Unformatted text preview: APPH 4200
Physics of Fluids
Review
October 21, 2010
1.!
! Review 2.! Problems from old Midterms 3.! Midterm 2008 1 Review
• Introduction
• Tensors, vectors, symmetric and antisymmetric tensors, vector calculus,
Gauss’ and Stokes’ Theorems, … • Streamlines, pathlines, convective
derivative • Deﬁnitions: strainrate tensor, vorticity, circulation, rigid rotation, stream function
2 Review
• Conservation of mass, NavierStokes, Newtonian ﬂuids, Stokes’ model for stress
tensor, deformation work, viscous
dissipation • Bernoulli’s principle for inviscid,
irrotational ﬂow • Rotating frame of reference, centrifugal
force, Coriolis force 3 Review
• Vorticity equation, Kelvin’s theorem,
vorticity dynamics • Potential ﬂow, 2D Euler ﬂow, complex
velocity potential, Blasius theorem • Dimensional analysis, Reynolds number
• Steady, laminar ﬂow with strong viscocity 4 Midterm 2006
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• 4 problems. Closed book
Conservation of mass
Bernoulli’s principle
Viscous ﬂow
Vorticity conservation 5 Midterm 2007
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• 4 problems. Closed book
Flow near a stagnation point
Viscous ﬂow (Couette)
Circulation, mass ﬂow, and energy
Bernoulli’s principle 6 Midterm Exam Problem 1 Applied Pher 19, 2006
ysics E4200
Octob
Physics of Fluids
Midterm Exam
Read the questions carefully. Notice that the ﬂuid is viscous in some problems,
inviscid in others. Inviscid means the viscosity is zero, µ = ν = 0. Good luck.
1. (20 points) October 19, 2006
(a) Determine the vertical velocity, w, such that the threedimensional ﬂow
ﬁeld whose other two components are
2 2 u = x the v = is x
Read the questions carefully. Notice that y z , ﬂuid −y viscous in some problems,
inviscid in others. Inviscid means the viscosity is zero, µ = ν = 0. Good luck. satisﬁes the incompressible mass conservation equation. You may neglect
any constants of integration. 1. (20 points) (b) An idealized velocity ﬁeld is given by the formula (a) Determine the vertical velocity, w, such that the threedimensional ﬂow
u = txy x − 2 ˆ
ˆ
z
ﬁeld whose other two components are t y y + xzˆ,
where x, y, ˆ are thu = x2 yvectors,=and 2t is time.
ˆˆz
e unit z , v −y x
component of the acceleration vector du .
dt Compute the x satisﬁes the incompressible mass conservation equation. You may neglect 2. (25 points) Fluid ﬁlls a tank to a level h1 above the bottom. The hose emits
any constants of tegration.
ﬂuid from a level hinabove the bottom. The pressure in the atmosphere outside
2
(b) the tank is pa , while theﬁeld is givenvby the ater surface is pt , diﬀerent from pa .
An idealized velocity pressure abo e the w formula
The force of gravity points downward, with constant gravitational acceleration
ˆ
z
g . Consider the ﬂuid to beu = txy x − t2 y y + xzˆ, density, and inviscid, and
incompressible,ˆ
constant
assume the ﬂow to be steady. Find the following quantities in terms of the given
where x, ˆ z
ˆ
constants: y, ˆ are the unit vectors, and t is time. Compute the x 7 Problem 2 component of the acceleration vector du
.
dt (a) The height to which the water stream rises, h3 in the ﬁgure. 2. (25 p(b) ts)he velocity of the waterastreamh1 abexitsthe bottom. The hose emits
oin T Fluid ﬁlls a tank to level as it ove the hose.
ﬂuid from a level h2 above the bottom. The pressure in the atmosphere outside
the tank is pa , while the pressure above the water surface is pt , diﬀerent from pa .
The force of gravity points downward, with constant gravitational acceleration
g . Consider the ﬂuid to be incompressible, constant density, and inviscid, and
assume the ﬂow to be steady. Find the 1 wing quantities in terms of the given
follo
constants:
(a) The height to which the water stream rises, h3 in the ﬁgure.
(b) The velocity of the water stream as it exits the hose. P 6 ~ i
1 hi rT
p~ ~ ( ~l_. 3. (35 points) Fluid with constant density, p, and constant, nonzero kinematic viscosity, lJ, flows between two rigid boundaries y = 0, y = h. The lower
8
boundary moves in the xdirection with speed U, the upper boundary is at rest. The boundaries are porous, and the vertical (y) velocity is Va at each one,
Va being a given positive constant, so that there is an imposed flow across the i
hi rT
Problem 3 3. (35 points) Fluid with constant ~ (y, ρ, and constant, nonzero kinematic
p~ densit
viscosity, ν , ﬂows between ~l_.
two rigid boundaries y = 0, y = h. The lower
3. (35 points)moves in the xdirection with , peed U , constant, nonzero kinematic
boundary Fluid with constant densitys ρ, and the upper boundary is at rest.
