By Peter C. Kendall

ISBN-10: 0080104401

ISBN-13: 9780080104409

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Additional resources for Magnetohydrodynamics with Hydrodynamics

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Figure 8 illustrates the effect of this mapping. 61) maps a contour Tz and its exterior in the x, j-plane into the contour Γζ and its exterior in the 1,77-plane (Fig. 9). 62) maps Γ, into part of a line ψ = constant in the w-plane. 62) thus represents the complex potential for streaming motion past the contour Γ, HYDROSTATICS AND HYDRODYNAMICS 61 t ér — rz *ζ FIG. 9. Transformation of flow round a contour Τζ in the ζ-plane to flow round a contour Γ , in the z-plane. Example. Show that the mapping ζ = \{z + 1/z) transforms the unit circle into the straight line joining the points ζ = ± 1 .

Here we discuss some general results concerning the motion of a continuous inviscid fluid and then briefly discuss the phenomena of hydrostatics, steady motions, surface and sound waves, shock waves and viscosity. We assume that the reader is familiar with the concepts of pressure, p, density p and velocity v of a fluid, together with the equations of motion of a fluid, equation of continuity and simple equations of state. ] 2:2 The equations of inviscidì hydrodynamics 1. The equation of motion of a liquid or gas takes the form(1) p^=-gnidp + pF.

Show further that the exterior of the unit circle maps onto the whole of the ζ-plane. On the unit circle, z = e*°, thus ζ ( = ξ -f ίή) = Ke<ô + e~<0) = cos 0. As the point z moves round the unit circle from 0 = 0 to 2π in an anti­ clockwise direction, the point cos Θ moves along the line η = 0 from ζ = 1 to ζ = — 1, and back. Therefore the unit circle maps onto the line joining the points ζ = ± 1 . Consider the mapping of any circle r = constant = r0 (say). The point z = rQQid maps into the point ζ = ξ 4- ίη, where S = 2(ro + —) cos 0> v = 2\ r ° ~ 7 " ) s i n ö · These are the parametric equations of an ellipse.

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Magnetohydrodynamics with Hydrodynamics by Peter C. Kendall


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