By I. Sturton
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Additional resources for All the World's Battleships - 1906 to the Present
This will always be the case in all applications considered herein. 74) to arrive at a formula for the velocity induced by an arbitrary distribution of vorticity. 84) 4TJV The practical application of this is to vorticity distributions along lines and surfaces. In line distributions £ is zero everywhere except along the line and hence the volume integral reduces to a line integral which must be closed or end normal to boundaries. The vorticity vector is everywhere tangent to the curve and is positive in the direction which produces circulation about the line element in the sense of the right-hand rule for rotation or moment vectors.
The reason is that they arise from conjugate functions. Thus we can immediately infer their behavior on the vortex sheet. 38) X- X Thus u depends only on the local value of 7 whereas v depends on the entire distribution. For |x| < a the integral is a Cauchy principal value. NON-PLANAR AND PLANAR DISTRIBUTIONS IN THREE DIMENSIONS Source Distributions Consider a curved surface over which sources are distributed having a strength density m cubic units per second per square unit of area. Thus m 34 Properties of Distributions of Singularities has the dimensions of velocity (linear units/sec).
From the foregoing we learn • The axial component of a source distribution along the x-axis is symmetric in y or an even function ofy. , all source elements contribute. e. it is an odd function of y. , on —a < x < a, y = 0 at first we may say zero because y -> 0 in the numerator. However at y = 0 the kernel function becomes (x — x1)"2 and the integrand tends to "blow up". Thus we suspect a "battle" going on between the vanishing of the numerator and the infinity of the integrand as x1 passes through x and so we have to "creep up" on this limit by the use of a transformation.