流体力学课后作业.doc

1.1 A pressure of 2´106N/m2 is applied to a mass of water that initially filled a 1,000cm3 volume. Estimate its volume after the pressure is applied.
将2´106N/m2 的压强施加于初始体积为1,000cm3的水上，计算加压后水的体积。
(999.1cm3)

1.2 As shown in Fig.1-9, in a heating system there is a dilatation water tank. The whole volume of the water in the system is 8m3. The largest temperature rise is 500C and the coefficient of volume expansion is av=0.0005 1/K, what is the smallest cubage of the water bank?
如图1-10所示，一采暖系统在顶部设一膨胀水箱，系统内的水总体积为8m3，最大温升500C，膨胀系数av=0.005 1/K，求该水箱的最小容积？(0.2m3) Fig. 1-9 Problem 1.2

1.3 When the increment of pressure is 50kPa, the density of a certain liquid is 0.02%. Find the bulk modulus of the liquid.
当压强增量为50kPa时，某种液体的密度增加0.02%。求该液体的体积模量。( 2.5´108Pa)

1.4 Fig.1-10 shows the cross-section of an oil tank, its dimensions are length a=0.6m, width b=0.4m, height H=0.5m. The diameter of nozzle is d=0.05m, height h=0.08m. Oil fills to the upper edge of the tank, find:
(1) If only the thermal expansion coefficient av=6.5´10-41/K of the oil tank is considered, what is the volume Fig.1-10 Problem 1.4
of oil spilled from the tank when the temperature of oil increases from t1=-200C to t2=200C?
(2) If the linear expansion coefficient al=1.2´10-51/K of the oil tank is considered, what is the result in this case?
图1-10为一油箱横截面，其尺寸为长a=0.6m、宽b=0.4m、高H=0.5m，油嘴直径d=0.05m，高h=0.08m。由装到齐油箱的上壁，求：
(1) 如果只考虑油液的热膨胀系数av=6.5´10-41/K时，油液从t1=-200C上升到
t2=200C时，油箱中有多少体积的油溢出？
(2) 如果还考虑油箱的线膨胀系数al=1.2´10-51/K，这时的情况如何？
((1)2.492´10-3m3 (2)2.32´10-3m3)

1.5 A metallic sleeve glides down by self weight, as shown in Fig. 1-11. Oil of n=3´10-5m2/s and r=850kg/m3 fills between the sleeve and spindle. The inner diameter of the sleeve is D=102mm, the outer diameter of the spindle is d=100mm, sleeve length is L=250mm, its weight is 100N. Find the maximum velocity when the sleeve glides down freely (neglect air resistance).
Fig. 1-11 Problem 1.5
有一金属套由于自重沿垂直轴下滑，如图 1-11所示。轴与套间充满了n=3´10-5m2/s、r=850kg/m3的油液。套的内径D=102mm，轴的外径d=100mm，套长L=250mm，套重100N。试求套筒自由下滑时的最大速度为多少(不计空气阻力)。
(50 m/s)

1.6 The velocity distribution for flow of kerosene at 200C (m=4´10-3N·s/m2) between two walls is given by u=1000y(0.01-y) m/s, where y is measured in meters and the spacing between the walls is 1 cm. Plot the velocity distribution and determine the shear stress at the walls.
在200C时，煤油(m=4´10-3N·s/m2)在两壁面间流动的速度分布由u=1000y(0.01-y) m/s确定，式中y的单位为m，壁面间距为1cm。画出速度分布图，并确定壁面上的剪应力。
(4´10-2Pa)

1.7 As shown in Fig.1-12, the velocity distribution for viscous flow between stationary plates is given as follows:

Fig. 1-12 Problem 1.7
If glycerin is flowing (T=200C) and the pressure gradient dp/dx is 1.6kN/m3, what is the velocity and shear stress at a distance of 12 mm from the wall if the spacing By is 5.0 cm? What are the shear stress and velocity at the wall?
如图1-12所示，两固定平板间粘性流动的速度分布由

给出。如果流体为甘油(T=200C) 且压强梯度dp/dx为1.6kN/m3，间距By为5.0 cm，距平板12mm处的速度与剪应力为多少？平板处的剪应力与速度为多少？(u12=0.59m/s；τ12=20.8N/m2；u0=0；τ0=40.4N/m2)

1.8 What is the ratio of the dynamic viscosity of air to that of water at standard pressure and T=200C? What is the ratio of the kinematic viscosity of air to water for the same conditions?
在标准大气压、T=200C时，空气与水的动力粘度之比为多少？同样条件下它们的运动粘度之比又为多少？
(mA/mW=0.0018；nA/nW=15.1)

