CONICAL POTENTIAL FLOW ABOUT BODIES OF REVOLUTION.

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Abstract

THE METHOD OF GREEN'S FUNCTIONS AND A FORMULATION IN TERMS OF CONAL FUNCTIONS ARE USED TO SOLVE THE AXISYMMETRIC LAPLACE EQUATION IN A DOMAIN BOUNDED EXTERNALLY OR INTERNALLY BY CONICAL BOUNDARIES.THE PARTICULAR APPLICATION OF INTEREST IS TO DETERMINE THE PRESSURE AND THE VELOCITY DISTRIBUTION ABOUT SLENDER BODIES LYING ON THE AXIS OF A CONICAL TUNNEL IN AN INCOMPRESSIBLE AND IRROTATIONAL FLOW.EXPRESSIONS ARE DERIVED FOR THE VELOCITY POTENTIAL AND THE STREAM FUNCTION OF BOTH AN ISOLATED SOURCE AND A RING SOURCE IN THE INTERIOR OF A CONICAL DOMAIN.THESE BASIC POTENTIAL FUNCTIONS ARE USED TO FORMULATE FREDHOLM INTEGRAL EQUATIONS OF THE FIRST AND SECOND KINDS FOR THE SOURCE DISTRIBUTION GENERATING A PRESCRIBED SLENDER BODY IN A CONICAL TUNNEL.THE INTEGRAL EQUATION FOR THE AXIAL SOURCE DISTRIBUTION IS SOLVED BY EXPANDING THE SOLUTION IN TERMS OF A CONVERGENT LEGENDRE SERIES.AN INTEGRAL EQUATION WHICH RENDERS DIRECTLY THE SURFACE VELOCITY DISTRIBUTION, BY USING A SURFACE DISTRIBUTION OF VORTEX RINGS, IS ALSO PRESENTED.NUMERICAL EXAMPLES ARE GIVEN FOR THE LAGALLY FORCE EXPERIENCED BY AN ISOLATED SOURCE LYING ON THE AXIS OF THE CONE, AND FOR THE VELOCITY DISTRIBUTION ON THE SURFACE OF A PROLATE SPHEROID INSIDE THE CONE.BOTH EXAMPLES INVOLVE A RADIAL UNDISTURBED FLOW.FINALLY, NEW EXPRESSIONS FOR THE CONAL FUNCTIONS, WHICH ARE MORE SUITABLE FOR NUMERICAL COMPUTATIONS, ARE ALSO PRESENTED TOGETHER WITH SOME NUMERICAL RESULTS.(A)

Original languageEnglish
Pages (from-to)FEB, 1976
JournalQuarterly Journal of Mechanics and Applied Mathematics
Volume28
Issue number1
StatePublished - 1976

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