A1 (25 points) (a) A fluid moves two-dimensionally so that its velocity u is given by u = e-'+e-21, where and ŷ are the

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A1 25 Points A A Fluid Moves Two Dimensionally So That Its Velocity U Is Given By U E E 21 Where And Y Are The 1 (42.26 KiB) Viewed 38 times

A1 (25 points) (a) A fluid moves two-dimensionally so that its velocity u is given by u = e-'+e-21, where and ŷ are the unit vectors for the Cartesian coordinates (x, y). Obtain equations in terms of x and y alone for the following: (i) the streamline through the point (1,1) at t = 0, (ii) the particle path for a particle released from (1,1) at t = 0, (iii) the streakline at t = 0 formed by particles released from (1,1) during the interval t <0. (b) What is the relationships between the slopes at (1,1) of each of the loci in part (a)? Explain why this relationship holds. (c) Sketch the three loci of part (a) on the same diagram in the (x, y) plane: (d) A fluid moves two-dimensionally so that its velocity u is given by u = (x – 3xy + x - y^2x + (y - 3xy - 2xy), where Ñ and ŷ are the unit vectors for the Cartesian coordinates (x, y). Obtain the corresponding complex velocity potential w(z), where z = x + iy. You may find it convenient to consider z" for appropriate values of n. 2