# Momentum equation

The integral form of Newton’s second law for a control volume that includes only one phase is expressed by

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

The surface integral terms in the above equation can be rewritten using the divergence theorem:

$\int_A {{{{\mathbf{\tau '}}}_{rel}} \cdot {\mathbf{n}}dA} = \int_V {\nabla \cdot {{{\mathbf{\tau '}}}_{rel}}dV} \qquad \qquad(1)$
$\int_A {\rho ({{\mathbf{V}}_{rel}} \cdot {\mathbf{n}}){{\mathbf{V}}_{rel}}dA} = \int_V {\nabla \cdot \rho {{\mathbf{V}}_{rel}}{{\mathbf{V}}_{rel}}dV} \qquad \qquad(2)$

Substituting eq. (1) and (2) into the integral momentum equation, and considering $\frac{d}{{dt}}\int_V {\rho \phi } dV = \int_V {\frac{{\partial (\rho \phi )}}{{\partial t}}} dV$, the entire equation can be rewritten as a volume integral:

$\int_V {\left[ {\nabla \cdot {{{\mathbf{\tau '}}}_{rel}} + \sum\limits_{i = 1}^N {{\rho _i}{{\mathbf{X}}_i}} - \frac{\partial }{{\partial t}}({\rho _k}{{\mathbf{V}}_{rel}}) - \nabla \cdot \rho {{\mathbf{V}}_{rel}}{{\mathbf{V}}_{rel}}} \right]dV} = 0 \qquad \qquad(3)$

As was the case for the continuity equation, the integrand must equal zero to assure the general validity of eq. (3); so, one obtains the desired differential form of the momentum equation:

$\frac{\partial }{{\partial t}}(\rho {{\mathbf{V}}_{rel}}) + \nabla \cdot \rho {{\mathbf{V}}_{rel}}{{\mathbf{V}}_{rel}} = \nabla \cdot {{\mathbf{\tau '}}_{rel}} + \sum\limits_{i = 1}^N {{\rho _i}{{\mathbf{X}}_i}} \qquad \qquad(4)$

The derivatives on the left-hand side of eq. (4) may be expanded to yield

${{\mathbf{V}}_{rel}}\left[ {\frac{{\partial \rho }}{{\partial t}} + \nabla \cdot \rho {{\mathbf{V}}_{rel}}} \right] + \rho \left[ {\frac{{\partial {{\mathbf{V}}_{rel}}}}{{\partial t}} + {{\mathbf{V}}_{rel}} \cdot \nabla {{\mathbf{V}}_{rel}}} \right] = \nabla \cdot {{\mathbf{\tau '}}_{rel}} + \sum\limits_{i = 1}^N {{\rho _i}{{\mathbf{X}}_i}} \qquad \qquad(5)$

The first bracketed term on the left vanishes, as required by the continuity equation. The second term may be written more simply in substantial derivative form, and the entire equation becomes

$\rho \frac{{D{{\mathbf{V}}_{rel}}}}{{Dt}} = \nabla \cdot {{\mathbf{\tau '}}_{rel}} + \sum\limits_{i = 1}^N {{\rho _i}{{\mathbf{X}}_i}} \qquad \qquad(6)$

The stress tensor, ${{\mathbf{\tau '}}_{rel}},$ is the sum of an isotropic thermodynamic stress, $- p{\mathbf{I}}$, and the viscous stress tensor, i.e.,

${{\mathbf{\tau '}}_{rel}} = - p{\mathbf{I}} + {{\mathbf{\tau }}_{rel}} \qquad \qquad(7)$

Substituting eq. (7) into eq. (6), the momentum equation becomes

$\rho \frac{{D{{\mathbf{V}}_{rel}}}}{{Dt}} = - \nabla p + \nabla \cdot {{\mathbf{\tau }}_{rel}} + \sum\limits_{i = 1}^N {{\rho _i}{{\mathbf{X}}_i}} \qquad \qquad(8)$

The viscous stress tensor measured in the reference frame, ${{\mathbf{\tau }}_{rel}}$, can be determined by using Newton’s law of viscosity (see Introduction to Momentum Transfer):

${{\mathbf{\tau }}_{rel}} = 2\mu {{\mathbf{D}}_{rel}} - \frac{2}{3}\mu (\nabla \cdot {{\mathbf{V}}_{rel}}){\mathbf{I}} \qquad \qquad(9)$

where Drel is the rate of strain tensor, i.e.,

${{\mathbf{D}}_{rel}} = \frac{1}{2}\left[ {\nabla {{\mathbf{V}}_{rel}} + {{\left( {\nabla {{\mathbf{V}}_{rel}}} \right)}^T}} \right] \qquad \qquad(10)$

and I in eq. (9) is the unit tensor that satisfies ${\mathbf{a}} \cdot {\mathbf{I}} = {\mathbf{I}} \cdot {\mathbf{a}} = {\mathbf{a}}$ for any tensor a. The diagonal components of I are equal to one and all other components are zero:

${I_{ij}} = \left\{ {\begin{array}{*{20}{c}} 1 & {i = j} \\ 0 & {i \ne j} \\ \end{array}} \right.\begin{array}{*{20}{c}} {} & {(i,j = 1,2,3)} \\ \end{array} \qquad \qquad(11)$

If the fluid is incompressible (ρ = const), the second term on the right-hand side of eq. (9) will be zero according to eq. $\nabla \cdot {\mathbf{V}} = 0$ from continuity equation. The momentum equation (6) then becomes

$\rho \frac{{D{{\mathbf{V}}_{rel}}}}{{Dt}} = \sum\limits_{i = 1}^N {{\rho _i}{{\mathbf{X}}_i}} - \nabla p + \nabla \cdot (\mu \nabla {{\mathbf{V}}_{rel}}) \qquad \qquad(12)$

where the left-hand side is the inertial term (mass per unit volume ρ times acceleration, DVrel / Dt). The three terms on the right-hand side represent body force per unit volume, pressure force per unit volume, and viscous force per unit volume, respectively.

