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The steady mixed convection boundary layer flow from a solid sphere in a micropolar fluid, generated by Newtonian heating in which the heat transfer from the surface is proportional to the local surface temperature, is considered. The governing boundary layer equations are first transformed into a system of nondimensional equations via the non-dimensional variables, and then into nonsimilar equations before they are solved numerically using an implicit finite-difference scheme known as the Keller-box method. Numerical solutions are obtained for the skin friction coefficient, wall temperature and heat transfer coefficient, as well as the velocity and temperature profiles with several parameters considered, namely the mixed convection parameter, the material or micropolar parameter, and the Prandtl number.

The essence of the theory of micropolar fluid flow lies in the extension of the constitutive equation for Newtonian fluid, so that more complex fluids such as particle suspensions, liquid crystal, animal blood, lubrication, and turbulent shear flows can be described by this theory. The theory of micropolar fluid was first proposed by Eringen [

Generally, in modeling the boundary layer flow and heat transfer of these problems, the boundary conditions that are usually applied are either a constant wall temperature (CWT) or a constant wall heat flux (CHF). However, there is a class of boundary layer flow and heat transfer problems in which the surface heat transfer depends on the surface temperature. Perhaps the simplest case of which is when there is a linear relation between the surface heat transfer and surface temperature. This situation arises in conjugate heat transfer problems (see, e.g., Merkin and Pop [

This configuration occurs in many important engineering devices, for example, in heat exchanger where the conduction in solid tube wall is greatly influenced by the convection in the fluid flowing over it. Furthermore, for conjugate heat transfer around fins where the conduction within the fin and the convection in the fluid surrounding it must be simultaneously analyzed in order to obtain the vital design information and also in convection flow setup when the bounding surfaces absorb heat by solar radiation [

Recently, Salleh et al. [

Therefore, the aim of the present paper is to study the problem of mixed convection boundary layer flow from a solid sphere with Newtonian heating in a micropolar fluid. The governing boundary layer equations are first transformed into a system of non-dimensional equations via the non-dimensional variables, and then into non-similar equations before they are solved numerically by the Keller-box method, as described in the books by Na [

Consider a heated sphere of radius

Physical model and coordinate system.

Let

We introduce now the following non-dimensional variables:

Further, the stream function

Equations (

At the lower stagnation point of the sphere,

The quantities of physical interest are the skin friction coefficient,

Equations in (

Reduce (

Write the difference equations using central differences.

Linearize the resulting algebraic equations by Newton’s method, and write them in the matrix-vector form.

Solve the linear system by the block tridiagonal elimination technique (see Salleh et al. [

