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

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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 bracketleftbigg 1 parenleftBig r R parenrightBig 2 bracketrightbigg , 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 θ (tangential coordinate). Assign numerical values to all components. 4. The flow of a fluid past a wedge is described by the potential ψ ( r, θ ) = − cr α sin αθ , where c and α are constants, and ( r, θ ) are the cylindrical coordinates of a point in the fluid (the potential is independent of z ). Verify that this function satisfies Laplace’s equation, ∇ 2 ψ = 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 ² = C, where C is a constant, subject to the boundary conditions v z = 0 at r = κR (0 < κ < 1) , v z = 0 at r = R. Here, κ and R are also constants. 1...
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