By H. R. Vallentine (auth.)
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Extra info for Applied Hydrodynamics
24---Acceleration. ¥·direction of a fluid particle as it moves from P1 toP1 in the interval 8t. There is, in addition, the local component, :~ 81 and the local change, due to change with the passage of time at the point x,y, z. Mathematically, 8t, u = A(x,y, z, t) au au au au au = -at+-ax+at ax ay ay+-az az au au auax auay auaz at = at+ axat + ayat + azat If at be considered to approach zero, the total acceleration in the x-direction is obtained du dt - = au au au au -+u-+v-+wat ax ay az 35 ....
22). The other streamlines plot as shown. Streamlines inside the circle can have no influence on the pattern outside the circle and this circle can be regarded therefore as the solid boundary of a cylinder of radius a, lying acroSs the flow. fo ar - v,- rIar/so9 - u(1- ~) •. "'Also cos 9 u(r+~) cos9+ft(9) r104>o9- v,- arts -or --u( + ~) 1 .. "'- u(r+ .... />-lines. (c). Veloci~ Distribution-The components of velocity, v, and v8 for any point in the flow are as determined above. At very large distances from the cylinder, afr approaches zero and v,~Ucos9, ..
16 Determination of :ftow patterns The problem of determining a particular pattern of irrotational flow amounts, in mathematical terms, to finding a stream function, or a potential function, which satisfies both the Laplace equation 49 FLOW OF AN IDEAL FLUID and the boundary conditions of the flow in question. These boundary conditions relate to the form and nature of the boundaries, including those across which the fluid flows. For example, in steady flow, solid boundaries must be streamlines and free surface boundaries must be streamlines at constant pressure.