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Quantification of Cardiomyocyte Alignment from Three-Dimensional (3D) Confocal Microscopy of Engineered Tissue
- William J. Kowalski, Fangping Yuan, Takeichiro Nakane, Hidetoshi Masumoto, Marc Dwenger, Fei Ye, Joseph P. Tinney, Bradley B. Keller
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- Journal:
- Microscopy and Microanalysis / Volume 23 / Issue 4 / August 2017
- Published online by Cambridge University Press:
- 19 June 2017, pp. 826-842
- Print publication:
- August 2017
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Biological tissues have complex, three-dimensional (3D) organizations of cells and matrix factors that provide the architecture necessary to meet morphogenic and functional demands. Disordered cell alignment is associated with congenital heart disease, cardiomyopathy, and neurodegenerative diseases and repairing or replacing these tissues using engineered constructs may improve regenerative capacity. However, optimizing cell alignment within engineered tissues requires quantitative 3D data on cell orientations and both efficient and validated processing algorithms. We developed an automated method to measure local 3D orientations based on structure tensor analysis and incorporated an adaptive subregion size to account for multiple scales. Our method calculates the statistical concentration parameter, κ, to quantify alignment, as well as the traditional orientational order parameter. We validated our method using synthetic images and accurately measured principal axis and concentration. We then applied our method to confocal stacks of cleared, whole-mount engineered cardiac tissues generated from human-induced pluripotent stem cells or embryonic chick cardiac cells and quantified cardiomyocyte alignment. We found significant differences in alignment based on cellular composition and tissue geometry. These results from our synthetic images and confocal data demonstrate the efficiency and accuracy of our method to measure alignment in 3D tissues.
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- By Mitchell Aboulafia, Frederick Adams, Marilyn McCord Adams, Robert M. Adams, Laird Addis, James W. Allard, David Allison, William P. Alston, Karl Ameriks, C. Anthony Anderson, David Leech Anderson, Lanier Anderson, Roger Ariew, David Armstrong, Denis G. Arnold, E. J. Ashworth, Margaret Atherton, Robin Attfield, Bruce Aune, Edward Wilson Averill, Jody Azzouni, Kent Bach, Andrew Bailey, Lynne Rudder Baker, Thomas R. Baldwin, Jon Barwise, George Bealer, William Bechtel, Lawrence C. Becker, Mark A. Bedau, Ernst Behler, José A. Benardete, Ermanno Bencivenga, Jan Berg, Michael Bergmann, Robert L. Bernasconi, Sven Bernecker, Bernard Berofsky, Rod Bertolet, Charles J. Beyer, Christian Beyer, Joseph Bien, Joseph Bien, Peg Birmingham, Ivan Boh, James Bohman, Daniel Bonevac, Laurence BonJour, William J. Bouwsma, Raymond D. Bradley, Myles Brand, Richard B. Brandt, Michael E. Bratman, Stephen E. Braude, Daniel Breazeale, Angela Breitenbach, Jason Bridges, David O. Brink, Gordon G. Brittan, Justin Broackes, Dan W. Brock, Aaron Bronfman, Jeffrey E. Brower, Bartosz Brozek, Anthony Brueckner, Jeffrey Bub, Lara Buchak, Otavio Bueno, Ann E. Bumpus, Robert W. Burch, John Burgess, Arthur W. Burks, Panayot Butchvarov, Robert E. Butts, Marina Bykova, Patrick Byrne, David Carr, Noël Carroll, Edward S. Casey, Victor Caston, Victor Caston, Albert Casullo, Robert L. Causey, Alan K. L. Chan, Ruth Chang, Deen K. Chatterjee, Andrew Chignell, Roderick M. Chisholm, Kelly J. Clark, E. J. Coffman, Robin Collins, Brian P. Copenhaver, John Corcoran, John Cottingham, Roger Crisp, Frederick J. Crosson, Antonio S. Cua, Phillip D. Cummins, Martin Curd, Adam Cureton, Andrew Cutrofello, Stephen Darwall, Paul Sheldon Davies, Wayne A. Davis, Timothy Joseph Day, Claudio de Almeida, Mario De Caro, Mario De Caro, John Deigh, C. F. Delaney, Daniel C. Dennett, Michael R. DePaul, Michael Detlefsen, Daniel Trent Devereux, Philip E. Devine, John M. Dillon, Martin C. Dillon, Robert DiSalle, Mary Domski, Alan Donagan, Paul Draper, Fred Dretske, Mircea Dumitru, Wilhelm Dupré, Gerald Dworkin, John Earman, Ellery Eells, Catherine Z. Elgin, Berent Enç, Ronald P. Endicott, Edward Erwin, John Etchemendy, C. Stephen Evans, Susan L. Feagin, Solomon Feferman, Richard Feldman, Arthur Fine, Maurice A. Finocchiaro, William FitzPatrick, Richard E. Flathman, Gvozden Flego, Richard Foley, Graeme Forbes, Rainer Forst, Malcolm R. Forster, Daniel Fouke, Patrick Francken, Samuel Freeman, Elizabeth Fricker, Miranda Fricker, Michael Friedman, Michael Fuerstein, Richard A. Fumerton, Alan Gabbey, Pieranna Garavaso, Daniel Garber, Jorge L. A. Garcia, Robert K. Garcia, Don Garrett, Philip Gasper, Gerald Gaus, Berys Gaut, Bernard Gert, Roger F. Gibson, Cody Gilmore, Carl Ginet, Alan H. Goldman, Alvin I. Goldman, Alfonso Gömez-Lobo, Lenn E. Goodman, Robert M. Gordon, Stefan Gosepath, Jorge J. E. Gracia, Daniel W. Graham, George A. Graham, Peter