# 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}

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## 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

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**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.

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**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