Bernoulli equation (the Bernoulli integral) in hydroaeromechanics [[after the Swiss scientist D. Bernoulli], one of the fundamental equations of hydromechanics, which for the steady motion of an incompressible ideal liquid in a uniform field of gravitational forces has the form:
Gh + p/ρ + v2/2 = C, (1)
where v is the velocity of the liquid, ρ is its density, p is the pressure in it, h is the height of a liquid particle above some horizontal plane, g is the acceleration of free fall, C is a quantity that is constant along each streamline but in the general case changes its value on passing from one streamline to another.
The sum of the first two terms on the left-hand side of equation (1) equals the total potential energy, and the third term the kinetic energy, both taken per unit mass of the liquid; the whole equation therefore expresses the law of conservation of mechanical energy for a moving liquid and establishes an important relation between v, p and h. For example, if at constant h the velocity of the flow along a streamline increases, the pressure falls, and conversely. This law is used in measuring velocity with the aid of measuring tubes and in other aerodynamic measurements. The Bernoulli equation is also written in the form
h + p/γ + v2/2g = C or
γh + p + ρv2/2 = C (2)
(where γ =ρg — the specific weight of the liquid). In the first equation all the terms have the dimension of length and are called, respectively, the geometric (levelling), the piezometric and the velocity height, and in the second — the dimensions of pressure, and are correspondingly named the weight, the static and the dynamic pressure. In the general case, when the liquid is compressible (a gas) but barotropic, that is, when p in it depends only on ρ, and when its motion takes place in any field of volume (mass) forces that is potential (see Force field), the Bernoulli equation follows from the Euler equations of hydromechanics and has the form:
П+∫ dp/ρ + v2/2 = C, (3)
where П — the potential energy (potential) of the field of volume forces, taken per unit mass of the liquid. In the flow of gases the value of П changes little along a streamline and can be included in the constant, so that (3) takes the form:
∫ dp/ρ + v2/2 = C. (4) In technical applications, for a flow averaged over the cross-section of a channel, the so-called generalised Bernoulli equation is used: keeping the form of equations (1) and (3), the left-hand side also includes the work of the frictional forces and of overcoming hydraulic resistances, as well as the mechanical work of the liquid or gas (the work of a compressor or of turbines) with the appropriate sign. The generalised Bernoulli equation is widely used in hydraulics in calculating the flow of liquids and gases in pipelines, and in mechanical engineering in calculating compressors, turbines, pumps and other hydraulic and gas machines.
Gh + p/ρ + v2/2 = C, (1)
where v is the velocity of the liquid, ρ is its density, p is the pressure in it, h is the height of a liquid particle above some horizontal plane, g is the acceleration of free fall, C is a quantity that is constant along each streamline but in the general case changes its value on passing from one streamline to another.
The sum of the first two terms on the left-hand side of equation (1) equals the total potential energy, and the third term the kinetic energy, both taken per unit mass of the liquid; the whole equation therefore expresses the law of conservation of mechanical energy for a moving liquid and establishes an important relation between v, p and h. For example, if at constant h the velocity of the flow along a streamline increases, the pressure falls, and conversely. This law is used in measuring velocity with the aid of measuring tubes and in other aerodynamic measurements. The Bernoulli equation is also written in the form
h + p/γ + v2/2g = C or
γh + p + ρv2/2 = C (2)
(where γ =ρg — the specific weight of the liquid). In the first equation all the terms have the dimension of length and are called, respectively, the geometric (levelling), the piezometric and the velocity height, and in the second — the dimensions of pressure, and are correspondingly named the weight, the static and the dynamic pressure. In the general case, when the liquid is compressible (a gas) but barotropic, that is, when p in it depends only on ρ, and when its motion takes place in any field of volume (mass) forces that is potential (see Force field), the Bernoulli equation follows from the Euler equations of hydromechanics and has the form:
П+∫ dp/ρ + v2/2 = C, (3)
where П — the potential energy (potential) of the field of volume forces, taken per unit mass of the liquid. In the flow of gases the value of П changes little along a streamline and can be included in the constant, so that (3) takes the form:
∫ dp/ρ + v2/2 = C. (4) In technical applications, for a flow averaged over the cross-section of a channel, the so-called generalised Bernoulli equation is used: keeping the form of equations (1) and (3), the left-hand side also includes the work of the frictional forces and of overcoming hydraulic resistances, as well as the mechanical work of the liquid or gas (the work of a compressor or of turbines) with the appropriate sign. The generalised Bernoulli equation is widely used in hydraulics in calculating the flow of liquids and gases in pipelines, and in mechanical engineering in calculating compressors, turbines, pumps and other hydraulic and gas machines.