### Abstract

The prediction of the microstructure of complex fluids, such as liquid crystalline polymers or concentrated rigid rod suspensions, is often based on a moment equation for the orientation dyad, 〈pp〉. Here the unit vector p represents the alignment of a constituent component of the dispersed phase. For homogeneous shear, the velocity gradient is ∇u - O_{-}, O _{-}, with unknown sign = constant. In a frame of reference relative to the local velocity, the orientation dyad is governed by the following ordinary differential equation (Doi and Edwards, 1986; Bird et al., 1987; Larson, 1999): d〈pp〉/dr - FO[O_{-}, O_{-}. 〈pp〉+〈pp〉.O_{-}, O_{-} - 2O_{-}, O_{-}: 〈pppp〉] + [1/3 l - 〈pp〉 + U (〈pp〉.〈pp〉-〈pp〉:〈pppp〉)] The above second order moment equation assumes that the aspect ratio of the dispersed phase is large compared with unity and that the rotary diffusion coefficient does not depend on the microstructure. The Péclet number FO = unknown sign compares the relative importance of a characteristic time scale associated with an external flow field and a characteristic time scale associated with microhydrodynamic fluctuations. The dimensionless group U compares the relative importance of the excluded volume potential and the potential for rotary diffusion. The dimensionless time t° 6D_{R}t, where DR represents a constant rotary diffusion coefficient in orientation space. Prediction of the low order statistical properties of the microstructure based on the foregoing moment equation requires knowledge of the local flow field and a closure model for the orientation tetrad, 〈pppp〉. Unfortunately, the widespread use of moment equations to describe the microstructure has been limited by the absence of a practical and accurate closure model for the orientation tetrad. A new closure approach for the orientation tetrad that explicitly retains the six-fold symmetry and six-fold contraction properties of the exact fourth order moment has been developed by researchers at Michigan State University (see Petty et al., 1999; Nguyen et al., 2001; Kini et al., 2003; Mandal et al., 2004). At high concentrations (i.e., large U) and in the absence of an external field (i.e., Pe =0), the new theory predicts that all realizable microstructures relax to multiple equilibrium realizable prolate states. Furthermore, in the presence of homogeneous shear flows and for large values of U, the orientation director may show realizable periodic behavior relative to the flow direction. This presentation will summarize recent results obtained by applying the new microstructure theory to a class of complex fluids subjected to homogeneous shear. The theory is used to predict the allowable microstructures and the concomitant rheological properties of the suspension induced by a simple shear flow. The proposed tetrad closure unifies previous theories of fiber suspensions and theories of complex fluids, such as liquid crystalline polymers.

Original language | English (US) |
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Title of host publication | AIChE Annual Meeting, Conference Proceedings |

Pages | 679-680 |

Number of pages | 2 |

State | Published - 2005 |

Externally published | Yes |

Event | 05AIChE: 2005 AIChE Annual Meeting and Fall Showcase - Cincinnati, OH, United States |

### Other

Other | 05AIChE: 2005 AIChE Annual Meeting and Fall Showcase |
---|---|

Country | United States |

City | Cincinnati, OH |

Period | 10/30/05 → 11/4/05 |

### Profile

### Keywords

- Closure approximation
- Fiber suspensions
- Flow induced alignment
- Liquid crystalline polymers
- Microstructure
- Orientation statistics
- Self-alignment
- Simple shear
- Suspensions

### ASJC Scopus subject areas

- Engineering(all)

### Cite this

*AIChE Annual Meeting, Conference Proceedings*(pp. 679-680)

**Microstructure of multiphase fluids in homogeneous shear flows.** / Kim, Yochan; Petty, Charles A.; Benard, Andre.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*AIChE Annual Meeting, Conference Proceedings.*pp. 679-680, 05AIChE: 2005 AIChE Annual Meeting and Fall Showcase, Cincinnati, OH, United States, 30-4 November.

}

TY - CHAP

T1 - Microstructure of multiphase fluids in homogeneous shear flows

AU - Kim,Yochan

AU - Petty,Charles A.

