# Integral momentum equation

### From Thermal-FluidsPedia

The integral form of Newton’s second law for a control volume that includes only one phase is expressed by eq. (2.10). The surface integral terms in eq. (2.10) can be rewritten using the divergence theorem: (2.62) (2.63) Substituting eq. (2.62) and (2.63) into eq. (2.10), and considering eq. (2.48), the entire equation can be rewritten as a volume integral: (2.64) As was the case for the continuity equation, the integrand must equal zero to assure the general validity of eq. (2.64); so, one obtains the desired differential form of the momentum equation: (2.65) The derivatives on the left-hand side of eq. (2.65) may be expanded to yield (2.66) The first bracketed term on the left vanishes, as required by the continuity eq. (2.51). The second term may be written more simply in substantial derivative form, and the entire equation becomes (2.67) The stress tensor, is the sum of an isotropic thermodynamic stress, , and the viscous stress tensor, [defined in eq. (1.56)], i.e., (2.68) Substituting eq. (2.68) into eq. (2.67), the momentum equation becomes (2.69) The viscous stress tensor measured in the reference frame, , can be determined by using Newton’s law of viscosity [see eq. (1.59)]: (2.70) where Drel is the rate of strain tensor, i.e., (2.71) and I in eq. (2.70) is the unit tensor that satisfies for any tensor a. The diagonal components of I are equal to one and all other components are zero: (2.72) If the fluid is incompressible ( ), the second term on the right-hand side of eq. (2.70) will be zero according to eq. (2.55). The momentum equation (2.67) then becomes (2.73) where the left-hand side is the inertial term (mass per unit volume 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 , we have Stokes’ flow or creep flow, and eq. (2.73) becomes elliptic and is similar to the steady-state conduction equation. In a Cartesian coordinate system, the vector form of the momentum equation, eq. (2.73), for incompressible and Newtonian fluid with constant viscosity can be written as three equations in the x-, y-, and z-directions: (2.74) (2.75) (2.76) where 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, , eq. (2.73) becomes (2.77) 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. (2.77); 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 ( ):

(2.78)

where is density at the reference point, is the coefficient of thermal expansion, and is the composition coefficient of volume expansion. Substituting eq. (2.78) into eq. (2.77), the momentum equation for natural convection is obtained

(2.79)

where the second and third terms on the right-hand side of eq. (2.79) describe the effect of buoyancy force due to temperature and composition variation within the system, respectively.