Symmetry Criteria for the Equality of Interior and Exterior Shape Factors With Exact Solutions
Lienhard (2019, “Exterior Shape Factors From Interior Shape Factors,” ASME J. Heat Mass Transfer-Trans. ASME, 141(6), p. 061301) reported that the shape factor of the interior of a simply-connected region (Ω) is equal to that of its exterior (ℝ2\Ω) under the same boundary conditions. In that study,...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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ASME International
2024
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| Online Access: | https://hdl.handle.net/1721.1/155554 |
| _version_ | 1811083879746895872 |
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| author | McKee, Kyle I. Lienhard, John H. |
| author_facet | McKee, Kyle I. Lienhard, John H. |
| author_sort | McKee, Kyle I. |
| collection | MIT |
| description | Lienhard (2019, “Exterior Shape Factors From Interior Shape Factors,” ASME J. Heat Mass Transfer-Trans. ASME, 141(6), p. 061301) reported that the shape factor of the interior of a simply-connected region (Ω) is equal to that of its exterior (ℝ2\Ω) under the same boundary conditions. In that study, numerical examples supported the claim in particular cases; for example, it was shown that for certain boundary conditions on circles and squares, the conjecture holds. In this paper, we show that the conjecture is not generally true, unless some additional condition is met. We proceed by elucidating why the conjecture does in fact hold in all of the examples analyzed by Lienhard. We thus deduce a simple criterion which, when satisfied, ensures the equality of interior and exterior shape factors in general. Our criterion notably relies on a beautiful and little-known symmetry method due to Hersch which we introduce in a tutorial manner. In addition, we derive a new formula for the shape factor of objects meeting our N-fold symmetry criterion, encompassing exact solutions for regular polygons and more complex shapes. |
| first_indexed | 2024-09-23T12:41:04Z |
| format | Article |
| id | mit-1721.1/155554 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T12:41:04Z |
| publishDate | 2024 |
| publisher | ASME International |
| record_format | dspace |
| spelling | mit-1721.1/1555542024-09-09T04:46:40Z Symmetry Criteria for the Equality of Interior and Exterior Shape Factors With Exact Solutions McKee, Kyle I. Lienhard, John H. Lienhard (2019, “Exterior Shape Factors From Interior Shape Factors,” ASME J. Heat Mass Transfer-Trans. ASME, 141(6), p. 061301) reported that the shape factor of the interior of a simply-connected region (Ω) is equal to that of its exterior (ℝ2\Ω) under the same boundary conditions. In that study, numerical examples supported the claim in particular cases; for example, it was shown that for certain boundary conditions on circles and squares, the conjecture holds. In this paper, we show that the conjecture is not generally true, unless some additional condition is met. We proceed by elucidating why the conjecture does in fact hold in all of the examples analyzed by Lienhard. We thus deduce a simple criterion which, when satisfied, ensures the equality of interior and exterior shape factors in general. Our criterion notably relies on a beautiful and little-known symmetry method due to Hersch which we introduce in a tutorial manner. In addition, we derive a new formula for the shape factor of objects meeting our N-fold symmetry criterion, encompassing exact solutions for regular polygons and more complex shapes. 2024-07-09T20:35:01Z 2024-07-09T20:35:01Z 2024-07-04 2024-07-09T20:31:02Z Article http://purl.org/eprint/type/JournalArticle 2832-8450 2832-8469 https://hdl.handle.net/1721.1/155554 McKee, K. I., and Lienhard, J. H. (July 4, 2024). "Symmetry Criteria for the Equality of Interior and Exterior Shape Factors With Exact Solutions." ASME. J. Heat Mass Transfer. November 2024; 146(11): 111401. en 10.1115/1.4065741 ASME Journal of Heat and Mass Transfer Creative Commons Attribution https://creativecommons.org/licenses/by/4.0/ application/pdf ASME International Author |
| spellingShingle | McKee, Kyle I. Lienhard, John H. Symmetry Criteria for the Equality of Interior and Exterior Shape Factors With Exact Solutions |
| title | Symmetry Criteria for the Equality of Interior and Exterior Shape Factors With Exact Solutions |
| title_full | Symmetry Criteria for the Equality of Interior and Exterior Shape Factors With Exact Solutions |
| title_fullStr | Symmetry Criteria for the Equality of Interior and Exterior Shape Factors With Exact Solutions |
| title_full_unstemmed | Symmetry Criteria for the Equality of Interior and Exterior Shape Factors With Exact Solutions |
| title_short | Symmetry Criteria for the Equality of Interior and Exterior Shape Factors With Exact Solutions |
| title_sort | symmetry criteria for the equality of interior and exterior shape factors with exact solutions |
| url | https://hdl.handle.net/1721.1/155554 |
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