Conservation of linear momentum

Conservation of linear momentum $\rho\dot{u}_i$ can be expressed as a global balance of surface tractions T_i and volumetric body forces f_i:

 

$$\int_V \rho\ddot{u}_i \,dV = \int_S T_i \,dS + \int_V f_i \,dV \,,$$ (5)

integrated over the bounding surface S and volume V. Note that in (5) I have already invoked conservation of mass in a quiescent fluid by assuming that the material derivative $D_t \rho$ vanishes identically. Since (5) is valid for an arbitrary volume, it can be re-expressed in local form:

 

$$\rho\ddot{u}_i = \d_j\sigma_{ij} + f_i \,.$$ (6)

The traction vector T_i has been replaced with the gradient of the stress tensor, $\d_j\sigma_{ij}$. Assuming a non-viscous fluid, which supports no shear stress, a linear isotropic stress-strain relation can be written as

 

$$\sigma_{ij} = \lambda\delta_{ij} \epsilon_{kk} \,.$$ (7)

Since pressure is identified with the normal compressional stress components, pressure and strain displacement can be related as

 

$$P = -\lambda\epsilon_{kk} = -\lambda\d_k u_k \,,$$ (8)

where the standard strain-displacement definition has been invoked:

 

$$\epsilon_{ij} = 1/2 ( \d_i u_j + \d_j u_i ) \,.$$ (9)

Substituting (7) into (6) and assuming body forces f_i (gravity, hydrostatic pressure, etc.) to be negligible or non-deviatoric:

 

$$\rho\ddot{u}_i = \d_j\lambda\delta_{ij}\epsilon_{kk} + 0 \,,$$ (10)

and using (8) results in

 

$$\rho\ddot{u}_i = \d_i\lambda\d_k u_k \,.$$ (11)

Taking the divergence of both sides of (11) to ensure compressional motion only,

 

$$\d_i\rho\ddot{u}_i = \d_i\d_i\lambda\d_k u_k \,,$$ (12)

and expanding gradient terms on the left results in

 

$$\rho\d_i\ddot{u}_i + \ddot{u}_i\d_i\rho= \d_i\d_i\lambda\d_k u_k \,.$$ (13)

Up until this point, the mass density and bulk modulus terms have been unrestricted with respect to spatial variability, i.e., $\rho=\rho({\bf \underline{x}})$, and $\lambda=\lambda({\bf \underline{x}})$.