Heat irreversibiility analysis for a couple stress fluid flow in an inclined channel with isothermal boundaries

Journal of the Taiwan Institute of Chemical Engineers 101 (2019) 251–258

Contents lists available at ScienceDirect

Journal of the Taiwan Institute of Chemical Engineers journal homepage: www.elsevier.com/locate/jtice

Heat irreversibiility analysis for a couple stress fluid flow in an inclined channel with isothermal boundaries Samuel O. Adesanya a,b, Basma Souayeh c, Mohammad Rahimi-Gorji d,e, M.N. Khan f,∗, O.G. Adeyemi g a

Environmental Hydrodynamics Research, Department of mathematical Sciences, Redeemer’s University, Ede, Nigeria Department of Mathematics, Vaal University of Technology, Vanderbijlpark, South Africa Physics department, college of Science, King Faisal University, PO Box 380, Alahsa 31982, Saudi Arabia d Experimental Surgery Lab, Faculty of Medicine and Health Science, Ghent University, Ghent, Belgium e Biofluid, Tissue and Solid Mechanics for Medical Applications Lab (IBiTech- bioMMeda), Ghent University, Ghent, Belgium f Department of Mechanical and Industrial Engineering, College of Engineering, Majmaah University, Al-Majmaah 11952, Kingdom of Saudi Arabia g Department of Chemical Sciences, Redeemer’s University, Ede, Nigeria b c

a r t i c l e

i n f o

Article history: Received 11 February 2019 Revised 3 April 2019 Accepted 26 April 2019 Available online 17 May 2019 Keywords: Isothermal heating Couple stresses Second law Inclined channel

a b s t r a c t Entropy generation in a fully developed couple stress fluid flow through an inclined channel is the focus of the present article. The steady flow of the non-Newtonian fluid via the inclined channel is assumed to be heated isothermally at the boundaries. The formulation of both the fluid flow and heat are based on hydrodynamics and thermodynamics laws. Exact solutions are constructed the dimensionless governing equations. The accuracy of the two solutions are established by direct comparison with the limiting case previously obtained in the current literature. Solution profiles are also presented to demonstrate the effect of variations in the parameter values followed closely by extensive discussions. © 2019 Published by Elsevier B.V. on behalf of Taiwan Institute of Chemical Engineers.

1. Introduction The last few years have witnessed increasing studies on the entropy generation in gravity-aided flows through an inclined channel. The tilt in channel encourages the influence of the earth’s gravitational force and this configuration is closer to reality in a good number of applications in the physical science and engineering. Some of the landmark achievement in building a more realistic model can be seen in a recent work by Makinde and Gbolagade [1], the authors presented a thermodynamics analysis based on the gravity effect and a two dimensional heat distribution. Havzali et al. [2] dealt with same problem with interest in the initial and entrance behavior of the channel. Baskaya et el. [3], considered the magnetic field effect on buoyancy and pressurized flow down an inclined channel. In the work of Abass et al. [4], the magnetized two-phased nanofluid slip flow experiencing thermal radiation was examined. The underlying fact about all the studies described above is the validity in the Newtonian class. There are ∗

Corresponding author. E-mail addresses: [email protected] (S.O. Adesanya), [email protected] (B. Souayeh), [email protected], [email protected] (M. Rahimi-Gorji), [email protected] (M.N. Khan), [email protected] (O.G. Adeyemi).

some research in this field which can be very useful and applicable in future researches and industries [5–23]. The main concern here is to extend the study to the nonNewtonian class where the effect of body couples and couple stresses can be captured, especially for commonly used industrial fluids where inclusion of tiny additives plays a dominant role. The couple stress model has been used recently in describing flow properties several fluids. For instance, Hayat et al. [24] implemented the model for a reacting three-dimensional fluid flow experiencing magnetic field. Reddy et al. [25] utilized the approach for fluids with temperature dependent density, Hayat et al. [26] presented the homotopy series solution to the developing three-dimensional flow endowed with Cattaneo–Christov condition. The non-Newtonian model was used to describe peristaltic flow in the work of Ramesh and Devakar [27] while the constitutive model was used for blood flow analysis in [28]. In short, the literature is unexhaustive on the theory and application of couple stresses. In the current study, a simple generalization of the study in [1] is presented that can describe a wider range of nonNewtonian fluids like the blood, pharmaceutical mixtures, especially lubricants used over a long period of time under intense heat and other fluid that flows steadily through an inclined channel with isothermal heating which has not be done in previous

https://doi.org/10.1016/j.jtice.2019.04.052 1876-1070/© 2019 Published by Elsevier B.V. on behalf of Taiwan Institute of Chemical Engineers.