AU - Benard,Andre

PY - 2005

Y1 - 2005

N2 - The prediction of the microstructure of complex fluids, such as liquid crystalline polymers or concentrated rigid rod suspensions, is often based on a moment equation for the orientation dyad, 〈pp〉. Here the unit vector p represents the alignment of a constituent component of the dispersed phase. For homogeneous shear, the velocity gradient is ∇u - O-, O -, with unknown sign = constant. In a frame of reference relative to the local velocity, the orientation dyad is governed by the following ordinary differential equation (Doi and Edwards, 1986; Bird et al., 1987; Larson, 1999): d〈pp〉/dr - FO[O-, O-. 〈pp〉+〈pp〉.O-, O- - 2O-, O-: 〈pppp〉] + [1/3 l - 〈pp〉 + U (〈pp〉.〈pp〉-〈pp〉:〈pppp〉)] The above second order moment equation assumes that the aspect ratio of the dispersed phase is large compared with unity and that the rotary diffusion coefficient does not depend on the microstructure. The Péclet number FO = unknown sign compares the relative importance of a characteristic time scale associated with an external flow field and a characteristic time scale associated with microhydrodynamic fluctuations. The dimensionless group U compares the relative importance of the excluded volume potential and the potential for rotary diffusion. The dimensionless time t° 6DRt, where DR represents a constant rotary diffusion coefficient in orientation space. Prediction of the low order statistical properties of the microstructure based on the foregoing moment equation requires knowledge of the local flow field and a closure model for the orientation tetrad, 〈pppp〉. Unfortunately, the widespread use of moment equations to describe the microstructure has been limited by the absence of a practical and accurate closure model for the orientation tetrad. A new closure approach for the orientation tetrad that explicitly retains the six-fold symmetry and six-fold contraction properties of the exact fourth order moment has been developed by researchers at Michigan State University (see Petty et al., 1999; Nguyen et al., 2001; Kini et al., 2003; Mandal et al., 2004). At high concentrations (i.e., large U) and in the absence of an external field (i.e., Pe =0), the new theory predicts that all realizable microstructures relax to multiple equilibrium realizable prolate states. Furthermore, in the presence of homogeneous shear flows and for large values of U, the orientation director may show realizable periodic behavior relative to the flow direction. This presentation will summarize recent results obtained by applying the new microstructure theory to a class of complex fluids subjected to homogeneous shear. The theory is used to predict the allowable microstructures and the concomitant rheological properties of the suspension induced by a simple shear flow. The proposed tetrad closure unifies previous theories of fiber suspensions and theories of complex fluids, such as liquid crystalline polymers.

AB - The prediction of the microstructure of complex fluids, such as liquid crystalline polymers or concentrated rigid rod suspensions, is often based on a moment equation for the orientation dyad, 〈pp〉. Here the unit vector p represents the alignment of a constituent component of the dispersed phase. For homogeneous shear, the velocity gradient is ∇u - O-, O -, with unknown sign = constant. In a frame of reference relative to the local velocity, the orientation dyad is governed by the following ordinary differential equation (Doi and Edwards, 1986; Bird et al., 1987; Larson, 1999): d〈pp〉/dr - FO[O-, O-. 〈pp〉+〈pp〉.O-, O- - 2O-, O-: 〈pppp〉] + [1/3 l - 〈pp〉 + U (〈pp〉.〈pp〉-〈pp〉:〈pppp〉)] The above second order moment equation assumes that the aspect ratio of the dispersed phase is large compared with unity and that the rotary diffusion coefficient does not depend on the microstructure. The Péclet number FO = unknown sign compares the relative importance of a characteristic time scale associated with an external flow field and a characteristic time scale associated with microhydrodynamic fluctuations. The dimensionless group U compares the relative importance of the excluded volume potential and the potential for rotary diffusion. The dimensionless time t° 6DRt, where DR represents a constant rotary diffusion coefficient in orientation space. Prediction of the low order statistical properties of the microstructure based on the foregoing moment equation requires knowledge of the local flow field and a closure model for the orientation tetrad, 〈pppp〉. Unfortunately, the widespread use of moment equations to describe the microstructure has been limited by the absence of a practical and accurate closure model for the orientation tetrad. A new closure approach for the orientation tetrad that explicitly retains the six-fold symmetry and six-fold contraction properties of the exact fourth order moment has been developed by researchers at Michigan State University (see Petty et al., 1999; Nguyen et al., 2001; Kini et al., 2003; Mandal et al., 2004). At high concentrations (i.e., large U) and in the absence of an external field (i.e., Pe =0), the new theory predicts that all realizable microstructures relax to multiple equilibrium realizable prolate states. Furthermore, in the presence of homogeneous shear flows and for large values of U, the orientation director may show realizable periodic behavior relative to the flow direction. This presentation will summarize recent results obtained by applying the new microstructure theory to a class of complex fluids subjected to homogeneous shear. The theory is used to predict the allowable microstructures and the concomitant rheological properties of the suspension induced by a simple shear flow. The proposed tetrad closure unifies previous theories of fiber suspensions and theories of complex fluids, such as liquid crystalline polymers.

KW - Closure approximation

KW - Fiber suspensions

KW - Flow induced alignment

KW - Liquid crystalline polymers

KW - Microstructure

KW - Orientation statistics

KW - Self-alignment

KW - Simple shear

KW - Suspensions

UR - http://www.scopus.com/inward/record.url?scp=33645580610&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33645580610&partnerID=8YFLogxK

M3 - Conference contribution

SP - 679

EP - 680

BT - AIChE Annual Meeting, Conference Proceedings

ER -