3.24 A symmetrical sprinkler is shown as in Fig. 3-40. The rotating radius is R=200mm, ?=450, the nozzle diameter is d=8mm, the total flow rate is qv=0.563L/s, if the frictional moment is 0.2N?m, find the rotating speed n. And what is the magnitude of the moment needed to hold the sprinkler at rest while it is in operation?

对称洒水器如图3-40所示。旋转半径 Fig. 3-40 Problem 3.24 R=200mm，?=450，喷口直径d=8mm，总

流量qv=0.563L/s，若已知摩擦阻力矩为0.2N?m，求转速n。若喷水时不让其旋转，应受到多大的力矩？(103 r/min，0.441 N?m)

Problems

1.1 The velocity field of a rotational flow is given by

Find the average angular rotating velocity at point (2,2,2).

已知有旋流动的速度场为

求在点（2,2,2）处平均旋转角速度。(?x=0.5，?y=?2，?z=?0.5)

1.2 Determine whether the following flow field is rotational flow or irrotational flow.

确定下列流场是有旋运动还是无旋运动：

(1) (2)

( (1)有旋，(2)无旋)

4.3 The velocity distribution of a flow field is described by v=x2yi?xy2j Is the flow irrotational?

流场的速度分布为

v=x2yi?xy2j

该流动是否无旋？(有旋)

4.4 For a certain incompressible, two-dimensional flow field the velocity component in the y direction is given by vy=x2+2xy

Determine the velocity component in the x direction so that the continuity equation is satisfied.

某一不可压缩平面流场在y方向的速度分量为 vy=x2+2xy

确定x方向的速度分量，以满足连续性方程。(vx=?x2+C)

4.5 For a certain incompressible flow field it is suggested that the velocity components are given by the equations

vx=x2y vy=4y3z vz=2z Is this a physically possible flow field?

某一不可压缩流场流场的速度分量由下列方程给出 vx=x2y vy=4y3z vz=2z 试问该流场在物理上是否可能？(不可能)

vvxxv

4.6 It is known that streamlines are concentric circles, and velocity distribution is 已知流线为同心圆族，其速度分布为

Find the velocity circulation along circle x2+y2=R2, where the radiuses of the circle are

(1) R=3；(2) R=5；(3) R=10 respectively.

求沿圆周x2+y2=R2的速度环量，其中圆的半径R分别为 (1) R=3；(2) R=5；(3) R=10。

((1) 18?/5，(2) 10?，(3) 10? )

4.7 Assume there is a vortex of ?=?0 locating at point (1，0), and another vortex of ?=-?0 at point (-1，0). Find the velocity circulation along the following routes: (1) x2+y2=4； (2) (x-1)2+y2=1；

(3) Square of x= ?2，y= ?2； (4) Square of x= ?0.5，y= ?0.5.

设在点(1，0)处置有?=?0的旋涡，在点(-1，0)处置有?=-?0的旋涡。试求下列路线的速度环量：

(1) x2+y2=4； (2) (x-1)2+y2=1；

(3) x= ?2，y= ?2的方形框；

(4) x= ?0.5，y= ?0.5的方形框。( (1) 0，(2) ?0，(3) 0，(4) 0 )

4.8 For incompressible fluid, determine if there exist stream functions in the following flow fields, where K is a constant.

对于不可压缩流体，试确定下列流场是否存在流函数？式中K为常数。 (1) vx=Ksin(xy)，vy=-K sin(xy)

(2) vx=Kln(xy)，vy=-Ky/x ( (1) 不存在，(2) 存在)

4.9 Demonstrate the following planar flow of an incompressible fluid

?satisfies continuity equation, and is a potential flow, then find the potential function. 试证明以下不可压缩流体平面流动

v

满足连续性方程，是有势流动，并求势函数。( ?=x2/2+x2y?y2/2?y3/3)

4.10 A velocity field is given by vx=x2y+y2，vy=x2?xy2，vz=0，questions： (1) If there exist stream function and potential function?

(2) Find the expressions of stream function and potential function if they exist. 给定速度场vx=x2y+y2，vy=x2?xy2，vz=0，问： (1) 是否存在流函数和势函数？ (2) 如果存在，求其具体表达式。

((1)存在流函数，不存在势函数；(2) ?=x2y2/2+y2/3?x2/3) 4.11 The velocity potential in a certain flow filed is ?=4xy

Determine the corresponding stream function. 某流场的速度势为

?=4xy 求相应的流函数。( ?=2x2?2y2)

4.12 The velocity potential for an incompressible, planar flow is ?=x2?y2+x Find its stream function.

不可压缩流体平面流动的势函数为 ?=x2?y2+x 试求流函数。( ?=2xy+y) 4.13 The stream function for an incompressible, planar flow is

?=xy+2x?3y

Find the potential function.

不可压缩流体平面流动的流函数为 ?=xy+2x?3y 试求势函数。( ?=(x2?y2)/2?3x?2y)

4.14 Demonstrate the following two flow fields are identical:

(1) the potential function is ?=x2+x ?y2 (2) the stream function is ?=2xy+y 证明下列两个流场是相同的。 (1) 势函数 ?=x2+x ?y2 (2) 流函数 ?=2xy+y

4.15

Given the velocity distribution of a flow field as

vx=Ax+By

vy=Cx+Dy

If the flow is incompressible and irrotational, find

vx