For $D{{\mathbf{V}}_{rel}}/Dt = 0$, we have Stokes’ flow or creep flow, and eq. (12) becomes elliptic and is similar to the steady-state conduction equation.

In a Cartesian coordinate system, the vector form of the momentum equation, eq. (12), for incompressible and Newtonian fluid with constant viscosity can be written as three equations in the x − ,y − , and z directions:

$\rho \frac{{D{u_{rel}}}}{{Dt}} = \sum\limits_{i = 1}^N {{\rho _i}{X_i}} - \frac{{\partial p}}{{\partial x}} + \mu \left( {\frac{{{\partial ^2}{u_{rel}}}}{{\partial {x^2}}} + \frac{{{\partial ^2}{u_{rel}}}}{{\partial {y^2}}} + \frac{{{\partial ^2}{u_{rel}}}}{{\partial {z^2}}}} \right) \qquad \qquad(13)$
$\rho \frac{{D{v_{rel}}}}{{Dt}} = \sum\limits_{i = 1}^N {{\rho _i}{Y_i}} - \frac{{\partial p}}{{\partial y}} + \mu \left( {\frac{{{\partial ^2}{v_{rel}}}}{{\partial {x^2}}} + \frac{{{\partial ^2}{v_{rel}}}}{{\partial {y^2}}} + \frac{{{\partial ^2}{v_{rel}}}}{{\partial {z^2}}}} \right) \qquad \qquad(14)$
$\rho \frac{{D{w_{rel}}}}{{Dt}} = \sum\limits_{i = 1}^N {{\rho _i}{Z_i}} - \frac{{\partial p}}{{\partial z}} + \mu \left( {\frac{{{\partial ^2}{w_{rel}}}}{{\partial {x^2}}} + \frac{{{\partial ^2}{w_{rel}}}}{{\partial {y^2}}} + \frac{{{\partial ^2}{w_{rel}}}}{{\partial {z^2}}}} \right) \qquad \qquad(15)$

where Xi,Yi,andZi are the components of body force per unit volume acting on the ith species in the x − ,y − , and z directions, respectively.

For the case that the only body force is gravity, ${{\mathbf{X}}_i} = {\mathbf{g}}$, eq. (12) becomes

$\rho \frac{{D{{\mathbf{V}}_{rel}}}}{{Dt}} = \rho {\mathbf{g}} - \nabla p + \nabla \cdot (\mu \nabla {{\mathbf{V}}_{rel}}) \qquad \qquad(16)$

For natural convection problem, it is often assumed that the fluid is incompressible except in the first term on the right-hand side of eq. (16); this is referred to as the Boussinesq assumption. The density of a mixture is a function of temperature and mass fractions of species. It can be expanded using a Taylor’s series near the vicinity of a reference point ($\bar T,{\rm{ }}{\bar \omega _1},{\rm{ }}{\bar \omega _2},{\rm{ }} \cdots {\bar \omega _N}$):

$\begin{array}{l} \rho = \bar \rho + \frac{{\partial \rho }}{{\partial T}}(T - \bar T) + \sum\limits_{i = 1}^N {\frac{{\partial \rho }}{{\partial {\omega _i}}}({\omega _i} - } {{\bar \omega }_i}) + \cdots \\ \approx \bar \rho - \bar \rho \beta (T - \bar T) - \bar \rho \sum\limits_{i = 1}^N {{\beta _m}({\omega _i} - } {{\bar \omega }_i}) \\ \end{array} \qquad \qquad(17)$

where $\bar \rho$ is density at the reference point, $\beta {\rm{ }} = - (\partial \rho /\partial T)/\bar \rho$ is the coefficient of thermal expansion, and ${\beta _m} = - (\partial \rho /\partial {\omega _i})/\bar \rho$ is the composition coefficient of volume expansion. Substituting eq. (17) into eq. (16), the momentum equation for natural convection is obtained

$\rho \frac{{D{{\mathbf{V}}_{rel}}}}{{Dt}} = \left( { - \nabla p + \bar \rho {\mathbf{g}}} \right) - \bar \rho {\mathbf{g}}\beta (T - \bar T) - \bar \rho {\mathbf{g}}\sum\limits_{i = 1}^N {{\beta _m}({\omega _i} - } {\bar \omega _i}) + \nabla \cdot (\mu \nabla {{\mathbf{V}}_{rel}}) \qquad \qquad(18)$

where the second and third terms on the right-hand side of eq. (18) describe the effect of buoyancy force due to temperature and composition variation within the system, respectively.

## References

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.