ; , Re, Ha,  , Ec, Pr, A1 , A2 , A3 , A4 )  1 

(18)

3 A2 Ec Pr (1   4 )  A3 (1   ) 2.5

 3 A2 Ec Pr Re A1   (9 4  38 2  19)(  1) 2 (  1) 2   O( 2 ) 560 A3  

(19)

The local Nusselt number Nu at the wall and heat transfer rate are

Nu 

rqw  

 f Ta

 1 T   , qw   nf    r  

(20)

Substitution of Eq. (9) into Eq. (20) gives

Nu  

1  nf

 f

 (1)

(21)

100


Expressions of skin friction coefficient and shear stress are respectively cf 

w

 f U max 

2

 1 u (r ,  )  ,  w  nf    r  

(22)

The Eq. (22) in view of Eq. (9) reduces to non-dimensional form as follows cf 

1 F (1) Re(1   ) 2.5

(23)

Applying differential and algebraic approximant methods to the series above, we establish the comparison between the present and previous published results and the effects of magnetic field and nanofluid on velocity field, temperature profile, entropy generation rate, skin friction coefficient and Nusselt number graphically. 4. Numerical Procedure: Hermite-Padé Approximants. To compute the criticality conditions and irreversibility in the system, we shall employ a very efficient solution method, known as Hermite-Padé approximants, which was first introduced by Padé [21] and Hermite [22]. Consider the partial sum N 1

S N 1 ( )   an n

as   1

(24)

n 0

The dominating behavior of the function S   represented by the series (24) may be written as

    B  A1    c S   ~      B  A1   c 

  



when   0, 1, 2, ..., (25)



   ln 1  c 

when   0, 1, 2,...,

as    c , where A and B are some constants and  c is the critical point with the critical exponent  . If  is a negative integer then the singularity is a pole; otherwise it represents a branch point singularity. Assume that the d  1 tuple of polynomials, where d is a positive integer: PN0 , PN1 , ..., PNd 

where,

deg PN0  deg PN1  ...  deg PNd   d  N ,

(26)

is a Hermite-Padé form of these series if d

 PNi   Si    O N  as

i 0

 1

(27)

Here S0 ( ), S1 ( ), ..., S d ( ) may be independent series or different form of a unique series. We

need to find the polynomials PNi  that satisfy the Eq. (26) and Eq. (27). These polynomials are completely determined by their coefficients. So, the total number of unknowns in Eq. (27) is d

 deg PNi   d  1  N  1

i0

(28)


101

Expanding the left hand side of Eq. (27) in powers of  and equating the first N equations of the system equal to zero, we get a system of linear homogeneous equations. To calculate the coefficients of the Hermite-Padé polynomials it requires some sort of normalization, such as

PNi  0  1 for some integer 0  i  d

(29)

It is important to emphasize that the only input required for the calculation of the Hermite-Padé polynomials are the first N coefficients of the series S0 ( ), S1 ( ), ..., S d ( ) . The Eq. (28) simply ensures that the coefficient matrix associated with the system is square. One way to construct the Hermite-Padé polynomials is to solve the system of linear equations by any standard method such as Gaussian elimination or Gauss-Jordan elimination. In practice, one usually finds that the dominant singularities as well as the possibility of multiple solution branches for the nonlinear problem are located at zeroes of the leading polynomial coefficients PN[ d ] ( ) of the Eq. (27). If the singularity is of algebraic type, then the exponent  may be approximated [23] by

N  d  2 

PNd 1  c, N 

(30)

DPNd   c, N 

Drazin and Tourigney [23] Approximants is a particular kind of algebraic approximants and Khan [24] introduced High-order differential approximant (HODA) as a special type of differential approximants. High-order partial differential approximants (HPDA) discussed in Rahman [25] is a partial differential approximant. More information about the above mentioned approximants can be found in the respective references. 5. Irreversibility of the System In the modern age, one of the major concerns of scientists and engineers is to find the methods which can control the wastage of useful energy in the system. The properties of flow in a convergent-divergent channel with isothermal walls in the presence of magnetic field and viscous dissipation are irreversible. The inequilibrium conditions arise due to the exchange of energy and momentum within the fluid and at solid boundaries which produces continuous entropy generation. Following Woods [26], the volumetric entropy generation rate for fully developed flow in cylindrical coordinates is given as 2

 nf  d T  nf   EG  2  Ta Ta  d 

2

2

 du   nf B0 2 u    Ta  d 

(31)