Equations in (

Due to the decoupled boundary layer equation (

Values of

Nazar et al. [ | Present | Nazar et al. [ | Present | |

1.0549 | 1.0462 | 1.5834 | 1.5681 | |

1.4465 | 1.4430 | 1.6726 | 1.6581 | |

0.0 | 1.7042 | 1.7020 | 1.7275 | 1.7127 |

3.0 | 2.1018 | 2.1089 | 1.8081 | 1.7944 |

10.0 | 2.9568 | 2.9554 | 1.9617 | 1.9434 |

Values of

Nazar et al. [ | Present | Nazar et al. [ | Present | |

0.3383 | 0.3035 | 1.6592 | 1.6781 | |

0.9790 | 0.9725 | 1.4674 | 1.4708 | |

0.0 | 1.7135 | 1.7112 | 1.3255 | 1.3274 |

2.0 | 2.2941 | 2.2923 | 1.2455 | 1.2471 |

10.0 | 4.0030 | 3.9911 | 1.0889 | 1.0909 |

Values of

Salleh et al. [ | Present | Salleh et al. [ | Present | |

0.05 | 3.5485 | 3.5485 | 65.4268 | 65.4268 |

1.0 | 5.2161 | 5.2162 | 8.7364 | 8.7367 |

5.0 | 8.1273 | 8.1273 | 3.9495 | 3.9495 |

7.0 | 9.1151 | 9.1151 | 3.4047 | 3.4047 |

10.0 | 10.3730 | 10.3730 | 2.9258 | 2.9258 |

Tables

Comparison on the values of

CWT | CHF | NH | ||

1.6670 | 0.5878 | — | — | |

1.6980 | 0.5847 | — | — | |

0.0 | 1.7272 | 0.5790 | 1.2748 | 2.2748 |

2.0 | 1.7805 | 0.5688 | 1.2410 | 2.2410 |

4.0 | 1.8286 | 0.5598 | 1.1750 | 2.1750 |

6.0 | 1.8725 | 0.5517 | 1.1190 | 2.1190 |

8.0 | 1.9159 | 0.5445 | 1.0745 | 2.0745 |

10.0 | 1.9509 | 0.5379 | 1.0378 | 2.0378 |

20.0 | 2.1101 | 0.5115 | 0.9165 | 1.9165 |

Comparison on the values of

CWT | CHF | NH | |

1.4748 | 1.6244 | ||

1.6179 | 1.6761 | ||

0.0 | 1.7568 | 1.7568 | 1.7568 |

2.0 | 2.0236 | 1.9102 | 2.1176 |

4.0 | 2.2785 | 2.0547 | 2.3937 |

6.0 | 2.5235 | 2.1918 | 2.6367 |

8.0 | 2.7603 | 2.3227 | 2.8568 |

10.0 | 2.9901 | 2.4483 | 3.0597 |

20.0 | 4.0590 | 3.0160 | 3.9136 |

The values of

Values of

0.05 | 1.7662 | 1.3736 | 1.4597 | 1.5677 | 1.2777 | 1.7382 |

1.0 | 1.9332 | 1.2988 | 1.5980 | 1.4970 | 1.3990 | 1.6551 |

5.0 | 2.4950 | 1.1527 | 2.0583 | 1.3055 | 1.7992 | 1.4333 |

7.0 | 2.7268 | 1.1019 | 2.2469 | 1.2450 | 1.9623 | 1.3643 |

10.0 | 3.0386 | 1.0378 | 2.5001 | 1.1755 | 2.1807 | 1.2853 |

Values of

0.05 | 5.3893 | 582.9966 | 3.3508 | 269.9892 | 1.4597 | 1.5677 |

1.0 | 5.9345 | 33.9767 | 4.0100 | 18.9700 | 1.5980 | 1.4970 |

5.0 | 7.5188 | 9.7701 | 5.5475 | 6.5872 | 2.0583 | 1.3055 |

7.0 | 8.1221 | 7.8312 | 6.0922 | 5.4627 | 2.2469 | 1.2450 |

10.0 | 8.9099 | 6.2842 | 6.7908 | 4.5284 | 2.5001 | 1.1755 |

Tables

Values of the local skin friction coefficient

0.005 | 0.02 | 0.05 | 1 | 5 | 7 | |

0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | |

2.2261 | 2.2280 | 2.2319 | 2.2995 | 2.5190 | 2.5862 | |

4.6769 | 4.6803 | 4.6870 | 4.8001 | 5.1915 | 5.3111 | |

7.0750 | 7.0797 | 7.0889 | 7.2419 | 7.7920 | 7.9602 | |

9.4615 | 9.4673 | 9.4789 | 9.6676 | 10.3657 | 10.5797 | |

11.6907 | 11.6976 | 11.7113 | 11.9338 | 12.7718 | 13.1966 | |

13.7553 | 13.7631 | 13.7788 | 14.0312 | 15.0807 | 15.8120 | |

15.7534 | 15.7622 | 15.7796 | 16.0600 | 17.4505 | 18.2886 | |

17.4679 | 17.4775 | 17.4965 | 17.8096 | 19.4869 | 20.3704 | |

19.0260 | 19.0362 | 19.0566 | 19.4335 | 21.2685 | 22.1885 | |

20.2155 | 20.2370 | 20.6934 | 22.5758 | 23.5257 | ||

21.1823 | 21.6963 | 23.5557 | 24.5200 | |||

