Question

Show (step by step) that the inertial terms of the conservative form of the momentum equation are equal to the density times

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Answer #1

Solution:

media%2Fcc6%2Fcc6f311e-4122-4e32-9879-75

Consider the LHS of the equation:

media%2Fda0%2Fda06a4c5-d55e-4c70-ab3b-7f

Let us expand the 2nd term media%2F63c%2F63cf0a04-043b-428a-89d0-47. We know that:

03

Assume, media%2F133%2F133af185-59b6-4518-ab7b-36. So V pV becomes:

media%2Fd60%2Fd6073302-3483-4a32-8583-c9

media%2F276%2F276027eb-2ef4-4eed-a7f6-05

media%2F117%2F11760974-dbe7-4c26-8dc8-b0

If we expand the partial derivative, we get:

media%2Fac6%2Fac6cc305-7140-4560-b22a-65

oy, oy,

Rearraigning the terms we get:

dxdydZ

So LHS becomes:

ro 4r(丽= tV + ρ.at) + a นที่ +의 yi-Opwi-a@ρυ +ay )ρν +av.и ду

Rearranging the terms we get:

dz continuity equation which is 0 total derivativeof

From the continuity equation, we know:

op opu ,δρν .apw

Hence we get:

media%2F890%2F89022f2d-fe80-463c-8ee1-07

hence proved.

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