– The fundamental equation of hydrodynamics of ideal fuid (Eulerian equation):
\[
p \frac{d v}{d i}=t-v_{P} \text {. }
\]

Where $p$ is the fuid density, $t$ is the volune density of mass forces ( $=$ pg is the ease of erovity), $
abla p$ is the prosure eradient.
– Bernoulli’, equation. In the steady now of an ideal fiuid
\[
\frac{p^{*}}{2}+\omega^{n}+p=\text { cosat }
\]
along any streamline.
– Reynolds number delining the fov patuen of a viscous fluid:
\[
\text { Re }=\rho o t / \eta \text {. }
\]
where $t$ is a charseleristie length, $n$ is the Duid viscosity.
– Poinuille’s law. The volume of tiguid towing through s eircular tube (is $\mathrm{m}^{2} / \mathrm{s}$ ):
\[
Q=\frac{n R^{4}}{8 m} \frac{h-h}{T},
\]
wherp $\boldsymbol{R}$ and $t$ are the tybe’s radius and lengh, $h-D_{1}$ is the preseure difference between itse sols of the tube.
– Stokes’ law. The Iriction force on the sphere of radius $r$ moving through a viscoes fivid:
\[
\boldsymbol{r}=6 \mathrm{sm} v .
\]
1.315. Ideal fluid flows along a flat tube of constant cross-section, located in a horizontal plane and bent as shown in Fig. 1.80 (tep view). The flow is steady. Are the pressures and velocities of the fluid equal at points $I$ and 2 ? What is the shape of the streamlines?
1.316. Two masometric tubes are mouated on a horizontal pipe of varying cross-section at the sections $S_{1}$ and $s_{3}$ (Fig. 1.81). Find
the volume of water flowing across the pipe’s section per unit time if the difference in water columns is equal to $\mathrm{Ah}$.
1.317. A Pitot tube (Fig. 1.82) is mounted along the axis of a gas pipeline whose cross-sectional area is equal to $S$. Assuming the viscosity to be negligible, find the volume of gas flowing across the
คie. 1.80 .
ค.g. 1.81.
section of the pipe per unit time, if the difference in the liquid columns is equal to $\Delta h$, and the densities of the liquid and the gas are $P_{0}$ and $P$ respectively,
1.318. A wide vessel with a small hole in the bottem is filled with water and kerosene. Neglecting the viscosity, find the velocity of the water flow, if the thickness of the water layer is equal to $h_{1}=30 \mathrm{~cm}$ and that of the kerosese layer to $h_{2}=20 \mathrm{~cm}$.
1.319. A wide cylindrical versel $50 \mathrm{~cm}$ in height is filled with water and rests on a table. Assuming the viscosity to be negligible, find at what height from the botton of the vessel a small hole should be perforated lor the water jet coming out of it to hit the surface of the table at the maximum distance $I_{\text {man }}$ from the vessel. Find $l_{\text {mex: }}$
1.320. A bent tube is lowered into a water stream as shown in Fig. 1.83 . The velocity of the stream relative to the tube is equal to $y=2.5 \mathrm{~m} / \mathrm{s}$. The closed upper end of the tube located at the height $h_{4}=12 \mathrm{~cm}$ has a small orifice. To what height $h$ will the water jet spurt?
1.321. The horisental bottom of a wide vessel with an ideal fluid has a round orifice of radius $R_{1}$ over which a round closed cylinder is monsted, whese radius $R_{1}>R_{1}$ (Fig. 1.84). The clearance between the cylinder and the bottem of the vessel is very small, the fluid density is $p$. Find the static pressure of the fluid in the clearance as a function of the distance $r$ from the axis of the orifice (and the cylinder), if the height of the fluid is equal to $h$.
1.322. What work should be done in order to squeete all water from a horizentally located cylisder (Fig. 1.85) during the time $t$ by means of a constant force acting on the plsten? The velume of water in the cylinder is equal to $V$, the eross-sectional area of the ori-
$\theta$

fice to $s$, with $s$ being considerably less than the piston area. The friction and viscosity are negligibly small.
1.323. A cylindrical vessel of height $h$ and base area $S$ is filled with water. An orifice of area $s \ll S$ is opened in the bottom of the vessel. Neglecting the viscosity of water, determine how soon all the water will pour out of the vessel.
1.324. A horizontally oriented tube $A B$ of length $l$ rotates with a constant angular velocity $\omega$ about a stationary vertical axis $O O^{\prime}$ passing through the end $A$ (Fig. 1.86). The tube is filled with an ideal fluid. The end $A$ of the tube is open, the closed end $B$ has a very small orifice. Find the velocity of the fluid relative to the tube as a function of the column \”height\” $h$.
1.325. Demonstrate that in the case of a steady flow of an ideal fluid Eq. (1.7a) turns into Bernoulli equation.
1.326. On the opposite sides of a wide vertical vessel filled with water two identical holes are opened, each having the cross-sectional
Fig. 1.84.
Fig. 1.85.
area $S=0.50 \mathrm{~cm}^{2}$. The height difference between them is equal to $\Delta h=51 \mathrm{~cm}$. Find the resultant force of reaction of the water flowing out of the vessel.
1.327. The side wall of a wide vertical cylindrical vessel of height $h=75 \mathrm{~cm}$ has a narrow vertical slit running all the way down to the bottom of the vessel. The length of the slit is $l=50 \mathrm{~cm}$ and the width $b=1.0 \mathrm{~mm}$. With the slit closed, the vessel is filled with water. Find the resultant force of reaction of the water flowing out of the vessel immediately after the slit is opened.
