3) Consider the problem of a fluid rotating in rigid body motion about the center of...

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3) Consider the problem of a fluid rotating in rigid body motion about the center of a cylindrical container of radius R that is rotating at a rate co. The container at rest is filled to a known height, ho, and it is open to the atmosphere. ho @ liquid Pill liquid Pill h In this case, the velocity components are V = v = 0 V = ro R Why? [5 points] The Navier-Stokes Equations in cylindrical coordinates are given below: Here, u is the viscosity, p is the density and gr, ge, and gz are the Cylindrical coordinates (r. 0, 2): 10 (80, OU v av, 0% av d 1 av, 2 ava op + + +0. (FV) + + +PS. ar of au av ar r BA ar 80 or r v.v av + + + V = " (ru) - r 60 ar 1 ave 602 2 av. I ap - + 200 r 20 r 60. OU VOU au. I a 00 Iv v. op + U, + U. + + = 1 +PS: of or r 80 20 ar r 80 02 r-,0-, and z-components of the acceleration due to gravity. 1. Simplify these equations and solve for the equation of the pressure field p(r,z) within the container. [50 points] 2. From this determine the equation for the surface of the liquid under constant rotation . You will find this solution up to an arbitrary constant, h. [25 points] 3. Find a relation to allow you to find this constant. DO NOT SOLVE [20 points] BONUS: Find the arbitrary constant in the expression for the surface of the rotating fluid. [15 points] 3) Consider the problem of a fluid rotating in rigid body motion about the center of a cylindrical container of radius R that is rotating at a rate co. The container at rest is filled to a known height, ho, and it is open to the atmosphere. ho @ liquid Pill liquid Pill h In this case, the velocity components are V = v = 0 V = ro R Why? [5 points] The Navier-Stokes Equations in cylindrical coordinates are given below: Here, u is the viscosity, p is the density and gr, ge, and gz are the Cylindrical coordinates (r. 0, 2): 10 (80, OU v av, 0% av d 1 av, 2 ava op + + +0. (FV) + + +PS. ar of au av ar r BA ar 80 or r v.v av + + + V = " (ru) - r 60 ar 1 ave 602 2 av. I ap - + 200 r 20 r 60. OU VOU au. I a 00 Iv v. op + U, + U. + + = 1 +PS: of or r 80 20 ar r 80 02 r-,0-, and z-components of the acceleration due to gravity. 1. Simplify these equations and solve for the equation of the pressure field p(r,z) within the container. [50 points] 2. From this determine the equation for the surface of the liquid under constant rotation . You will find this solution up to an arbitrary constant, h. [25 points] 3. Find a relation to allow you to find this constant. DO NOT SOLVE [20 points] BONUS: Find the arbitrary constant in the expression for the surface of the rotating fluid. [15 points]