hw1b - v x and v y , as well as the cylindrical components...

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CBE 320 September 3, 2010 Introductory Transport Phenomena Problem Session I Part B: Review of Mathematical Topics Review Appendices A and C in Transport Phenomena before doing this assignment. 1. Sketch the following functions; be sure to label the axes carefully. a. y ( x ) = cosh bx cosh b , −∞ < x < , b = constant b. y ( x ) = e bx , −∞ < x < , b = constant c. y ( x ) = tanh x x , −∞ < x < d. v z ( r ) = v 0 b 1 p r R P 2 B , 0 r R, v 0 , R = constant e. v z ( r ) = v 0 ln( r/R ) ln κ , κR r R, v 0 , R, κ = constant , 0 < κ < 1 2. A particle is located at x = 4 , y = 2 , z = 1 in Cartesian (rectangular) coordinates. a. What are the cylindrical coordinates of the particle ( r, θ, z ) ? b. What are the spherical coordinates of the particle ( r, θ, φ ) ? 3. A particle, located at x = 2 , y = 4 , z = 0 , is moving in a Northeasterly direction with speed 8 cm/sec (“North” = direction of the + y -axis; “East” = direction of the + x -axis). By means of a carefully labelled sketch, show the vector v (assume one unit in distance is numerically equal to one cm/sec). Then show the Cartesian components
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Unformatted text preview: v x and v y , as well as the cylindrical components v r (radial component) and v (tangential coordinate). Assign numerical values to all components. 4. The flow of a fluid past a wedge is described by the potential ( r, ) = cr sin , where c and are constants, and ( r, ) are the cylindrical coordinates of a point in the fluid (the potential is independent of z ). Verify that this function satisfies Laplaces equation, 2 = 0 . 5. Expand the function y ( x ) = e x sin x in a Taylor series about x = 0 , keeping the first 2 nonzero terms. 6. Solve the ordinary differential equation (cylindrical coordinates) 1 r d d r r d v z d r = C, where C is a constant, subject to the boundary conditions v z = 0 at r = R (0 &lt; &lt; 1) , v z = 0 at r = R. Here, and R are also constants. 1...
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