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In order to find the weak form of the Navier–Stokes equations, firstly, consider the momentum equation [20] + + = multiply it for a test function , defined in a suitable space , and integrate both members with respect to the domain : [20] + + = Counter-integrating by parts the diffusive and the pressure terms and by using the Gauss' theorem ...
A general momentum equation is obtained when the conservation relation is applied to momentum. When the intensive property φ is considered as the mass flux (also momentum density), that is, the product of mass density and flow velocity ρu, by substitution into the general continuity equation:
All acceleration requires an exchange of momentum, which can be thought of as the "unit of movement". Momentum is related to mass and velocity, as given by the formula P = mv, where P is the momentum, m the mass, and v the velocity. The velocity of a body is easily changeable, but in most cases the mass is not, which makes it important.
There are two main descriptions of motion: dynamics and kinematics.Dynamics is general, since the momenta, forces and energy of the particles are taken into account. In this instance, sometimes the term dynamics refers to the differential equations that the system satisfies (e.g., Newton's second law or Euler–Lagrange equations), and sometimes to the solutions to those equations.
The previous equations for planar motion can be used here: corollaries of momentum, angular momentum etc. can immediately follow by applying the above definitions. For any object moving in any path in a plane, = = ^ the following general results apply to the particle.
This shows how the effective momentum can be seen as composed of two parts, ^ = +, a standard momentum and a crystal momentum . More precisely the crystal momentum is not a momentum but it stands for the momentum in the same way as the electromagnetic momentum in the minimal coupling , and as part of a canonical transformation of the momentum.
Defining equation (physical chemistry) List of electromagnetism equations; List of equations in classical mechanics; List of equations in gravitation; List of equations in nuclear and particle physics; List of equations in quantum mechanics; List of photonics equations; List of relativistic equations; Table of thermodynamic equations
A continuum version of the conservation of momentum leads to equations such as the Navier–Stokes equations for fluids or the Cauchy momentum equation for deformable solids or fluids. Classical Momentum is a vector quantity : it has both magnitude and direction.