J. Graham, Richard E. Grandy, I. Grattan-Guinness, John Greco, Philip T. Grier, Nicholas Griffin, Nicholas Griffin, David A. Griffiths, Paul J. Griffiths, Stephen R. Grimm, Charles L. Griswold, Charles B. Guignon, Pete A. Y. Gunter, Dimitri Gutas, Gary Gutting, Paul Guyer, Kwame Gyekye, Oscar A. Haac, Raul Hakli, Raul Hakli, Michael Hallett, Edward C. Halper, Jean Hampton, R. James Hankinson, K. R. Hanley, Russell Hardin, Robert M. Harnish, William Harper, David Harrah, Kevin Hart, Ali Hasan, William Hasker, John Haugeland, Roger Hausheer, William Heald, Peter Heath, Richard Heck, John F. Heil, Vincent F. Hendricks, Stephen Hetherington, Francis Heylighen, Kathleen Marie Higgins, Risto Hilpinen, Harold T. Hodes, Joshua Hoffman, Alan Holland, Robert L. Holmes, Richard Holton, Brad W. Hooker, Terence E. Horgan, Tamara Horowitz, Paul Horwich, Vittorio Hösle, Paul Hoβfeld, Daniel Howard-Snyder, Frances Howard-Snyder, Anne Hudson, Deal W. Hudson, Carl A. Huffman, David L. Hull, Patricia Huntington, Thomas Hurka, Paul Hurley, Rosalind Hursthouse, Guillermo Hurtado, Ronald E. Hustwit, Sarah Hutton, Jonathan Jenkins Ichikawa, Harry A. Ide, David Ingram, Philip J. Ivanhoe, Alfred L. Ivry, Frank Jackson, Dale Jacquette, Joseph Jedwab, Richard Jeffrey, David Alan Johnson, Edward Johnson, Mark D. Jordan, Richard Joyce, Hwa Yol Jung, Robert Hillary Kane, Tomis Kapitan, Jacquelyn Ann K. Kegley, James A. Keller, Ralph Kennedy, Sergei Khoruzhii, Jaegwon Kim, Yersu Kim, Nathan L. King, Patricia Kitcher, Peter D. Klein, E. D. Klemke, Virginia Klenk, George L. Kline, Christian Klotz, Simo Knuuttila, Joseph J. Kockelmans, Konstantin Kolenda, Sebastian Tomasz Kołodziejczyk, Isaac Kramnick, Richard Kraut, Fred Kroon, Manfred Kuehn, Steven T. Kuhn, Henry E. Kyburg, John Lachs, Jennifer Lackey, Stephen E. Lahey, Andrea Lavazza, Thomas H. Leahey, Joo Heung Lee, Keith Lehrer, Dorothy Leland, Noah M. Lemos, Ernest LePore, Sarah-Jane Leslie, Isaac Levi, Andrew Levine, Alan E. Lewis, Daniel E. Little, Shu-hsien Liu, Shu-hsien Liu, Alan K. L. Chan, Brian Loar, Lawrence B. Lombard, John Longeway, Dominic McIver Lopes, Michael J. Loux, E. J. Lowe, Steven Luper, Eugene C. Luschei, William G. Lycan, David Lyons, David Macarthur, Danielle Macbeth, Scott MacDonald, Jacob L. Mackey, Louis H. Mackey, Penelope Mackie, Edward H. Madden, Penelope Maddy, G. B. Madison, Bernd Magnus, Pekka Mäkelä, Rudolf A. Makkreel, David Manley, William E. Mann (W.E.M.), Vladimir Marchenkov, Peter Markie, Jean-Pierre Marquis, Ausonio Marras, Mike W. Martin, A. P. Martinich, William L. McBride, David McCabe, Storrs McCall, Hugh J. McCann, Robert N. McCauley, John J. McDermott, Sarah McGrath, Ralph McInerny, Daniel J. McKaughan, Thomas McKay, Michael McKinsey, Brian P. McLaughlin, Ernan McMullin, Anthonie Meijers, Jack W. Meiland, William Jason Melanson, Alfred R. Mele, Joseph R. Mendola, Christopher Menzel, Michael J. Meyer, Christian B. Miller, David W. Miller, Peter Millican, Robert N. Minor, Phillip Mitsis, James A. Montmarquet, Michael S. Moore, Tim Moore, Benjamin Morison, Donald R. Morrison, Stephen J. Morse, Paul K. Moser, Alexander P. D. Mourelatos, Ian Mueller, James Bernard Murphy, Mark C. Murphy, Steven Nadler, Jan Narveson, Alan Nelson, Jerome Neu, Samuel Newlands, Kai Nielsen, Ilkka Niiniluoto, Carlos G. Noreña, Calvin G. Normore, David Fate Norton, Nikolaj Nottelmann, Donald Nute, David S. Oderberg, Steve Odin, Michael O’Rourke, Willard G. Oxtoby, Heinz Paetzold, George S. Pappas, Anthony J. Parel, Lydia Patton, R. P. Peerenboom, Francis Jeffry Pelletier, Adriaan T. Peperzak, Derk Pereboom, Jaroslav Peregrin, Glen Pettigrove, Philip Pettit, Edmund L. Pincoffs, Andrew Pinsent, Robert B. Pippin, Alvin Plantinga, Louis P. Pojman, Richard H. Popkin, John F. Post, Carl J. Posy, William J. Prior, Richard Purtill, Michael Quante, Philip L. Quinn, Philip L. Quinn, Elizabeth S. Radcliffe, Diana Raffman, Gerard Raulet, Stephen L. Read, Andrews Reath, Andrew Reisner, Nicholas Rescher, Henry S. Richardson, Robert C. Richardson, Thomas Ricketts, Wayne D. Riggs, Mark Roberts, Robert C. Roberts, Luke Robinson, Alexander Rosenberg, Gary Rosenkranz, Bernice Glatzer Rosenthal, Adina L. Roskies, William L. Rowe, T. M. Rudavsky, Michael Ruse, Bruce Russell, Lilly-Marlene Russow, Dan Ryder, R. M. Sainsbury, Joseph Salerno, Nathan Salmon, Wesley C. Salmon, Constantine Sandis, David H. Sanford, Marco Santambrogio, David Sapire, Ruth A. Saunders, Geoffrey Sayre-McCord, Charles Sayward, James P. Scanlan, Richard Schacht, Tamar Schapiro, Frederick F. Schmitt, Jerome B. Schneewind, Calvin O. Schrag, Alan D. Schrift, George F. Schumm, Jean-Loup Seban, David N. Sedley, Kenneth Seeskin, Krister Segerberg, Charlene Haddock Seigfried, Dennis M. Senchuk, James F. Sennett, William Lad Sessions, Stewart Shapiro, Tommie Shelby, Donald W. Sherburne, Christopher Shields, Roger A. Shiner, Sydney Shoemaker, Robert K. Shope, Kwong-loi Shun, Wilfried Sieg, A. John Simmons, Robert L. Simon, Marcus G. Singer, Georgette Sinkler, Walter