1.9 The device shown in Fig. 1-13 consists of a disk that is rotated by a shaft. The disk is positioned very close to a solid boundary. Between the disk and boundary is viscous oil.
(1) If the disk is rotated at a rate of 1 rad/s, what will be the ratio of the shear stress in the oil at r=2cm to Fig. 1-13 Problem 1.9
the shear stress at r=3cm?
(2) If the rate of rotation is 2 rad/s, what is the speed of oil in contact with the disk at r=3cm?
(3) If the oil viscosity is 0.01 N·s/m2 and the spacing y is 2mm, what is the shear stress for the condition noted in (b)?
图1-13所示装置由绕一根轴旋转的圆盘构成。圆盘放置在与固体边界很近的位置。圆盘与边界间为粘性油。
(1) 如果圆盘的旋转速率为1 rad/s，问半径为r=2cm与r=3cm处的剪应力之比为多少？
(2) 如果旋转速率为2 rad/s，r=3cm处与圆盘接触的油层的速度为多少？
(3) 如果油的粘度为0.01 N·s/m2 、且间距y为2mm，(b)情况下的剪应力为多少？
((1) 2:3；(2) 6cm/s；(3) 0.3Pa)

1.10 As shown in Fig. 1-14, a cone rotates around its vertical center axis at uniform velocity. The gap between two cones is d=1mm. It filled with lubricant which m=0.1Pa·s. In the Figure, R=0.3m, H=0.5m, Fig. 1-14 Problem 1.10
w=16 rad/s. What is the moment needed to rotate the cone?
如图1-14所示，一圆锥体绕竖直中心轴等速旋转，锥体与固定的外锥体之间的隙缝d=1mm，其中充满m=0.1Pa·s的润滑油。已知锥体顶面半径R=0.3m，锥体高度H=0.5m，当旋转角速度 w=16 rad/s 时，求所需要的旋转力矩。
(39.6N·m)

2.1 Two pressure gauges are located on the side of a tank that is filled with oil. One gauge at an elevation of 48m above ground level reads 347 kPa. Another at elevation 2.2m reads 57.5 kPa. Calculate the specific weight and density of the oil.
两个测压计位于一充满油的油箱的一侧。一个测压计高于地面的位置高度为48m，读数57.5 kPa。另一个位置高度为2.2m，读数347kPa。计算油的重度与密度。(g=6.32kN/m3,r=644kg/m3)

2.2 Two hemispherical shells are perfectly sealed together, and the internal pressure is reduced to 20 kPa, the inner radius is 15 cm and the outer radius is 15.5 cm. If the atmospheric pressure is 100 kPa, what force is required to pull the shells apart?
两半球壳完美密闭在一起，内压减至20 kPa，内径15 cm，外径15.5 cm。如果大气压强为100 kPa，求要将半球拉开所需的力为多少。(24.5kN)

2.3 As shown in the figure, there is a quantity of oil with density of 800 kg/m3, and a quantity of water below it in a closed container. If h1=300mm, h2=500mm, and h=400mmHg, find the pressure at the free surface of the oil.
如图所示，密闭容器中油的密度为800 kg/m3，其下方为水。如果h1=300mm, h2=500mm, 及 h=400mmHg，求油的自由表面上的压强。(46.1kPa) Problem 2.3
2.4 According to the diagram, one end of a tube connected to an evacuated container and the other end is put into a water pool whose surface is exposed to normal atmospheric pressure. If hv=2m, what is the pressure inside of container A?
如图，一根管子一端与一抽空的容器相连，另一端插入暴露于大气的水池中。如hv=2m，容器内的压强是多少？(81.7kPa)

P Problem 2.4

2.5 A pressure gauge is placed under sea level. If the gauge pressure at a point 300m below the free surface of the ocean, it registers 309 kPa, find the average specific weight of sea water.
一测压计置于海平面下，如果在自由表面下300m深处测压计的读数为309 kPa，求海水的平均重度。(1.03kN/m3)
2.6 If the local atmospheric pressure is given by 98.1 kPa absolute, find the relative pressure at points a, b and c in water. (see attached figure)
如果当地大气压的绝对压强为98.1 kPa，求水中a、b 及 c 点的相对压强。（见附图）(Pa=68.6kPa，Pb=31.3kPa，Pc=-29.4kPa)

Problem 2.6

2.7 There is an applied load of 5788N on the piston within a cylindrical container, in which is filled with oil and water. The oil column height is h1=30cm when the water column h2=50cm. The diameter of the container is giver as d=40cm. The density of oil is r1=800kg/m3 and that of mercury as r3=13 600kg/m3. Compute the height H(cm) of the mercury column in the U-tube.
圆柱形容器的活塞上作用一5788N的力，容器内充有水和油。当水柱高h2=50cm时，油柱高h1=30cm。容器直径d=40cm，油的密度r1=800kg/m3，水银密度r3=13 600kg/m3。求U形管内的汞柱高H(cm) 。(14.07cm)

Problem 2.7

2.8 According to the diagram, a closed tank contains water that has a relative pressure on the water surface of po= - 44.5kN/m2.
(1) What is the distance h?
(2) What is the pressure at point M with 0.3m below water surface?
(3) What is the piezometric head of point M relative to datum plane1-1?

