# Numerical Investigation of Thermal Radiation and Viscous Effects on Entropy Generation in Forced Convection Blood Flow over an Axisymmetric Stretching Sheet

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Governing Equations

_{w}, respectively. The temperature difference between the medium temperature, T

_{∞}, and the stretching sheet, T

_{w}, is also assumed to be high enough to induce heat transfer due to radiation. Since an infinitely large medium is assumed, the flow velocity at far distances from the plate is assumed to be zero.

_{zz}≫ v

_{rr}) equations are:

^{n}and the power law index for blood is n = 0.708. For n = 1 the fluid is Newtonian, with a dynamic coefficient of viscosity k. For n > 1 the behavior of the fluid is dilatant or shear-thickening and for 0 < n < 1 the behavior is shear-thinning. In this work we shall restrict our study to the blood flow and the values of the power law around n = 0.708 which can represent the effect of perturbations near this point an variation of steady state value exposed to impact [25,26,27]. The energy equation for thermal radiative absorbance of thermal radiation in a semi-transparent gray medium is considered here that as ruled by the Rosseland approximation [41] for conductive radiative heat flux. The diffusion approximation is extremely convenient to use. The equation of energy is:

_{r}in Equation (3) is modeled as:

^{4}can be expressed as a linear function of temperature, ${T}^{4}={T}_{\infty}^{4}+4{T}_{\infty}^{3}\left(T-{T}_{\infty}^{}\right)+\mathrm{...}$, expanding T

^{4}in a Taylor series about T

_{∞}and neglecting higher order terms thus:

_{R}is 16k

_{R}where k

_{R}is the radiation conductivity:

_{s}is the dimensionless form of local entropy generation rate in Equation (12). From Equation (23), it is obvious that the Bejan number ranges from 0 < Be < 1. While Be = 0 represents the limit case of fluid friction dominated irreversibility, Be = 1 corresponds to the limit case of heat transfer dominated irreversibility. The contribution of both heat transfer and fluid friction to entropy production in the flow system is the same when Be = 0.5. These expressions may be evaluated as follows:

## 3. Results and Discussions

_{exact}= 0.6888095535). As shown the HAM has the acceptable results for the series with more than 10 terms. Figure 4 illustrates the effect of the power-law index on the non-dimensional velocity profile. For all of the curves, the η

_{∞}is considered as 20 which Figure 2 proves has enough accuracy for numerical computations. As noted before the current research is focused on blood flow for which the flow consistency index and power law index is given (n = 0.708). Although the flow consistency index appears in the similarity variable the final solution is independent of the flow consistency index and is just a function of the power law index. As shown, by increasing the power-law index the boundary layer thickness is smaller and the effect of the moving surface is just sensed at near distances.

_{∞}is considered as 20 and by increasing the power-law index the thermal boundary layer thickness is smaller and the effect of the moving surface just sensed at near distances. The corresponding derivative of the dimensionless temperature which is a measure of the Nusselt number for the power-law indexes of 0.01, 0.1, and 0.708 is illustrated in Figure 7. As shown by increasing the power-law index the thinner thermal boundary layer leads to a higher rate of heat exchange. In Figure 8 the dimensionless thermal entropy generated versus similarity variable for various power-law indexes is presented. As shown most of the thermal entropy generation occurs near the stretching wall and it increases with the increase of power law index.

_{∞}equal to 20) the thermal boundary layer depth is increased by increasing the thermal radiation parameter (in η

_{∞}< 1) and the effect of a stretching sheet surface affects the temperature in the adjacent fluid. The matching derivative of the dimensionless temperature is proportional to Nusselt number for the thermal radiation parameters in range of 0.01 through 10 in logarithmic scale as exemplified in Figure 13. As discovered by intensification of the thermal radiation parameter the lean thermal boundary layer causes a better convective heat transfer. In Figure 14 the dimensionless thermal entropy produced versus similarity variable for numerous thermal radiation parameters is revealed. As shown most of the thermal entropy generation occurs near the stretching sheet and it decreases quickly as the thermal radiation parameter increases.

_{u}than S

_{θ}, but similarly it is increased by increases of Br.

