19. Let us first consider an open cylindrical vessel containing a liquid and revolving about its vertical axis with an angular velocity ω. The liquid gradually attains the same angular

19. Let us first consider an open cylindrical vessel containing a liquid and revolving about its vertical axis with an angular velocity ω. The liquid gradually attains the same angular velocity as the vessel and its free surface becomes a concave surface of revolution (Fig. 20). Acting on the liquid are two body forces, gravity and a centrifugal force, respectively equal to g and ω 2r when referred to unit mass, due to its second component, the resultant body force j increases ith the radius while its inclination to the horizontal decreases. inclination of the surface increases with the radius.

From the stated conditions it follows that at the intersection of curve AOB with the rotation axis C = h, whence we finally have

A more common case in practice is when a vessel containing a liquid revolves about a horizontal or arbitrary axis and the angular velocity ω is so great that gravity can be neglected as compared with the centrifugal forces.

The pressure gradient in the liquid can easily be obtained by considering the equilibrium equation for an elementary volume of base area S and height taken along the radius (Fig. 21). The volume is subjected to pressure and centrifugal forces.

So, or

The constant C is found from the condition that, at r = r₀, p=p₀, whence

Finally, we obtain the relation between p and r in the form

It is frequently necessary to calculate the thrust of a liquid rota­ting with a vessel exerted on the wall normal to the axis of rotation

.

20. (along a stream tube) (4.6)

This is the equation of continuity for an elementary stream tube.

Introducing the mean velocity instead of the actual velocity for a finite stream contained in impermeable walls, we obtain:

(along a stream). (4.7)

Equation (4.7) shows that for an incompressible liquid the mean velocity varies inversely as the cross-sectional area of stream:

(4.7`)

The equation of continuity is thus a special case of the law of conservation of matter.

22. The continuity equation is simply a mathematical expression of the principle of conservation of mass. For a control volume that has a single inlet and a single outlet, the principle of conservation of mass states that, for steady-state flow, the mass flow rate into the volume must equal the mass flow rate out. The continuity equation for this situation is

23. Euler equations describing the flow of an ideal fluid.

,

де —velocity of fluid, ρ — її густина, p — тиск.

Ideal fluid called liquid for which the non-essential processes of heat conduction and viscosity.

In the case of mass forces, such as fluid in a gravitational field, the Euler equation is written

,

24. A streamline is a line in a flowing fluid the tangent to which at any point shows the direction of the velocity vector of the fluid particles at that point (Fig. 23). In steady flow, evidently, streamlines and pathlines coincide and do not change with time.

25. A stream tube is a tubular space bounded by a surface consisting of streamlines (Fig. 24). When the cross-section of a stream tube is contracted to zero, a streamline is obtained in the limit.

The velocity vectors at all points of the surface of a stream tube are tangential to the surface. There being no normal components of the velocity, no particle of the liquid can enter or leave the stream tube, except at the ends. Thus, a stream tube can be regarded as surrounded by impenetrable walls and therefore treated as an elementary stream.

26.

This is Bernoulli's equation for an ideal liquid. It was developed by Daniel Bernoulli in 1738.

The terms in Eq. (4.12) represent linear quantities; they are called:

z = elevation head, or potential head, or geodetic head;

= pressure head, or static head;

= velocity head.

The sum

is called the total head.

Equation (4.12) is written for two arbitrary cross-sections of a stream tube and it expresses the equality of the total head H at the sections. As the sections were chosen arbitrarily, it follows that the total head is the same at any other section of the same stream tube, i. e.,

(along a stream tube).

Thus, the sum of the elevation, pressure and velocity heads of an ideal liquid is constant along a stream tube.

29.

This is Bernoulli's equation for real flow. It differs from the equa­tion for a differential stream tube in an ideal liquid by the term rep­resenting the loss of energy (head) and the coefficient which takes into account the nonuniform velocity distribution. Furthermore, the equation contains the mean velocities across the respective sec­tions.

Whereas Bernoulli's equation for an ideal fluid stream tube is an expression of the law of conservation of mechanical energy, for real flow it represents an energy-balance equation taking losses into account. The energy dissipated by the flow does not vanish, of course. It changes into. another, form of energy, heat, which raises the tem­perature of the liquid.

27. The

flow in which in each point occupied by

fluid its velocity doesn’t change in time is called

stationary flow.

28. if velocity

vectors components of fluid elements are not the fu

nctions of the time, the flow is called

non-

stationary

.

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