### Abstract

We consider the existence of solutions for a non-Newtonian fluid that is based upon a nonlinear dumb-bell model. It is shown that a rigorous existence proof can be obtained for solutions on a bounded domain at arbitrary values of Deborah number provided the model includes a spatial diffusion term that is usually neglected in the derivation of the model by assuming that the structure is spatially homogeneous. Although this diffusion term is critical to the existence proof, it is expected to be numerically small compared to other terms in the constitutive model, except possibly in the vicinity of very large stress gradients, which it will tend to smooth out. The proof also requires that the stress always remain bounded. Although it is likely that this will be true for a model with a nonlinear (FENE) spring, it is difficult to prove rigorously. Hence, in the existence proof we resort to an ad hoc assumption that is equivalent to asserting that the polymer breaks (degrades) if the end-to-end distance exceeds some prescribed values that is less than the full cotour length but is otherwise arbitrary.

Original language | English (US) |
---|---|

Pages (from-to) | 257-287 |

Number of pages | 31 |

Journal | Journal of Non-Newtonian Fluid Mechanics |

Volume | 33 |

Issue number | 3 |

DOIs | |

State | Published - 1989 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Fluid Flow and Transfer Processes

### Cite this

**Existence of solutions for all Deborah numbers for a non-Newtonian model modified to include diffusion.** / Fox, Ardith El-Kareh; Gary Leal, L.

Research output: Contribution to journal › Article

*Journal of Non-Newtonian Fluid Mechanics*, vol. 33, no. 3, pp. 257-287. https://doi.org/10.1016/0377-0257(89)80002-3

}

TY - JOUR

T1 - Existence of solutions for all Deborah numbers for a non-Newtonian model modified to include diffusion

AU - Fox, Ardith El-Kareh

AU - Gary Leal, L.

PY - 1989

Y1 - 1989

N2 - We consider the existence of solutions for a non-Newtonian fluid that is based upon a nonlinear dumb-bell model. It is shown that a rigorous existence proof can be obtained for solutions on a bounded domain at arbitrary values of Deborah number provided the model includes a spatial diffusion term that is usually neglected in the derivation of the model by assuming that the structure is spatially homogeneous. Although this diffusion term is critical to the existence proof, it is expected to be numerically small compared to other terms in the constitutive model, except possibly in the vicinity of very large stress gradients, which it will tend to smooth out. The proof also requires that the stress always remain bounded. Although it is likely that this will be true for a model with a nonlinear (FENE) spring, it is difficult to prove rigorously. Hence, in the existence proof we resort to an ad hoc assumption that is equivalent to asserting that the polymer breaks (degrades) if the end-to-end distance exceeds some prescribed values that is less than the full cotour length but is otherwise arbitrary.

AB - We consider the existence of solutions for a non-Newtonian fluid that is based upon a nonlinear dumb-bell model. It is shown that a rigorous existence proof can be obtained for solutions on a bounded domain at arbitrary values of Deborah number provided the model includes a spatial diffusion term that is usually neglected in the derivation of the model by assuming that the structure is spatially homogeneous. Although this diffusion term is critical to the existence proof, it is expected to be numerically small compared to other terms in the constitutive model, except possibly in the vicinity of very large stress gradients, which it will tend to smooth out. The proof also requires that the stress always remain bounded. Although it is likely that this will be true for a model with a nonlinear (FENE) spring, it is difficult to prove rigorously. Hence, in the existence proof we resort to an ad hoc assumption that is equivalent to asserting that the polymer breaks (degrades) if the end-to-end distance exceeds some prescribed values that is less than the full cotour length but is otherwise arbitrary.

UR - http://www.scopus.com/inward/record.url?scp=0024922206&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0024922206&partnerID=8YFLogxK

U2 - 10.1016/0377-0257(89)80002-3

DO - 10.1016/0377-0257(89)80002-3

M3 - Article

AN - SCOPUS:0024922206

VL - 33

SP - 257

EP - 287

JO - Journal of Non-Newtonian Fluid Mechanics

JF - Journal of Non-Newtonian Fluid Mechanics

SN - 0377-0257

IS - 3

ER -