## 4. Conclusions

- (1)
- By increasing n, the boundary layer of fluid flow and heat flow is decreased and the heat transfer rate and fluid friction on the stretching sheet is increased.
- (2)
- By increasing Pr, the boundary layer of fluid flow is decreased and the heat transfer rate on the stretching sheet is increased.
- (3)
- By increasing N
_{R}, the boundary layer of fluid flow is increased and the convective heat transfer rate on the stretching sheet is decreased. - (4)
- By increasing Br, the boundary layer of fluid flow and the convective heat transfer rate on the stretching sheet are increased.
- (5)
- By increasing n, Pr the heat transfer part of the entropy increases dramatically while the viscous part of the entropy is not changed.
- (6)
- By increasing Br, the total heat transfer entropy generation and its component increase.
- (7)
- By increasing N
_{R}, the thermal heat transfer entropy generation decreases.

## Author Contributions

## Conflicts of Interest

## References

- Crane, L.J. Flow past a stretching plate. Z. Angew. Math. Phys.
**1970**, 21, 645–647. [Google Scholar] [CrossRef] - Gupta, P.S.; Gupta, A.S. Heat and mass transfer on a stretching sheet with suction or blowing. Can. J. Chem. Eng.
**1977**, 55, 744–746. [Google Scholar] [CrossRef] - Grubka, J.; Bobba, K.M. Heat transfer characteristics of a continuous stretching surface with variable temperature. J. Heat Transf.
**1985**, 107, 248–250. [Google Scholar] [CrossRef] - Ali, M.E. On thermal boundary layer on a power law stretched surface with suction or injection. Int. J. Heat Fluid Flow
**1995**, 16, 280–290. [Google Scholar] [CrossRef] - Chen, C.H. Laminar mixed convection adjacent to vertical, continuously stretching sheets. Heat Mass Transf.
**1998**, 33, 471–476. [Google Scholar] [CrossRef] - Datta, B.K.; Roy, P.; Gupta, A.S. Temperature field in a flow over a stretching that within uniform heat flux. Int. Commun. Heat Transf.
**1985**, 12, 89–94. [Google Scholar] [CrossRef] - Chen, C.K.; Char, M.I. Heat transfer on a continuous stretching surface with suction or blowing. J. Math. Anal. Appl.
**1988**, 135, 568–580. [Google Scholar] [CrossRef] - Elbashbeshy, E.M.A. Heat transfer over a stretching surface with variable heat flux. J. Phys. D Appl. Phys.
**1998**, 31. [Google Scholar] [CrossRef] - Makhmalbaf, M.H.M. Experimental study on convective heat transfer coefficient around a vertical hexagonal rod bundle. Heat Mass Transf.
**2012**, 48, 1023–1029. [Google Scholar] [CrossRef] - Makhmalbaf, H.; Liu, T.; Merati, P. Experimental Simulation of Buoyancy-Driven Vortical Flow in Jupiter Great Red Spot. In Proceedings of the 68th Annual Meeting of the APS Division of Fluid Dynamics, Boston, MA, USA, 22–24 November 2015.
- Cortell, R. Flow and heat transfer of a fluid through a porous medium over a stretching surface with internal heat generation/absorption and suction/blowing. Fluid Dyn. Res.
**2005**, 37, 231–245. [Google Scholar] [CrossRef] - Liao, S.J. An analytic solution of unsteady boundary-layer flows caused by an impulsively stretching plate. Commun. Nonlinear Sci. Numer. Simul.
**2006**, 11, 326–339. [Google Scholar] [CrossRef] - Mehmood, A.; Ali, A. Analytic homotopy solution of generalized three dimensional channel flow due to uniform stretching of the plate. Acta Mech. Sin.
**2007**, 23, 502–510. [Google Scholar] [CrossRef] - Ishak, A.; Nazar, R.; Pop, I. Hydromagnetic flow and heat transfer adjacent to a stretching vertical sheet. Heat Mass Transf.
**2008**, 44, 921–927. [Google Scholar] [CrossRef] - Crane, L.J. Boundary layer flow due to stretching cylinder. Z. Angew. Math. Phys.
**1975**, 25, 619–622. [Google Scholar] [CrossRef] - Wang, C.Y. Fluid flow due to stretching cylinder. Phys. Fluids
**1988**, 31, 466–468. [Google Scholar] [CrossRef] - Burde, H.I. On the motion of fluid near a stretching circular cylinder. J. Appl. Math. Mech.
**1988**, 53, 271–273. [Google Scholar] [CrossRef] - Ishak, A.; Nazar, R. Laminar boundary layer flow along a stretching cylinder. Eur. J. Sci. Res.
**2009**, 36, 22–29. [Google Scholar] - Ishak, A.; Nazar, R.; Pop, I. Magnetohydrodynamic (MHD) flow and heat transfer due to a stretching cylinder. Energy Convers. Manag.
**2008**, 49, 3265–3269. [Google Scholar] [CrossRef] - Mastroberardino, A.; Paullet, J.E. Existence and priori bounds for steady stagnation flow toward a stretching cylinder. J. Math. Anal. Appl.
**2010**, 365, 701–710. [Google Scholar] [CrossRef] - Weidman, P.D.; Ali, M.E. Aligned and nonaligned radial stagnation flow on a stretching cylinder. Eur. J. Mech. B Fluids
**2011**, 30, 120–128. [Google Scholar] [CrossRef] - Wang, C.Y.; Ng, C.O. Slip flow due to a stretching cylinder. Int. J. Nonlinear Mech.
