Q.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 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.
Solution. 328. Let the velocity of water flowing through the tube at a certain instant of time be u, then where Q is the rate of flow of water and πr^{2} is the cross section area of the tube.
From impulse momentum theorem, for the stream of water striking the tube comer, in xdirection in the time interval dt,
Therefore, the force exerted on the water stream by the tube,
According to third law, the reaction force on the tube’s wall by the stream equals
Hence, the sought moment of force about 0 becomes
Q.329. A side wall of a wide open tank is provided with a narrowing tube (Fig. 1.88) through which water flows out. The crosssectional area of the tube decreases from S = 3.0 cm^{2} to s = 1.0 cm^{2}. 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.
Solution. 329. Suppose the radius at A is R and it decreases uniformaly to r at B where S = πR^{2} and s = πr^{2}. Assume also that the semi vectical angle at 0 is α. Then
So
where y is the radius at the point P distant x from the vertex O. Suppose the velocity with which the liquid flows out is V at A, v at B and u at P. Then by the equation of continuity
The velocity v of efflux is given by
and Bernoulli's theorem gives
where p_{p} is the pressure at P and p_{0} is the atmospheric pressure which is the pressure just outside of B. The force on the nozzle tending to pull it out is then
We have subtracted p_{0} which is the force due to atmosphenic pressure the factor sinθ gives horizontal component of the force and ds is the length of the element of nozzle surface, ds = dx sec θ and
Thus
Note : If we try to calculate F from the momentum change of the liquid flowing out will be wrong even as regards the sign of the force.
There is of course the effect of pressure at S and s but quantitative derivation of F fron Newton's law is difficult.
Q.330. A cylindrical vessel with water is rotated about its vertical axis with a constant angular velocity ω. 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_{o}.
Solution. 330. The Euler’s equation is in the space fixed frame where downward. We assume incompressible fluid so p is constant.
Then where z is the height vertically upwards from some fixed origin. We go to rotating frame where the equation becomes
the additional terms on the right are the well known coriolis and centrifugal forces. In the frame rotating with the liquid
or
On the free surface p = constant, thus
If we choose the origin at point r = 0 fi.e. the axis) of the free surface then “cosntant” = 0 and
At the bottom z = constant
So
If p = p_{0} on the axis at the bottom, then
Q.331. A thin horizontal disc of radius R = 10 cm is located within a cylindrical cavity filled with oil whose viscosity η = 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 ω = 60 rad/s. The end effects are to be neglected.
Solution. 331. When the disc rotates the fuild in contact with, corotates but the fluid in contact with the walls of the cavity does not rotate. A velocity gradient is then set up leading to viscous forces. At a distance r from the axis the linear velocity is ω r so there is a velocity gradient both in the upper and lower clearance. The corresponding force on the element whose radial width is dr is
The torque due to this force is
2nd the net torque considering both the upper and lower clearance is
So power developed is
(A s instructed end effects i.e. rotation of fluid in the clearance r > R has been neglected.)
Q.332. A long cylinder of radius R_{1} is displaced along its axis with a constant velocity v_{0} inside a stationary coaxial cylinder of radius R_{2}. 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.
Solution. 332. Let us consider a coaxial cylinder of radius r and thickness dr. then force of friction or viscous force on this elemental layer
This force must be constant from layer to layer so that steady motion may be possible.
or,
Integrating,
or,
Putting
From (2) by (3) we get,
Note : The force F is supplied by the agency which tries to carry the inner cylinder with velocity v_{0}.
Q.333. A fluid with viscosity η fills the space between two long coaxial cylinders of radii R_{1} and R_{2}, with R_{1} < R_{2}. The inner cylinder is stationary while the outer one is rotated with a constant angular velocity ω_{2}. 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 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.
Solution. 333. (a) Let us consider an elemental cylinder of radius r and thickness dr then from Newton’s formula
and moment of this force acting on the element,
As in the previous problem N is constant when conditions are steady
Integrating,
or, (3)
Putting
From (3) and (4),
(b) From Eq. (4),
Q.334. A tube of length 1 and radius R carries a steady flow of fluid whose density is p and viscosity η. The fluid flow velocity de pends on the distance r from the axis of the tube as 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.
Solution. 334. (a) Let dV be the volume flowing per second through the cylindrical shell of thickness dr then,
and the total volume,
(b) Let, dE be the kinetic energy, within the above cylindrical shell. Then
Hence, total energy of the fluid,
(c) Here frictional force is the shearing force on the tube, exerted by the fluid, which equals
Given,
So,
And at
Then, viscous force is given by,
(d) Taking a cylindrical shell of thickness dr and radius r viscous force,
Let Δp be the pressure difference, then net force on the element
But, since the flow is steady,
or,
Q.335. In the arrangement shown in Fig. 1.90 a viscous liquid whose density is p = 1.0 g/cm^{3} flows along a tube out of a wide tank A. Find the velocity of the liquid flow, if h_{1} = 10 cm, h_{2 }= 20 cm, and h_{3} = 35 cm. All the distances l are equal.
Solution. 335. The loss of pressure head in travelling a distance l is seen from the middle section to be h_{2}  h_{1} = 10 cm. Since h_{2}  h_{1} = h_{1} in our problem and h_{3}  h_{2} = 15 cm = 5 + h_{2} h_{1}, we see that a pressure head of 5 cm remains incompensated and must be converted into kinetic energy, the liquid flowing out. Thus
Thus
Q.336. The crosssectional radius of a pipeline decreases gradually as is the distance from the pipeline inlet. Find the ratio of Reynolds numbers for two crosssections separated by Δx = 3.2 m.
Solution. 336. We know that, Reynold’s number (R_{e}) is defined as, where v is the velocity / is the characteristic length and r\ the coefficient of viscosity. In the case of circular cross section the chracteristic length is the diameter of crosssection d, and v is taken as average velocity of flow of liquid.
Now, (Reynold’s number at x_{1} from the pipe end) is the velocity at distance x_{1}
and similarly,
From equation of continuity,
or,
Thus
Q.337. When a sphere of radius r_{1} = 1.2 mm moves in glycerin, the laminar flow is observed if the velocity of the sphere does not exceed v_{1} = 23 cm/s. At what minimum velocity v_{2} of a sphere of radius r_{2} = 5.5 cm will the flow in water become turbulent? The viscosities of glycerin and water are equal to η_{1} = 13.9 P and η_{2} = 0.011 P respectively.
Solution. 337. We know that Reynold’s number for turbulent flow is greater than that on laminar flow.
on putting the values.
Q.338. A 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.)
Solution. 338.
(p = density of lead, p_{0} = density of glycerine.)
Thus
and mm on putting the values,
Q.339. A 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 steadystate velocity by n = 1.0%?
Solution. 339.
or
or
or
or
Since
So
Thus
The steady state velocity is g/k.
or
Thus
We have neglected buoyancy in olive oil.
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