3. (35 points) Fluid with constant
p, and constant,
viscosity, ν , ﬂolJ, areetween two density, bertical y y )0,vy= h. Theis= v0 at eaclower
ws b porous, andrigid voundaries nonzero kinematic
0,
The boundaries flows between two rigid boundaries ( = y elocity y lower . The h one,
the
−h
viscosity,
=
boundary a gives in ositive constant, so that theU , the an impbat rest.ﬂow is at rest.
boundary moves in the xdirection with speed eed upper boundary oun
v0 being mov en p the xdirection with spU, there is upperisosed dary across the
The boundaries p the system vertical (y) velocity x and , one,
The boundaries yreakeporous, and the to be uniform )in is Va at yisassume steadyone,
system. You maa t areorous, and the vertical (y velocity each−v0 at each ﬂow,
Va being a given positive constant, so that there is an imposed flow across the
v0 being a givenvity.take e constant, so thatin x and y, assume compflow, w acrosst: do
positiv the the to w (both there and steady onen
and neglect gra may Findsystemﬂobe uniform the x is anyimposed ﬂots). Hin the
system. You
system.andouonent t(v )eﬁrst.system to bethe x and y components). Hint: do steady ﬂow,
Y neglect ak Find
the y comp maygravity.the the flow (both uniform in x and y , assume
the ycomponent (v) first. the ﬂow (b oth the x and y comp onents). Hint: do
and neglect gravity. Find
the y component (v ) ﬁrst.
l Vo 0 r. '0 c: '= 'c i= L: ~ w 0 D Cl 0 c: CJ G' l' h
L l C7 Va c/ \,j .  Vi
l :: (! J ~ ç7Y ¡V
9 4. (20 points) Consider a large lake, initially at rest. In the center, the water is
stirred by a solid object inserted into it (e.g., a spoon or egg beater). The object
is then removed. Problem 4 (a) If the fluid is inviscid, explain why no vorticity can be generated by the stirring.
4. (20 points) Consider a large lake, initially at rest. In the center, the water is
stirred by a solid ob ject inserted into it (e.g., a spoon or egg beater). The ob ject
4. (20then removed.
is points) Consider a large lake, initially at rest. In the center, the water is
stirred by a solid ob ject inserted in2to it (e.g., a spoon or egg beater). The ob ject
is(a) If remoﬂuid is inviscid, explain why no vorticity can be generated by the
then the ved.
stirring.
(a) If the ﬂuid is inviscid, explain why no vorticity can be generated by the
stirring.
(b) If the ﬂuid is viscous, vorticity can be created. Assume that this occurs,
but only in a small region around2the ob ject. Explain why after the removal
of the ob ject, the net vorticity created must still be zero, that is, the areal
2
integral of the vertical vorticity over the entire lake must vanish, ω dA ω= ∂v ∂u
−
∂x ∂y lake where is the vertical vorticity.
In both parts, you may assume that the water has constant density.
10 Problem 5 11 Problem 6 12 Problem 7 13 Midterm 2008
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• 3 problems. Closed book
2D Potential Flow
Bernoulli’s principle
Poiseuille Flow 14 This is a closed book exam. Please read each question carefully. Please also show your
work and explain your reasoning as this helps in understanding your answer and the award
of partial credit. Formula are attached. Problem 1 1. (30 Points) For twodimensional potential ﬂow, the divergence and curl of the ﬂow
vanishes, i.e. ∇ · U = ∇ × U = 0. The steady ﬂow can be represented by a
ˆ
velocity potential, U ≡ ∇φ, or a stream function, U ≡ −z × ∇ψ . Since φ and
ψ satisfy Laplace’s equation, the ﬂow can be represented on the complex plane,
z = x + iy = reıθ , by an analytic function called the “complex velocity potential”,
w(z ) ≡ φ + iψ . When w(z ) = Az n , w(z ) represents the ﬂow around or inside a corner located at
(x, y ) = (0, 0). Show that (i) when n < 1, the ﬂow passes over the “outside” of a
corner having an angle greater than 180◦ , and (ii) when n > 1, the ﬂow passes on
the “inside” of a corner having an angle less than 180◦ .
For each case, n > 1 and n < 1, determine whether the ﬂuid velocity increases or
decreases when approaching the corner. J Q fl .. o0 .) .. 'i ~ .. oJ 01 .J ~ \. " 0 h IN i'~ V ii" ;:' '"', ," 3 0 ,J
U. q,
" :i
\)  ~ l\  " ~ Figure 1: For Question 1. Twodimensional potential ﬂow near corners.
15 Problem 2
1 2. (35 points) An incompressible liquid of density, ρ1 , is ﬂowing through a tube at a
volumetric rate of Q. The tube crosssectional area as two parts. On the entrance
side, the area is A1 , and on the exit side the area is A2 (with A2 > A1 .)
A “Utube” is connected to the tube as shown in the ﬁgure below. The Utube is
partially ﬁlled with a liquid of density ρ2 , which is denser than the ﬂuid ﬂowing
through the tube, ρ2 > ρ1 . The two ﬂuids do not mix.
The diﬀererence between the heights of the upstream and downstream sides of the
Utube is h. Your problem : Derive an expression for h in terms of Q, A1 , A2 , ρ1 , ρ2 ,
and g , the acceleration of gravity.
[Hint: The ﬂuid in the Utube is in a state of hyrdostatic equilibrium, where pressure
gradients balance gravitational forces. The pressure must be constant where the Utube connects to the tube with ﬂowing ﬂuid.] 16 Figure 2: For Question 2. Tube with Utube connected. Problem 3
3. (35 points) A viscous ﬂuid ﬂows at a steady rate through a tube with an elliptical
crosssection. See ﬁgure below.
What is the relation between the volumetric ﬂow rate, Q, through the tube and
the axial rate of pressure drop, dP/dz , in terms of the dimensions of the tube and
the ﬂuid viscosity. (You may leave integrals unevaluated, provided they are deﬁned
fully.)
Show that the pressure gradient is balanced by a viscous force.
[Note: You do not have to evaluate any expressions for any integral. Just deﬁne
them fully.] y z
2c x 2b
17 Figure 3: For Question 3. Tube with elliptical crosssection for steady ﬂuid ﬂow. 3 ...
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