**2011**, 45, 1191–1194. [Google Scholar] [CrossRef] [Green Version] - Munawar, S.; Mehmood, A.; Ali, A. Unsteady flow of viscous fluid over the vacillate stretching cylinder. Int. J. Numer. Methods Fluids
**2011**, 70, 671–681. [Google Scholar] [CrossRef] - Vajravelu, K.; Prasad, K.V.; Santhi, S.R. Axisymmetric magneto-hydrodynamic (MHD) flow and heat transfer at a non-isothermal stretching cylinder. Appl. Math. Comput.
**2012**, 219, 3993–4005. [Google Scholar] [CrossRef] - Joodaki, H.; Forman, J.; Forghani, A.; Overby, B.; Kent, R.; Crandall, J.; Beahlen, B.; Beebe, M.; Bostrom, O. Comparison of Kinematic and Dynamic Behavior of a First Generation Obese Dummy and Obese PMHS in Frontal Sled Tests. In Proceedings of the 11th Ohio State University Injury Biomechanics Symposium, Columbus, OH, USA, 17–19 May 2015.
- Foreman, J.L.; Joodaki, H.; Forghani, A.; Riley, P.O.; Bollapragada, V.; Lessley, D.J.; Overby, B.; Heltzel, S.; Kerrigan, J.R.; Crandall, J.R.; et al. Whole-Body Response for Pedestrian Impact with a Generic Sedan Buck. Stapp Car Crash J.
**2015**, 59, 401–444. [Google Scholar] - Forman, J.L.; Joodaki, H.; Forghani, A.; Riley, P.; Bollapragada, V.; Lessley, D.; Overby, B.; Heltzel, S.; Crandall, J. Biofidelity Corridors for Whole-Body Pedestrian Impact with a Generic Buck. IRCOBI Conf.
**2015**, 49, 356–372. [Google Scholar] - Chen, X.; Pavlish, K.; Benoit, J.N. Myosin phosphorylation triggers actin polymerization in vascular smooth muscle. Am. J. Physiol. Heart Circ. Physiol.
**2008**, 295, H2172–H2177. [Google Scholar] [CrossRef] [PubMed] - Jones, E.A.V. The initiation of blood flow and flow induced events in early vascular development. Semin. Cell Dev. Biol.
**2011**, 22, 1028–1035. [Google Scholar] [CrossRef] [PubMed] - Das, A.; Paul, A.; Taylor, M.D.; Banerjee, R.K. Pulsatile arterial wall-blood flow interaction with wall pre-stress computed using an inverse algorithm. BioMed. Eng. Online
**2015**, 14, S1–S18. [Google Scholar] - Munawar, S.; Mehmood, A.; Ali, A. Time-dependent flow and heat transfer over a stretching cylinder. Chin. J. Phys.
**2012**, 50, 828–848. [Google Scholar] - Mukhopadhyay, S.; Ishak, A. Mixed convection flow along a stretching cylinder in a thermally stratified medium. J. Appl. Math.
**2012**, 8. [Google Scholar] [CrossRef] - Shateyi, S.; Marewo, G.T. A new numerical approach for the laminar boundary layer flow and heat transfer along a stretching cylinder embedded in a porous medium with variable thermal conductivity. J. Appl. Math.
**2013**, 7. [Google Scholar] [CrossRef] - Si, X.; Li, L.; Zheng, L.; Zhang, X.; Liu, B. The exterior unsteady viscous flow and heat transfer due to a porous expanding stretching cylinder. Comput. Fluids
**2014**, 105, 280–284. [Google Scholar] [CrossRef] - Vajravelu, K.; Prasad, K.V.; Santhi, S.R.; Umesh, V. Fluid flow and heat transfer over a permeable stretching cylinder. J. Appl. Fluids Mech.
**2014**, 7, 111–120. [Google Scholar] - Odat, M.Q.A.; Damseh, R.A.; Nimr, M.A.A. Effect of magnetic field on entropy generation due to laminar forced convection past a horizontal flat plate. Entropy
**2004**, 4, 293–303. [Google Scholar] [CrossRef] - Makinde, O.D.; Osalusi, E. Entropy generation in a liquid film falling along an inclined porous heated plate. Mech. Res. Commun.
**2006**, 33, 692–698. [Google Scholar] [CrossRef] - Makinde, O.D. Irreversibility analysis for a gravity driven non-Newtonian liquid film along an inclined isothermal plate. Phys. Scr.
**2006**, 74, 642–645. [Google Scholar] [CrossRef] - Munawar, S.; Ali, A.; Mehmood, A. Thermal analysis of the flow over an oscillatory stretching cylinder. Phys. Scr.
**2012**, 86. [Google Scholar] [CrossRef] - Butt, A.S.; Ali, A. Entropy analysis of magnetohydrodynamic flow and heat transfer due to a stretching cylinder. J. Taiwan Inst. Chem. Eng.
**2014**, 45, 780–786. [Google Scholar] [CrossRef] - Rosseland, S. Theoretical Astrophysics: Atomic Theory and the Analysis of Stellar Atmospheres and Envelopes; Clarendon Press: Oxford, UK, 1936. [Google Scholar]
- Press, W.H.; Teukolsky, S.A.; Vetterling, W.T.; Flannery, B.P. Numerical Recipes: The Art of Scientific Computing; Cambridge University Press: New York, NY, USA, 2007. [Google Scholar]
- Makinde, O.D.; Mabood, F.; Khanc, W.A.; Tshehla, M.S. MHD flow of a variable viscosity nanofluid over a radially stretching convective surface with radiative heat. J. Mol. Liquids
**2016**, 219, 624–630. [Google Scholar] [CrossRef] - Motsa, S.S.; Sibanda, P. On the solution of MHD flow over a nonlinear stretching sheet by an efficient semi-analytical technique. Int. J. Numer. Methods Fluids
**2012**, 68, 1524–1537. [Google Scholar] [CrossRef] - Abdollahzadeh Jamalabadi, M.Y. Entropy generation in boundary layer flow of a micro polar fluid over a stretching sheet embedded in a highly absorbing medium. Front. Heat Mass Transf.
**2015**, 6, 1–13. [Google Scholar] [CrossRef] - Liao, S.J. The Proposed Homotopy Analysis Technique for the Solution of Nonlinear Problems. Ph.D. Thesis, Shanghai Jiao Tong University, Shanghai, China, 1992. [Google Scholar]

