Integral momentum equation

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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. Mathematically, Newton’s second law of motion for fixed-mass in a reference frame that moves at a constant velocity Vref is written as

	\sum {{\mathbf{F}} = \frac{{d{{(m{\mathbf{V}})}_{rel}}}}{{dt}}}     \qquad \qquad(1)

where the left-hand side is the net force vector acting on the fixed-mass system, and the right-hand side is the rate of momentum change.

For control volumes that contain only one phase, the integral form of Newton’s second law can be obtained by using eq.

{\left. {\frac{{d\Phi }}{{dt}}} \right|_{system}} = \frac{\partial }{{\partial t}}\int_V {\rho \phi dV + \int_A {\rho ({{\mathbf{V}}_{rel}} \cdot {\mathbf{n}})\phi dA} }

from Transformation formula.

With the applicable value of Φ and φ defined as \Phi  = m{{\mathbf{V}}_{rel}} and \phi  = {{\mathbf{V}}_{rel}}, one obtains

\sum {\mathbf{F}}  = \frac{\partial }{{\partial t}}\int_V {\rho {{\mathbf{V}}_{rel}}dV}  + \int_A {\rho ({{\mathbf{V}}_{rel}} \cdot {\mathbf{n}}){{\mathbf{V}}_{rel}}dA}     \qquad \qquad(2)

which is in the vector form and is valid in all three directions.

Forces acting on the control volume include body forces and contact forces that act on its surface. For example, for a multicomponent system that contains N components, if the body force per unit volume acting on the ith species is Xi, the total body force acting on the control volume is \int_V {\left[ {\sum\limits_{i = 1}^N {{\rho _i}{{\mathbf{X}}_i}} } \right]} dV, where ρi is the mass concentration (kg/m3) of the ith species. If the body force per unit mass is the same for different species (as is the case with gravity), the body force term is reduced to \int_V {\rho {\mathbf{X}}} dV, where ρ is the density of the mixture. The stress tensor acts on the surface of a fluid control volume, and includes both normal and shear stresses. The net force may be written as

\sum {{\mathbf{F}} = \int_V {\left[ {\sum\limits_{i = 1}^N {{\rho _i}{{\mathbf{X}}_i}} } \right]} dV + \int_A {{{{\mathbf{\tau '}}}_{rel}} \cdot {\mathbf{n}}dA} }     \qquad \qquad(3)

where {\mathbf{\tau '}} is the total stress tensor and n is the local normal unit vector on surface A. The dot product of a tensor of rank two, {\mathbf{\tau '}}, and a vector, n, is a vector that represents the force acting on the surface of the control volume per unit area. Combining eqs. (2) and (3), we obtain the momentum equation for the control volume of a single-phase system:

\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(4)

where the two terms on the left-hand side represent, respectively, the body force and stress on the control volume, and the two terms on the right-hand side represent, respectively, the rate of momentum change in the control volume and the rate of the momentum flow into or out of the control volume. Vrel is the bulk velocity of the mixture that contains N components.

When the control volume includes multiple phases, integrations must be performed for each subvolume. In that case, the momentum equation becomes (Faghri and Zhang, 2006)

\begin{array}{l} \sum\limits_{k = 1}^\Pi  {\left[ {\int_{{V_k}(t)} {\left[ {\sum\limits_{i = 1}^N {{\rho _{k,i}}{{\mathbf{X}}_{k,i}}} } \right]} dV + \int_{{A_k}(t)} {{{{\mathbf{\tau '}}}_{k,rel}} \cdot {{\mathbf{n}}_k}dA} } \right]}  \\ = \sum\limits_{k = 1}^\Pi  {\left[ {\frac{\partial }{{\partial t}}\int_{{V_k}(t)} {{\rho _k}{{\mathbf{V}}_{k,rel}}dV}  + \int_{{A_k}(t)} {{\rho _k}({{\mathbf{V}}_{k,rel}} \cdot {{\mathbf{n}}_k}){{\mathbf{V}}_{k,rel}}dA} } \right]}  \\ \end{array} \qquad \qquad(5)

Equations (4) and (5) are momentum equations in a coordinate system that is attached to and moves with an inertial reference frame. For a fixed coordinate system that does not move with the reference frame (while the control volume still moves with the reference frame at velocity Vref), one can substitute the general variables \Phi  = m{\mathbf{V}} and \phi  = {\mathbf{V}} to obtain the momentum equation.


Faghri, A., and Zhang, Y., 2006, Transport Phenomena in Multiphase Systems, Burlington, MA.

Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.

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