Sinnott-Armstrong, Matti T. Sintonen, Lawrence Sklar, Brian Skyrms, Robert C. Sleigh, Michael Anthony Slote, Hans Sluga, Barry Smith, Michael Smith, Robin Smith, Robert Sokolowski, Robert C. Solomon, Marta Soniewicka, Philip Soper, Ernest Sosa, Nicholas Southwood, Paul Vincent Spade, T. L. S. Sprigge, Eric O. Springsted, George J. Stack, Rebecca Stangl, Jason Stanley, Florian Steinberger, Sören Stenlund, Christopher Stephens, James P. Sterba, Josef Stern, Matthias Steup, M. A. Stewart, Leopold Stubenberg, Edith Dudley Sulla, Frederick Suppe, Jere Paul Surber, David George Sussman, Sigrún Svavarsdóttir, Zeno G. Swijtink, Richard Swinburne, Charles C. Taliaferro, Robert B. Talisse, John Tasioulas, Paul Teller, Larry S. Temkin, Mark Textor, H. S. Thayer, Peter Thielke, Alan Thomas, Amie L. Thomasson, Katherine Thomson-Jones, Joshua C. Thurow, Vzalerie Tiberius, Terrence N. Tice, Paul Tidman, Mark C. Timmons, William Tolhurst, James E. Tomberlin, Rosemarie Tong, Lawrence Torcello, Kelly Trogdon, J. D. Trout, Robert E. Tully, Raimo Tuomela, John Turri, Martin M. Tweedale, Thomas Uebel, Jennifer Uleman, James Van Cleve, Harry van der Linden, Peter van Inwagen, Bryan W. Van Norden, René van Woudenberg, Donald Phillip Verene, Samantha Vice, Thomas Vinci, Donald Wayne Viney, Barbara Von Eckardt, Peter B. M. Vranas, Steven J. Wagner, William J. Wainwright, Paul E. Walker, Robert E. Wall, Craig Walton, Douglas Walton, Eric Watkins, Richard A. Watson, Michael V. Wedin, Rudolph H. Weingartner, Paul Weirich, Paul J. Weithman, Carl Wellman, Howard Wettstein, Samuel C. Wheeler, Stephen A. White, Jennifer Whiting, Edward R. Wierenga, Michael Williams, Fred Wilson, W. Kent Wilson, Kenneth P. Winkler, John F. Wippel, Jan Woleński, Allan B. Wolter, Nicholas P. Wolterstorff, Rega Wood, W. Jay Wood, Paul Woodruff, Alison Wylie, Gideon Yaffe, Takashi Yagisawa, Yutaka Yamamoto, Keith E. Yandell, Xiaomei Yang, Dean Zimmerman, Günter Zoller, Catherine Zuckert, Michael Zuckert, Jack A. Zupko (J.A.Z.)
- Edited by Robert Audi, University of Notre Dame, Indiana
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- Book:
- The Cambridge Dictionary of Philosophy
- Published online:
- 05 August 2015
- Print publication:
- 27 April 2015, pp ix-xxx
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- By Rose Teteki Abbey, K. C. Abraham, David Tuesday Adamo, LeRoy H. Aden, Efrain Agosto, Victor Aguilan, Gillian T. W. Ahlgren, Charanjit Kaur AjitSingh, Dorothy B E A Akoto, Giuseppe Alberigo, Daniel E. Albrecht, Ruth Albrecht, Daniel O. Aleshire, Urs Altermatt, Anand Amaladass, Michael Amaladoss, James N. Amanze, Lesley G. Anderson, Thomas C. Anderson, Victor Anderson, Hope S. Antone, María Pilar Aquino, Paula Arai, Victorio Araya Guillén, S. Wesley Ariarajah, Ellen T. Armour, Brett Gregory Armstrong, Atsuhiro Asano, Naim Stifan Ateek, Mahmoud Ayoub, John Alembillah Azumah, Mercedes L. García Bachmann, Irena Backus, J. Wayne Baker, Mieke Bal, Lewis V. Baldwin, William Barbieri, António Barbosa da Silva, David Basinger, Bolaji Olukemi Bateye, Oswald Bayer, Daniel H. Bays, Rosalie Beck, Nancy Elizabeth Bedford, Guy-Thomas Bedouelle, Chorbishop Seely Beggiani, Wolfgang Behringer, Christopher M. Bellitto, Byard Bennett, Harold V. Bennett, Teresa Berger, Miguel A. Bernad, Henley Bernard, Alan E. Bernstein, Jon L. Berquist, Johannes Beutler, Ana María Bidegain, Matthew P. Binkewicz, Jennifer Bird, Joseph Blenkinsopp, Dmytro Bondarenko, Paulo Bonfatti, Riet en Pim Bons-Storm, Jessica A. Boon, Marcus J. Borg, Mark Bosco, Peter C. Bouteneff, François Bovon, William D. Bowman, Paul S. Boyer, David Brakke, Richard E. Brantley, Marcus Braybrooke, Ian Breward, Ênio José da Costa Brito, Jewel Spears Brooker, Johannes Brosseder, Nicholas Canfield Read Brown, Robert F. Brown, Pamela K. Brubaker, Walter Brueggemann, Bishop Colin O. Buchanan, Stanley M. Burgess, Amy Nelson Burnett, J. Patout Burns, David B. Burrell, David Buttrick, James P. Byrd, Lavinia Byrne, Gerado Caetano, Marcos Caldas, Alkiviadis Calivas, William J. Callahan, Salvatore Calomino, Euan K. Cameron, William S. Campbell, Marcelo Ayres Camurça, Daniel F. Caner, Paul E. Capetz, Carlos F. Cardoza-Orlandi, Patrick W. Carey, Barbara Carvill, Hal Cauthron, Subhadra Mitra Channa, Mark D. Chapman, James H. Charlesworth, Kenneth R. Chase, Chen Zemin, Luciano Chianeque, Philip Chia Phin Yin, Francisca H. Chimhanda, Daniel Chiquete, John T. Chirban, Soobin Choi, Robert Choquette, Mita Choudhury, Gerald Christianson, John Chryssavgis, Sejong Chun, Esther Chung-Kim, Charles M. A. Clark, Elizabeth A. Clark, Sathianathan Clarke, Fred Cloud, John B. Cobb, W. Owen Cole, John A Coleman, John J. Collins, Sylvia Collins-Mayo, Paul K. Conkin, Beth A. Conklin, Sean Connolly, Demetrios J. Constantelos, Michael A. Conway, Paula M. Cooey, Austin Cooper, Michael L. Cooper-White, Pamela Cooper-White, L. William Countryman, Sérgio Coutinho, Pamela Couture, Shannon Craigo-Snell, James L. Crenshaw, David Crowner, Humberto Horacio Cucchetti, Lawrence