### Abstract

Original language | English |
---|---|

Pages (from-to) | 1094-1114 |

Number of pages | 20 |

Journal | Numerical Methods for Partial Differential Equations |

Volume | 24 |

Issue number | 4 |

DOIs | |

Publication status | Published - Nov 2007 |

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### Keywords

- free surface flow
- pseudospectral methods
- rotating elliptical cylinder
- differential equations
- numerical mathematics

### Cite this

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*Numerical Methods for Partial Differential Equations*, vol. 24, no. 4, pp. 1094-1114. https://doi.org/10.1002/num.20307

**Numerical solution of the free-surface viscous flow on a horizontal rotating elliptical cylinder.** / Hunt, R.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Numerical solution of the free-surface viscous flow on a horizontal rotating elliptical cylinder

AU - Hunt, R.

PY - 2007/11

Y1 - 2007/11

N2 - The numerical solution of the free-surface fluid flow on a rotating elliptical cylinder is presented. Up to the present, research has concentrated on the circular cylinder for which steady solutions are the main interest. However, for noncircular cylinders, such as the ellipse, steady solutions are no longer possible, but there will be periodic solutions in which the solution is repeated after one full revolution of the cylinder. It is this new aspect that makes the investigation of noncircular cylinders novel. Here we consider both the time-dependent and periodic solutions for zero Reynolds number fluid flow. The numerical solution is expedited by first mapping the fluid film domain onto a rectangle such that the position of the free-surface is determined as part of the solution. For the time-dependent case a simple time-marching method of lines approach is adopted. For the periodic solution the discretised nonlinear equations have to be solved simultaneously over a time period. The resulting large system of equations is solved using Newton's method in which the form of the Jacobian enables a straightforward decomposition to be implemented, which makes matrix inversion manageable. In the periodic case all derivatives have been approximated pseudospectrally with the time derivative approximated by a differentiation matrix which has been specially derived so that the weight of fluid is algebraically conserved. Of interest is the solution for which the weight of fluid is at its maximum possible value, and this has been obtained by increasing the weight until a consistency break-down occurs. Time-dependent solutions do not produce the periodic solution after a long time-scale but have protuberances which are constantly appearing and disappearing. Periodic solutions exhibit spectral accuracy solutions and maximum supportable weight solutions have been obtained for ranges of eccentricity and angular velocity. The maximum weights are less than and approximately proportional to those obtained for the circular case. The shapes of maximum weight solutions is distinctly different from sub-maximum weight solutions.

AB - The numerical solution of the free-surface fluid flow on a rotating elliptical cylinder is presented. Up to the present, research has concentrated on the circular cylinder for which steady solutions are the main interest. However, for noncircular cylinders, such as the ellipse, steady solutions are no longer possible, but there will be periodic solutions in which the solution is repeated after one full revolution of the cylinder. It is this new aspect that makes the investigation of noncircular cylinders novel. Here we consider both the time-dependent and periodic solutions for zero Reynolds number fluid flow. The numerical solution is expedited by first mapping the fluid film domain onto a rectangle such that the position of the free-surface is determined as part of the solution. For the time-dependent case a simple time-marching method of lines approach is adopted. For the periodic solution the discretised nonlinear equations have to be solved simultaneously over a time period. The resulting large system of equations is solved using Newton's method in which the form of the Jacobian enables a straightforward decomposition to be implemented, which makes matrix inversion manageable. In the periodic case all derivatives have been approximated pseudospectrally with the time derivative approximated by a differentiation matrix which has been specially derived so that the weight of fluid is algebraically conserved. Of interest is the solution for which the weight of fluid is at its maximum possible value, and this has been obtained by increasing the weight until a consistency break-down occurs. Time-dependent solutions do not produce the periodic solution after a long time-scale but have protuberances which are constantly appearing and disappearing. Periodic solutions exhibit spectral accuracy solutions and maximum supportable weight solutions have been obtained for ranges of eccentricity and angular velocity. The maximum weights are less than and approximately proportional to those obtained for the circular case. The shapes of maximum weight solutions is distinctly different from sub-maximum weight solutions.

KW - free surface flow

KW - pseudospectral methods

KW - rotating elliptical cylinder

KW - differential equations

KW - numerical mathematics

UR - http://dx.doi.org/10.1002/num.20307

U2 - 10.1002/num.20307

DO - 10.1002/num.20307

M3 - Article

VL - 24

SP - 1094

EP - 1114

JO - Numerical Methods for Partial Differential Equations

JF - Numerical Methods for Partial Differential Equations

SN - 0749-159X

IS - 4

ER -