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You are tracking the x-component of momentum in terms 1, 2, and 4, while you are tracking the unspecified j component of momentum in term 3. The latter you envision as the surface force according to your discussion, so term 3 would ostensibly be tracking the vector momentum in your understanding, though your mixing of tensor index and boldface vector notation is ambiguous and error-prone. The final equation you obtain by "bringing all the terms together" is actually the correct integral form of the x-momentum equation, provided you set j=1 or j=x in the surface force term. This does correspond to using 3 out of 9 components of the stress tensor, namely Tau_11, Tau_21 and Tau_31, as you suggest, because the other six terms appear in the y-momentum and z-momentum equations (three in each). Note that by convention, it is Tau_1j, Tau_2j and Tau_3j that appear in the dot product, and not Tau_i1, Tau_i2 and Tau_i3 as you have suggested, which would matter if the stress tensor were not symmetric. The second term encapsulates the spatial rate of change of x-momentum (because of the u) in all directions (because of the vector V).
I would suggest doing the derivation in the component (scalar) notation, and then combining terms to get the vector form, in order to get a clear picture of what each term or factor represents. |

Thanks very much for your reply it has cleared up alot of the questions that I had about the equation I derived. I can see now what the stress vector (or tensor) should consist of in the third term and why and I can see how the second term represents the spatial rate of change of momentum in the x momentum alone.
By deconstructing the equation into its components I can also see that the terms match up with the terms in the differential form of the momentum equation. Again thanks alot for your help, in your explanation. It really did clarifiy the equation and what each term means. |

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