S. Cunningham, Elizabeth Mason Currier, Emmanuel Cutrone, Mary L. Daniel, David D. Daniels, Robert Darden, Rolf Darge, Isaiah Dau, Jeffry C. Davis, Jane Dawson, Valentin Dedji, John W. de Gruchy, Paul DeHart, Wendy J. Deichmann Edwards, Miguel A. De La Torre, George E. Demacopoulos, Thomas de Mayo, Leah DeVun, Beatriz de Vasconcellos Dias, Dennis C. Dickerson, John M. Dillon, Luis Miguel Donatello, Igor Dorfmann-Lazarev, Susanna Drake, Jonathan A. Draper, N. Dreher Martin, Otto Dreydoppel, Angelyn Dries, A. J. Droge, Francis X. D'Sa, Marilyn Dunn, Nicole Wilkinson Duran, Rifaat Ebied, Mark J. Edwards, William H. Edwards, Leonard H. Ehrlich, Nancy L. Eiesland, Martin Elbel, J. Harold Ellens, Stephen Ellingson, Marvin M. Ellison, Robert Ellsberg, Jean Bethke Elshtain, Eldon Jay Epp, Peter C. Erb, Tassilo Erhardt, Maria Erling, Noel Leo Erskine, Gillian R. Evans, Virginia Fabella, Michael A. Fahey, Edward Farley, Margaret A. Farley, Wendy Farley, Robert Fastiggi, Seena Fazel, Duncan S. Ferguson, Helwar Figueroa, Paul Corby Finney, Kyriaki Karidoyanes FitzGerald, Thomas E. FitzGerald, John R. Fitzmier, Marie Therese Flanagan, Sabina Flanagan, Claude Flipo, Ronald B. Flowers, Carole Fontaine, David Ford, Mary Ford, Stephanie A. Ford, Jim Forest, William Franke, Robert M. Franklin, Ruth Franzén, Edward H. Friedman, Samuel Frouisou, Lorelei F. Fuchs, Jojo M. Fung, Inger Furseth, Richard R. Gaillardetz, Brandon Gallaher, China Galland, Mark Galli, Ismael García, Tharscisse Gatwa, Jean-Marie Gaudeul, Luis María Gavilanes del Castillo, Pavel L. Gavrilyuk, Volney P. Gay, Metropolitan Athanasios Geevargis, Kondothra M. George, Mary Gerhart, Simon Gikandi, Maurice Gilbert, Michael J. Gillgannon, Verónica Giménez Beliveau, Terryl Givens, Beth Glazier-McDonald, Philip Gleason, Menghun Goh, Brian Golding, Bishop Hilario M. Gomez, Michelle A. Gonzalez, Donald K. Gorrell, Roy Gottfried, Tamara Grdzelidze, Joel B. Green, Niels Henrik Gregersen, Cristina Grenholm, Herbert Griffiths, Eric W. Gritsch, Erich S. Gruen, Christoffer H. Grundmann, Paul H. Gundani, Jon P. Gunnemann, Petre Guran, Vidar L. Haanes, Jeremiah M. Hackett, Getatchew Haile, Douglas John Hall, Nicholas Hammond, Daphne Hampson, Jehu J. Hanciles, Barry Hankins, Jennifer Haraguchi, Stanley S. Harakas, Anthony John Harding, Conrad L. Harkins, J. William Harmless, Marjory Harper, Amir Harrak, Joel F. Harrington, Mark W. Harris, Susan Ashbrook Harvey, Van A. Harvey, R. Chris Hassel, Jione Havea, Daniel Hawk, Diana L. Hayes, Leslie Hayes, Priscilla Hayner, S. Mark Heim, Simo Heininen, Richard P. Heitzenrater, Eila Helander, David Hempton, Scott H. Hendrix, Jan-Olav Henriksen, Gina Hens-Piazza, Carter Heyward, Nicholas J. Higham, David Hilliard, Norman A. Hjelm, Peter C. Hodgson, Arthur Holder, M. Jan Holton, Dwight N. Hopkins, Ronnie Po-chia Hsia, Po-Ho Huang, James Hudnut-Beumler, Jennifer S. Hughes, Leonard M. Hummel, Mary E. Hunt, Laennec Hurbon, Mark Hutchinson, Susan E. Hylen, Mary Beth Ingham, H. Larry Ingle, Dale T. Irvin, Jon Isaak, Paul John Isaak, Ada María Isasi-Díaz, Hans Raun Iversen, Margaret C. Jacob, Arthur James, Maria Jansdotter-Samuelsson, David Jasper, Werner G. Jeanrond, Renée Jeffery, David Lyle Jeffrey, Theodore W. Jennings, David H. Jensen, Robin Margaret Jensen, David Jobling, Dale A. Johnson, Elizabeth A. Johnson, Maxwell E. Johnson, Sarah Johnson, Mark D. Johnston, F. Stanley Jones, James William Jones, John R. Jones, Alissa Jones Nelson, Inge Jonsson, Jan Joosten, Elizabeth Judd, Mulambya Peggy Kabonde, Robert Kaggwa, Sylvester Kahakwa, Isaac Kalimi, Ogbu U. Kalu, Eunice Kamaara, Wayne C. Kannaday, Musimbi Kanyoro, Veli-Matti Kärkkäinen, Frank Kaufmann, Léon Nguapitshi Kayongo, Richard Kearney, Alice A. Keefe, Ralph Keen, Catherine Keller, Anthony J. Kelly, Karen Kennelly, Kathi Lynn Kern, Fergus Kerr, Edward Kessler, George Kilcourse, Heup Young Kim, Kim Sung-Hae, Kim Yong-Bock, Kim Yung Suk, Richard King, Thomas M. King, Robert M. Kingdon, Ross Kinsler, Hans G. Kippenberg, Cheryl A. Kirk-Duggan, Clifton Kirkpatrick, Leonid Kishkovsky, Nadieszda Kizenko, Jeffrey Klaiber, Hans-Josef Klauck, Sidney Knight, Samuel Kobia, Robert Kolb, Karla Ann Koll, Heikki Kotila, Donald Kraybill, Philip D. W. Krey, Yves Krumenacker, Jeffrey Kah-Jin Kuan, Simanga R. Kumalo, Peter Kuzmic, Simon Shui-Man Kwan, Kwok Pui-lan, André LaCocque, Stephen E. Lahey, John Tsz Pang Lai, Emiel Lamberts, Armando Lampe, Craig Lampe, Beverly J. Lanzetta, Eve LaPlante, Lizette Larson-Miller, Ariel Bybee Laughton, Leonard Lawlor, Bentley Layton, Robin A. Leaver, Karen Lebacqz, Archie Chi Chung Lee, Marilyn J. Legge, Hervé LeGrand, D. L. LeMahieu, Raymond Lemieux, Bill J. Leonard, Ellen M. Leonard, Outi Leppä, Jean Lesaulnier, Nantawan Boonprasat Lewis, Henrietta Leyser, Alexei Lidov, Bernard Lightman, Paul Chang-Ha Lim, Carter Lindberg, Mark R. Lindsay, James R. Linville, James C. Livingston, Ann Loades, David Loades, Jean-Claude Loba-Mkole, Lo Lung Kwong, Wati Longchar, Eleazar López, David W. Lotz, Andrew Louth, Robin