流體力學

1. Consider a steady, fully developed laminar flow in an annulus with inside radius

R2 and outside radius R1. Find a relation between the pressure gradient dp/dx, the

volume flow rate Q, the fluid viscosity μ, R1 and R2/R1.

2. (a) Demonstrate that when a cylindrical can of liquid rotates like a solid body

about its vertical axis with uniform angular velocity, , the free surface is a

paraboloid of revolution.

(b) Demonstrate that the pressure difference between any two points in the fluid is

given by 2/2

1

2

2

2

1212 rrzzgpp , where z is elevation and r is the

radial distance from axis.

(c) How would the results differ if the can were of square cross-section?

3. Please derive the momentum integral equation for the boundary layer flow on a

flat plate such as dx

dUw

2 , in which w is the shear stress on the surface

of the flat plate θ, is the boundary layer momentum thickness, is the upstream

uniform flow velocity along the x direction, and ρ is the fluid density.

4. Plot the Moody chart and explain which related parameters should be used to

determine friction factor for (a) a laminar flow, (b) a turbulent flow,and (c)

a wholly turbulent flow, respectively.

5. A capillary tube with a small radius a is held vertically in air with its bottom

immersed in a large body of liquid. The surface tension of the air-liquid

combination is σ, and the contact angle of the air-liquid-tube wall combination is

α.

(a) Show that if l >> a, the capillary rise l is given by l = 2σcosα/ρga.

(b) How would this expression change if the system were comprised of two

plates separated by 2a, instead of being a tube with radius a?

Fluid

r

x

R2 R1

6. Derive the loss coefficient for the pipe fluid flowing through a sudden

expansion.。

7. The open U-tube of Fig.1 is partially filled with a liquid. When this device is

accelerated with a horizontal acceleration, a, a differential reading, h, develops

between the manometer legs which are spaced a distance l apart. Determine the

relationship between a, h, and l.

8. Water flowing from the oscillating slit shown below produces a velocity field

given by 0 0 0sin /V u t y v i v j , where u0, v0, and are constants. Thus,

the y component of velocity remains constant (v = v0) and the x component of

velocity at y = 0 coincides with the velocity of the oscillating sprinkler head [u =

u0sin(wt) at y = 0].

(a) Determine the streamline that passes through the origin at t = π/(2w).

(b) Determine the pathline of the particle that was at the origin at t = 0.

(c) Discuss the shape of the streakline that passes through the origin.

9. Consider a region of potential flow:

(a) What is the governing equation of the flow? Where does it come from?

(b) Is the solution unique? Under what conditions? Justify your answer.

(c) Does the velocity field, u, obtained from the potential flow solutions satisfy

the Navier-Stokes equations? Why?

10. (a) A rocket is launched vertically upwards with an acceleration of 5g. Find the

pressure difference p2-p1 ( in bars ) between the bottom and the top of the liquid

fuel tank, if the fuel has a density of 900 kg/m3.

(b) After reaching an altitude of 200 km, 20% of the fuel remains, and the power

plant is shut off. Find the pressure difference between the top and bottom of the

tank during the free fall of the rocket, assuming no air friction.

11. Consider an incompressible fluid flowing past a circular cylinder, define the

pressure coefficient (Cp) on the surface of the cylinder, plot and

explain the Cp distribution along the surface from the forward stagnation point to

the rear region for the cases of (a) an inviscid flow, (b) a viscous laminar flow,

and (c) a viscous turbulent flow, respectively.

。

12. The sketch shows a liquid emulsion (a finely-divided mixture of two liquids) of

mean density ρ1 entering a reaction zone of a constant-area reactor with speed V1.

The components of the emulsion react chemically, and leave the reaction zone as a

liquid at the density ρ2. Pitot tubes are installed upstream and downstream of the

reaction zone. ( Pressure inside a pitot tube is stagnation pressure, p0 = p + ρ

V2/2 ). It is agreed to assume that the flow is invisid, steady and one-dimensional,

that the original emulsion is incompressible, and that the liquid leaving the

reaction zone is incompressible. Calculate the value of 2/2

110201 Vpp in

terms of the density ratio ρ2/ρ1.