252

S.O. Adesanya, B. Souayeh and M. Rahimi-Gorji et al. / Journal of the Taiwan Institute of Chemical Engineers 101 (2019) 251–258

(a)

5, 10, 15 u

0.008

0.006

0.004

0.002

1.0

0.5

x

(b)

0.1,

0.5

1.0

0.5

1.0

y

5, 10, 15

1.000 0.995 0.990 0.985 0.980 0.975 0.970 1.0

0.5

0.0

y

Fig. 1. (a) Velocity with large values of couple stress parameter (b) Temperature with large values of couple stress parameter (c) Entropy generation rate with large values of couple stress parameter (d) Bejan number with large values of couple stress parameter.

models despite the huge studies available in the literature, hence, the study is original and new.

∂T ∂ 2T = α 2 ∂ x ∂y

(2)

2. Model formulation

Subject to no-slip, symmetric, the inlet, and isothermal wall conditions

As shown in the geometry of the problem, we consider the steady, incompressible flow of a couple stress fluid through a channel tilted at an angle ϑ to the horizontal. The channel walls are distanced 2 h apart and maintained at a constant temperature. Following the Cartesian system, the x-axis is taken along the inclined wall while the y-axis is chosen perpendicular to it. Therefore, including the effect of couple stresses [24–28], the extended work in [1] can be written as:

u = 0, T (0, y ) = T0 , T (x, h ) = Tw on

0 = ρ gSinϑ + μ

2

4

d u d u −η 4 dy¯ 2 dy¯

(1)

du dy

= 0,

∂T (x, 0 ) = 0 on y = 0 ∂ y

y = h (3)

The expression for the thermodynamic irreversibility in the process is due to the combination of heat transfer and frictional effect can be written as

k EG = 2 T0



∂T ∂ x

2



∂T + ∂ y

2 

+

 2  2 μ ∂ u η ∂ 2 u + (4) T0 ∂ y T0 ∂ y2

S.O. Adesanya, B. Souayeh and M. Rahimi-Gorji et al. / Journal of the Taiwan Institute of Chemical Engineers 101 (2019) 251–258

x

(c)

0.1,

5, 10, 15 , Br

0.3, Pe

7.1,

253

1

NS 0.005

0.004

0.003

0.002

0.001

1.0

0.5

(d)

x

0.1,

0.5

5, 10, 15 , Br

0.3, Pe

y

1.0

7.1,

1

Be

0.8

0.6

0.4

0.2

1.0

0.5

0.5

y

1.0

Fig. 1. Continued

Here EG represents entropy generation, k-thermal conductivity, (T, θ )-fluid temperature in dimensional and dimensionless form, (T0 ,Tw )-referenced temperatures, (u , u)-dimensional and non-dimensional axial velocity and (y , x )- the cartesian coordinates. (μ, η)-dynamic viscosity and couple stress coefficient, (ρ , g)-density, gravitational force, and α thermal diffusivity. Introducing

y μ0 u  y= , u= 2 , h h ρ gSinϑ

θ

T − T0 x αμ = , x= 4 , κ2 = Tw − T0 h ρ gSinϑ

η μh 2

(5)

1 NS = 2 Pe



∂θ ∂x

∂θ ∂ θ ∂θ = ; θ (0, y ) = 0; θ (x, 1 ) = 1; (x, 0 ) = 0 ∂ x ∂ y2 ∂y

(6) = (7)

2  +

Br





∂u ∂y

2

 + κ2

∂ 2 u ∂ y2

2 

here Nx , Ny are the entropy generation due to heat transfer in axial and transverse direction, respectively, while Nf is the entropy produced due to frictional interaction of the fluid particles.the heat irreversibility ratio is then given by

Be =

2

∂θ + ∂y

(8)

Br

d2 u d4 u d2 u 1+ − κ 2 4 ; u(±1 ) = 0 = (±1 ) 2 dy dy d y2



= Nx + Ny + N f

we get

u

2

1 P e2

  2 ∂θ ∂x



  2 ∂u ∂y

 ∂θ 2

+ ∂y



2  + κ 2 ∂∂ yu2

+ Br

2

  2 ∂u ∂y



2  + κ 2 ∂∂ yu2

Nf Nx + Ny + N f

The exact solution of the BVP (6) is given by

2

254

S.O. Adesanya, B. Souayeh and M. Rahimi-Gorji et al. / Journal of the Taiwan Institute of Chemical Engineers 101 (2019) 251–258

