Integral governing equations

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Transformation Formula

A fixed-mass system describes an amount of matter that can move, flow and interact with the surroundings, but the control volume approach depicts a region or volume of interest in a flow field, which is not unique and depends on the user. So conservation laws for a fixed-mass system need to be transformed to apply to a control volume. The transformation Formula allows one to mathematically and physically link the conservation laws for a control volume with that of a fixed-mass system.

See Main Article Transformation formula

Continuity Equation

The law of the conservation of mass dictates that mass may be neither created nor destroyed. For a control volume that contains only one phase, the integral continuity equation is

\frac{\partial }{{\partial t}}\int_V {\rho dV + \int_A {\rho ({{\mathbf{V}}_{rel}} \cdot {\mathbf{n}})dA} }  = 0     \qquad \qquad(1)

See Main Article Integral continuity equation


Newton’s second law states that, in an inertial reference frame, the time rate of momentum change of a fixed mass system is equal to the net force acting on the system, and it takes place in the direction of the net force. The momentum equation for the control volume of a single-phase system is

\begin{array}{l} \int_V {\left[ {\sum\limits_{i = 1}^N {{\rho _i}{{\mathbf{X}}_i}} } \right]} dV + \int_A {{{{\mathbf{\tau '}}}_{rel}} \cdot {\mathbf{n}}dA}  \\  = \frac{\partial }{{\partial t}}\int_V {\rho {{\mathbf{V}}_{rel}}dV}  + \int_A {\rho ({{\mathbf{V}}_{rel}} \cdot {\mathbf{n}}){{\mathbf{V}}_{rel}}dA}  \\ \end{array} \qquad \qquad(2)

See Main Article Integral momentum equation


See Main Article Energy


See Main Article Entropy

Conservation of Mass Species

See Main Article Conservation of Mass Species