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By H. R. Vallentine (auth.)

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0 lS. the two- d'rmensmna carteSian co-ordinate form of what is known as the Laplace equation. The d' . lr . ()2 F ()2 F (}2 F 0 d h . h correspond mgt ree- 1mensmna _10rmis ox2 + oy2 + oz2 = an t e general form, in vector notation, is V2 F = 0. This Lapla~ equation is met in several other fields of physical science, for example, in electrostatics, as well as in hydrodynamics. Whilst it is evident that only certain functions of x andy satisfy Eq. 17, the above reasoning indicates that the stream functions, 1/J, of all possible two-dimensional irrotational flow patterns satisfy it or, in other words, are solutions of the Laplace equation.

P - C or xy - C, which equation represents a family of rectangular hyperbolae. Considering the first quadrant only, the pattern represents flow at a 90° corner (Fig. 16). The pattern of the first and second quadrants together represents flow towards a flat plate, or stagnation flow (Fig. 21). p - ax ~-function ay ax• ay• 0 the flow is irrotational and therefore a will exist for the pattern. p ax ay --u---% .. , - ixl+ft{y) Also .... p a~ ay- v- -ax- -y .. (x) .... p- i(x'-y 1 ) +a constant, satisfies both (a) and (b).

Consideration is given to the velocities and accelerations of particles as they pass through the general point rather than to the variations of the velocities and accelerations of particles as they follow their various paths. If u, v and w are, in order, the components of velocity in the x-, yand z-directions at the point (x,y, z) at time t, then u, v and ware functions of position (x,y, z) and time (t). For a particular value of t, they define the motion at all points in the fluid; and for a particular point, (x,y, z), they are simply functions of time, providing a history of the velocity variations at the point.

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