Where the first term on the right side of Eq. (31) is the irreversibility due to heat transfer, the second term is the irreversibility due to viscous dissipation and the third term is the local entropy generation due to the effect of the magnetic field (Joule heating). The entropy generation number can be expressed in dimensionless form as,

 2 EG  nf NS   f f

2

2

2

 d  A2 Pr Ec  d 2 F   dF   2   A2 A4 Pr Ec  2 Ha2       N1  N 2  N 3 2.5   d  (1   )  d   d  2

2

 dF   d  A2 Pr Ec  d 2 F   2  , N3  A2 A4 Pr Ec  2 Ha2    , N 2  Where N1  A3  2.5   (1   )  d   d   d 

(32)

2

In order to obtain an idea of whether the entropy generation due to the heat transfer dominates over the entropy generation due to the fluid friction and the magnetic field, the Bejan number Be is defined to be the ratio of entropy generation due to the heat transfer to the entropy generation number as Bejan [18]

102


N1 N1 1   N S N1  N 2  N3 1  L N  N3 Where L  2 is called the irreversibility ratio. N1 It is notable that the Bejan number ranges from 0 to 1 and Be  0 is the limit where the irreversibility is dominated by fluid friction effects. The irreversibility due to heat transfer dominates the flow system at the limit Be  1 . The contributions of heat transfer and fluid friction to entropy generation are equal when Be  1 . Moreover, the behavior of the Bejan number Be is 2 studied for the optimum values of the parameters at which the entropy generation takes its minimum. Be 

6. Results and Discussion We have investigated the effects of five parameters of interest in the present problem, which are, nanoparticles volume fraction  , channel angle  , Eckert number Ec, Reynolds number Re and Hartman number Ha for different values on fluid velocity, temperature and entropy generation of the system. The value of the Prandtl number for the base fluid is kept at Pr  7.1 (room temperature) and the effect of solid volume fraction of Cu-nanoparticles is investigated in the range of 0    0.2 with Ha  1 . 6.1 Stability Analysis As mentioned earlier, we have computed for investigation the centre line axial velocity and radial velocity as two series in powers of  , Re,  , A1 , A4 and Ha by differentiating series (18) at   0 and for all  respectively in the following functional form.

F (  0;  , Re, Ha,  , A1, A4 )

(33)

F (; , Re, Ha,  , A1, A4 )

(34)

The series (33) is analyzed by High-order differential approximant method [24] to show the comparison between present results and the results of Fraenkel [7] and Makinde [12] in Tables 2-3 and the variation in the critical values  c and Re c with critical exponent  for various values of nanoparticles solid volume fraction significantly. Table 2 exhibits the decreases of critical channel semi-angle  c for four different increasing values of   0, 0.05, 0.1, 0.2 as Cu-nanoparticles with water is the base fluid by considering d  4 and N  18 . The values of  confirm that  c is a branch point using HODA. Moreover, Table 3 implies that Re c decreases as significantly and uniformly for different increasing values of  and Re c is a branch point verified by the values of  . The results of Tables 2-3 show a good agreement with those results of Fraenkel [7] and Makinde [12] for   0 . Also the obtained results indicate that the presence of nanofluids forms early development in the instability of the flow process. Table 2 Numerical values of critical angles  c and corresponding exponent  at Re  20 and Ha  1for various values of  .

 0 0.05 0.1 0.2

Present study

c 0.2691819115000 0.2122678984825 0.1963360739593 0.1828175409234

 0.49515872313 0.49785814583 0.50387522948 0.49803815751

Fraenkel [7]

Makinde [12]

c

c

0.269 _ _ _

0.269162 _ _ _


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Table 3 Numerical values of critical Re c and corresponding exponent  at   0.1 and Ha  1for various values of  .

 0 0.05 0.1 0.2

Present study Re c 54.47285679258 44.31499529952 39.49239826450 36.73956809792

 0.5071160433 0.4985570114 0.4996886803 0.4990504907

Fraenkel [7] Re c 54.61 _ _ _

Makinde [12] Re c 54.4717 _ _ _

Employing the algebraic approximation method to the series (33) we have obtained the bifurcation graphs of skin friction coefficient against  and Re. Fig. 2(a) shows the bifurcation diagram of skin friction versus  with the effect of Cu-water nanofluid. We say that there is a simple turning point, fold or a saddle-node bifurcation at    c . It is interesting to notice that there are two solution branches of skin friction when    c , one marginal solution when    c , and no solution when

   c , where  c is the critical value of  for which the solution exists. The stability analysis indicates that the lower solution branch (I) is stable and physically realizable. For different values of  , the upper solution branch (II) is unstable and physically unacceptable shown in Fig. 2(a). It can be also noted here that the bifurcation points change for different values of nanoparticles volume fraction respectively at Re  20, Ha  1 . Moreover, from Fig. 2 (b) it is observed that the coefficient of skin friction also bifurcates in a similar way at Re  Rec and the bifurcation point decreases uniformly for four different values of nanoparticles volume fraction at   0.1, Ha  1 . Moreover, an increase in the skin friction coefficient is observed when solid volume fraction increases.

