\[\sum F_x = \frac{\partial}{\partial t} \int_{CV} u \,\rho\, d\forall \, + \, \int_{CS} u \,\rho \,\vec{V} \cdot \, \vec{n}\,dA \]

In this equation, \(\sum F_x\) means the sum of all external forces acting on the control volume in the \(x\)-direction (units of Newtons); \(u\) is the x-component of velocity (units of m/s); \(\rho\) is the density of the fluid (units of kg/m³); and \(\vec{V}\,\cdot \,\vec{n}\) represents the dot product of the velocity vector and the outward-facing unit normal (units of m/s). The notation “CV” and “CS” stand for “control volume” and “control surface”. \(d\forall\) and \(dA\) represent differential volume and area elements within the control volume and the control surface. Similar equations can be written for the \(y\)- and \(z\)-directions:

\[\sum F_y = \frac{\partial}{\partial t} \int_{CV} v \,\rho\, d\forall \, + \, \int_{CS} v \,\rho \,\vec{V} \cdot \, \vec{n}\,dA \]

\[\sum F_z = \frac{\partial}{\partial t} \int_{CV} w \,\rho\, d\forall \, + \, \int_{CS} w \,\rho \,\vec{V} \cdot \, \vec{n}\,dA \]

\[\dot{m} = \rho \,\vec{V} \cdot \,\vec{n} A\]

where \(\dot{m}\) is the mass flow rate of the fluid (in units of kg/s).