Elements of the Mathematical Theory of Fluid Motion: Wave and Vortex Motion |
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Elements of the Mathematical Theory of Fluid Motion: Wave and Vortex Motion Thomas Craig, Sir No preview available - 2016 |
Common terms and phrases
2π τ a₁ amplitudes angular velocity assume axes becomes center of gravity co-ordinates consequently const constant cross section d(Pp denote df dW differential equation direction displacement distance dq dq dt db dt dt dv dw dx dt dx dx dx dy dz dy dx dy dy dz dw dz dx dz dz element ellipse equa equal to zero equation of continuity exact differential expression fila finite flow fluid motion gives Green's Theorem indefinitely small infinitely integral mass obtained Ot+i parallelopiped particles of fluid plane waves quantity radius right angles rings sin² Substituting Suppose surface surface integral tion values velocity potential vertical vortex filament vortex line vortex motion wave function wave length write wt+x Απ αρ ατ ε 2π ε ε σα τ 2π ΦΩ င်
Popular passages
Page 30 - Therefore differentiating the first equation with respect to y and the second with respect to x, and subtracting we eliminate ca and the impressed forces, and have d dv d du du( dv du \ °~~dx dt.
Page 21 - I (udx + vdy + wdz) = 0 ............... (4) ; or, in words, The line-integral of the tangential component velocity round any closed curve of a moving fluid remains constant throughout all time. The line-integral in question is appropriately called the circulation, and the proposition may be stated : — The circulation in any closed line moving with the fluid remains constant. 1 Vortex Motion. Edinburgh Transaction, 1869.
Page 123 - From this equation we see that the magnitude of this induced t velocity is directly proportional to the volume of the first element, its angular velocity and the sine of the angle between the line...
Page 113 - ... dz From which we have — if there is no difference of phase between the waves from the parallel axes — at the plane of symmetry there is no displacement in the direction of the axis of X, ie in the direction perpendicular to this plane ; also that the displacement parallel to the plane and the vertical displacement are twice as great as they would be if there was but one wave. The reader who is interested in the subject of wave motion will do well to read an article on the subject by Lord...
Page 80 - The highest points of the crests of two waves are like phases; the highest point of the crest and the lowest point of the trough are opposite phases.