如图所示，一密闭容器内盛有自由表面相对 Problem 2.8
压强为po=- 44.5kN/m2的水。
(1) 求距离h；
(2) 水面下0.3m处的M点
的压强为多少？
(3) M点相对于基准面1-1的测压管水头是多少？
(h=4.54m，Pm=-41.6kPa，hm=-4.24m)

2.9 An upright U-tube is fixed on the hood of a car which traveling in a straight-line, with a constant acceleration a=0.5m/s2. The length L=500mm. Find the height difference of the two free surfaces in the U-tube.
一竖直的U形管安装在以匀加速度a=0.5m/s2作直线运行的车罩上，长度L=500mm。求U形管内两自由液面的高度差。
(25.5mm)

Problem 2.9
2.10 According to the diagram, an open tank containing water moves with an acceleration a=3.6m/s2, along a slope of 30o. What is the inclined angle q with the horizontal plane and the equation of water pressure p at the free surface?
如图所示，一盛水的开口容器以加速度a=3.6m/s2沿30o的斜面运动。求自由液面与水平面的夹角q，并写出水的压强p的分布方程。
(q=15o，p=pa+rgh)

Problem 2.10

2.11 As shown in the figure, a gate of 2m wide, extends out of the plane of the diagram toward the reader. The gate is pivoted at hinge H, and weighs 500kg. Its center of gravity is 1.2m to the right and 0.9m above H. For what values of water depth x above H will the gate remain closed? (Neglect friction at the pivot point and neglect the thickness of the gate)
如图所示，闸门宽2m，重500kg，绕铰链H转动。闸门重心在距右端1.2m处且高于H 0.9m。问铰链H上方水深x为多少时，闸门将关闭？（忽略铰链摩擦及闸门的厚度）(1.25m) Problem 2.11
2.12 A flat 1 m high gate is hinged at point O, and can rotate around the point (see attached diagram). The height of point O is a=0.4m. What is the water depth when the gate can automatically open around point O?
一1米高的平板闸门在O点处用铰链连接，且可绕铰链转动（见附图）。O点的高度为a=0.4m。水深多少时，闸门将自动开启？
(h=1.33m)

Problem 2.12

2.13 As shown in the diagram, a closed tank forming a cube is half full of water, find: (a) the absolute pressure on the bottom of the tank, (b) the force exerted by the fluids on a tank wall as a function of height, (c) the location of the center of pressure by water on the tank wall.

如图所示，一密闭立方体水箱装了一半的水，求：(a) 箱底的绝对压强；(b) 将流体对一侧箱壁的作用力表示为高度的函数；(c) 水对箱壁的压力中心位置。 Problem 2.13
(17.8kPa，F=2h(8000+0.5rgh)，2/3)

2.14 A circular sluice gate of diameter d=1m is submerged in water as shown in the figure. The slant angle a=60o, the submerged depth hc=4.0m, and the weight of the gate FG=1kN. Determine the magnitude of the vertical force T so as to make the gate rotate upward about axis a (neglect friction at axis a).
如图所示的一直径d=1m圆形闸门淹没于水中。倾角a=60o，淹深hc=4.0m，闸门重FG=1kN。求使闸门绕a轴向上转动的铅直力T的大小。(230kN)
Problem 2.14

2.15 The gate M shown in the figure rotates about an axis through N. If a=33m, b=13m, d=20m and the width perpendicular to the plane of the figure is 3m, what torque applied to the shaft through N is required to hold the gate closed?
图示闸门M绕N轴转动。如a=33m、 b=13m、d=20m且与图形垂直的宽度为3m，要使闸门关闭，需要施加多大的力矩在转轴上？(666´103kN·m)
Problem 2.15

2.16 Find the horizontal and vertical components of the force exerted by fluids on the fixed circular cylinder shown in the figure, if:(1) the fluid to the left of the cylinder is a gas confined in a closed tank at a pressure of 35kPa, and (2) the fluid to the left of the cylinder is water with a free surface at an elevation coincident with the uppermost part of the cylinder. Assuming in both cases that normal atmospheric pressure conditions exist to the right and top of the cylinder.
求下述流体对图示圆柱体所施加的水平与铅直方向的分力。(1)圆柱体左边是压强为35kPa的密闭气体；(2)圆柱体左边是自由液面刚好与柱顶平齐的水。设两种情况下圆柱体的右边与顶部为标准大气压。 Problem 2.16
((1) Fx=130.5kN，Fz=35kN；(2) Fx=68.1kN，Fz=100.5kN)