1.328. Water flows out of a big tank along a tube bent at right angles; the inside radius of the tube is equal to $r=0.50 \mathrm{~cm}$ (Fig. 1.87). The length of the horizontal section of the tube is equal to $l=22 \mathrm{~cm}$. The water flow rate is $Q=0.50$ litres per second. Find the moment of reaction forces of flowing water, acting on the tube’s walls, relative to the point $O$.
64
1.329. A side wall of a wide open tank is provided with a narrowing tube (Fig. 1.88) through which water flows out. The cross-sectional area of the tube decreases from $S=3.0 \mathrm{~cm}^{2}$ to $s=1.0 \mathrm{~cm}^{2}$. The water level in the tank is $h=4.6 \mathrm{~m}$ higher than that in the tube.
Fig. 1.86.
Neglecting the viscosity of the water, find the horizontal component of the force tending to pull the tube out of the tank.
Fig. 1.87.
Fig. 1.88.
1.330. A cylindrical vessel with water is rotated about its vertical axis with a constant angular velocity $\omega$. Find:
(a) the shape of the free surface of the water;
(b) the water pressure distribution over the bottom of the vessel along its radius provided the pressure at the central point is equal to $p_{0}$.
1.331. A thin horizontal disc of radius $R=10 \mathrm{~cm}$ is located within a cylindrical cavity filled with oil whose viscosity $\eta=0.08 \mathrm{P}$ (Fig. 1.89). The clearance between the disc and the horizontal planes
Fig. 1.89.
of the cavity is equal to $h=1.0 \mathrm{~mm}$. Find the power developed by the viscous forces acting on the disc when it rotates with the angular velocity $\omega=60 \mathrm{rad} / \mathrm{s}$. The end effects are to be neglected.
1.332. A long cylinder of radius $R_{1}$ is displaced along its axis with a constant velocity $v_{0}$ inside a stationary co-axial cylinder of radius $h_{3}$. The space between the eylinders is filled with viscous liquid. Find the velocity of the liquid as a function of the distance $r$ from the axis of the cylinders. The flow is laminar.
1.333. A fluid with viscosity in fills the space between two long. co-axial eylinders of radii $R_{1}$ and $R_{n}$, with $R_{1}<R_{7}$. The inner cylinder is stationary while the outer ene is rotated with a constant angular velocity $\omega_{\text {, }}$. The fuid flow is laminar. Taking inte account that the friction force acting on a unit area of a cylindrical surface of radius $r$ is defined by the formula $\sigma=
abla(\partial \omega / \partial r)$, find:
(a) the angular velocity of the retating fluid as a function of radius $r$;
(b) the moment of the friction forces acting on a unit length of the outer eylinder.
1.334. A tube of length $l$ and radius $A$ carries a steady flow of fluid whose density is $p$ and viscosity $\eta$. The fuid Alow velocity depends on the distance $r$ irom the axis of the tube as $v=v_{0}\left(1-r^{2} / R^{7}\right)$. Find:
(a) the volume of the fluid flowing across the section of the tube per wait time;
(b) the kinetic energy of the fluid within the tube’s volume;
(c) the friction force exerted on the tube by the fluid;
(d) the pressure difference at the ends of the tube.
1.335. In the arrangement shown in Fig. 1.90 a viscous liquid whose density is $p-1.0 \mathrm{~g} / \mathrm{cm}^{3}$ ‘ llows along a tube out of a wide tank
Fie. 1.90 .
A. Find the velocity of the liquid flow, if $h_{1}=10 \mathrm{~cm}, h_{2}=20 \mathrm{~cm}$, and $h_{3}=35 \mathrm{~cm}$. All the distances $l$ are equal.
1.336. The eross-sectional radius of a pipeline decreases gradually as $r=r_{0} e^{-a x}$, where $a=0.50 \mathrm{~m}^{-1}, x$ is the distance from the pipeline inlet. Find the ratio of Reynolds numbers for twe eress-sections separated by $\Delta x=3.2 \mathrm{~m}$.
1.337. When a sphere of radius $r_{1}=1.2 \mathrm{~mm}$ moves in glycerin, the laminar fow is ebeerved if the velocity of the sphere does not exceed $v_{1}=23 \mathrm{~cm} / \mathrm{s}$. At what minimum velocity $v_{2}$ of a sphere of radius $r_{2}=5.5 \mathrm{~cm}$ will the flow in water become turbulent? The
viscosities of glycerin and water are equal to $\eta_{1}=13.9 P$ and $\eta_{4}=$ $=0.011 \mathrm{P}$ respectively.
1.338. A lead sphere is steadily sinking in glycerin whose viscesity is equal to $\mathrm{n}=13.9 \mathrm{P}$. What is the maximum diameter of the sphere at which the flow around that sphere still remains laminar? It is known that the transition to the furbulent flow corresponds to Reynolds number $\mathrm{Re}=0.5$. (Here the characteristic length is taken to be the sphere diameter.)
1.339. A steel ball of diameter $d=3.0 \mathrm{~mm}$ starts sinking with zere initial velocity in olive oil whose viscosity is $\eta=0.90 \mathrm{P}$. How soon after the beginaing of motion will the velocity of the ball differ from the steady-state velocity by $n=1.0 \%$ ?