13. As shown below, a horizontal jet of water exists a nozzle with a uniform speed of

V1 = 10 ft/s, strikes a vane, and is turned through an angle θ. Determine the

anchoring force needed to hold the vane stationary. Neglect gravity and viscous

effects.

14. Consider a steady, incompressible, inviscid flow passing a circular cylinder of

radius, a, as shown in Fig.3. Find out the pressure variation and the acceleration

experienced by the fluid particles as they flow along the streamline A-B.

15. A sluice gate across a channel of width b is shown in the closed and open

positions in the figures below. Is the anchoring force required to hold the gate in

place larger when the gate is closed or when it is open? Explain.

16. Consider a spherical ball with a uniform flow, when the Reynolds number

corresponding to the uniform flow increases, please sketch and describe the

variation of the flow field near the spherical ball..

17. Fig. 5 shows the velocity profiles before and after a two-dimensional body in a

wind tunnel. The upstream (section(1) ) velocity is uniform at 100 ft/s. The static

pressures are given by p1 = p2 = 14.7 psia. The downstream velocity distribution

which is symmetrical about the centerline is given by

ftyu

ftyy

u

3100

33

130100

where u is the velocity in ft/s and y is the distance on either side of the centerline

in feet. Calculate the drag force exerted on the air by the body per unit length

normal to the plane of the sketch.

18. Consider the steady frictionless flow of a perfect gas through a pipe of constant,

uniform cross-sectional area. Heat is added to this flow through the pipe walls so

that the total temperature, T0, of the gas increases by an amount dT0 over a small

length of the pipe. Find a relation for the correspondingly small change in the

Mach number (denoted by dM ) in terms of dT0, the Mach number, M, and the

temperature, T, of the flow. The expression also contains the ratio of the specific

heats, γ. (The total temperature, T0, is defined as the total enthalpy divided by Cp.)

19. The rough surface of an automobile tire consists of roughness of size, ε. Consider

the following Couette flow which models the hydroplaning of the tire on a smooth

road：

The speed of the tire is U, the mean liquid film thickness is h, and the kinematic

viscosity of the liquid is v. If the magnitudes of the unsteady turbulent velocities,

u’ and v’, generated by the roughness are both given approximately by Uεy/h2

where y is the distance above the smooth road, find the ratio of the “effective”

dynamic viscosity of the film of liquid to actual liquid dynamic viscosity. The

answer includes U, ε, h and v.

20.Air flows steadily between two sections in a long, straight portion of 4-in. inside

diameter pipe as shown below. The uniformly distributed temperature and

pressure at each section are given. If the average air velocity (of the nonuniform

velocity distribution) at section (2) is 1000 ft/s, calculate the average air velocity

at section (1).

21.A constant force F is applied to a simple cylindrical bellows of diameter D1. The

air flows out of the bellows, via a nozzle of diameter D2, to the ambient

atmosphere.

(a) If the air flow is incompressible (density ) and invisid, derive an expression

for the time it takes to exhaust a volume V of air from the bellows.

(b) Compute this time for STP (1 atm, 25oC) air if V = 1 liter, D1 = 10 cm, D2 = 1

cm, and F = 2 kgf.

22.Derive the velocity profile of the fully developed laminar flow in a circular pipe,

22

14 D

rDru w

where τw is the wall shear stress, D is the pipe

diameter, and μ is the fluid viscosity.