(a)

0.1, 0.2, 0.3 u

0.5

0.4

0.3

0.2

0.1

1.0

0.5

0.5

x

(b)

0.1,

1.0

y

0.1, 0.2, 0.3 1.0 0.9 0.8 0.7 0.6 0.5 0.4

1.0

0.5

0.5

1.0

y

Fig. 2. (a) Velocity with small values of couple stress parameter (b) Temperature with small values of couple stress parameter, (c) Entropy generation rate with small values of couple stress parameter, (d) Bejan number with small values of couple stress parameter.

u (y ) =







 1 y 1 1 − y2 − κ 2 1 − Cosh Sech 2 κ κ

(9)

By using separation of variables method, the analytical solution of (7) becomes

θ (x, y ) =



24x





 

⎛ Nx =

 

1 ⎜ ⎜−  P e2 ⎝

576x



(

For validation purpose, it is easy to see that in the asymptotic case when κ → 0the exact solutions (9) and (10) becomes that obtained in [1], that is,

 1 1 − y2 2

(11)

and

θ (x, y ) =

24x 5 + 24x − 6y2 + y4

+

)



5+24x−6y2 +y4 +12κ 2 y2 −1 +24κ 4 1−Cosh

5 + 24x − 6y2 + y4 + 12 y2 − 1 κ 2 + 24κ 4 1 − Cosh κy Sech κ1

(10)

u (y ) =

therefore, the accuracy of the solutions has been established. Now substituting (9) and (10), we get



24



y

κ



1

Sech

2

κ



5 + 24x − 6y2 + y4 + 12κ 2 y2 − 1 + 24κ 4 1 − Cosh

⎞

y κ

κ

(13) 

576x2 −12y + 4y3 + 24yκ 2 − 24κ 3 Sinh

y κ

Sech

 1 2 κ

Ny =   y  1 4   5 + 24x − 6y2 + y4 + 12κ 2 y2 − 1 + 24κ 4 1 − Cosh Sech

κ

(12)

Sech

⎟  1 ⎠2

κ

(14)

S.O. Adesanya, B. Souayeh and M. Rahimi-Gorji et al. / Journal of the Taiwan Institute of Chemical Engineers 101 (2019) 251–258

(c)

x

0.1,

0.1, 0.2, 0.3 , Br

0.3, Pe

7.1,

255

1

NS

3

2

1

1.0

(d)

0.5

x

0.1,

0.5

0.1, 0.2, 0.3 , Br

0.3, Pe

7.1,

1.0

y

1

Be 0.25

0.20

0.15

0.10

0.05

1.0

0.5

0.5

1.0

y

Fig. 2. Continued

and

  y  1 2 N f = Br −1 κ 2 −1 + Cosh Sech κ κ  1  1 2  + −y + κ Sech Sinh κ κ

(15)

3. Discussion of results One benefit of exact solution is the valid for both small and large parameter values, in which case the solution profiles differ. Fig. 1a shows the decreasing velocity as the couple stress parameter increases due to fluid thickening as the additive increases. The result in Fig. 1b is seen to satisfy the boundary conditions with rising temperature. The result in well behaved since temperature rises with viscous heating. In Fig. 1c, entropy produced decreases with couple stress parameter thus promoting energy conservation and longevity of the machineries. Also, Fig. 1d revealed the mini-

mum effect of heat irreversibility from the heat transfer process as the couple stress parameters grows. The behavior of the system when the couple stress parameter is small is presented as Fig. 2. A comparison between the couple stress parameter values for the velocity profiles revealed that they are both well-behaved, but the magnitude of the velocity component is higher when couple stress parameter is small than when it is large. Fig. 2c showed that temperature is much lower in the core region when couple stress parameter is small as against the large values. In this regard, entropy is much higher since polymer additive is minimal and interparticle collision is much higher. Therefore, irreversibility associated with heat transfer rises in the core region of the inclined channel as shown in Fig. 2d. Fig. 3a–d represents the axial length effect on the flow structure. The plots revealed that temperature increases with axial length while fluid friction irreversibility rises as the axial length inside the inclined channel. Finally, the effect of viscous dissipation on the entropy generation rate is seen in Fig. 4a to be on the increase as the viscous heating increases. This promotes

256

S.O. Adesanya, B. Souayeh and M. Rahimi-Gorji et al. / Journal of the Taiwan Institute of Chemical Engineers 101 (2019) 251–258

x

(a)

0.1, 0.2, 0.3 ,

1

1.0

0.9

0.8

0.7

1.0

0.5

x

(b)

0.1, 0.2, 0.3 ,

0.5

1, Br

0.3, Pe

1.0

7.1,

y

1

NS 0.5

0.4

0.3

0.2

0.1

1.0

(c)

0.5

x

0.1, 0.2, 0.3 ,

0.5

1, Br

0.3, Pe

7.1,

1.0

y

1

Be

0.8

0.6

0.4

0.2

1.0

0.5

0.5

1.0

y

Fig. 3. (a) Temperature with axial length, (b) Entropy generation rate with axial length (c): Bejan number with values of axial length.