W. Lovin, William Luis, Frank D. Macchia, Diarmaid N. J. MacCulloch, Kirk R. MacGregor, Marjory A. MacLean, Donald MacLeod, Tomas S. Maddela, Inge Mager, Laurenti Magesa, David G. Maillu, Fortunato Mallimaci, Philip Mamalakis, Kä Mana, Ukachukwu Chris Manus, Herbert Robinson Marbury, Reuel Norman Marigza, Jacqueline Mariña, Antti Marjanen, Luiz C. L. Marques, Madipoane Masenya (ngwan'a Mphahlele), Caleb J. D. Maskell, Steve Mason, Thomas Massaro, Fernando Matamoros Ponce, András Máté-Tóth, Odair Pedroso Mateus, Dinis Matsolo, Fumitaka Matsuoka, John D'Arcy May, Yelena Mazour-Matusevich, Theodore Mbazumutima, John S. McClure, Christian McConnell, Lee Martin McDonald, Gary B. McGee, Thomas McGowan, Alister E. McGrath, Richard J. McGregor, John A. McGuckin, Maud Burnett McInerney, Elsie Anne McKee, Mary B. McKinley, James F. McMillan, Ernan McMullin, Kathleen E. McVey, M. Douglas Meeks, Monica Jyotsna Melanchthon, Ilie Melniciuc-Puica, Everett Mendoza, Raymond A. Mentzer, William W. Menzies, Ina Merdjanova, Franziska Metzger, Constant J. Mews, Marvin Meyer, Carol Meyers, Vasile Mihoc, Gunner Bjerg Mikkelsen, Maria Inêz de Castro Millen, Clyde Lee Miller, Bonnie J. Miller-McLemore, Alexander Mirkovic, Paul Misner, Nozomu Miyahira, R. W. L. Moberly, Gerald Moede, Aloo Osotsi Mojola, Sunanda Mongia, Rebeca Montemayor, James Moore, Roger E. Moore, Craig E. Morrison O.Carm, Jeffry H. Morrison, Keith Morrison, Wilson J. Moses, Tefetso Henry Mothibe, Mokgethi Motlhabi, Fulata Moyo, Henry Mugabe, Jesse Ndwiga Kanyua Mugambi, Peggy Mulambya-Kabonde, Robert Bruce Mullin, Pamela Mullins Reaves, Saskia Murk Jansen, Heleen L. Murre-Van den Berg, Augustine Musopole, Isaac M. T. Mwase, Philomena Mwaura, Cecilia Nahnfeldt, Anne Nasimiyu Wasike, Carmiña Navia Velasco, Thulani Ndlazi, Alexander Negrov, James B. Nelson, David G. Newcombe, Carol Newsom, Helen J. Nicholson, George W. E. Nickelsburg, Tatyana Nikolskaya, Damayanthi M. A. Niles, Bertil Nilsson, Nyambura Njoroge, Fidelis Nkomazana, Mary Beth Norton, Christian Nottmeier, Sonene Nyawo, Anthère Nzabatsinda, Edward T. Oakes, Gerald O'Collins, Daniel O'Connell, David W. Odell-Scott, Mercy Amba Oduyoye, Kathleen O'Grady, Oyeronke Olajubu, Thomas O'Loughlin, Dennis T. Olson, J. Steven O'Malley, Cephas N. Omenyo, Muriel Orevillo-Montenegro, César Augusto Ornellas Ramos, Agbonkhianmeghe E. Orobator, Kenan B. Osborne, Carolyn Osiek, Javier Otaola Montagne, Douglas F. Ottati, Anna May Say Pa, Irina Paert, Jerry G. Pankhurst, Aristotle Papanikolaou, Samuele F. Pardini, Stefano Parenti, Peter Paris, Sung Bae Park, Cristián G. Parker, Raquel Pastor, Joseph Pathrapankal, Daniel Patte, W. Brown Patterson, Clive Pearson, Keith F. Pecklers, Nancy Cardoso Pereira, David Horace Perkins, Pheme Perkins, Edward N. Peters, Rebecca Todd Peters, Bishop Yeznik Petrossian, Raymond Pfister, Peter C. Phan, Isabel Apawo Phiri, William S. F. Pickering, Derrick G. Pitard, William Elvis Plata, Zlatko Plese, John Plummer, James Newton Poling, Ronald Popivchak, Andrew Porter, Ute Possekel, James M. Powell, Enos Das Pradhan, Devadasan Premnath, Jaime Adrían Prieto Valladares, Anne Primavesi, Randall Prior, María Alicia Puente Lutteroth, Eduardo Guzmão Quadros, Albert Rabil, Laurent William Ramambason, Apolonio M. Ranche, Vololona Randriamanantena Andriamitandrina, Lawrence R. Rast, Paul L. Redditt, Adele Reinhartz, Rolf Rendtorff, Pål Repstad, James N. Rhodes, John K. Riches, Joerg Rieger, Sharon H. Ringe, Sandra Rios, Tyler Roberts, David M. Robinson, James M. Robinson, Joanne Maguire Robinson, Richard A. H. Robinson, Roy R. Robson, Jack B. Rogers, Maria Roginska, Sidney Rooy, Rev. Garnett Roper, Maria José Fontelas Rosado-Nunes, Andrew C. Ross, Stefan Rossbach, François Rossier, John D. Roth, John K. Roth, Phillip Rothwell, Richard E. Rubenstein, Rosemary Radford Ruether, Markku Ruotsila, John E. Rybolt, Risto Saarinen, John Saillant, Juan Sanchez, Wagner Lopes Sanchez, Hugo N. Santos, Gerhard Sauter, Gloria L. Schaab, Sandra M. Schneiders, Quentin J. Schultze, Fernando F. Segovia, Turid Karlsen Seim, Carsten Selch Jensen, Alan P. F. Sell, Frank C. Senn, Kent Davis Sensenig, Damían Setton, Bal Krishna Sharma, Carolyn J. Sharp, Thomas Sheehan, N. Gerald Shenk, Christian Sheppard, Charles Sherlock, Tabona Shoko, Walter B. Shurden, Marguerite Shuster, B. Mark Sietsema, Batara Sihombing, Neil Silberman, Clodomiro Siller, Samuel Silva-Gotay, Heikki Silvet, John K. Simmons, Hagith Sivan, James C. Skedros, Abraham Smith, Ashley A. Smith, Ted A. Smith, Daud Soesilo, Pia Søltoft, Choan-Seng (C. S.) Song, Kathryn Spink, Bryan Spinks, Eric O. Springsted, Nicolas Standaert, Brian Stanley, Glen H. Stassen, Karel Steenbrink, Stephen J. Stein, Andrea Sterk, Gregory E. Sterling, Columba Stewart, Jacques Stewart, Robert B. Stewart, Cynthia Stokes Brown, Ken Stone, Anne Stott, Elizabeth Stuart, Monya Stubbs, Marjorie Hewitt Suchocki, David Kwang-sun Suh, Scott W. Sunquist, Keith Suter, Douglas Sweeney, Charles H. Talbert, Shawqi N. Talia, Elsa Tamez, Joseph B. Tamney, Jonathan