22.1532 | 23.9418 | 24.8979 |

Values of the wall temperature distribution

0.005 | 0.02 | 0.05 | 1 | 5 | 7 | |

3972.1601 | 997.6703 | 402.7344 | 24.9385 | 7.7892 | 6.3404 | |

14946.0423 | 3741.4930 | 1500.5887 | 78.2596 | 18.0482 | 13.3545 | |

15962.9645 | 3995.3036 | 1601.7763 | 82.9188 | 18.8185 | 13.8678 | |

16578.9876 | 4149.1161 | 1663.1461 | 85.8007 | 19.3253 | 14.2146 | |

17114.5642 | 4282.9142 | 1716.5875 | 88.3705 | 19.8096 | 14.5542 | |

17663.7427 | 4420.1691 | 1771.4575 | 91.0592 | 20.3424 | 14.8667 | |

18250.9445 | 4566.9709 | 1830.1793 | 93.9733 | 20.9019 | 15.0610 | |

18940.4194 | 4739.3755 | 1899.1698 | 97.4257 | 21.4484 | 15.3222 | |

19744.1335 | 4940.3723 | 1979.6231 | 101.4657 | 22.0515 | 15.7276 | |

20692.2273 | 5177.4964 | 2074.5533 | 106.1772 | 22.8095 | 16.2955 | |

5460.6204 | 2187.9111 | 111.7045 | 23.7877 | 17.0402 | ||

2324.9537 | 118.3370 | 25.0444 | 17.9891 | |||

126.5161 | 26.6599 | 19.1931 |

Tables

Values of the local skin friction coefficient

0.7 | 1 | 7 | |

0.0000 | 0.0000 | 0.0000 | |

2.2995 | 1.5053 | 0.1490 | |

4.8001 | 3.1609 | 0.3806 | |

7.2419 | 4.7598 | 0.5211 | |

9.6679 | 6.3723 | 0.7668 | |

11.9338 | 7.8549 | 0.8887 | |

14.0312 | 9.2817 | 1.0657 | |

16.0600 | 10.7771 | 1.3207 | |

17.8096 | 12.0495 | 1.4135 | |

19.4335 | 13.2069 | 1.6429 | |

20.6934 | 14.0239 | 1.6813 | |

21.6963 | 14.6790 | 1.8646 | |

22.1532 | 14.9131 | 1.9266 |

Values of the wall temperature distribution

0.7 | 1 | 7 | |

24.9385 | 13.7810 | 1.2988 | |

78.2596 | 47.5798 | 3.4913 | |

82.9188 | 50.5583 | 3.8423 | |

85.8007 | 52.3905 | 4.0603 | |

88.3705 | 54.0104 | 4.2166 | |

91.0592 | 55.6998 | 4.3814 | |

93.9733 | 57.4598 | 4.5348 | |

97.4257 | 59.2523 | 4.6891 | |

101.4657 | 61.1798 | 4.8683 | |

106.1772 | 63.4915 | 5.0758 | |

111.7045 | 66.3902 | 5.3383 | |

118.3370 | 70.0617 | 5.6635 | |

126.5161 | 74.7520 | 6.0838 |

The velocity and temperature profiles near the lower stagnation point of the sphere,

Velocity profiles for various values of

Temperature profiles for various values of

Figures

Velocity profiles for various values of

Temperature profiles for various values of

Figures

Velocity profiles for various values of

Temperature profiles for various values of

In this paper, we have numerically studied the problem of mixed convection boundary layer flow from a solid sphere in a micropolar fluid, generated by Newtonian heating. It is shown that the mixed convection or buoyancy parameter

when

when

when

when

near the lower stagnation point of the sphere, the velocity profiles decrease while the temperature profiles increase when the parameter

near the lower stagnation point of the sphere, the velocity profiles increase while the temperature profiles decrease when the parameter

near the lower stagnation point of the sphere, both the velocity and temperature profiles decrease when the Prandtl number

The authors gratefully acknowledge the financial supports received from the Ministry of Higher Education, Malaysia (UKM-ST-07-FRGS0036-2009) and a research grant (RDU 090308) from the Universiti Malaysia Pahang.