**Figure 2.**Prantdl boundary layer equation for various boundary layer thicknesses versus Crocco transformed equation

**Figure 8.**Dimensionless thermal entropy generated versus similarity variable for various power-law index.

**Figure 11.**Dimensionless thermal entropy generated versus similarity variable for various Prantdl numbers.

**Figure 14.**Dimensionless thermal entropy generated versus similarity variable for various thermal radiation parameters.

**Figure 17.**Dimensionless thermal entropy generated versus similarity variable for various Brinkman numbers.

**Figure 18.**Dimensionless flow entropy generated versus similarity variable for various Brinkman numbers.

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Abdollahzadeh Jamalabadi, M.Y.; Hooshmand, P.; Hesabi, A.; Kwak, M.K.; Pirzadeh, I.; Keikha, A.J.; Negahdari, M.
Numerical Investigation of Thermal Radiation and Viscous Effects on Entropy Generation in Forced Convection Blood Flow over an Axisymmetric Stretching Sheet. *Entropy* **2016**, *18*, 203.
https://doi.org/10.3390/e18060203

**AMA Style**

Abdollahzadeh Jamalabadi MY, Hooshmand P, Hesabi A, Kwak MK, Pirzadeh I, Keikha AJ, Negahdari M.
Numerical Investigation of Thermal Radiation and Viscous Effects on Entropy Generation in Forced Convection Blood Flow over an Axisymmetric Stretching Sheet. *Entropy*. 2016; 18(6):203.
https://doi.org/10.3390/e18060203

**Chicago/Turabian Style**

Abdollahzadeh Jamalabadi, Mohammad Yaghoub, Payam Hooshmand, Ashkan Hesabi, Moon K. Kwak, Isma’il Pirzadeh, Ahmad Jamali Keikha, and Mohammadreza Negahdari.
2016. "Numerical Investigation of Thermal Radiation and Viscous Effects on Entropy Generation in Forced Convection Blood Flow over an Axisymmetric Stretching Sheet" *Entropy* 18, no. 6: 203.
https://doi.org/10.3390/e18060203