S.O. Adesanya, B. Souayeh and M. Rahimi-Gorji et al. / Journal of the Taiwan Institute of Chemical Engineers 101 (2019) 251–258

(a)

Br

0.1, 0.2, 0.3 ,

1, x

0.3, Pe

7.1,

257

1.

NS 0.12

0.10

0.08

0.06

0.04

0.02 1.0

(b)

0.5

Br

0.1, 0.2, 0.3 ,

0.5

1, x

0.3, Pe

7.1,

1.0

y

1.

Be

0.8

0.6

0.4

0.2

1.0

0.5

0.5

1.0

y

Fig. 4. (a) Entropy generation rate with values of Brinkman number, (b) Bejan number with values of Brinkman number.

frictional heat irreversibility over that of heat transfer as seen in Fig. 4b.

4. Conclusion In this study, the effect of size-dependent particles on the entropy generation rate of a Newtonian fluid flow through an inclined channel is presented. The exact solutions of the flow equations are obtained, presented and validated with previous result in literature in a special case. Salient outcomes from the present analysis are listed as follows: i. The velocity of the non-Newtonian fluid inclined channel decreases with increasing values of the couple stress parameter while the temperature rises. ii. Entropy generation rate and Bejan number decreases at different rates with small and large couple stress parameter

Acknowledgement The author would like to thank Deanship of Scientific Research at Majmaah University for supporting this work under Project Number NO. 1440-100. Also, they are grateful to the Physics Department, College of Science, King Faisal University for their technical and moral support. References [1] Makinde OD, Gbolagade AW. Second law analysis of incompressible viscous flow through an inclined channel with isothermal walls. Rom J Phys 2005;50(9–10):923–30. [2] Havzali M, Arikoglu A, Komurgoz G, Keser H I, Ozkol I. Analytical–numerical analysis of entropy generation for gravity-driven inclined channel flow with initial transition and entrance effects. Phys Scr 2008;78:045401. [3] Baskaya E, Komurgoz G, Ozkol I. Investigation of oriented magnetic field effects on entropy generation in an inclined channel filled with ferrofluids. Entropy 2017;19:377. doi:10.3390/e19070377. [4] Abbas Z, Rahim T, Hasnain J. Slip flow of magnetite-water nanomaterial in an inclined channel with thermal radiation. Int J Mech Sci 2017;122:288–96.

258

S.O. Adesanya, B. Souayeh and M. Rahimi-Gorji et al. / Journal of the Taiwan Institute of Chemical Engineers 101 (2019) 251–258

