Solved: Flow through a circle Suppose water flows in a

Chapter 11, Problem 72

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Flow through a circle Suppose water flows in a thin sheet over the xy-plane with a uniform velocity given by the vector v = 81, 29; this means that at all points of the plane, the velocity of the water has components 1 m>s in the x-direction and 2 m>s in the y-direction (see figure). Let C be an imaginary unit circle (that does not interfere with the flow). x C y a. Show that at the point 1x, y2 on the circle C, the outwardpointing unit vector normal to C is n = 8x, y9. b. Show that at the point 1cos u, sin u2 on the circle C, the outwardpointing unit vector normal to C is also n = 8cos u, sin u9. c. Find all points on C at which the velocity is normal to C. d. Find all points on C at which the velocity is tangential to C. e. At each point on C, find the component of v normal to C. Express the answer as a function of 1x, y2 and as a function of u. f. What is the net flow through the circle? That is, does water a ccumulate inside the circle? 7