### Abstract

The Boeder differential equation is solved in this work over a wide range of α, yielding the probability density functions (PDF), that describe the average orientations of rod-like macromolecules in a flowing liquid. The quantity α is the ratio of the hydrodynamic shear rate to the rotational diffusion coefficient. It characterises the coupling of the motion of the macromolecules in the hydrodynamic flow to their thermal diffusion. Previous analytical work is limited to approximate solutions for small values of α. Special analytical as well as numerical methods are developed in the present work in order to calculate accurately the PDF for a range of α covering several orders of magnitude, 10^{-6} ≤ α ≤ 10^{8}. The mathematical nature of the differential equation is revealed as a singular perturbation problem when α becomes large. Scaling results are obtained over the differential equation for α ≥ 10^{3}. Monte Carlo Brownian simulations are also constructed and shown to agree with the numerical solutions of the differential equation in the bulk of the flowing liquid, for an extensive range of α. This confirms the solidity of the developed analytical and numerical methods.

Original language | English |
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Pages (from-to) | 393-399 |

Number of pages | 7 |

Journal | Computational Materials Science |

Volume | 18 |

Issue number | 3-4 |

Publication status | Published - Sep 2000 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Materials Science(all)

### Cite this

*Computational Materials Science*,

*18*(3-4), 393-399.

**Exact solutions of the Boeder differential equation for macromolecular orientations in a flowing liquid.** / Khater, A.; Tannous, C.; Hijazi, A.

Research output: Contribution to journal › Article

*Computational Materials Science*, vol. 18, no. 3-4, pp. 393-399.

}

TY - JOUR

T1 - Exact solutions of the Boeder differential equation for macromolecular orientations in a flowing liquid

AU - Khater, A.

AU - Tannous, C.

AU - Hijazi, A.

PY - 2000/9

Y1 - 2000/9

N2 - The Boeder differential equation is solved in this work over a wide range of α, yielding the probability density functions (PDF), that describe the average orientations of rod-like macromolecules in a flowing liquid. The quantity α is the ratio of the hydrodynamic shear rate to the rotational diffusion coefficient. It characterises the coupling of the motion of the macromolecules in the hydrodynamic flow to their thermal diffusion. Previous analytical work is limited to approximate solutions for small values of α. Special analytical as well as numerical methods are developed in the present work in order to calculate accurately the PDF for a range of α covering several orders of magnitude, 10-6 ≤ α ≤ 108. The mathematical nature of the differential equation is revealed as a singular perturbation problem when α becomes large. Scaling results are obtained over the differential equation for α ≥ 103. Monte Carlo Brownian simulations are also constructed and shown to agree with the numerical solutions of the differential equation in the bulk of the flowing liquid, for an extensive range of α. This confirms the solidity of the developed analytical and numerical methods.

AB - The Boeder differential equation is solved in this work over a wide range of α, yielding the probability density functions (PDF), that describe the average orientations of rod-like macromolecules in a flowing liquid. The quantity α is the ratio of the hydrodynamic shear rate to the rotational diffusion coefficient. It characterises the coupling of the motion of the macromolecules in the hydrodynamic flow to their thermal diffusion. Previous analytical work is limited to approximate solutions for small values of α. Special analytical as well as numerical methods are developed in the present work in order to calculate accurately the PDF for a range of α covering several orders of magnitude, 10-6 ≤ α ≤ 108. The mathematical nature of the differential equation is revealed as a singular perturbation problem when α becomes large. Scaling results are obtained over the differential equation for α ≥ 103. Monte Carlo Brownian simulations are also constructed and shown to agree with the numerical solutions of the differential equation in the bulk of the flowing liquid, for an extensive range of α. This confirms the solidity of the developed analytical and numerical methods.

UR - http://www.scopus.com/inward/record.url?scp=0001078552&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0001078552&partnerID=8YFLogxK

M3 - Article

VL - 18

SP - 393

EP - 399

JO - Computational Materials Science

JF - Computational Materials Science

SN - 0927-0256

IS - 3-4

ER -