2.17 The cross section of a tank is as shown in the figure. BC is a cylindrical surface with r=6m, and h=10m. If the tank contains gas at a pressure of 8kPa, determine the magnitude and location of the horizontal and vertical force components acting on unit width of tank wall ABC.
图示为一容器的剖面，BC是半径r=6m的圆柱面。容器高h=10m。如果容器装有压强为8kPa的气体，求作用在容器壁ABC上单位宽度的水平分力与铅直分力的大小与作用位置。
(Fx=80kN，Fz=48kN) Problem 2.17
2.18 Find the minimum value of z for which the gate in the figure will rotate counterclockwise if the gate is: (a) rectangular 5m high by 4m wide; (b) triangular, 4m base as axis, height 5m. neglect friction in bearings.

Problem 2.18
当下述情况时，求使图示闸门沿逆时针方向转动的z的最小值：(a) 闸门为高5m、宽4m的矩形；(b) 闸门为上底4m（转轴）、高5m的三角形。
((a) z=307.5m，(b)305m)

2.19 A curved surface is formed as a quarter of a circular cylinder with R=0.75m, as shown in the figure. The cylinder surface is w=3.55m wide (out of the plane of the figure toward the reader). Water standstill to the right of the curved surface to a depth of H=0.65m. Determine
(1) The magnitude of the hydrostatic force on the surface.
(2) The direction of the hydrostatic force. Problem 2.19
一曲面由半径为R=0.75m的四分之一的圆柱体组成，如图所示。圆柱面宽w=3.55m（垂直于图形平面）。曲面右侧静水深度H=0.65m。试求：
(1) 作用于曲面上静水压力的大小；
(2) 静水压力的方向。
(Fx=7357N，Fz=6013N)

2.20 As shown in the diagram, there is a cylinder with diameter D=4m and length L=12m in water. The depth of water on the right and left side of the cylinder are 4m and 2m respectively. Find the magnitude and direction of the force on the cylinder exerted by water.

如图所示，直径D=4m、长L=12m Problem 2.20
的圆柱体放于水中，圆柱体左右两边水的深度分别为4m 和 2m。求水对圆柱体的作用力的大小与方向。
(Fx=706kN，Fz=1579kN)

2.21 As shown in the figure, determine the pivot location y of the rectangular gate, so that the gate will just open.
如图所示，求闸门刚好开启时转轴的位置y为多少。(y=0.44m)

Problem 2.21

Problems
3.1 A two-dimensional, incompressible flow field is given by

Find the velocity and acceleration at point (1,2).
二维不可压缩流动由 确定。试求点（1，2）处的速度与加速度。(vx=5，vy=-30；ax=75，ay=150)

3.2 Suppose velocity distribution of a flow field is given by

Find:(1) the expression of local acceleration; (2) the acceleration of the fluid particle at point (1,1) when t=0.
设流场的速度分布为。求：（1）当地加速度的表达式；（2）t=0时在点（1，1）处流体质点的加速度。
((1) ¶vx/¶t=4, ¶vy/¶t=0；(2) ax=3, ay=-1)

3.3 The velocity components of a flow field is

Determine the streamline equation through point (x0,y0) at t=t0.
一流场的速度分量为

确定在t=t0时刻通过点(x0,y0)的流线方程。(x2-y-Aty+C=0)

3.4 A two-dimensional velocity field is given by

What is the streamline equation in this flow field?
已知二维速度场，求流线方程。(x1+t=cy)

3.5 It is known the velocity field is

Try to find the streamline equation passing through point（2，1，1）.
已知速度场，求通过点（2，1，1）的流线方程。
(x=2，5-z=2z)
3.6 An oil transportation pipeline, the velocity at the section of diameter 20cm is 2m/s, what is the velocity and mass flow rate at the section of diameter 5cm? The density of the oil is 850kg/m3.
有一输油管道，在内径为20cm的截面上流速为2m/s，求在另一内径为5cm的截面上的流速以及管内的质量流量。油的密度为850kg/m3。(32m/s，53.4kg/s)

3.7 In a pipeline of inner diameter 5cm, the mass flow rate of air is 0.5kg/s, pressure at a certain is 5´105Pa, the temperature is 1000C. Find the average air flow velocity on this section.
在内径为5cm的管道中，流动空气的质量流量为0.5kg/s，在某一截面上压强为5´105Pa，温度为1000C。求该截面上气流的平均速度。(54.5m/s)

3.8 The velocity distribution of an incompressible fluid is
，
Try to deduce the expression of vz by adopting continuity e