By H. R. Vallentine (auth.)
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4. Show that the variable ellipsoid x• a1 k1 t' + kt• (yb + Clz 1 1 1) - 1 is a possible form of boundary surface of a liquid at time t. Determine the velocity components u, v and w for fluid particles on the boundary surface and show that the requirements of continuity are satisfied. Ans. u- ~ •. 5. fsec. parallel to the x-axis, in the positive x-direction. /sec. parallel to they-axis in the positivey-direction. fsec. at an angle of 30° to the x-axis. p- - 5x1 represents flow parallel to they-axis with a velocity which varies linearly with distance from they-axis.
As an example, the characteristics of the stream function for steady, two-dimensional, irrotational flow at a 90° corner can be stated (Fig. 16). It will be seen from subsequent considerations that, for this pattern, the stream function is r/J = axy 20 STREAM FUNCTIONS and the general equation for the streamlines is t/J = constant, that is axy = C the streamlines being rectangular hyperbolae. The coefficient, a, determines the scale or magnitude of the flow and different values of C define different streamlines.
E. the rate of contraction of the fluid in the region. A useful expression for the kinetic energy can be obtained if P, Q and R are defined as = rpu = rp ~:; where rp is the velocity p potential. The velocity along the inward normal, n, to an element of surface is§ Vn = ~: and it equals the sum of the components of u, v and w in the n-direction aq, aq, aq, aq, - = 1-+m-+n- t See Appendix A (i). on oy OX oz ~ A region of space is said to be • connected' if it is possible to pass from any point to any other point in the region without leaving the region.