lecture exam practice

8/6/2019 Lecture Exam Practice

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K&C page 77, exercise 3

y,v

C (r=1)

x,u


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x,u

y,v u= ay = a r sinCirculation:

= C u d s= 02

a sin2

d

=[ a 12

x cos x sin x ]0

2

= a

d s r [ i sin j cos ]d

ds, d

C (r=1)

Stokes theorem: surface integral of vorticity = line integral of velocitySometimes one of the two is far easier to calculate!


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x,u

y,v

Circulation: = C dA= 12 a = a

u= ay = a r sinCirculation:

Vorticity:

z= a

= C u d s= 02

a sin2

d

=[ a 12

x cos x sin x ]0

2

= a

d s r [ i sin j cos ]d

z= v x

u y

ds, d

C (r=1)

Stokes theorem: surface integral of vorticity = line integral of velocitySometimes one of the two is far easier to calculate!


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u i

x j= 1

2

u i

x j

u j

xi

1

2

u i

x j

u j

xir ij = ije ij


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e11 = u1

x1= u

x= 0

e 22= u 2 x2

= v y

= 0

Linear strain rates:

ui x j

= 12

ui x j

u j xi

12

ui x j

u j

xi

r ij = ije ij

Shear strain rates:

e12 = 12 u1 x2

u 2 x1

= 12

U b

Vorticity:

z= r 21= 12 u2 x1

u1 x2

= U b

e 21=12

U b

z= r 12=U b


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u i x j

= 12

u i x j

u j xi

12

ui x j

u j xi

r ij = ije ij

strain along principal axis (e12

); followed by rotation ( z)


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u= Uy /b= y= U

2

2bC


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Streamlines:dxu

= dyv

1 t x

dx= 2 t 2y

dy

1 t log x =12

2 t log 2y log f t NB: instant. streamlines, e.g. t = const.; ln f(t) for convenience

log x1 t =log 2y 1 t /2 log f t We combine last 2 terms in one log, so it becomes a multiplication

x1 t = f t 2y 1 t /2 This is our answer for the streamlines: x expressed as a function y (or vice versa)


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Pathlines: x t = 0t dx

dt dt = 0

t u dt

dxdt

= x1 t

dx x

= dt 1 t

log x= log 1 t log f x0

x= f x0 1 t

x= x0 1 t

t = 0 x= x0,so f x0 = x0


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Same for y: y t = 0t dy

dt dt = 0

t v dt

dydt

= 2y2 t

dy y

= 2dt2 t

log y= 2log 2 t 2log g y0

y= g y02 2 t 2

y= y02 /4 2 t 2

t = 0 y= y0, so g y02= 1 /4 y0

2


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y= y02 /4 2 t 2

x= x0 1 t

The pathline for a particle that is at x0,y

0at t=0 is given by:


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const. acceleration a

Steady, so no viscous deformation;only hydrostatic forces; Navier-Stokesequation reduce to:

0= 1 p x

a

g x

x

= tan

0= 1 p z

gcompareclassichydrostatics:

tan = ag


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xy


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D u D t

= 1 P g 2 u

0= g sin 2 u

y2

Fully developed, steady, no driving pressure

0= g sin y u

yC 1

0= g2

sin y2 u C 1 y C 2

u= g y2

2sin C 1 y C 2


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D u D t

= 1 P g 2 u

0= g sin 2 u

y2

Fully developed, steady, no driving pressure

0= g sin y u

yC 1

0= g2

sin y2 u C 1 y C 2

u= g y2

2sin C 1 y C 2

No slip: u = 0 at y = h

0= g h2

2sin h2 C 2

Stress-free at surface:

du /dy = 0 at y= 0

C 1= 0

C 2=g h 2

2sin


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u=g y2

2 sin C 1 y C 2 C 1= 0 C 2= g h2

2 sin

u= g y2

2sin g h

2

2sin

u= g

2sin h2 y2 QED


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u= g2

sin h2 y2

Q= 0h

udy = 0h g

2sin h2 y2 dy =[ g sin

2h2 y y

3

3]0

h

= g sin h3

3


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u= g2

sin h2 y2

W = u y y= h

W =h g sin = gh sin NB : / =

u y = g 2ysin2 = g y sin

NB: shear stress is related to column of water that drives the flow

y = h


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u= g2

sin h2 y2

W = u y y= h

W =h g sin = gh sin NB : / =

u y = g 2ysin2 = g y sin

NB: shear stress is related to column of water that drives the flow

Question: would this also hold for rivers?

y = h


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Outer flow is irrotational,So no net viscous forces:Inviscid Bernoulli

Show ( x u =0)or recognize inviscid vortex


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Pressure in outer flow:(Bernoulli, compare withfar away u=0, p = p 0)

p F(z) =p 0 + g(h-z) u 2

The stress is continuous atsurface, so pressure fromboth sides should be equal(no pressure drop):

p F = p c

p 0+ g(h-z)=p 0+ cg(h-z) u2

Solving for u 2 gives answer


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Compare the two expressions:

u =2 r

u2= 2g ' h z

2 r

2

= 2g ' h z

outside

surface

R z 2=2

8 2 g ' 1

h z

Surface r=R, solve:


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Velocity scale: V =h

Froude influence of hydrostatic pressureversus flow-induced pressure (e.g. Y>>1, hardlyany influence of gravity)

u2

V 2= u

2

2 /h2= 2 g ' h z2 /h2

u ' = 2Y

1 z , with Y =2

g ' h 3


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SEE KUNDU:Stokes' second problem


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Surface tension creates a pressuredifference; Pressure differenceused to fill plenum cannot be higherthan the pressure due to surfacetension; so: calculate surfacetension, compare with Poiseuille


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lecture exam practice

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