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In mathematics, an Euler–Cauchy equation, or Cauchy–Euler equation, or simply Euler's equation, is a linear homogeneous ordinary differential equation with variable coefficients. It is sometimes referred to as an equidimensional equation. Because of its particularly simple equidimensional structure, the differential equation can be solved ...
The Euler equations can be applied to incompressible and compressible flows. The incompressible Euler equations consist of Cauchy equations for conservation of mass and balance of momentum, together with the incompressibility condition that the flow velocity is divergence-free.
In mathematics a Cauchy–Euler operator is a ... It is named after Augustin-Louis Cauchy and Leonhard Euler. The simplest example is ... Cauchy–Euler equation ...
The next three years Cauchy was mainly on unpaid sick leave; he spent his time fruitfully, working on mathematics (on the related topics of symmetric functions, the symmetric group and the theory of higher-order algebraic equations). He attempted admission to the First Class of the Institut de France but failed on three different occasions ...
By expressing the shear tensor in terms of viscosity and fluid velocity, and assuming constant density and viscosity, the Cauchy momentum equation will lead to the Navier–Stokes equations. By assuming inviscid flow, the Navier–Stokes equations can further simplify to the Euler equations. The divergence of the stress tensor can be written as
In mathematics and computational science, the Euler method (also called the forward Euler method) is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value.
represents body accelerations acting on the continuum, for example gravity, inertial accelerations, electrostatic accelerations, and so on. In this form, it is apparent that in the assumption of an inviscid fluid – no deviatoric stress – Cauchy equations reduce to the Euler equations.
We would like boundary conditions to ensure that exactly one (unique) solution exists, but for second-order partial differential equations, it is not as simple to guarantee existence and uniqueness as it is for ordinary differential equations. Cauchy data are most immediately relevant for hyperbolic problems (for example, the wave equation) on ...