^{1}

^{*}

^{2}

^{*}

^{3}

^{*}

The rotationally symmetric flow of a micropolar fluid in the presence of an infinite rotating disk has been studied numerically. The equations of motion are reduced to a system of ordinary differential equations, which in turn are solved numerically using SOR method and Simpson’s (1/3) rule. The results are calculated for different values of the parameter
*s* (the ratio of angular velocities of disc and fluid) and the suction parameter
*a*. Moreover, three different sets of the values of non-dimensional material constants related to micropolar behavior of the fluid have been chosen arbitrarily. The calculations have been carried out using three different grid sizes to check the accuracy of the results. The research concludes that the micropolar fluids flow resembles with that of Newtonian fluids when the material constants become close to zero. The comparison of these results is presented for possible values of the parameter
*s*.

Eringen [

The laminar flow due to an infinite rotating disk was first theoretically investigated with an approximate method by Von Karman [

In this research, the numerical solutions of the rotationally symmetric slow of micropolar fluids in the presence of an infinite rotating disk have been discussed. In order to find the numerical solution of the problem, the Navier Stokes equations are reduced to ordinary differential equations by using similarity transformations [

The purpose of using these numerical techniques for numerical solution is that, the finite difference approximations are found to be discrete techniques wherein the domain of interest is represented by a set of points or nodes and information among these points is commonly obtained by using Taylor series expansions while the finite element method employs piecewise continuous polynomials to interpolate among nodal points. The finite difference techniques are very easy to understand and straight forward for computational analysis.

The cylindrical polar coordinates

where

The following similarity transformations are used:

where

where primes denote differentiation with respect to_{1}, C_{2}, C_{3}, C_{4}, C_{5} and C_{6} all are non dimensional.

The boundary conditions are

In order to obtain the numerical solution of nonlinear ordinary differential Equations (6) to (10), we approximate these equations by central difference approximation at a typical point

where h denotes a grid size,

We now solve numerically the finite difference Equations (12) to (16) by using SOR method subject to the appropriate boundary conditions (11). The first order ordinary differential Equation (5) integrate by Simpson’s (1/3) rule subject to the initial condition

The computation has been checked for different of the relaxation parameter between

where n denotes the number of iterations and U stands for each of F, G, L, M and N. The above procedure is repeated for higher grid levels

Numerical results have been found to observe the effect of parameters s and a on velocity field and microrotation. In order to check the accuracy of the results for velocity components F, G and H and the microrotation components L, M and N, the calculations have been carried out on three different grid sizes namely h = 0.1, 0.05 and 0.025. The three different sets of the material constants C_{1}, C_{2}, C_{3}, C_{4}, C_{5} and C_{6} in the

The velocity derivatives at the surface of the disc are given in

Cases | C_{1 } | C_{2 } | C_{3 } | C_{4 } | C_{5 } | C_{6 } |
---|---|---|---|---|---|---|

I | 0.1 | 0.3 | 0.4 | 0.5 | 0.7 | 0.8 |

II | 0.5 | 1.5 | 2.0 | 3.0 | 3.5 | 4.0 |

III | 0.3 | 0.5 | 1.5 | 2.5 | 3.0 | 3.5 |

s | ||||
---|---|---|---|---|

Micropolar fluids | Newtonian fluids [ | Micropolar fluids | Newtonian fluids [ | |

0.0 | 0.51022801 | 0.51022912 | −0.61592027 | −0.61591916 |

−0.10 | 0.49130449 | 0.49130550 | −0.60825160 | −0.60825056 |

−0.15 | 0.47627299 | 0.47627301 | −0.58762407 | −0.58761507 |

−0.16 | 0.47332786 | 0.47332988 | −0.57766843 | −0.57766748 |

h | F | G | H | L | M | N | |
---|---|---|---|---|---|---|---|

0.05 | 0.000 | 0.000000 | 1.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

1.000 | 0.179240 | 0.475672 | −0.264748 | −0.132547 | −0.005065 | 0.061811 | |

2.000 | 0.116682 | 0.198637 | −0.569258 | −0.100132 | −0.017006 | 0.030972 | |

3.000 | 0.055710 | 0.079067 | −0.736567 | −0.055573 | −0.015614 | 0.009449 | |

4.000 | 0.023615 | 0.030491 | −0.811870 | −0.027335 | −0.010107 | 0.001537 | |

5.000 | 0.009307 | 0.011294 | −0.842782 | −0.012569 | −0.005525 | −0.000480 | |

6.000 | 0.003340 | 0.003853 | −0.854550 | −0.005392 | −0.002665 | −0.000683 | |

7.000 | 0.000936 | 0.001029 | −0.858460 | −0.001912 | −0.001024 | −0.000438 | |

8.000 | 0.000000 | 0.000000 | −0.859246 | 0.000000 | 0.000000 | 0.000000 | |

0.025 | 0.000 | 0.000000 | 1.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

