ME 106 Hwk 3 solutions

ME 106 Hwk 3 solutions - University of California Berkeley...

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Unformatted text preview: University of California, Berkeley Mechanical Engineering Prof S. Morris ME106 Fluid Mechanics, Problem Set 3 1. For the velocity ¯eld V = k ( y i + x j ) with k constant and positive, ¯nd and sketch the streamlines. On your sketch, indicate the direction of motion. 2. A long sealed tube contains compressed air. At t = 0, a gate is opened at one end O , and air °ows leftwards with velocity V = u ( x;t ) i where u ( x;t ) = ½ 2 ° +1 ( x t ¡ c ) ; if x · c t ; ; if x > c t . Here ° and c are constants. Find the acceleration a ( x; t ). Is this °ow steady or unsteady? 3. In plane polar coordinates ( r , Á ), the position vector r = r ^ r , where r is distance from the origin to the point of interest. The velocity ¯eld V has the form V = u ( r;Á; t ) ^ r + v ( r; Á; t ) ^ Á , so that u and v are respectively the radial and circumferential components of V . (a) By using a sketch, show that the unit vectors ^ r , ^ Á are given by ^ r = i cos Á + j sin Á; ^ Á = ¡ i sin Á + j cos Á (b) By using the product rule for the material derivative, and the fact that i and j are constant in magnitude and direction, show that d ^ r d t = ^ Á d Á d t ; d ^ Á d t = ¡ ^ r d Á d t Using a sketch, explain what the derivatives on the left hand side represent, and why they are not zero in general. (c) By using the de¯nition V = d r = d t , and using the product rule, show that the radial and circumferential components of V are given by u = d r d t ; v = r d Á d t : Interpret these results geometrically. (d) By using the product rule, and the above results, show that d V d t = ^ r ³ d u d t ¡ v 2 r ´ + ^ Á ³ d v d t + uv r ´ : ( A ) (e) Use equation ( A ) to calculate the °uid acceleration for rigid{rotation, for which V = Kr ^ Á where K is a positive constant. 4. Do problem 4.41, p. 188 of the text. Explain why you could also use equation ( A ) to answer the question posed in the text. DUE: Thursday 9 Feb 06 BEGINNING OF CLASS SOLUTIONS are posted on the due date at \www.me.berkeley.edu/courses.html" Problem 1 (3 points) : Using the equation for streamlines dx u = dy v and given V = k ( y i + x j ) we get: dx ky = dy kx ⇒ xdx- ydy = 0 ⇒ 1 2 d ( x 2- y 2 ) = 0 Therefore x 2- y 2 = const. is the equation for the streamlines where each streamline takes a different...
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This note was uploaded on 04/29/2008 for the course ME 106 taught by Professor Morris during the Spring '08 term at Berkeley.

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ME 106 Hwk 3 solutions - University of California Berkeley...

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