(a) (b) Fig. 2 Approximate bifurcation diagrams of skin friction coefficient against (a) divergent angle  at Re  20, Ha  1 and (b) flow Reynolds number Re at   0.1, Ha  1 with different values of  for Cu-water nanofluid. 6.2 Effect of Channel Angle Fig. 3(a) shows the effect of channel angle on the velocity profiles in divergent channel for nanofluid. In Fig. 3(a), the presence of Cu-nanoparticles   0.05 accelerates the increment of centerline velocities more rapidly, while there occurs major backflow near the walls at large value of    4 . The flow breaks the symmetry, with most of the fluid going in a thin layer along the walls as channel angle increases. The fluid is prevented from utilizing the whole area of the expanding channel by a recirculation vortex which blocks the exit. In addition, secondary instabilities driven by this vertical motion develop in this flow. The effect of increasing  on the

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temperature of the fluid is seen in the Fig. 3(b) in a way that the temperature raises massively at the centerline of the channel due to the faster flow rate. The variation of Nusselt number with channel angle  for different solid volume fraction is shown in the Fig. 3(c). As expected that in divergent channel, the Nusselt number increases when  increases and the rate of increase also accelerates with the rising values solid volume fraction. As mentioned in many related textbooks and papers, nanofluid enhances the rate of heat transfer properties within the flow region. The entropy generation rate N s by the effect of  is shown in Fig. 4 (a) such that the irreversibility of the system is absolutely zero at the center of the channel whether symmetrically increases towards the two fixed hot walls. The maximum velocity at the center leads to approximately zero velocity gradient, also the temperature distribution observed in Fig. 3 (b) is very high at the center which turns the temperature gradient to zero. As a result, the contribution of fluid friction N 2 , magnetic field N 3 and heat transfer N1 to the total entropy N s become zero at the center of the channel. Moreover, the increasing values of  accelerates N s near the walls because of the formation of back flow over there. For the same reason it is noticed from Fig. 4 (b) in the Bejan profile that fluid friction irreversibility is dominant at the center and heat transfer irreversibility is dominant at the two hot walls. Increases of  has a significant influence on the dominance effect of heat transfer irreversibility at the walls.

(a)

(b)

(c)

Fig. 3 (a) Velocity profiles, (b) temperature distributions and (c) Nusselt number in divergent channel with different values of  and  respectively at Ec  0.1, Pr  7.1, Re  7, and Ha  1 for Cu-water nanofluid.


105

(a) (b)

Fig. 4 Effect of channel angle on (a) Entropy rates and (b) Bejan profiles respectively at Ec  0.1, Pr  7.1, Re  7, and Ha  1 . 6.3 Effect of Reynolds Number The effect of Reynolds number on temperature distribution is noticed in Fig. 5 (a) that the temperature at the center of the channel increases rapidly with increases of Re due to the faster flow rate at that region. The effect of flow Reynolds number Re on Nusselt number in divergent channel is demonstrated in Fig. 5(b) in such a way that as Re increases, the rate of heat transfer increases rapidly due to the faster flow rate. It is seen from Fig. 6 (a) for divergent channel that the entropy generation rate goes faster at the two walls for the rising values of Re, which is consistent with the results that flow reversal produces at that region. A symmetrical behaviour of Bejan profile for Reynolds number is observed in Fig. 6 (b) in comparison of channel angle that the dominance of fluid friction irreversibility occurs at the centerline whereas the heat transfer irreversibility is dominant at the two hot walls.

(a)

(b)

Fig. 5 Effect of Reynolds number on (a) temperature distributions and (b) Nusselt number respectively at Ec  0.1, Pr  7.1,   10 , and Ha  1 .

