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The law can be formulated mathematically in the fields of fluid mechanics and continuum mechanics, where the conservation of mass is usually expressed using the continuity equation, given in differential form as + =, where is the density (mass per unit volume), is the time, is the divergence, and is the flow velocity field.
All of the conservation laws listed above are local conservation laws. A local conservation law is expressed mathematically by a continuity equation, which states that the change in the quantity in a volume is equal to the total net "flux" of the quantity through the surface of the volume. The following sections discuss continuity equations in ...
The first law of thermodynamics is a version of the law of conservation of energy, adapted for thermodynamic processes. In general, the conservation law states that the total energy of an isolated system is constant; energy can be transformed from one form to another, but can be neither created nor destroyed.
The exact conservation law used in the analysis of the system depends on the context of the problem, but all revolve around mass conservation, i.e., that matter cannot disappear or be created spontaneously. [2]: 59–62 Therefore, mass balances are used widely in engineering and environmental analyses.
The conservation laws may be applied to a region of the flow called a control volume. A control volume is a discrete volume in space through which fluid is assumed to flow. The integral formulations of the conservation laws are used to describe the change of mass, momentum, or energy within the control volume.
Noether's theorem states that every continuous symmetry of the action of a physical system with conservative forces has a corresponding conservation law.This is the first of two theorems (see Noether's second theorem) published by mathematician Emmy Noether in 1918. [1]
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The movement of a fluid through porous media is described by the combination of Darcy's law with the principle of conservation of mass in order to express the capillary force or fluid velocity as a function of various other parameters including the effective pore radius, liquid viscosity or permeability. [3]