9.6.1 Motion on an infinite plane

To get started, it is useful to consider fluid motion on an infinite flat plane. In the absence of external forces which act to make a particular horizontal direction special, the environment maintains translational symmetry in either of the horizontal directions. Hence, the total horizontal momentum in either direction is conserved. Mathematically, this result means that the momentum of a fluid parcel takes the form of a conservation equation. That is, the forces affecting the time tendency of this momentum are represented as a total divergence. These statements take their mathematical form as the time tendency for the momentum density

 

$$(\rho \, u^{m})_{,t} (T^{mn} - \rho \, u^{m} \, u^{n})_{,n} + \rho \, f^{m}$$ = (9.143)

where the tensor labels extend over the horizontal Cartesian coordinates x,y, and the conservation of mass was used in the form

$$\rho_{,t} + (\rho \, u^{n})_{,n} = 0.$$ (9.144)

Additionally, the stress tensor has been written in the form

$$T^{mn} = \tau^{mn} - \delta^{mn} \, p,$$ (9.145)

which is the sum of the symmetric frictional stress tensor and diagonal pressure stress tensor. In the absence of external forces f^m, or in the case when these forces can be derived as the divergence of a scalar, the total horizontal momentum per unit volume $\int \rho \, u^{m} \, dV$ is a constant in time. In addition to momentum in a particular direction, the discussion in Section 9.2.3 showed that so long as the stress tensor is symmetric and there is an absence of external forces, there is an angular momentum conservation law. For motion on the plane, this conservation law arises from symmetry of the unforced motion under rotations about the vertical axis. That is, angular momentum about the vertical direction is conserved in the absence of external forces or boundary effects. Mathematically, the conservation of angular momentum can be derived from the momentum equation in a similar manner to that used in Section 9.2.3. For completeness, the derivation is summarized. Recall that the angular momentum per unit volume is given by

$$\rho \, \mathcal{M}_{m} = \rho \, \epsilon_{mnp} \, x^{n} \, u^{p}.$$ (9.146)

Using the conservation of mass, the momentum equations, and symmetry of the stress tensor, it is straightforward to determine the conservation law

$$(\rho \, \mathcal{M}_{m})_{t} + (\rho \, \mathcal{M}_{m} \, u^{p})_{,p} \epsilon_{mnp} \, [ (x^{n} \, T^{pq})_{,q} + x^{n} \, \rho \, f^{p} ].$$ = (9.147)

The first term on the right hand side takes the form of a total divergence, and the second term represents external torques. In the absence of external torques and boundary effects, $\int \rho \, \mathcal{M}_{m} \, dV$ is a constant in time.