Y. Tan, Yak-Hwee Tan, Kathryn Tanner, Feiya Tao, Elizabeth S. Tapia, Aquiline Tarimo, Claire Taylor, Mark Lewis Taylor, Bishop Abba Samuel Wolde Tekestebirhan, Eugene TeSelle, M. Thomas Thangaraj, David R. Thomas, Andrew Thornley, Scott Thumma, Marcelo Timotheo da Costa, George E. “Tink” Tinker, Ola Tjørhom, Karen Jo Torjesen, Iain R. Torrance, Fernando Torres-Londoño, Archbishop Demetrios [Trakatellis], Marit Trelstad, Christine Trevett, Phyllis Trible, Johannes Tromp, Paul Turner, Robert G. Tuttle, Archbishop Desmond Tutu, Peter Tyler, Anders Tyrberg, Justin Ukpong, Javier Ulloa, Camillus Umoh, Kristi Upson-Saia, Martina Urban, Monica Uribe, Elochukwu Eugene Uzukwu, Richard Vaggione, Gabriel Vahanian, Paul Valliere, T. J. Van Bavel, Steven Vanderputten, Peter Van der Veer, Huub Van de Sandt, Louis Van Tongeren, Luke A. Veronis, Noel Villalba, Ramón Vinke, Tim Vivian, David Voas, Elena Volkova, Katharina von Kellenbach, Elina Vuola, Timothy Wadkins, Elaine M. Wainwright, Randi Jones Walker, Dewey D. Wallace, Jerry Walls, Michael J. Walsh, Philip Walters, Janet Walton, Jonathan L. Walton, Wang Xiaochao, Patricia A. Ward, David Harrington Watt, Herold D. Weiss, Laurence L. Welborn, Sharon D. Welch, Timothy Wengert, Traci C. West, Merold Westphal, David Wetherell, Barbara Wheeler, Carolinne White, Jean-Paul Wiest, Frans Wijsen, Terry L. Wilder, Felix Wilfred, Rebecca Wilkin, Daniel H. Williams, D. Newell Williams, Michael A. Williams, Vincent L. Wimbush, Gabriele Winkler, Anders Winroth, Lauri Emílio Wirth, James A. Wiseman, Ebba Witt-Brattström, Teofil Wojciechowski, John Wolffe, Kenman L. Wong, Wong Wai Ching, Linda Woodhead, Wendy M. Wright, Rose Wu, Keith E. Yandell, Gale A. Yee, Viktor Yelensky, Yeo Khiok-Khng, Gustav K. K. Yeung, Angela Yiu, Amos Yong, Yong Ting Jin, You Bin, Youhanna Nessim Youssef, Eliana Yunes, Robert Michael Zaller, Valarie H. Ziegler, Barbara Brown Zikmund, Joyce Ann Zimmerman, Aurora Zlotnik, Zhuo Xinping
- Edited by Daniel Patte, Vanderbilt University, Tennessee
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- The Cambridge Dictionary of Christianity
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- 05 August 2012
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- 20 September 2010, pp xi-xliv
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Asymptotic evaluation of oscillatory sums
- Joseph B. Keller, Charles Knessl
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- European Journal of Applied Mathematics / Volume 4 / Issue 4 / December 1993
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- 26 September 2008, pp. 361-379
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A stationary phase method is developed for the asymptotic evaluation, as R → ∞, of oscillatory sums of the form
It is extended to multidimensional sums. Numerical comparisons demonstrate the accuracy of the asymptotic approximations. The results are applied to the practical estimation of the number of lattice points in large domains in ℝ2.
Embryonic ventricular diastolic and systolic pressure-volume relations
- Bradley B. Keller, Joseph P. Tinney, Norman Hu
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- Cardiology in the Young / Volume 4 / Issue 1 / January 1994
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- 19 August 2008, pp. 19-27
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The embryonic heart and vascular bed interact dynamically to support rapid growth of the embryo during cardiovascular development. Pressure-volume relations define ventricular function during alterations in loading conditions. We analyzed these relationships in the embryonic heart in order to define ventricular function and the response of the ventricle and vascular bed to acute changes in preload. We simultaneously measured ventricular pressure and recorded 60 video images per second in n≥6 stage 16, 18 and 21 white Leghorn chick embryos at baseline and during the infusion of 1–2 microliters of physiologic buffer into the venous sinus (sinus venosus). Ventricular tetany was then induced with the topical application of 2 Molar sodium chloride. Video fields were traced for ventricular pressure and epicardial cross-sectional area. Cross-sectional area was converted to volume assuming ellipsoid geometry, and cavity volume was calculated as total volume minus wall volume derived from the tetanized heart. We defined end-diastole at the onset of ventricular contraction and end-systole at maximum pressure/volume ratio. Stroke volume increased linearly with end-diastolic volume. End-diastolic pressure-volume relations were positive and linear, and end-systolic pressure-volume relations were curvilinear downward. Arterial elastance decreased with growth of the embryo and with volume infusion. Pressure-volume loop area, an index of consumption of energy, doubled between the embryonic stages. Thus, embryonic ventricular pressure-volume relations define diastolic and systolic function at rest and in response to altered preload.
