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Uni. Oulu

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School: Uni. Oulu
Course: Hydrodynamics
= i cos + j sin = i sin + j cos r i j r r = cos i + sin j = r r/ r = cos i + sin j,  r/ r r/ r sin i + r cos j = = = sin i + cos j.  r/ r sin cos sin + cos = (sin2 + cos2 )j = j r cos sin cos sin = (cos2 + sin2 )i
School: Uni. Oulu
Course: Hydrodynamics
763654S HYDRODYNAMICS Solutions 3 Autumn 2011 1. Verify that the example ow v = (ax, ay, 0) satises the continuity equation with constant a and constant density. Determine the stream function . Discuss the particle paths based on . Solution: Continuity eq
School: Uni. Oulu
Course: Hydrodynamics
763654S HYDRODYNAMICS Solutions 2 Autumn 2011 1. Show that ( 2 2 ( ) dS . ) dV = V S Solution: In the previous exercise (4.b), it was shown that 2 ( ) = ( ) ( ) + By reordering this identity: = ( ) ( ) ( ) by interchanging = ( ) ( ) ( ) dierence of t
School: Uni. Oulu
Course: Hydrodynamics
763654S HYDRODYNAMICS Solutions 4 Autumn 2011 1. For applications later in this course, go through the solution of the Laplace equation given in the appendix C of the lecture notes. Find (x, y ) for the cases a) f (y ) = C (y a ), 2 b) f (y ) = C sin y .
School: Uni. Oulu
Course: Hydrodynamics
763654S HYDRODYNAMICS Solutions 5 Autumn 2011 1. A frame of reference which is accelerating (with respect to inertial frames) is used to describe an experiment. The acceleration has the constant value f in the i direction. (a) Show that an inertial force
School: Uni. Oulu
Course: Hydrodynamics
763654S 1. Solutions 7 HYDRODYNAMICS Autumn 2011 Microscopic model of viscosity The viscosity of a gas can be estimated as = 1 v, where v = 3kb T /m is the average 3 velocity of molecules and is the mean free path. Estimate numerically for air (use the re
School: Uni. Oulu
Course: Hydrodynamics
= i cos + j sin = i sin + j cos r i j r r = cos i + sin j = r r/ r = cos i + sin j,  r/ r r/ r sin i + r cos j = = = sin i + cos j.  r/ r sin cos sin + cos = (sin2 + cos2 )j = j r cos sin cos sin = (cos2 + sin2 )i
School: Uni. Oulu
Course: Hydrodynamics
763654S HYDRODYNAMICS Solutions 3 Autumn 2011 1. Verify that the example ow v = (ax, ay, 0) satises the continuity equation with constant a and constant density. Determine the stream function . Discuss the particle paths based on . Solution: Continuity eq
School: Uni. Oulu
Course: Hydrodynamics
763654S HYDRODYNAMICS Solutions 2 Autumn 2011 1. Show that ( 2 2 ( ) dS . ) dV = V S Solution: In the previous exercise (4.b), it was shown that 2 ( ) = ( ) ( ) + By reordering this identity: = ( ) ( ) ( ) by interchanging = ( ) ( ) ( ) dierence of t
School: Uni. Oulu
Course: Hydrodynamics
763654S HYDRODYNAMICS Solutions 4 Autumn 2011 1. For applications later in this course, go through the solution of the Laplace equation given in the appendix C of the lecture notes. Find (x, y ) for the cases a) f (y ) = C (y a ), 2 b) f (y ) = C sin y .
School: Uni. Oulu
Course: Hydrodynamics
763654S HYDRODYNAMICS Solutions 5 Autumn 2011 1. A frame of reference which is accelerating (with respect to inertial frames) is used to describe an experiment. The acceleration has the constant value f in the i direction. (a) Show that an inertial force
School: Uni. Oulu
Course: Hydrodynamics
763654S 1. Solutions 7 HYDRODYNAMICS Autumn 2011 Microscopic model of viscosity The viscosity of a gas can be estimated as = 1 v, where v = 3kb T /m is the average 3 velocity of molecules and is the mean free path. Estimate numerically for air (use the re
School: Uni. Oulu
Course: Hydrodynamics
763654S HYDRODYNAMICS Solutions 6 Autumn 2011 1. Derive the formulae: a) S f dS = V f dV, and b) S Aij dSj = V Aij dV . [Hint: Mulxj tiply by a constant vector and use the divergence theorem.] Solution: The (Gauss') divergence theorem states: A dS = (1)
School: Uni. Oulu
Course: Hydrodynamics
763654S 1. HYDRODYNAMICS Solutions 10 Autumn 2011 Transient ow between parallel planes (double points) Fluid is at rest in a long channel with rigid walls t = 0. velocity U (y, t)i y = a when a pressure gradient G is suddenly imposed at a) Show that the s
School: Uni. Oulu
Course: Hydrodynamics
763654S HYDRODYNAMICS Solutions 9 Autumn 2011 1. Dimensioless Euler's equation The Euler equation for incompressible ow in a rotating system was given in the form v + v t v + 2 v = p in the lectures. Write this in a dimensionless form. Solution: Followin
School: Uni. Oulu
Course: Hydrodynamics
763654S HYDRODYNAMICS Solutions 8 Autumn 2011 1. Plane Couette ow Consider uid between parallel planes. The wall at y = 0 is xed, and the wall at y = a moves with steady speed V in its own plane. Solve the NavierStokes equations for the case = constant t
School: Uni. Oulu
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