23.Explain the physical principle behind the flow rate measurements by (a) the

orifice meter, (b) the nozzle meter, and (c) the Venturi meter. Considering a

steady, inviscid and incompressible fluid flowing through these meters, derive

the equation to determine the corresponding flow rate.。

24.A watertank has an orifice in the bottom of the tank:

The height, h, of water in the tank is kept constant by a supply of water which is

shown. A jet of water emerges from the orifice; the cross-sectional area of the

jet, A(z), is a function of the vertical distance, z. Neglecting friction (viscous

effects) and surface tension find an expression for A(z) in terms of A(0), h and z

where A(0) is the cross-sectional area at z = 0. Assume that the area of the tank

free surface is very large compared with A(0).

25.Fluid flows from the fire extinguisher tank shown below. Discuss the

differences between dBsys/dt and dBCV/dt if B represents mass.

26.Water at 60 ℉ flows from the basement to the second floor through the 0.75-in.

(0.0625-ft)-diameter copper pipe (a drawn tubing) at a rate of Q = 12.0 gal/min

= 0.0267 ft3/s and exits through a faucet of diameter 0.50 in. as shown in Fig.1.

The pressure variations are determined and shown in Fig.2. as (a) all loses are

neglected and (b) all loses are included.

(A) Write down the energy equation for this incompressible, steady flow between

point (1) and (2), and show expressions of the major loses and the minor loses.

(B) Use Fig.2 and the energy equation, describe the physical meaning for pressure

variations of case (a) and case (b).

27.Consider turbulent flow of an incompressible fluid past a flat plate. The boundary

layer velocity profile is assumed to be 7171/ YyUu for 1/ yY and

Uu for 1Y as shown in Fig.8. This is a reasonable approximation of

experimentally observed profiles, except very near the plate where this formula

gives yu / at y = 0. Note the differences between the assumed turbulent

profile and the laminar profile. Also assume that the shear stress agrees with the

experimentally determined formula:

41

20225.0

U

vUw

Determine the boundary layer thickness δ,δ*, and θand the wall shear stress,

τw, as a function of x. Determine the friction drag coefficient, CDf.

28.A liquid drop is held at the end of a straw, as sketched. Its volume is controlled by

the position of the piston. Gravity is negligible. Show that if the drop’s volume is

changed by a (slow) displacement of the piston, the net work done on the system

comprised of the liquid and its bounding surface is equal to the product of the

surface tension coefficient and the incremental change in the system’s surface area.

Explain why it follows that the surface tension coefficient can be interpreted as an

internal energy per unit surface area.

29.Consider the frictionless, steady flow of a compressible fluid in an infinitesimal

stream tube.

(c) Demonstrate by the continuity and momentum theorems that

0V

dV

A

dAd

0 gdzVdVdp

(d) Determine the integrated forms of these equations for an incompressible

fluid.

(e) Derive the appropriate equations for unsteady frictionless, compressible flow,

in a steam tube of cross-sectional area which depends on both space and

time.

30.Consider two cases, (a) water drains from a bathtub, (b) a liquid contained in a

tank that is rotated about its axis with angular velocity , please explain the

physical differences between two cases, and which one is rotational?

31.Determine the streamlines for two-dimensional steady flow jyixlVV

0 .

32.A flat plate is hinged at one side to the floor, as shown, and held at a small angle

θ(θ<<1) relative to the floor. The entire system is submerged in a liquid of

density ρ. At t = 0, a vertical force is applied and adjusted continually so that it

produces a constant rate of decrease of the plate angleθ,

tconsdtd tan/

Assuming that the flow is incompressible and inviscid,

(a) derive an expression for the velocity u(x,t) at point x and time t.

(b) Find the horizontal force F(t) exerted by the hinge in the floor (assume the

plate has neglible mass).

33.A syringe (shown below) is used to inoculate a cow. The plunger has a face area of

500 mm2. If the liquid in the syringe is to be injected steadily at a rate of 300

cm3/min, at what speed should the plunger be advanced? The leakage rate past the

plunger is 0.10 times the volume flowrate out of the needle.