[5] Hayat T, Nawaz S, Alsaedi A, Rafiq M. Analysis of entropy generation in mixed convective peristaltic flow of nanofluid. Entropy 2016;18(10):355. [6] Hayat T, Aslam N, Rafiq M, EAlsaadi F. Hall and joule heating effects on peristaltic flow of powell–eyring liquid in an inclined symmetric channel. Results Phys 2017;7:518–28. [7] Hayat T, Rafiq M, Ahmad B, Asghar S. Entropy generation analysis for peristaltic flow of nanoparticles in a rotating frame. Int J Heat Mass Transf 2017;108(Part B):1775–86. [8] Hayat T, Farooq S, Ahmad B, Alsaedi A. Effectiveness of entropy generation and energy transfer on peristaltic flow of Jeffrey material with Darcy resistance. Int J Heat Mass Transf 2017;106:244–52. [9] Farooq S, Hayat T, Alsaedi A, Asghar S. Mixed convection peristalsis of carbon nanotubes with thermal radiation and entropy generation. J Mol Liq 2018;250:451–67. [10] Ijaz Khan M, Qayyum S, Hayat T, Imran Khan M, Alsaedi A, Ahmad Khan T. Entropy generation in radiative motion of tangent hyperbolic nanofluid in presence of activation energy and nonlinear mixed convection. Phys Lett A 2018;382(31):2017–26. [11] M. Waqas, M. Ijaz Khan, S. Farooq, T. Hayat, A. Alsaedi, Significance of improved Fourier-Fick laws in nonlinear convective micropolar material stratified flow with variable properties, 2018, doi:10.2298/TSCI171027285W. [12] Hayat T, Ijaz Khan M, Farooq M, Alsaedi A, Waqas M, Yasmeende T. Impact of Cattaneo–Christov heat flux model in flow of variable thermal conductivity fluid over a variable thicked surface. Int J Heat Mass Transf 2016;99:702–10. [13] Hayat T, Ijaz Khan M, Farooq M, Yasmeend T, Alsaedi A. Stagnation point flow with Cattaneo–Christov heat flux and homogeneous-heterogeneous reactions. J Mol Liq 2016;220:49–55. [14] Krishna ChM, Reddy GV, Souayeh B, Raju CSK, Rahimi-Gorji M, Raju SSK. Thermal convection of MHD Blasius and Sakiadis flow with thermal convective conditions and variable properties. Microsyst Technol, 2019. doi:10.1007/ s00542- 019- 04353- y. [15] Raju SS, Kumar KG, Rahimi-Gorji M, Khan I. Darcy–Forchheimer flow and heat transfer augmentation of a viscoelastic fluid over an incessant moving needle in the presence of viscous dissipation. Microsyst Technol, 2019. doi:10.1007/ s00542- 019- 04340- 3. [16] Ahmed N, Shah NA, Ahmad B, Shah SIA, Ulhaq S, Rahimi-Gorji M. Transient MHD convective flow of fractional nanofluid between vertical plates. J Appl Comput Mech, 2018. doi:10.22055/JACM.2018.26947.1364.

[17] Hussanan A, Khan I, Rahimi Gorji M, Khan WA. CNTS-water–based nanofluid over a stretching sheet. BioNanoScience, 2019. doi:10.07/s12668- 018- 0592- 6. [18] Pourmehran O, Rahimi-Gorji M, Ganji DD. Analysis of nanofluid flow in a porous media rotating system between two permeable sheets considering thermophoretic and brownian motion. Therm Sci 2017;21:5. [19] Bezi S, Souayeh B, Ben-Cheikh N, Ben-Beya B. Numerical simulation of entropy generation due to unsteady natural convection in a semi-annular enclosure filled with nanofluid. Int J Heat Mass Transf 2018;124:841–59. [20] Hammami F, Ben-Cheikh N, Ben-Beya B, Souayeh B. Combined effects of the velocity and the aspect ratios on the bifurcation phenomena in a two-sided lid-driven cavity flow. Int J Numer Methods Heat Fluid Flow 2018;28(4):943–62. [21] Hammami F, Souayeh B, Ben-Cheikh N, Ben-Beya B. Computational analysis of fluid flow due to a two-sided lid driven cavity with a circular cylinder. Comput Fluids 2017;156:317–28. [22] Souayeh B, Ben-Cheikh N, Ben-Beya B. Periodic behavior flow of three-dimensional natural convection in a titled obstructed cubical enclosure. Int J Numer Methods Heat Fluid Flow 2017;27(9):2030–52. [23] Souayeh B, Ben-Cheikh N, Ben-Beya B. Effect of thermal conductivity ratio on flow features and convective heat transfer. Part Sci Technol 2017;35(5):565–74. [24] Hayat T, Asghar Sabia, Tanveer Anum, Alsaedi Ahmed. Chemical reaction in peristaltic motion of MHD couple stress fluid in channel with Soret and Dufour effects. Results Phys 2018;10:69–80. [25] Reddy GJ, Kumar M, Kethireddy B, Chamkha AJ. Colloidal study of unsteady magnetohydrodynamic couple stress fluid flow over an isothermal vertical flat plate with entropy heat generation. J Mol Liq February 2018;252:169–79. [26] Hayat T, Muhammad T, Alsaedi A. On three-dimensional flow of couple stress fluid with Cattaneo–Christov heat flux. Chin J Phys June 2017;55(3):930–8. [27] Ramesh K, Devakar M. Magnetohydrodynamic peristaltic transport of couple stress fluid through porous medium in an inclined asymmetric channel with heat transfer. J Magn Magn Mater 2015;394:335–48 15 November. [28] Srinivasacharya D, Madhava Rao G. Computational analysis of magnetic effects on pulsatile flow of couple stress fluid through a bifurcated artery. Comput Methods Programs Biomed December 2016;137:269–79.