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Assuming conservation of mass, with the known properties of divergence and gradient we can use the mass continuity equation, which represents the mass per unit volume of a homogenous fluid with respect to space and time (i.e., material derivative) of any finite volume (V) to represent the change of velocity in fluid media ...
The compressible Euler equations consist of equations for conservation of mass, balance of momentum, and balance of energy, together with a suitable constitutive equation for the specific energy density of the fluid. Historically, only the equations of conservation of mass and balance of momentum were derived by Euler.
This equation is called the Cauchy momentum equation and describes the non-relativistic momentum conservation of any continuum that conserves mass. σ is a rank two symmetric tensor given by its covariant components. In orthogonal coordinates in three dimensions it is represented as the 3 × 3 matrix:
The foundational axioms of fluid dynamics are the conservation laws, specifically, conservation of mass, conservation of linear momentum, and conservation of energy (also known as the first law of thermodynamics). These are based on classical mechanics and are modified in quantum mechanics and general relativity.
In continuum mechanics, the most general form of an exact conservation law is given by a continuity equation. For example, conservation of electric charge q is ∂ ρ ∂ t = − ∇ ⋅ j {\displaystyle {\frac {\partial \rho }{\partial t}}=-\nabla \cdot \mathbf {j} \,} where ∇⋅ is the divergence operator, ρ is the density of q (amount per ...
With the help of these equations the head developed by a pump and the head utilised by a turbine can be easily determined. As the name suggests these equations were formulated by Leonhard Euler in the eighteenth century. [1] These equations can be derived from the moment of momentum equation when applied for a pump or a turbine.
Flux F through a surface, dS is the differential vector area element, n is the unit normal to the surface. Left: No flux passes in the surface, the maximum amount flows normal to the surface.
The Navier–Stokes equations form a vector continuity equation describing the conservation of linear momentum. If the fluid is incompressible (volumetric strain rate is zero), the mass continuity equation simplifies to a volume continuity equation: [3] =, which means that the divergence of the velocity field is zero everywhere. Physically ...