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Volumetric flow rate is defined by the limit [3] = ˙ = =, that is, the flow of volume of fluid V through a surface per unit time t.. Since this is only the time derivative of volume, a scalar quantity, the volumetric flow rate is also a scalar quantity.
Often a vector field is drawn by curves (field lines) following the "flow"; the magnitude of the vector field is then the line density, and the flux through a surface is the number of lines. Lines originate from areas of positive divergence (sources) and end at areas of negative divergence (sinks).
In this sense, the function is the flow and both induces and is induced by the vector field. From a point, the rate of change of the i-th component with respect to the parametrization of the flow (“how much the flow has acted”) is described by the i-th component of the field.
Mathematically, mass flux is defined as the limit =, where = = is the mass current (flow of mass m per unit time t) and A is the area through which the mass flows.. For mass flux as a vector j m, the surface integral of it over a surface S, followed by an integral over the time duration t 1 to t 2, gives the total amount of mass flowing through the surface in that time (t 2 − t 1): = ^.
Informally, a flow may be viewed as a continuous motion of points over time. More formally, a flow is a group action of the real numbers on a set. The idea of a vector flow, that is, the flow determined by a vector field, occurs in the areas of differential topology, Riemannian geometry and Lie groups.
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.
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. Right: The reduction in flux passing through a surface can be visualized by reduction in F or d S equivalently (resolved into components , θ is angle to ...
For example, for a macroscopic scalar field φ(x, t) and a macroscopic vector field A(x, t) the definition becomes: +, +. In the scalar case ∇ φ is simply the gradient of a scalar, while ∇ A is the covariant derivative of the macroscopic vector (which can also be thought of as the Jacobian matrix of A as a function of x ).