By Eds. Theodore Y. Wu & John W. Hutchinson
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Extra info for Advances in Applied Mechanics, Vol. 25
We will focus our survey on work done in these subtopics. The basic difficulty in these problems is exhibited in the waveguide example of Love’s treatment of the free longitudinal vibration of a finitelength, circular section, cylindrical rod (Love, 1927). The rod here has a stress free lateral surface and edges (or ends), hence stress free corners. Its edge conditions are of the nonmixed type, that is, stress components (present case) or displacement components specified. Love shows that attempts to treat this problem with a classical separation technique, aided b y Pochhammer’s frequency equation for the infinite rod (Pochhammer, 1876) failed (hence nonseparability), leading to a solution in which the normal stress on the rod edges vanishes but not the shear stress.
We first direct attention to slender shapes, those whose two transverse dimensions are both small compared with their longitudinal dimension. The centerline can meander through three dimensions so that (as for a helix) it has torsion or twist as well as curvature, and the cross section can vary slowly in shape, proportion, size, and orientation. In unpublished work Slow Variations in Continuum Mechanics 25 Todd (1978, 1979, 1980) has studied steady laminar flow through a slender pipe of such general slowly varying shape.
Then the continuity equation becomes a -(fu) dX + -a( f v ) aY = 0. 5) -$x. If the flow is irrotational, all three components of vorticity must vanish, but to a first approximation only the principal component u, - uy normal to the plane of symmetry need vanish. This gives the equation governing $: ($x/fL + (Gy/fIy = 0. 6) This equation is exact for both plane flow (when f = const) and axisymmetric flow (when f = y , the cylindrical radius from the x axis of rotation), . polar coordinates, but in general it involves a relative error of order E ~ In with x = r cos 8 and y = r sin 8, it becomes As an application we consider uniform potential flow past a sphere tangent to a plane, which is equivalent to flow normal to the line of centers of two equal spheres in contact.
Advances in Applied Mechanics, Vol. 25 by Eds. Theodore Y. Wu & John W. Hutchinson