Q.6 A stream function is given by  = 2x2-2y2. Determine the velocity and velocity potential function at (1, 2) (Dec 2013, jan 2015) Ans: Given:  = 2x2-2y2 y   4  v;  4  (1,2), x v x Velocity at x Units v  4 4    y  u; u  4 y  Velocity at Units (1,2),  8 u Resultant velocity V(1,2) = 8.94  Units   u  (4  4    4    ( ( y) y x y C f x   x   v 4 4 ( (  (4 x)  x     x  y  C f y   x  C    C  C C 4 4  x    x   (4  )  x 4 4 0     x x    y y y  0 C From Eq.1 0 C  Integrating y y y  x  y 4 4 2 8    x  y    1  Units Q. 7 In a flow the velocity vector is by V = 3xi + 4yj -7zk. Determine the equation of the stream line passing through a point M (1, 4, 5). (July2013, july 2015) Ans: Given the Velocity vector V = 3xi+4yj -7zk u = 3x ; v = 4y; w = -7z  To determine the equation of the stream line passing through a point M (1, 4, 5) The 3-D equation of streamline is given by, dx  dy  dz u v w dx  dy  dz Eq.1 3x 4 7z y  The streamline equation at M (1, 4, 5), x =1, y = 4, z = 5 Substituting the values of x, y, and z in Eq.1 dx  dy  dz
Q.6 A stream function is given by  = 2x2-2y2. Determine the velocity and velocity potential function at (1, 2) (Dec 2013, jan 2015) Ans: Given:  = 2x2-2y2   4 x  v; v  4 x  Velocity at (1,2), x    4 y  u; u  4 y  Velocity at (1,2), y v  4 u  8 Units Units Resultant velocity V(1,2) =  8.94 Units   u x   v y   x   x  (4 y)  4 y    4  x  y  C ( f (  (4 x)  4 x    4  x  y  C ( f (  C C  C    C  (4 x  )   4 x    4 x       4 x  4 x  0 From Eq.1 y y C  0 Integrating y y y y C  0  x  y   4  x  y    4 1 2  8 Units Q. 7 In a flow the velocity vector is by V = 3xi + 4yj -7zk. Determine the equation of the stream line passing through a point M (1, 4, 5). (July2013, july 2015) Ans: Given the Velocity vector V = 3xi+4yj -7zk  u = 3x ; v = 4y; w = -7z To determine the equation of the stream line passing through a point M (1, 4, 5) The 3-D equation of streamline is given by, dx  dy  dz u v w dx  dy  dz Eq.1 3x 4 y  7z The streamline equation at M (1, 4, 5), x =1, y = 4, z = 5 Substituting the values of x, y, and z in Eq.1 dx  dy  dz
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