exam1 - Solution to MT 2 Problem#1(a Streamlines passing...

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Solution to MT 2: Problem#1 (a) Streamlines passing through A=(1,1): => equation of the streamline is the hyperbola: Streamlines passing through B=(3,2): => equation of the streamline is: (b) Flow rate through a line across points A and B: This flow rate is per unit z, so it has the dimension L 2 /T. (c) Material acceleration (use material derivative): and One contribution is the rate of change of V 0 with time; the second contribution is from the convective derivative, change following the velocity vector: Velocity vector: Radial vector: . Comparing the acceleration expressions with the above vectors: One component lines up with vector V, the other lines up With the radial vector from the origin. (d) To make the material acceleration in the y-direction vanish, we need: for all y, which is possible. Hence the condition: However, the x-acceleration cannot be made to vanish. (e) The differential equation satisfied by V 0 is: This equation can be solved separating the V o & t variables: . Integrating both sides and using the initial condition V 0 (0)=K , we get: or y l = 6 ( x / l ) a x = u t + ( V . ) u = x l ( V 0 + V 0 2 / l ) a y = v t + ( V . ) v = y l ( V 0 + V 0 2 / l ) V = u ˆ e x + v ˆ e y = x l V 0 ˆ e x y l V 0 ˆ e y a = V 0 V 0
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