On unsymmetrically impinging jets
- Joseph B. Keller
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- Journal of Fluid Mechanics / Volume 211 / February 1990
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- 26 April 2006, pp. 653-655
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Two impinging two-dimensional incompressible inviscid fluid jets of known widths and velocities produce two outgoing jets. The speeds of the outgoing jets are readily determined from the Bernoulli equation. Their two widths and two directions (four quantities) are related by conservation of mass and conservation of two components of momentum (three relations). Because these three conservation relations do not suffice to determine the four unknowns, Milne-Thomson (1968) states on p. 302 that ‘a unique solution is, in general, not possible’. He incorrectly attributes this indeterminateness to disregard of ‘the initial conditions from which this steady motion is supposed to arise’.
Stability of rotating shear flows in shallow water
- Charles Knessl, Joseph B. Keller
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- Journal:
- Journal of Fluid Mechanics / Volume 244 / November 1992
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- 26 April 2006, pp. 605-614
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Asymptotic methods are used to determine the dispersion equation for disturbances of rotating parallel flows in shallow water. From this equation the unstable modes and their growth rates are determined. The solution involves seven asymptotic expansions which are matched together. The results supplement and extend those which have been obtained previously using numerical methods by Griffiths, Killworth & Stern (1982) and by Hayashi & Young (1987).
Stability of linear shear flows in shallow water
- Charles Knessl, Joseph B. Keller
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- Journal of Fluid Mechanics / Volume 303 / 25 November 1995
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- 26 April 2006, pp. 203-214
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The stability or instability of various linear shear flows in shallow water is considered. The linearized equations for waves on the surface of each flow are solved exactly in terms of known special functions. For unbounded shear flows, the exact reflection and transmission coefficients R and T for waves incident on the flow, are found. They are shown to satisfy the relation |R|2= 1+ |T|2, which proves that over reflection occurs at all wavenumbers. For flow bounded by a rigid wall, R is found. The poles of R yield the eigenvalue equation from which the unstable mides can be found. For flow in a channel, with two rigid walls, the eigenvalue equation for the modes is obtained. The results are compared with previous numerical results.
Free surface flow due to a sink
- Jean-Marc Vanden-Broeck, Joseph B. Keller
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- Journal:
- Journal of Fluid Mechanics / Volume 175 / February 1987
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- 21 April 2006, pp. 109-117
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Two-dimensional free surface flows without waves, produced by a submerged sink in a reservoir, are computed numerically for various configurations. For a sink above the horizontal bottom of a layer of fluid, there are solutions for all values of the Froude number F greater than some particular value. However, when the fluid is sufficiently deep, there is an additional solution for one special value of F < 1. The results for a sink at the vertex of a sloping bottom, treated by Craya and by Hocking, and for a sink in fluid of infinite depth, treated by Tuck & Vanden-Broeck, are confirmed and extended. In particular it is shown that as the bottom tends to the horizontal, the solution for a sink at the vertex of a sloping bottom approaches a solution for a horizontal bottom with F = 1. However solutions are found for all values of the Froude number F [ges ] 1 for a sink on a horizontal bottom.
Surfing on solitary waves
- J. M. Vanden-Broeck, Joseph B. Keller
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- Journal:
- Journal of Fluid Mechanics / Volume 198 / January 1989
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- 21 April 2006, pp. 115-125
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The steady motion of a flat surfboard propelled by a solitary wave is considered. The shape of the free surface and the flow of the fluid are determined numerically by series truncation for flows without spray or splash. These flows all bifurcate from the uniform horizontal flow at the critical value of the Froude number. Various limiting cases of these special flows are described analytically. Flows past submerged hydrofoils are discussed also.
Weir flows
- Jean-Marc Vanden-Broeck, Joseph B. Keller
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- Journal of Fluid Mechanics / Volume 176 / March 1987
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- 21 April 2006, pp. 283-293
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The flow of a liquid with a free surface over a weir in a channel is calculated numerically for thin weirs in channels of various depths, and for broad-crested weirs in channels of infinite depth. The results show that the upstream velocity, as well as the entire flow, are determined by the height of the free surface far upstream and by the geometry of the weir and channel, in agreement with observation. The discharge coefficient is computed for a thin weir, and a formula for it is given that applies when the height of the weir is large compared to the height of the upstream free surface above the top of the weir. The coefficients in this formula are close to those found empirically.
Resonantly interacting water waves
- Joseph B. Keller
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- Journal:
- Journal of Fluid Mechanics / Volume 191 / June 1988
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- 21 April 2006, pp. 529-534
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Coupled nonlinear equations are derived for the amplitudes of two small-amplitude resonantly interacting gravity waves in water of non-uniform depth. Such resonance is possible only for wavelengths long compared to the depth. It is shown that the same equations are obtained from the exact Euler equations, from the nonlinear shallow water theory, and from the Boussinesq equations.
Axisymmetric bubble or drop in a uniform flow
- Michael Miksis, Jean-Marc Vanden-Broeck, Joseph B. Keller
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- Journal:
- Journal of Fluid Mechanics / Volume 108 / July 1981
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- 20 April 2006, pp. 89-100
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The deformation of an axisymmetric bubble or drop in a uniform flow of constant velocity U is computed numerically. The flow is assumed to be inviscid and incompressible. The problem is formulated as a nonlinear integrodifferential system of equations for the bubble surface and for the potential function on the surface. These equations are discretized and the resulting algebraic system is solved by Newton's method. For U = 0 the bubble is a sphere. The results show that as U increases the bubble becomes oblate, spreading out in the direction normal to the flow and contracting in the direction of the flow. Then the poles get pushed in and ultimately they touch each other. The results also show that there is a maximum value of the Weber number above which there is no steady axially symmetric bubble. This value is somewhat smaller than the approximate value obtained by Moore (1965) but close to that found by El Sawi (1974). We also compute the added mass, the drag on the bubble, and its terminal velocity in a gravitational field, for large Reynolds numbers.
Capillary waves on a vertical jet
- Joseph B. Keller
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- Journal of Fluid Mechanics / Volume 135 / October 1983
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- 20 April 2006, pp. 171-173
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Effective viscosity of a periodic suspension
- Kevin C. Nunan, Joseph B. Keller
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- Journal of Fluid Mechanics / Volume 142 / May 1984
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- 20 April 2006, pp. 269-287
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The effective viscosity of a suspension is defined to be the four-tensor that relates the average deviatoric stress to the average rate of strain. We determine the effective viscosity of an array of spheres centred on the points of a periodic lattice in an incompressible Newtonian fluid. The formulation involves the traction exerted on a single sphere by the fluid, and an integral equation for this traction is derived. For lattices with cubic symmetry the effective viscosity tensor involves just two parameters. They are computed numerically for simple, body-centred and face-centred cubic lattices of spheres with solute concentrations up to 90% of the close-packing concentration. Asymptotic results for high concentrations are obtained for arbitrary lattice geometries, and found to be in good agreement with the numerical results for cubic lattices. The low-concentration asymptotic expansions of Zuzovsky also agree well with the numerical results.
