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

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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 bracketleftbigg y h parenleftBig y h parenrightBig 2 bracketrightbigg , 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
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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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