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Problem 1 (4 Points) In problems 2 and 3, we will consider the Navier-Stokes equation in cylindrical geometries. To be ready for these two problems, write down the three components (i.e. r, ?, and z)...

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Problem 1 (4 Points) In problems 2 and 3, we will consider the Navier-Stokes equation in cylindrical geometries. To be ready for these two problems, write down the three components (i.e. r, ?, and z) of the Navier-Stokes equation for an incompressible viscous fluid which is not subject to gravity. Hint: Be mindful of the fact that er and e? are not fixed vectors. Problem 2 (4 Points) In this problem, we assume that the fluid occupies the region 0 < z < h between two rigid boundaries z = 0 and z = h. The lower boundary is at rest, and the upper boundary rotates with constant angular velocity ? about the z-axis. Show that a steady solution of the Navier-Stokes equation written in Problem 1 which is of the form u = u?(r, z)e? is not possible, so that any circular motion u?(r, z) in this system must be accompanied by a secondary flow with ur 6= 0, uz 6= 0. Problem 3 (8 Points) 1. Write the three components of the Navier-Stokes equation for the special case in which u = u?(r, t)e? 2. Show that in that case, the equations imply that ?p/?? = 0. 3. Prove that u? then satisfies ?u? ?t = ? r ? ?r  r ?u? ?r  - ?u? r 2 4. Viscous fluid inside an infinitely long circular cylinder with radius r = a is rotating with angular velocity ?, so that u? = ?r for r = a. The cylinder is suddenly brought to rest at t = 0. Use the equation from 3. to show that dE dt + 2? a 2 E = 0 where E = 1 2 Z a 0 u 2 ? rdr which is proportional to the kinetic energy of the flow. Conclude that E ? 0, as t ? 8. 1 Problem 4 (4 Points) Consider the gravitational potential ? such that g = -??. For an inviscid fluid, we have Euler’s equation ?u ?t + ? × u + 1 2 ? u 2  = - 1 ? ?p - ?? where ? = ? × u is the vorticity of the flow. We also have the equation for the conservation of mass of the fluid: ?? ?t + ? · (?u) = 0 Show that D Dt  ? ?  =  ? ? · ? u - 1 ? ?  1 ?  × ?p We thus see that if p is a function of ? alone (in which case the flow is called barotropic), the vorticit
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Introduction to Fluid Dynamics { Problem Set 6 March 28, 2017 Due April 6, 2017 after class Problem 1 (4 Points) In problems 2 and 3, we will consider the Navier-Stokes equation in cylindrical geometries. To be ready for these two problems, write down the three components (i.e. r,, andz) of the Navier-Stokes equation for an incompressible viscous uid which is not subject to gravity. Hint: Be mindful of the fact that e and e are not xed vectors. r  Problem 2 (4 Points) In this problem, we assume that the uid occupies the region 0

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Answered Same Day Dec 25, 2021

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David answered on Dec 25 2021
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