Chaplygin's equation
Encyclopedia
In mathematics
, Chaplygin's equation, named after Sergei Alekseevich Chaplygin
, is a partial differential equation
useful in the study of transonic
flow
. It is
Here,
is the speed of sound
, determined by the equation of state
of the fluid and Bernoulli's principle
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Chaplygin's equation, named after Sergei Alekseevich Chaplygin
Sergei Alekseevich Chaplygin
Sergey Alexeyevich Chaplygin was a Russian and Soviet physicist, mathematician, and mechanical engineer. He is known for mathematical formulas such as Chaplygin's equation and for a hypothetical substance in cosmology called Chaplygin gas, named after him. He graduated in 1890 from Moscow...
, is a partial differential equation
Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
useful in the study of transonic
Transonic
Transonic speed is an aeronautics term referring to the condition of flight in which a range of velocities of airflow exist surrounding and flowing past an air vehicle or an airfoil that are concurrently below, at, and above the speed of sound in the range of Mach 0.8 to 1.2, i.e. 600–900 mph...
flow
Fluid mechanics
Fluid mechanics is the study of fluids and the forces on them. Fluid mechanics can be divided into fluid statics, the study of fluids at rest; fluid kinematics, the study of fluids in motion; and fluid dynamics, the study of the effect of forces on fluid motion...
. It is
Here,
is the speed of soundSpeed of sound
The speed of sound is the distance travelled during a unit of time by a sound wave propagating through an elastic medium. In dry air at , the speed of sound is . This is , or about one kilometer in three seconds or approximately one mile in five seconds....
, determined by the equation of state
Equation of state
In physics and thermodynamics, an equation of state is a relation between state variables. More specifically, an equation of state is a thermodynamic equation describing the state of matter under a given set of physical conditions...
of the fluid and Bernoulli's principle
Bernoulli's principle
In fluid dynamics, Bernoulli's principle states that for an inviscid flow, an increase in the speed of the fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy...
.