1.000 | 0.179756 | 0.475490 | −0.265459 | −0.132669 | −0.005076 | 0.061869 | |

2.000 | 0.116687 | 0.198008 | −0.570529 | −0.100185 | −0.017102 | 0.030837 | |

3.000 | 0.055505 | 0.078594 | −0.737561 | −0.055501 | −0.015655 | 0.009322 | |

4.000 | 0.023472 | 0.030247 | −0.812501 | −0.027248 | −0.010100 | 0.001480 | |

5.000 | 0.009235 | 0.011179 | −0.843204 | −0.012505 | −0.005508 | −0.000501 | |

6.000 | 0.003306 | 0.003803 | −0.854869 | −0.005355 | −0.002651 | −0.000689 | |

7.000 | 0.000925 | 0.001015 | −0.858737 | −0.001898 | −0.001017 | −0.000438 | |

8.000 | 0.000000 | 0.000000 | −0.859515 | 0.000000 | 0.000000 | 0.000000 | |

0.012 | 0.000 | 0.000000 | 1.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

1.000 | 0.179545 | 0.475495 | −0.265239 | −0.132672 | −0.005128 | 0.061799 | |

2.000 | 0.116731 | 0.198355 | −0.570108 | −0.100151 | −0.017086 | 0.030901 | |

3.000 | 0.055642 | 0.078881 | −0.737367 | −0.055528 | −0.015651 | 0.009401 | |

4.000 | 0.023554 | 0.030398 | −0.812532 | −0.027287 | −0.010111 | 0.001519 | |

5.000 | 0.009270 | 0.011246 | −0.843348 | −0.012534 | −0.005518 | −0.000486 | |

6.000 | 0.003321 | 0.003830 | −0.855060 | −0.005372 | −0.002658 | −0.000685 | |

7.000 | 0.000930 | 0.001023 | −0.858946 | −0.001904 | −0.001020 | −0.000437 | |

8.000 | 0.000000 | 0.000000 | −0.859728 | 0.000000 | 0.000000 | 0.000000 |

h | F | G | H | L | M | N | |
---|---|---|---|---|---|---|---|

0.05 | 0.000 | 0.000000 | 1.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

1.000 | 0.163850 | 0.477984 | −0.248418 | −0.136301 | −0.006879 | 0.061170 | |

2.000 | 0.091781 | 0.186209 | −0.511450 | −0.107574 | −0.017863 | 0.026623 | |

3.000 | 0.029148 | 0.040950 | −0.625998 | −0.065370 | −0.014898 | −0.000802 | |

4.000 | 0.001462 | −0.032056 | −0.651904 | −0.037765 | −0.008380 | −0.014413 | |

5.000 | −0.006261 | −0.068608 | −0.645015 | −0.022444 | −0.003547 | −0.020480 | |

6.000 | −0.006178 | −0.086359 | −0.631914 | −0.014246 | −0.000984 | −0.022865 | |

7.000 | −0.004098 | −0.094583 | −0.621554 | −0.009695 | 0.000058 | −0.023092 | |

8.000 | −0.002131 | −0.098170 | −0.615414 | −0.006718 | 0.000315 | −0.021139 | |

9.000 | −0.000784 | −0.099592 | −0.612605 | −0.003921 | 0.000229 | −0.015200 | |

10.000 | 0.000000 | −0.100000 | −0.611900 | 0.000000 | 0.000000 | 0.000000 | |

0.025 | 0.000 | 0.000000 | 1.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

1.000 | 0.163982 | 0.477946 | −0.248625 | −0.136390 | −0.006934 | 0.061146 | |

2.000 | 0.091849 | 0.186099 | −0.511859 | −0.107607 | −0.017903 | 0.026584 | |

3.000 | 0.029187 | 0.040848 | −0.626508 | −0.065366 | −0.014914 | −0.000831 | |

4.000 | 0.001495 | −0.032127 | −0.652484 | −0.037750 | −0.008385 | −0.014430 | |

5.000 | −0.006233 | −0.068650 | −0.645656 | −0.022430 | −0.003549 | −0.020489 | |

6.000 | −0.006157 | −0.086381 | −0.632605 | −0.014236 | −0.000986 | −0.022869 | |

7.000 | −0.004083 | −0.094593 | −0.622282 | −0.009688 | 0.000056 | −0.023094 | |

8.000 | −0.002124 | −0.098174 | −0.616163 | −0.006714 | 0.000313 | −0.021140 | |

9.000 | −0.000781 | −0.099593 | −0.613365 | −0.003919 | 0.000228 | −0.015201 | |

10.000 | 0.000000 | −0.100000 | −0.612663 | 0.000000 | 0.000000 | 0.000000 | |

0.012 | 0.000 | 0.000000 | 1.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |

1.000 | 0.164012 | 0.477938 | 0.248672 | 0.136410 | 0.006948 | 0.061140 | |

2.000 | 0.091861 | 0.186080 | −0.511947 | −0.107613 | −0.017915 | 0.026576 | |

3.000 | 0.029189 | 0.040832 | −0.626607 | −0.065365 | −0.014920 | −0.000836 | |

4.000 | 0.001493 | −0.032137 | −0.652584 | −0.037747 | −0.008387 | −0.014431 | |

5.000 | −0.006235 | −0.068656 | −0.645753 | −0.022427 | −0.003549 | −0.020488 | |