34.An air fan has a bladed rotor of 12-in. outside diameter and 10-in. inside diameter

as illustrated below. The height of each rotor blade is constant at 1 in. from blade

inlet to outlet. The flowrate is steady, on a time-average basis, at 230ft3/min and

the absolute velocity of the air at blade inlet 1V

is radial. The blade discharge

angle is 30°from the tangential direction. If the rotor rotates at a constant speed of

1725 rpm, estimate the power required to run the fan. Takeρair = 2.38×10-3

slug/ft3.

35.Air flowing into a 2-m-square duct with a uniform velocity of 10 m/sec forms a

boundary layer on the walls as shown in Fig.7. The fluid within the core region

(outside the boundary layers) flows as if it were inviscid. From advanced

calculations it is determined that the boundary layer displacement thickness is

given by

x007.0*

Where δ*and x are in feet. Determine the velocity U=U(x) of the air within the

duct but outside of the boundary layer.

36.Pressures are sometimes determined by measuring the height of a column of liquid

in a vertical tube (for example, a barometer). What diameter of clean glass tubing

is required so that the rise of water at 20℃ (with surface tension coefficient, σ=

0.0728N/m, in contact with air) in a tube due to capillary action is less than 1 mm?

37.Explain the physical meaning of fluid viscosity based on the molecular structure of

the fluid, and describe its differences between liquid and gas. How the viscosity

depends on the temperature and pressure for the cases of liquid and

gas,respectively?

38.Focusing on pressure gradient, inertia force and viscous force, discuss and explain

their balancing relations for the following flows.

(f) the entrance region of a pipe flow.

(g) A fully developed pipe flow.

(h) A steady viscous flow past a flat plate.

(i) A steady viscous flow past a cylinder.

39.A sluice gate is installed in a steady water stream of depth h1 and speed V1 (as

measured far upstream of the gate). Downstream of the gate the stream has a

depth h2 which is less than h1.The flow is incompressible and inviscid.

(j) Assuming uniform velocities at (1) and (2), derive an expression for the

horizontal force F, per unit width, required to hold the gate in place, given ρ,

V1, h1 and h2. Check your result by showing that it is zero when h2 = h1 and

equal to the hydrostatic result when h2 = 0.

(k) Also obtain an expression for V2. Show that as h2 approaches zero, V2

approaches 12gh , and F approaches 2

1 / 2gh . Explain.

40.For a uniform flow passing a circular cylinder, its potential flow field may be considered as

a doublet (K is the strength) combining with a uniform flow (U is the free stream velocity),

and its corresponding stream function and velocity potential can be expressed as

rUr

sinsin and

rUr

coscos

,

respectively. Find the velocity distribution of the flow field, and the pressure distribution on the

cylinder surface.

41.A static thrust stand as sketched below is to be designed for testing a jet engine. The following

conditions ate known for a typical test:

‧ Intake air velocity = 200 m/s, exhaust gas velocity = 500 m/s.

‧ Intake cross-sectional area = 1 m2

‧ Intake static pressure = -22.5 kPa (gage) = 78.5 kPa (abs), exhaust static pressure = 0 kPa

(gage) = 101 kPa (abs)

‧ Intake static temperature = 268 K

Estimate the nominal thrust for which to design. You may want to know that the ideal-gas constant for

air is Rair = 286.9 J/kg．K.

42.In a fuel injection system small droplets are formed due to the breakup of the

liquid jet. Assume the droplet diameter, d, is a function of the liquid density, ,

viscosity, surface tension, , the jet velocity, V, and diameter, D. Form an

appropriate set of dimensionless parameters using , V, and D as repeating

variables.

43.Consider the steady, two-dimensional flow field ))(/( 0 jyixlVV

. Determine

the acceleration field for this flow.

44. As shown in the figure below, the fluid moves through the pipe from entrance region

and gradually becomes fully developed. Discuss the relationships among inertial force,

pressure force and viscous force in the process of (1) →(2) →(3). Plot the pressure

variation along x direction.

