KFU. Card publication. A Modified Formulation of the Problem of the Steady Finger in a Hele-Shaw Cell

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FEDERAL UNIVERSITY

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A MODIFIED FORMULATION OF THE PROBLEM OF THE STEADY FINGER IN A HELE-SHAW CELL

Form of presentation

Articles in international journals and collections

Year of publication

2019

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Authors, employees of KFU

Alimov Mars Myasumovich, author

Bibliographic description in the original language

Alimov M. M., A Modified Formulation of the Problem of the Steady Finger in a Hele-Shaw Cell//UCHENYE ZAPISKI KAZANSKOGO UNIVERSITETA-SERIYA FIZIKO-MATEMATICHESKIE NAUKI. - 2018. - Vol.160, Is.3. - P.462-476.

Annotation

The problem of the motion of steady fingers in a Hele-Shaw cell in case of nonzero surface tension was posed more than 60 years ago. However, it has not been completely solved. The analysis in the classical formulation by Saffman and Taylor has revealed that the solution has a singularity at the infinity point where the base of the finger lies. This makes it difficult to realize a correct numerical analysis of the problem. A new modified formulation of the problem is proposed. In this formulation, we postulate that the distance at which the finger is completely formed is limited, which removes the potential source of solution singularity. This distance is, in fact, an additional parameter of the modified formulation. For the case of zero surface tension, it has been established that, in the limit, when this distance tends to infinity, the solution of the problem in the modified formulation tends to the Saffman and Taylor finger solution. Specific examples show that it is sufficient to choose a quantity of order 1 as the value of the additional parameter of the modified formulation. This will already ensure the practical coincidence of the finger shape with the form that corresponds to the classical solution for the case of zero surface tension. It has been also established that the principle of minimum energy dissipation rate formulated in the local form can be used when dissipation is estimated only in a small neighborhood of the free boundary. It has been shown that this principle selects a finger of thickness 0.5 from the family of solutions of the problem in the new formulation in full agreement with the results of the experiments in the case of zero surface tension. The revealed qualities of the modified formulation make it promising from the point of view of its generalization to the case of nonzero surface tension and the possibility of its correct numerical analysis.

Keywords

Hele-Shaw flow, free boundary problems, exact solution, minimum energy dissipation rate

The name of the journal

UCHENYE ZAPISKI KAZANSKOGO UNIVERSITETA-SERIYA FIZIKO-MATEMATICHESKIE NAUKI

Please use this ID to quote from or refer to the card

https://repository.kpfu.ru/eng/?p_id=202374&p_lang=2

Full metadata record

Field DC	Value	Language
dc.contributor.author	Alimov Mars Myasumovich	ru_RU
dc.date.accessioned	2019-01-01T00:00:00Z	ru_RU
dc.date.available	2019-01-01T00:00:00Z	ru_RU
dc.date.issued	2019	ru_RU
dc.identifier.citation	Alimov M. M., A Modified Formulation of the Problem of the Steady Finger in a Hele-Shaw Cell//UCHENYE ZAPISKI KAZANSKOGO UNIVERSITETA-SERIYA FIZIKO-MATEMATICHESKIE NAUKI. - 2018. - Vol.160, Is.3. - P.462-476.	ru_RU
dc.identifier.uri	https://repository.kpfu.ru/eng/?p_id=202374&p_lang=2	ru_RU
dc.description.abstract	UCHENYE ZAPISKI KAZANSKOGO UNIVERSITETA-SERIYA FIZIKO-MATEMATICHESKIE NAUKI	ru_RU
dc.description.abstract	The problem of the motion of steady fingers in a Hele-Shaw cell in case of nonzero surface tension was posed more than 60 years ago. However, it has not been completely solved. The analysis in the classical formulation by Saffman and Taylor has revealed that the solution has a singularity at the infinity point where the base of the finger lies. This makes it difficult to realize a correct numerical analysis of the problem. A new modified formulation of the problem is proposed. In this formulation, we postulate that the distance at which the finger is completely formed is limited, which removes the potential source of solution singularity. This distance is, in fact, an additional parameter of the modified formulation. For the case of zero surface tension, it has been established that, in the limit, when this distance tends to infinity, the solution of the problem in the modified formulation tends to the Saffman and Taylor finger solution. Specific examples show that it is sufficient to choose a quantity of order 1 as the value of the additional parameter of the modified formulation. This will already ensure the practical coincidence of the finger shape with the form that corresponds to the classical solution for the case of zero surface tension. It has been also established that the principle of minimum energy dissipation rate formulated in the local form can be used when dissipation is estimated only in a small neighborhood of the free boundary. It has been shown that this principle selects a finger of thickness 0.5 from the family of solutions of the problem in the new formulation in full agreement with the results of the experiments in the case of zero surface tension. The revealed qualities of the modified formulation make it promising from the point of view of its generalization to the case of nonzero surface tension and the possibility of its correct numerical analysis.	ru_RU
dc.language.iso	ru	ru_RU
dc.subject	Hele-Shaw flow	ru_RU
dc.subject	free boundary problems	ru_RU
dc.subject	exact solution	ru_RU
dc.subject	minimum energy dissipation rate	ru_RU
dc.title	A Modified Formulation of the Problem of the Steady Finger in a Hele-Shaw Cell	ru_RU
dc.type	Articles in international journals and collections	ru_RU