41 What principle is a base for obtaining differential equation of inviscid gas motion? Name and formulate it.

Inviscid flow is a schematic representation of the motion of mobile media. from similarity criteria .

Theoretical analysis of inviscid flows have been obtained for stationary and irrotational flows. The acceleration of a fluid particle, entered into the Euler equation, can be written as follows:

where   = 1/2 curl   is the velocity vector curl. For a steady-state flow (   = 0), when   = 0 or   ||   , the Euler equation allows a general integral. Since the mass forces have a potential in a majority of cases—i.e., can be written as   = −grad Φ, where Φ is the mass force potential or the force function—the general integral of Euler's equation takes the form known as Bernoulli's integral:

For a heavy incompressible fluid Φ = gZ, and ρ = const,

where Z is the vertical coordinate. For weightless, incompressible fluid,

The equations of an inviscid flow are reduced in this case to a system of two ordinary differential equations:

where γ = c_p/c_v is the ratio of specific heat capacities of the gas, and the prime means differentiation with respect to θ.

Depending on whether the motion is subsonic or supersonic, the differential equation is an elliptic or hyperbolic one. Inviscid flows of an incompressible fluid form a large and important class because with the velocity of flow much lesser than the velocity of sound, the velocity potential equation takes the form of the Laplace linear equation, which is well-studied in mathematics:

42. Which forces act in flow and are taken into account during obtaining differential equation of inviscid gas motion?

Euler’s equation, which characterize the pressure force acting on elementary volume off a liquid and inertia forces

differential from of a continuity equation. Express the law of mass conservation.

The law of energy conservation – a change in the internal and kinetic energy of a fluid particle is due to an action of impressed mass forces and surface (pressure p), and to an inflow of heat with intensity q from an external source/

43. Write vector form of Euler’s equation for inviscid gas motion and call equation members.

𝞺 ;

V- is the fluid flow speed ( at point on a stream line) g- is a acceleration due to gravity - ia a external forse

P – is the pressure

-the velocity at points in space where they are continuous

44. Call flow condition where Euler’s equation has closed form solution.

Conditions of fluid motion in a bounded container, can be incorporated with minimal difficulty. Other generalizations to flow on

manifolds or in higher dimensions offer no additional complications,

although the classification of symmetries and conservation laws does depend

on the specific geometry and boundary conditions.

45. Write Euler’s equation solution, which is named Bernoulli’s integral. Call its members.

V - is a velocity

P- is a pressure

𝞺- density

Ф – the mass force potential

46 Write Bernoulli’s equation for incompressible gas and call its members.

V- is the fluid flow speed ( at point on a stream line) g- is a acceleration due to gravity y – is a the elevation of the point above a reference plane, with the positive direction pointing upward.

P – is the pressure

47 Write Bernoulli’s equation for compressible gas in follow forms:

1. when equation contains pressure and density;

2. when equation contains temperature;

;

3. when equation contains speed of sound;

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

4. when equation contains enthalpy.

48. What assumption with respect to relation between pressure and density are used obtaining of Bernoulli’s equation for compressible gas?

The higher the pressure of the gas, the higher the density. The lower the pressure of the gas, the lower the density.

49. Draw oblique shock wave velocity triangles, call elements of these triangles

49. Write momentum change theorem for unit cross section area stream, when it cross the shock wave (for normal and tangential directions). Call physical values used in this formulas.

From the Bernoulli’s equation we can get the change of momentum in x direction:

Y direction:

p- pressure ; V –velocity; 𝞺- density ; a- Step of the cross-cection; U- local energy density (flow velocity of vector field)

49. Write formula for normal component of the velocity relation behind and ahead shock wave, call physical values used in this formula. The same for tangential component of the velocity

for normal component of the velocity behind and ahead shock wave

V1 behind shock wave; V2 – ahead shock wave; M- Mach number.

For tahgential velocity behind and ahead shock wave

V1 behind shock wave; V2 – ahead shock wave, 𝞺 density

52 Write formula for pressures relation behind and ahead shock wave, call physical values used in this formula

the mass flow that enters the system on the left is - ρ1u1

The mass flow that leaves the system on the right is ρ2u2

is the energy entering the system every second

is the energy , which leaving the system

53 Write formula for temperatures relation behind and ahead shock wave, call physical values used in this formula, call physical values used in this formula

54. Write the formula of relation between flow deflection angle, shock wave angle and Mach number ahead shock wave.

55. Relation between velocity coefficients behind and ahead normal shock wave.

56 Shock adiabat, its formula, call physical values used in this formula.

W –enthalpy densities

P – pressure

N – baryon number densities

57. Explain the difference between adiabatic compression and compression in shock wave.

Abiabatic compression. A reduction in volume of a substance without heat flow, in or out.  Any process that occurs without gain or loss of heat.

Compression in shock wave

 A large-amplitude compression wave, as that produced by an explosion or by supersonic motion of a body in a medium.

 A violent disruption, disturbance, or reaction: Shock waves of revolution shattered the government.