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

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CBE 320 September 2, 2009 Introductory Transport Phenomena Problem Session I Part B: Review of Mathematical Topics 1. Sketch the following functions; be sure to label the axes carefully. a. y ( x ) = tanh bx tanh b , −∞ < x < , b = constant b. y ( x ) = cxe - bx , −∞ < x < , b, c = constant c. y ( x ) = sinh x x , −∞ < x < d. v z ( y ) = v 0 b y h p y h P 2 B , 0 y h, v 0 , h = constant e. v z ( r ) = v 0 ln( R/r ) ln(1 ) , 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 = 1, 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 θ (tangential coordinate). Assign numerical values to all components. 4. Show that ∇ 2 ψ = 0, where the function ψ ( r, θ, φ ) (i.e., a function of the spherical coordinates) is given by ψ ( r, θ, φ ) = ψ cos θ r 2 , where ψ is a constant. 5. Consider the function y ( x ) = x sinh x . Perform a Taylor series expansion about x = 0, keeping the Frst two non-zero terms. 6. Solve the di±erential equation d 2 v x dy 2 = C, where C is a constant, subject to the boundary conditions v x = 0 at y = 0 v x = 0 at y = h to determine the function v x ( y ). 1...
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This note was uploaded on 05/06/2010 for the course CBE 320 taught by Professor Staff during the Fall '08 term at University of Wisconsin.

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