Jets rising and falling under gravity
- Jean-Marc Vanden-Broeck, Joseph B. Keller
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- Journal:
- Journal of Fluid Mechanics / Volume 124 / November 1982
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- 20 April 2006, pp. 335-345
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Steady two-dimensional jets of inviscid incompressible fluid, rising and falling under gravity, are calculated numkrically. The shape of each jet depends upon a single parameter, the Froude number $\lambda = q_{r\m c}(Qg)^{-\frac{1}{3}}$, which ranges from zero to infinity. Here qc is the velocity at the crest of the jet, i.e. the highest point of the upper surface, Q is the flux in the jet. and g is the acceleration of gravity. For λ = ∞ the jet is slender and parabolic. It becomes thicker as λ decreases, and reaches a limiting form at λ = 0. Then there is a stagnation point at the crest, where the surface makes a 120° angle with itself. This angle is predicted by the same argument Stokes used in his study of water waves.
The problem is formulated as an integro-differential equation for the two free surfaces of the jet, This equation is dlscretized to yield a set of nonlinear equations, which are solved numerically by Newton's method. In addition, asymptotic results for large λ are obtained analytically. Graphs of the results are presented.
Rising bubbles
- Michael J. Miksis, Jean-Marc Vanden-Broeck, Joseph B. Keller
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- Journal:
- Journal of Fluid Mechanics / Volume 123 / October 1982
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- 20 April 2006, pp. 31-41
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The shape of a rising bubble, or of a falling drop, in an incompressible viscous fluid is computed numerically, omitting the condition on the tangential traction at the bubble or drop surface. When the bubble is sufficiently distorted, its top is found to be spherical and its bottom is found to be rather flat. Then the radius of its upper surface is in fair agreement with the formula of Davis & Taylor (1950). This distortion occurs when the effect of gravity is large while that of surface tension is small. When the effect of surface tension is large, the bubble is nearly a sphere.
The shape is found, together with the flow of the surrounding fluid, by assuming that both are steady and axially symmetric, with the Reynolds number being large. The flow is taken to be a potential flow. The boundary condition on the normal component of normal stress, including the viscous stress, is satisfied, but not that on the tangential component. The problem is converted into an integro-differential set of equations, reduced to a set of algebraic equations by a difference method, and solved by Newton's method together with parameter variation.
Wave propagation in a viscoelastic tube containing a viscous fluid
- Sol I. Rubinow, Joseph B. Keller
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- Journal:
- Journal of Fluid Mechanics / Volume 88 / Issue 1 / 13 September 1978
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- 19 April 2006, pp. 181-203
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Small amplitude, axially symmetric waves in a thin-walled viscoelastic tube containing a viscous compressible fluid are considered. Previous authors have found two modes of propagation for such waves but have studied them only in the low frequency, long wavelength limit. We show that there are infinitely many modes and study them at all frequencies. The appropriate dispersion equation was derived previously (Rubinow & Keller 1971) and analysed for an inviscid fluid. Now it is analysed for a viscous fluid. Asymptotic formulae for the propagation constant k are obtained for both low and high frequencies and for various ranges of the parameters characterizing the tube and the fluid. Special attention is paid to the case of a rigid tube and to parameter values that characterize the flow of blood in mammalian arteries. In addition, numerical results are obtained which complement the asymptotic formulae. Graphs of the velocity c vs. the frequency ω are presented for various modes and for various ranges of the parameters. Transmission-line equations and formulae for the impedance and compliance of the fluid-tube system are obtained, together with asymptotic and numerical results.
A new family of capillary waves
- Jean-Marc Vanden-Broeck, Joseph B. Keller
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- Journal:
- Journal of Fluid Mechanics / Volume 98 / Issue 1 / 15 May 1980
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- 19 April 2006, pp. 161-169
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A new family of finite-amplitude periodic progressive capillary waves is presented. They occur on the surface of a fluid of infinite depth in the absence of gravity. Each pair of adjacent waves touch at one point and enclose a bubble at pressure P. P depends upon the wave steepness s, which is the vertical distance from trough to crest divided by the wavelength. Previously Crapper found a family of waves without bubbles for 0 [les ] s [les ] s* = 0·730. Our solutions occur for all s > s** = 0·663, with the trough taken to be the bottom of the bubble. As s → ∞, the bubbles become long and narrow, while the top surface tends to a periodic array of semicircles in contact with one another. The solutions were obtained by formulating the problem as a nonlinear integral equation for the free surface. By introducing a mesh and difference method, we converted this equation into a finite set of nonlinear algebraic equations. These equations were solved by Newton's method. Graphs and tables of the results are included. These waves enlarge the class of phenomena which can occur in an ideal fluid, but they do not seem to have been observed.
Deformation of a bubble or drop in a uniform flow
- Jean-Marc Vanden-Broeck, Joseph B. Keller
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- Journal:
- Journal of Fluid Mechanics / Volume 101 / Issue 4 / 29 December 1980
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- 19 April 2006, pp. 673-686
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Steady potential flow around a two-dimensional bubble with surface tension, either free or attached to a wall, is considered. The results also apply to a liquid drop. The flow and the bubble shape are determined as functions of the contact angle β and the dimensionless pressure ratio γ = (pb − ps)/½ρU2. Here pb is the pressure in the bubble, ps = p∞ + ½ρU2 is the stagnation pressure, p∞ is the pressure at infinity, ρ is the fluid density and U is the velocity at infinity. The surface tension σ determines the dimensions of the bubble, which are proportional to 2σ/ρU2. As γ tends to ∞, the bubble surface tends to a circle or circular arc, and as γ decreases the bubble elongates in the direction normal to the flow. When γ reaches a certain value γ0(β), opposite sides of the bubble touch each other. The problem is formulated as an integrodifferential equation for the bubble surface. This equation is discretized and solved numerically by Newton's method. Bubble profiles, the bubble area, the surface energy and the kinetic energy are presented for various values of β and γ. In addition a perturbation solution is given for γ large when the bubble is nearly a circular arc, and a slender-body approximation is presented for β ∼ ½π and γ ∼ γ0(β), when the bubble is slender.