45. Water is siphoned from the tank through a hose of constant diameter, as shown in the

figure below. The hose has a valve at the exit and a small hole at location (1) as

indicated. Plot the energy line (EL) and hydraulic grade line (HGL) for the cases of the

valve being (a) opened, and (b) closed, respectively. At location (1), will water leak out

of the hose, or will air leak into the hose, discuss for both cases.

46.The flow in the neighborhood of a corner in a rectangular ventilation duct is to be

modeled as a planar potential flow of an incompressible, inviscid fluid:

This flow is then changed by withdrawing fluid through pipes connected to the

walls at the origin, O; fluid is thereby withdrawn at a volumetric rate of q per unit

depth normal to the sketch. Construct the velocity potential for the modified flow

and find expressions for the velocity components in terms of x, y, A and q.

A piece of thread is attached by one end to a point, C, which is at a distance, H,

from the origin. The flow will extend the free end of this thread either toward the

origin or toward x = ∞. Find the condition under which it will extend toward the

origin.

47.A capillary tube of internal diameter 10-3

m is placed vertically in a bucket of water.

How high will the level in the capillary rise above the level in the bucket if the

contact angle at the inner walls of the tube is 100 and the surface tension is 0.07

kg/sec2?

Consider a smaller capillary with the same contact angle and surface tension. If

the water will vaporize below a pressure of 0.017 bar, what is the maximum

capillary height which can, in principle, be achieved and what size of capillary in

necessary to achieve this elevation?

48. What is the Buckingham Pi theorem? Consider a pipe flow with a functional relation

expressed as ΔPl= f(D, ρ, μ, V),from which we choose ΔPl( pressure drop per unit

length) and μ(the fluid viscosity) as two nonrepeating variables, use D (the pipe

diameter), ρ(the fluid density) and V(the mean flow velocity) to determine the final

set of Pi terms.

49.Water enters a rotating lawn sprinkler through its base at the steady rate of 1000 ml/s as sketched below.

The exit area of each of the two nozzles is 30 mm2 and the flow leaving each nozzle is in the tangential

direction. The radius from the axis of rotating to the centerline of each nozzle is 200mm. Determine the

resisting torque associated with the sprinkler rotating with a constant speed of 500rev/min.

50.Small spherical pollen grains of diameter D = 0.04 mm and specific gravity SG =

0.80 drift down from the top of a 20 m tall pine tree. If there is a constant, uniform

breeze of Uwind = 1 m/sec, determine the distance from the tree that the pollen will

travel before it reaches the ground. Assume the horizontal velocity of the pollen is

equal to Uwind. (For standard atmospheric conditions,ρair = 1.23 kg/m3 and μ =

1.79×10-5

Nsec/m2.)(The drag coefficient as a function of Reynolds no. is shown

as follows.)

51.When a circular cylinder is placed in a uniform stream, a stagnation point is

created on the cylinder as is shown in Fig.a. If a small hole is located at this point,

the stagnation pressure, Pstag, can be measured and used to determine the approach

velocity, U. (a) Show how Pstag and U are related. (b)If the cylinder is misaligned

by an angle α(Fig.b), but the measured pressure still interpreted as the stagnation

pressure, determine an expression for the ratio of the true velocity, U, to the

predicted velocity, U’.

52.An ancient device for measuring time is shown in Fig.6. The axisymmetric vessel

is shaped so that the water level falls at a constant rate. Determine the shape of the

vessel, R=R(z), if the water level is to decrease at a rate of 0.1 m/hr and the drain

hole is 5.0 mm in diameter. The device is to operate for 12 hours without needing

refilling.

53.Assume that the drag, D, acting on a spherical particle that falls very slowly

through a viscous fluid is a function of the particle diameter, d, the particle

velocity, V, and the fluid viscosity, μ. Determine, with the aid of dimensional

analysis, how the drag depends on the particle velocity.

