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In classical mechanics, Euler's rotation equations are a vectorial quasilinear first-order ordinary differential equation describing the rotation of a rigid body, using a rotating reference frame with angular velocity ω whose axes are fixed to the body. They are named in honour of Leonhard Euler. Their general vector form is
In the physical science of dynamics, rigid-body dynamics studies the movement of systems of interconnected bodies under the action of external forces.The assumption that the bodies are rigid (i.e. they do not deform under the action of applied forces) simplifies analysis, by reducing the parameters that describe the configuration of the system to the translation and rotation of reference ...
In fluid dynamics, the Kirchhoff equations, named after Gustav Kirchhoff, describe the motion of a rigid body in an ideal fluid. = + + +, = + +, = (~ +) = ^, = ^ where and are the angular and linear velocity vectors at the point , respectively; ~ is the moment of inertia tensor, is the body's mass; ^ is a unit normal vector to the surface of the body at the point ; is a pressure at this point ...
Traditionally the Newton–Euler equations is the grouping together of Euler's two laws of motion for a rigid body into a single equation with 6 components, using column vectors and matrices. These laws relate the motion of the center of gravity of a rigid body with the sum of forces and torques (or synonymously moments) acting on the rigid body.
where D / Dt is the material derivative operator, u is the flow velocity, ρ is the local fluid density, p is the local pressure, τ is the viscous stress tensor and B represents the sum of the external body forces. The first source term on the right hand side represents vortex stretching.
For < <, (= represents solid body rotation, the whole fluid rotates at the same speed) the solution reaches the solid body rotation at infinity in an oscillating manner from the plate. The axial velocity is negative w < 0 {\displaystyle w<0} for 0 ≤ γ < 1 {\displaystyle 0\leq \gamma <1} and positive w > 0 {\displaystyle w>0} for 1 < γ < ∞ ...
Rigid-body-like vortex v ∝ r: Parallel flow with shear Irrotational vortex v ∝ 1 / r where v is the velocity of the flow, r is the distance to the center of the vortex and ∝ indicates proportionality. Absolute velocities around the highlighted point: Relative velocities (magnified) around the highlighted point Vorticity ≠ 0 ...
Movements of air in the atmosphere and water in the ocean are notable examples of this behavior: rather than flowing directly from areas of high pressure to low pressure, as they would on a non-rotating planet, winds and currents tend to flow to the right of this direction north of the equator, and to the left of this direction south of the ...