106


(a) (b)

Fig. 6 Effect of Reynolds number on (a) Entropy rates and (b) Bejan profiles respectively at Ec  0.1, Pr  7.1,   10 , and Ha  1 . 6.4 Effect of Eckert number The effect of Eckert number on temperature field within the channel for convergent and divergent is discussed in Figs. 7 (a, b). Eckert number describes the effects due to the dissipation term in energy equation. Since Eckert number is the ratio of the square of maximum velocity and specific heat. As a result, when the value of Eckert number increases, the fluid flow rate along the centerline becomes faster. It is noticed from Figs. 7 (a, b) that the fluid temperature increases successively with increasing values of viscous heating parameter Ec for both convergent and divergent channels. Furthermore, due to the higher thermal conductivity coefficient of the nanofluid, the heat is more intensely transferred. The consequences of Eckert number on the irreversibility of the system are analysed in Figs. 8 (a, b). The entropy generation rate N s increases promptly and symmetrically near the two hot walls with the growing values of Ec due to viscous dissipation effect as seen in Fig. 8 (a). Moreover, the positive variation of Eckert number has a key impact on the dominant effect of heat transfer irreversibility at the two heated walls shown in Fig. 8(b).

(a)

(b)

Fig. 7 Temperature profiles for different values of Eckert number Ec (a) in divergent channel and (b) in convergent channel at Pr  7.1, Re  7 and Ha  1 .


(a)

107

(b)

Fig. 8 Effect of Eckert number Ec on (a) Entropy rates and (b) Bejan profiles respectively at   10 , Pr  7.1, and Ha  1 . 6.4 Effect of Nanoparticles volume fraction Figs. 9 (a, b) explain the variations of dimensionless temperature with increasing nanoparticle volume fraction  for both divergent and convergent channel respectively. A definite and uniform rise in temperature is seen in Fig. 9(a) for divergent channel because of an increase in the gap between the walls. This explains also the influences of higher thermal conductivity and specific heat of nanofluid. Moreover, copper nanoparticles are much more efficient to control the rise in temperature for various practical situations as compared to pure base fluid. The temperature field in convergent channel presented by Fig. 9 (b) is decreasing uniformly by the positive change of  . Since the gap between the walls reduces rapidly towards the sink at r  0 , which produces reduction of temperature at the centerline region.

(a)

(b)

Fig. 9 Temperature profiles for different values of  (a) in divergent channel and (b) in convergent channel at Pr  7.1, Ec  0.1, Re  7 and Ha  1 .

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(a)

(b)

Fig. 10 Effect of nanoparticles volume fraction on (a) Entropy rates and (b) Bejan profiles respectively at Ec  0.1, Pr  7.1, and Ha  1 . Figs. 10 (a, b) depict the entropy generation with Bejan number with the control of nanoparticles volume fraction correspondingly. The properties of total entropy rate N s in Fig. 10 (a) is almost similar as in Fig. 4(a) and Fig. 6(a), whereas the effect of  on N s is significant at the two hot walls in the divergent channel. It can be noted from Fig. 10(b) that the dominant effect of heat transfer irreversibility enhances due to the escalating values of  . As it is already discussed previously that the higher thermal conductivity and specific heat of nanofluid generate more heat transfer. 7. Conclusion The entropy generation in a magnetohydrodynamic flow of a viscous incompressible electrically conducting nanofluid through convergent-divergent channel with convectively heating is investigated using a special type of Hermite-Padé approximation technique. A comparison is made between the available results and the present approximate solutions. The influences of various physical parameters on the velocity field, temperature distribution and entropy generation rate with Bejan profile are discussed in detail. Based on the obtained graphical and tabular results, the following conclusions can be drawn:  The solution of shear stress at the wall has two branches bifurcating at the critical channel angle and critical Reynolds number namely an upper branch and a lower branch. It is found that at the lower solution branch which is physically acceptable, the value of skin friction coefficient enhances with the increase in the nanoparticles volume fraction.  Increasing channel semi angle and Reynolds number leads to enrichment of fluid centerline velocity and flow reversal near the walls in the divergent channel.  Behaviour of the flow for changing of the physical parameters in convergent channel is quite opposite to the one seen in divergent channel.  The increasing values of the pertinent physical parameters namely channel angle, flow Reynolds number and Eckert number enhances temperature field along the channel centerline region. Also temperature increases uniformly as nanoparticles volume fraction increases.  The heat transfer rate at the wall against channel angle and flow Reynolds number enlarges with the increase in the nanoparticles volume fraction.  The channel heated walls act as strong source of entropy and heat transfer irreversibility. Fluid friction irreversibility dominates the entropy generation absolutely near the channel centerline region.  Nanofluid enhances the dominant effect of heat transfer irreversibility in the entropy generation of the system.  The optimum design and efficient performance of a thermal system can be improved by choosing the appropriate values of the physical parameters.


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