54.A thin rectangular plate having a width w and a height h is located so that it is

normal to a moving stream of fluid. Assume the drag, D, that the fluid exerts on

the plate is a function of w and h, the fluid viscosity and density, μand ρ,

respectively, and the velocity V of the fluid approaching the plate. Determine a

suitable set of pi terms to study this problem experimentally.

55.Water flows through a pipe reducer as is shown in Fig.a. The static pressure at (1)

and (2) are measured by the inverted U-tube manometer containing oil of specific

gravity, SG, less than one. Determine the manometer reading, h.

56. Plot the energy line (EL) and hydraulic grade line (HGL) for the pipe flow flowing

from a large tank, as shown in the figure below. If a small hole is found in the hose,

specify in which region the fluid will leak out and which region air will leak into.

57.Incompressible, laminar water flow develops in a straight pipe having radius R as

indicated in the figure below. At section (1), the velocity profile is uniform; the

velocity is equal to a constant value U and is parallel to the pipe axis everywhere.

At section (2), the velocity profile is axisymmetric and parabolic, with zero

velocity at the pipe wall and a maximum value of umax at the centerline.

(l) How are U and umax related?

(m) How are the average velocity at section (2) 2V

and umax related?

(n) If he flow is vertically upward, develop an expression for the fluid pressure

drop that occurs between section (1) and (2).

58.Water enters a rotating lawn sprinkler through its base at steady rate of 1000 ml/s

as shown in Fig.9. The exit area of each of the two nozzles is 30 mm2 and the flow

leaving each nozzle is in the tangential direction. The radius from the axis of

rotation to the centerline of each nozzle is 200 mm.

(o) Determine the resisting torque required to hold the sprinkler head stationary.

(p) Determine the resisting torque associated with the sprinkler rotating with a

constant speed of 500 rev/min.

(q) Determine the speed of the sprinkler if no resisting torque is applied.

59.Consider the fully developed pipe flow of an incompressible, non-Newtonian fluid:

This fluid is such that the normal stress in the x direction is equal to –p where p is

the pressure and the shear stress, s, is related to the velocity gradient by

2

dr

duC

where C is a known constant. Find the friction factor, f, for this pipe flow in terms

of C, r (the fluid density) and R (the radius of the pipe).

[Note: Remember the definition:

dx

dp

u

Rf

4, where u is the average

velocity of flow in the pipe.]

60. Explain the physical principle behind the flow velocity measurement by using the Pitot

tube.Plot agraph to show the pressure variation along the surface of a Pitot tube.

Explain how the flow velocity is measured and how to judge the direction of the flow.

61.A strong explosion (like an atomic bomb) causes a spherically symmetric shock wave to

move through the air radially out from the origin. As the shock sweeps by, it causes a

sudden rise in pressure and sets the initially static air into radially outward motion.

It can be argued from strong shock wave theory that if the undisturbed atmosphere is

homogeneous at a density ρa, the velocity Vs of the shock, as well as the pressure Ps

and the wind speed just behind the shock wave, should depend only on the density ρa,

the distance rs of the shock wave from the origin, and the total energy E released by the

explosion.

(r) Show that :

3

2

3

2

1

.

)/(.

ss

sas

rEconstp

rEconstv

(s) Obtain an expression for the shock’s radial position as a function of time (the

expression may involve one unknown dimensionless constant). Show how the

strengths of two different bomb explosions, as measured by their energy releases,

can be compared based on film information about their shock wave positions as

functions of time.

62.Discuss the differences between laminar flow and turbulent flow in (a) flow

characteristics, (b) flow structures, and (c) momentum transfer.

63.It is often conjectured that the earth was, at one time, comprised of molten material.

If the acceleration due to gravity, g(r), at a radius, r, within this fluid sphere

(radius, R = 6440 km) varied linearly with r, if the density of the fluid was

uniformly 5600 kg/m3 and if g(R) = 9.81 m/s

2, find the pressure at the center of

this fluid earth.

