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The Quarterly Journal of Mechanics and Applied Mathematics 1952 5(4):441-446; doi:10.1093/qjmam/5.4.441
© 1952 by Oxford University Press
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ON THE INTEGRATION OF SOME VECTOR DIFFERENTIAL EQUATIONS

II. APPLICATION TO THE LINEARIZED THEORY OF STEADY COMPRESSIBLE FLUID FLOW

G. N. WARD

( Department of Mathematics, The University Manchester )

It has been shown in Part I (1) that solutions of a special system of vector differential equations can be constructed by using a suitable integral vector identity. A special case of the system arises in the linearized theory of the steady flow of an inviscid fluid, and in this paper the general solutions of Part I are adapted to the flow problem.

Special interest is attached to the particular integral of the equations which gives the perturbation velocity in terms of the source and vorticity distributions. These are easily modified to give expressions for the velocity fields due to source layers, vortex sheets, and vortex lines. The formulae obtained are the generalizations of the formulae of Helmholtz and Stokes for incompressible flow. The perturbation velocity is discontinuous at source layers and vortex sheets, and the discontinuities are expressed in terms of the source and vorticity densities and the normals to the surfaces of discontinuity.


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