. A stream function is given by: y sin where L and K are constants, 0xL...
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. A stream function is given by: y sin where L and K are constants, 0≤x≤™L and y ≥ 0. a) Does y represent a potential flow, does it obey Laplace's equation? b) What are the velocity components Vx and vy? c) If L = K = 1, locate any possible stagnation points (where x-velocity is 0). d) What do you think happens to the kinetic energy of the fluid if it is decelerated in the presence of a stationary object? What is it transformed into? To be a potential flow we must obey Laplace's equation 2² ² + dx² dy² ay 1 əx L =-COS ay 1 = 1 KL Əv. Əv ????х Vx =0 ду K The velocity components would be X L -sin cosh COS sinh + =0 dy K Əy 1 (1) (+)- cosh V₁ = ay ду 1 K KL Əy K L COS 2²y 1 əx² Ľ X y = = sin( 7 ) cosh ( 2 )=0 K 2² y 1 = Əy² K² Vy = So, if we add up, unless K = L, the system does not obey Laplace's equation and so the flow is not a potential flow. For the flow to be incompressible, it must obey the continuity equation. V₁ = sin cosh sin ay Əx (7) si 1 v₂ =dy=sin() cosh (1) Vx K dy 1 dx = COS L sinh =0 sinh (7) si K y K If L = K = 1, locate any possible stagnation points. Look where vx = 0 because the cosh function can never be 0. sinh 2 (²) Incompressible x=±ntL or ±лL . A stream function is given by: y sin where L and K are constants, 0≤x≤™L and y ≥ 0. a) Does y represent a potential flow, does it obey Laplace's equation? b) What are the velocity components Vx and vy? c) If L = K = 1, locate any possible stagnation points (where x-velocity is 0). d) What do you think happens to the kinetic energy of the fluid if it is decelerated in the presence of a stationary object? What is it transformed into? To be a potential flow we must obey Laplace's equation 2² ² + dx² dy² ay 1 əx L =-COS ay 1 = 1 KL Əv. Əv ????х Vx =0 ду K The velocity components would be X L -sin cosh COS sinh + =0 dy K Əy 1 (1) (+)- cosh V₁ = ay ду 1 K KL Əy K L COS 2²y 1 əx² Ľ X y = = sin( 7 ) cosh ( 2 )=0 K 2² y 1 = Əy² K² Vy = So, if we add up, unless K = L, the system does not obey Laplace's equation and so the flow is not a potential flow. For the flow to be incompressible, it must obey the continuity equation. V₁ = sin cosh sin ay Əx (7) si 1 v₂ =dy=sin() cosh (1) Vx K dy 1 dx = COS L sinh =0 sinh (7) si K y K If L = K = 1, locate any possible stagnation points. Look where vx = 0 because the cosh function can never be 0. sinh 2 (²) Incompressible x=±ntL or ±лL
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Related Book For
Thomas Calculus Early Transcendentals
ISBN: 9780321884077
13th Edition
Authors: Joel R Hass, Christopher E Heil, Maurice D Weir
Posted Date:
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