64.Describe the basic structure and the function of the Pitot-static tube. What is its

physical principle and what are the key-points in designing it?

65.A soap bubble (surface tension σ) is attached to a narrow glass tube of the

dimensions shown. The initial radius of the bubble is R0. At t = 0 the end of the

tube is abruptly opened.

(a)Obtain a solution for R(t), assuming that the flow is : (i) incompressible and (ii)

inviscid, that (iii) gravitational effects are negligible, and that (iv) the temporal

acceleration term in Euler’s or Bernoulli’s equation is negligible (we are referring

to the term involving the partial derivative of the velocity with time).

(b)Derive a criterion for when assumption (iv) is satisfied.

66.An incompressible fluid is contained between two infinite parallel plates as

illustrated in Fig.a. Under the influence of a harmonically varying pressure

gradient in the x direction, the fluid oscillates harmonically with a frequency, w.

The differential equation describing the fluid motion is

2

2

cosy

uwtX

t

u

where X is the amplitude of the pressure gradient. Express this equation in

nondimensional form using h and w as reference parameters.

67.When a cup of tea is stirred, the tea leaves (which are denser than water) collect at

the center of the bottom of the cup. Give a qualitative explanation of this.

68.Estimate the projected area of a parachute (viewed from below) required to being a

man (of mass 70 kg) down to earth at a vertical descent velocity of 3 m/sec.

(Assume the drag coefficient for a parachute is CD = 1.2, density of air = 1.2

kg/m3)

69.A plane wall is immersed in a large body of liquid of density ρ which is at rest:

The surface tension of the liquid surface is denoted by S and the contact angle

with the wall by θ. Find the equation of the water surface in the form y = f(x);

the function should contain the quantities S,θ,ρand the acceleration due to

gravity, g. To simplify the problem assume that the curvature of the water surface

can be approximated by 22 dxyd . Find the height, h, in terms of S,θ,ρand g.

70.Determine the anchoring force required to hold in place a conical nozzle attached

to the end of a laboratory sink faucet (see the sketch below) when the water

flowrate is 0.6 liter/s. The nozzle mass is 0.1 kg. The nozzle inlet and exit

diameters are 16 mm and 5 mm, respectively. The nozzle axis is vertical and the

axial distance between sections (1) and (2) is 30 mm. The pressure at section (1) is

464 kPa.

71.The small fan shown in Fig. 4 moves air at a mass flow rate of 0.1 kg/min.

Upstream of the fan, the pipe diameter is 60 mm, the flow is laminar with

parabolic velocity distribution and the kinetic energy coefficient, α1, is equal to

2.0. Downstream of the fan, the pipe diameter is 30 mm, the flow is turbulent, the

velocity profile is quit uniform, and the kinetic energy coefficient, α2, is equal to

1.08. If the rise in static pressure across the fan is 0.1 kPa and the fan motor draws

0.14 W, compare the value of loss calculated: (a) assuming uniform velocity

distributions, (b) considering actual velocity distributions.

72.A hurricane can be visualized as a planar incompressible flow consisting of a

rotating circular core surrounded by a potential flow:

A particular hurricane has a core of radius 40 m and air is sucked into this core at

a volume flow rate per meter depth perpendicular to the diagram of 5000 m3/sec.

Furthermore the pressure difference between the air far away from the hurricane

and the air at the edge of the core is assumed to be negligible. The density of the

air is assumed uniform and constant at 1.2 kg/m3. Find the angular rate of rotation

of the hurricane.