6.000 | −0.006158 | −0.086385 | −0.632699 | −0.014233 | −0.000986 | −0.022868 | |

7.000 | −0.004084 | −0.094595 | −0.622374 | −0.009687 | 0.000056 | −0.023093 | |

8.000 | −0.002124 | −0.098174 | −0.616255 | −0.006714 | 0.000313 | −0.021139 | |

9.000 | −0.000781 | −0.099593 | −0.613456 | −0.003919 | 0.000228 | −0.015201 | |

10.000 | 0.000000 | −0.100000 | −0.612754 | 0.000000 | 0.000000 | 0.000000 |

h | F | G | H | L | M | N | |
---|---|---|---|---|---|---|---|

0.05 | 0.000 | 0.000000 | 1.000000 | −1.500000 | 0.000000 | 0.000000 | 0.000000 |

1.000 | −0.002528 | 0.088033 | −1.522923 | −0.082486 | −0.002232 | −0.010100 | |

2.000 | −0.008317 | −0.109105 | −1.507162 | −0.049570 | −0.000038 | −0.035059 | |

3.000 | −0.003198 | −0.150499 | −1.495820 | −0.029503 | 0.000588 | −0.040513 | |

4.000 | −0.000582 | −0.158720 | −1.492485 | −0.020994 | 0.000374 | −0.040488 | |

5.000 | 0.000199 | −0.160254 | −1.492282 | −0.017493 | 0.000139 | −0.039504 | |

6.000 | 0.000329 | −0.160529 | −1.492864 | −0.015606 | 0.000020 | −0.038042 | |

7.000 | 0.000272 | −0.160560 | −1.493479 | −0.013809 | −0.000022 | −0.035449 | |

8.000 | 0.000170 | −0.160498 | −1.493923 | −0.011234 | −0.000028 | −0.030365 | |

9.000 | 0.000069 | −0.160330 | −1.494159 | −0.007036 | −0.000018 | −0.020215 | |

10.000 | 0.000000 | −0.160000 | −1.494221 | 0.000000 | 0.000000 | 0.000000 | |

0.025 | 0.000 | 0.000000 | 1.000000 | −1.500000 | 0.000000 | 0.000000 | 0.000000 |

1.000 | −0.011205 | 0.060019 | −1.506979 | −0.089021 | −0.000175 | −0.016609 | |

2.000 | −0.005664 | −0.144366 | −1.484961 | −0.057573 | 0.003504 | −0.044292 | |

3.000 | 0.006414 | −0.180500 | −1.487086 | −0.035908 | 0.003608 | −0.049458 | |

4.000 | 0.009843 | −0.179762 | −1.504564 | −0.025014 | 0.002249 | −0.047525 | |

5.000 | 0.008359 | −0.173125 | −1.523221 | −0.019566 | 0.001041 | −0.044332 | |

6.000 | 0.005517 | −0.167406 | −1.537139 | −0.016459 | 0.000315 | −0.041014 | |

7.000 | 0.002973 | −0.163677 | −1.545512 | −0.014051 | −0.000021 | −0.037073 | |

8.000 | 0.001260 | −0.161617 | −1.549599 | −0.011245 | −0.000115 | −0.031068 | |

9.000 | 0.000352 | −0.160591 | −1.551095 | −0.006991 | −0.000082 | −0.020325 | |

10.000 | 0.000000 | −0.160000 | −1.551378 | 0.000000 | 0.000000 | 0.000000 | |

0.012 | 0.000 | 0.000000 | 1.000000 | −1.500000 | 0.000000 | 0.000000 | 0.000000 |

1.000 | −0.067561 | 0.195665 | −1.454176 | −0.078957 | −0.009610 | 0.012609 | |

2.000 | −0.108889 | −0.008283 | −1.266693 | −0.045617 | −0.003901 | −0.010169 | |

3.000 | −0.108214 | −0.076914 | −1.045830 | −0.025831 | 0.001519 | −0.019670 | |

4.000 | −0.093475 | −0.109525 | −0.842828 | −0.018374 | 0.004688 | −0.024761 | |

5.000 | −0.073748 | −0.129694 | −0.675170 | −0.016381 | 0.006288 | −0.028789 | |

6.000 | −0.053074 | −0.143288 | −0.548445 | −0.016000 | 0.006704 | −0.032064 | |

7.000 | −0.034184 | −0.152062 | −0.461664 | −0.015355 | 0.006121 | −0.033907 | |

8.000 | −0.018827 | −0.157114 | −0.409328 | −0.013376 | 0.004737 | −0.032798 | |

9.000 | −0.007588 | −0.159491 | −0.383588 | −0.008934 | 0.002729 | −0.024962 | |

10.000 | 0.000000 | −0.160000 | −0.376529 | 0.000000 | 0.000000 | 0.000000 |