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One can measure a given property by either carrying out the measurement on a fixed point in space as particles of the fluid pass by, or by following a parcel of fluid along its streamline. The derivative of a field with respect to a fixed position in space is called the Eulerian derivative, while the derivative following a moving parcel is ...
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 ...
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.
For constant fluid density, the incompressible equations can be written as a quasilinear advection equation for the fluid velocity together with an elliptic Poisson's equation for the pressure. On the other hand, the compressible Euler equations form a quasilinear hyperbolic system of conservation equations .
Second derivatives of thermodynamic potentials generally describe the response of the system to small changes. The number of second derivatives which are independent of each other is relatively small, which means that most material properties can be described in terms of just a few "standard" properties.
The vorticity equation of fluid dynamics describes the evolution of the vorticity ω of a particle of a fluid as it moves with its flow; that is, the local rotation of the fluid (in terms of vector calculus this is the curl of the flow velocity). The governing equation is:
The convection–diffusion equation can be derived in a straightforward way [4] from the continuity equation, which states that the rate of change for a scalar quantity in a differential control volume is given by flow and diffusion into and out of that part of the system along with any generation or consumption inside the control volume: + =, where j is the total flux and R is a net ...
Mass flow rate is defined by the limit [3] [4] ˙ = =, i.e., the flow of mass through a surface per time .. The overdot on ˙ is Newton's notation for a time derivative.Since mass is a scalar quantity, the mass flow rate (the time derivative of mass) is also a scalar quantity.