73.The working section of a water tunnel consists of a duct with a rectangular

cross-section. The width of the cross-section, b (perpendicular to sketch), is

constant but the height, h(x), may vary with longitudinal distance, x, measured

along the centerline of the duct:

Laminar boundary layers form on the upper and lower surface of the working

section and would cause an acceleration of the flow if the height h were constant

with x. (A similar effect would be caused by the front and back surfaces but we

ignore thus for the purposes of this problem and assume that there are no

boundary layers on the front and back surface.) A water tunnel designer wishes to

select the function h(x) in order to ensure that the pressure and velocity outside the

boundary layer (say, on the centerline) do not vary with x. The designer decides to

use a function, h(x), of the form kHxhxh 0)( , where h0, H and k are constants

and the boundary layers begin at x=0. Find: (a) The value of k. (b) The expression

for H in terms of the kinematic viscosity, v, and the velocity, U, of the flow on the

centerline.

74.A wide moving belt passes through a container of viscous liquid. The belt moves

vertically upward with a constant velocity, V0, as illustrated in Fig.10. Because of

viscous forces the belt picks up a film of fluid of thickness h. Gravity tends to

make the fluid drain down the belt. Use the Navier-Stokes equation to determine

an expression for the average velocity of the fluid film as it is dragged up the belt.

Assume that the flow is laminar, steady, and uniform.

75.Consider the steady, planar flow of an inviscid (frictionless), incompressible fluid

(density r) in a right-angle corner:

(t) Show that this flow is irrotational.

(u) Find an expression for the pressure, p, at any point in the flow assuming that

the pressure at the origin, p0, is known. The y-axis is vertically upward and

the only body force is that due to gravity, g.

(v) If the x-axis is a thin wall with a uniform pressure, p0, on its underside, find

the vertical force on that portion of the wall between x = 0 and x = 1. Assume

unit depth perpendicular to the page.

76.An arctic hut in the shape of a half-circular cylinder has a radius, R0. A wind of

velocity, V0, batters the hut and threatens to raise it off its foundations due to the

lift of the wind. This lift is partly due to the fact that the entrance to the hut is at

ground level at the location of the stagnation pressure. A clever occupant sized up

the situation quickly and relocated the entrance at an angle, θ0, from the ground

level, which caused the net force on the hut to vanish. What is the angle, θ0? For

the purpose of this problem the opening will be assumed to be very small

compared to the radius, R0. Assume incompressible potential flow and observe

that the static pressure inside the hut, ps, will depends on, θ0.

77.Consider a spherical ball with a uniform flow, when the Reynolds number

corresponding to the uniform flow increases, please sketch and describe the

variation of the flow field near the spherical ball.

78.Please use molecular structure to explain (viscosity)，please explain the viscosity

difference of liquid and gas，and relation between temp and pressure。

79.Flow over a cylinder, stream function and velocity potential can be refer as

rUr

sinsin and

rUr

coscos (as a uniform flow and a doublet)，

please find out flow velocity distribution and cylinder surface pressure distribution.

Consider the sketch below，from entrance region to (fully developed)。discuss(1)

→(2)→(3)，pipe flow inertial force、pressure and viscous force correlation，please

draw form(1)→(2)→(3) pressure along X-direction 。

80.Use Bernoulli Equation to analyze siphonic。Consider siphonvalve open and close

situation(EL, energy line) and(HGL, hydraulic grad line)，if there is a hole at

(1),please describe when valve open/close,(1) water will in or out?

81.What is Pi theory? Consider a possible relation △ Pi = f(𝐷, 𝜌, 𝜇, 𝑉)，The pressure

difference △ P and viscousμare(nonrepeating variable)，utilized D，densityρ，

velocity V，through Pi theory to prove the relation.

82.Fluid flow through resivor and pipe。Please mark Energy Line and Hydraulic

Grade Line，please describe when pipe has a hole，which section liquid will spurt，

and which section air will into the pipe.

83.Please describe (Pitot tube) theory to measure velocity。Please sketch a pictureto

explain when fluid flow over a Pitot tube the static pressure change along pitot

tube, and how does Pitot tube determine flow direction.

84.Please describe the difference between laminar flow and turbulent flow

characteristic , flow structure, momentum transfer.