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321The horizontal bottom of a wide vessel with an ideal fluid has a round orifice of radius R1 over which a round closed cylinder is mounted, whose radius R2 >R1 ( Fig. 1.84). The clearance between the cylinder and the bottom 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.
322What work should be done in order to squeeze all water from a horizontally located cylinder (Fig. 1.85) during the time t by means of a constant force acting on the piston? The volume of water in the cylinder is equal to V, the cross-sectional area of the orifice to s, with s being considerably less than the piston area. The friction and viscosity are negligibly small.
323A cylindrical vessel of height h and base area S is filled with water. An orifice of area s << 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.
324A horizontally oriented tube the bottom of the AB of length L rotates with a constant angular velocity co about a stationary vertical axis OO' passing through the end A (Fig. 1.86). The tube is filled with an ideal fluid. The A the tube is --- end of 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.
325Demonstrate that in the case of a steady flow of an ideal fluid Eq. Fig. 1.83. (1.7a) turns into Bernoulli equation.
326On the opposite sides of a wide vertical vessel filled with water two identical holes are opened, each having the cross-sectional area S = 0.50 cm2. The height difference between them is equal to dh = 51 cm. Find the resultant force of reaction of the water flowing out of the vessel.
327The side wall of a wide vertical cylindrical vessel of height h = 75 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 cm and the width b = 1.0 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.
328Water 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 cm (Fig. 1.87). The length of the horizontal section of the tube is equal to l = 22 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.
329A 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 cm2 to s = 1.0 cm2. The water level in the tank is h = 4.6 m higher than that in the tube. Neglecting the viscosity of the water, find the horizontal component of the force tending to pull the tube out of the tank.
330A cylindrical vessel with water is rotated about its vertical axis with a constant angular velocity w. 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 p0.
331A thin horizontal disc of radius R = 10 cm is located within a cylindrical cavity filled with oil whose viscosity n = 0.08 P (Fig. 1.89). The clearance between the disc and the horizontal planes of the cavity is equal to h = 1.0 mm. Find the power developed by the viscous forces acting on the disc when it rotates with the angular velocity w = 60 rad/s. The end effects are to be neglected.
332A long cylinder of radius R1 is displaced along its axis with a constant velocity vo inside a stationary co-axial cylinder of radius R2. The space between the cylinders 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.
333A fluid with viscosity n fills the space between two long co-axial cylinders of radii R1 and R2, with R1 < R2. The inner cylinder is stationary while the outer one is rotated with a constant angular velocity w2. The fluid flow is laminar. Taking into account that the friction force acting on a unit area of a cylindrical surface of radius r is defined by the formula s = nr(dw/dr), find: (a) the angular velocity of the rotating fluid as a function of radius r; (b) the moment of the friction forces acting on a unit length of the outer cylinder.
334A tube of length l and radius R carries a steady flow of fluid whose density is ρ and viscosity η. The fluid flow velocity depends on the distance r from the axis of the tube as v = v0 (1 - r2/R2). Find: (a) the volume of the fluid flowing across the section of the tube per unit 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.
335In the arrangement shown in Fig. 1.90 a viscous liquid whose density is p = 1.0 g/cm3 flows along a tube out of a wide tank A. Find the velocity of the liquid flow, if h1 = 10 cm, h2 = 20 cm, and h3 = 35 cm. All the distances L are equal.
336The cross-sectional radius of a pipeline decreases gradually as r = r0·e^(-ax), where a = 0.50 m-1, x is the distance from the pipeline inlet. Find the ratio of Reynolds numbers for two cross-sections separated by dx = 3.2 m.
337When a sphere of radius r1 = 1.2 mm moves in glycerin, the laminar flow is observed if the velocity of the sphere does not exceed v1 = 23 cm/s. At what minimum velocity v2 of a sphere of radius r2 = 5.5 cm will the flow in water become turbulent? The viscosities of glycerin and water are equal to n1 = 13.9 P and n2 = = 0.011 P respectively.
338A lead sphere is steadily sinking in glycerin whose viscosity is equal to η = 13.9 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 turbulent flow corresponds to Reynolds number Re = 0.5. (Here the characteristic length is taken to be the sphere diameter.)
339A steel ball of diameter d = 3.0 mm starts sinking with zero initial velocity in olive oil whose viscosity is η = 0.90 P. How soon after the beginning of motion will the velocity of the ball differ from the steady-state velocity by n = 1.0%?
340A rod moves lengthwise with a constant velocity v relative to the inertial reference frame K. At what value of v will the length of the rod in this frame be η = 0.5% less than its proper length?
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