Online study resources available anywhere, at any time
Highquality Study Documents, expert Tutors and Flashcards
Everything you need to learn more effectively and succeed
Uni. Oulu

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
Course: The Role Of Social Media In Customer Communication In Businesstobusiness
thesis on social media
Highest Degree Offered  Continuing Professional Programs  Academic and Career Counseling Services  Employment Services for Students  Placement Services for Graduates  Study Abroad 

Unknown  Unknown  Unknown  Unknown  Unknown  Unknown 
Source: National Center for Education Statistics (NCES), Institute of Education Sciences, 20122013
Course Hero, Inc. does not independently verify the accuracy of the information presented above.