Homework_1 - Fall 2014 Professor Gagnon 540.304 Tranport Phenomena II Homework#1 Due These first few problems should serve as a review for the vector

Homework_1 - Fall 2014 Professor Gagnon 540.304 Tranport...

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Unformatted text preview: Fall 2014 Professor Gagnon 540.304 Tranport Phenomena II Homework #1 Due 9/9/14 These first few problems should serve as a review for the vector calculus material you have learned in the past, which we will use extensively this term in developing and solving important transport equations. 1.) Calculate the angle between the following pairs of vectors: a. (1,0,1), (1,0,0) b. (1,0,1), (0,1,0) c. (1,2,3), (3,2,1) 2.) Calculate the following quantities: a. (1,0,1)⋅ (1,0,0) b. (1,0,1)⋅ (0,1,0) c. (1,2,3) × (3,2,1) 3.) For the scalar potential function φ = (x2 + y2 + z2)2 and the velocity vector field u=(y2, z, x2) calculate the following vector quantities: a.) ∇φ ; ∇⋅ u 2 b.) ∇ φ = (∇⋅ ∇)φ ; ∇ 2 u c.) ∇ × u 4.) Show that in a one-­‐dimensional steady flow the following equation is valid. dA du dρ + + = 0 A u ρ 5.) In class we showed that the integral expression for the mass balance over a general control volume is ∂ ˜ ˜ ∫ ρ(u⋅ n)dA + ∂t ∫ s V ρdV = 0 . Using the symbol M for the total mass in the control volume, show that the above equation may be written as: ⋅ ⋅ ∂M + ∫ d m = 0 , where m =mass flow rate ∂t s Fall 2014 Professor Gagnon 6.) A shock wave moves down a pipe as shown below. The fluid properties change as the shock wave passes through the control volume, but they are not functions of time. The velocity of the shock wave is uw. Write the continuity equation and obtain the relation between ρ1, ρ2, u2, and uw. The mass in the control volume (defined by the dotted rectangle) at any time is M = ρ 2 Ax + ρ1 Ay , where A is the surface area of the pipe. Hint: Imagine you are in a moving frame of reference, i.e. the control volume (CV) is moving with the shock wave -­‐ what is the net velocity entering the CV? 7.) The velocity profile in a circular pipe is given by u = umax (1 − r /R)1/ 7 , where R is the pipe radius. Find the average velocity in the pipe. ...
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