NS_PracticeProblems

# NS_PracticeProblems - Practice Problems on the...

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Practice Problems on the Navier-Stokes Equations C. Wassgren, Purdue University Page 1 of 13 Last Updated: 2010 Oct 13 ns_02 A viscous, incompressible, Newtonian liquid flows in steady, laminar, planar flow down a vertical wall. The thickness, , of the liquid film remains constant. Since the liquid free surface is exposed to atmospheric pressure, there is no pressure gradient in the liquid film. Furthermore, the air provides a negligible resistance to the motion of the fluid. 1. Determine the velocity distribution for this gravity driven flow. Clearly state all assumptions and boundary conditions. 2. Determine the shear stress acting on the wall by the fluid. 3. Determine the maximum velocity of the fluid. Answer(s) : --- wall air liquid y x g

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Practice Problems on the Navier-Stokes Equations C. Wassgren, Purdue University Page 2 of 13 Last Updated: 2010 Oct 13 ns_03 An incompressible, viscous fluid is placed between horizontal, infinite, parallel plates as is shown in the figure. The two plates move in opposite directions with constant velocities, U 1 and U 2 . The pressure gradient in the x direction is zero and the only body force is due to gravity which acts in the y -direction. 1. Derive an expression for the velocity distribution between the plates assuming laminar flow. 2. Determine the volumetric flow rate of the fluid between the plates. 3. Determine the magnitude of the shear stress the fluid exerts on the upper plate (at y = b ) and clearly indicate its direction on a sketch. Answer(s) : --- b y x U 1 U 2 g fluid with density, , and dynamic viscosity,
Practice Problems on the Navier-Stokes Equations C. Wassgren, Purdue University Page 3 of 13 Last Updated: 2010 Oct 13 ns_05 In cylindrical coordinates, the momentum equations for an inviscid fluid (Euler’s equations) become: 2 1 r r r z z u Du p f Dt r r Du u u p f Dt r r Du p f Dt z        where u r , u , and u z are the velocities in the r , and z directions, p is the pressure, is the fluid density, and f r , f , and f z are the body force components. The Lagrangian derivative is: rz u D uu Dt t r r z    A cylinder is rotated at a constant angular velocity denoted by . The cylinder contains a compressible fluid which rotates with the cylinder so that the fluid velocity at any point is u = r ( u r = u z =0). If the density of the fluid, , is related to the pressure, p , by the polytropic relation: k p A where A and k are known constants, find the pressure distribution p ( r ) assuming that the pressure, p 0 , at the center ( r =0) is known. Neglect all body forces.

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## This note was uploaded on 10/01/2011 for the course ME 509 taught by Professor Wereley during the Spring '11 term at Purdue.

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NS_PracticeProblems - Practice Problems on the...

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