Divergence of the Navier-Stokes Equation

[FluidStu]

The Navier-Stokes equation is:

(DU_j/Dt) = v [(∂²U_i/∂x_j∂x_i) + (∂²U_j/∂x_i∂x_i)] – 1/ρ (∇p)

where D/Dt is the material (substantial) derivative, v is the kinematic viscosity and ∇p is the modified pressure gradient (taking into account gravity and pressure). Note that the velocity field is non-solenoidal (∇⋅U ≠ 0).

How, then, can we take the divergence of this equation and get the following result?:

I can follow all of the terms other than the one underlined in blue. I know that it comes from the blue section of the Navier-Stokes written above, since I can easily get all the other terms.

Thanks in advance

 

[Chestermiller]

The Navier-Stokes equation is:

(DU_j/Dt) = v [(∂²U_i/∂x_j∂x_i) + (∂²U_j/∂x_i∂x_i)] – 1/ρ (∇p)

where D/Dt is the material (substantial) derivative, v is the kinematic viscosity and ∇p is the modified pressure gradient (taking into account gravity and pressure). Note that the velocity field is non-solenoidal (∇⋅U ≠ 0).

How, then, can we take the divergence of this equation and get the following result?:

View attachment 100112

I can follow all of the terms other than the one underlined in blue. I know that it comes from the blue section of the Navier-Stokes written above, since I can easily get all the other terms.

Thanks in advance

The terms in blue can not come from the blue section of the NS. They are non-linear in velocity, and blue section of the NS equation is linear in velocity. The blue terms in your relationship must come from the left side of the NS equation.

My advice to you is to write out the NS equation for each of the three components. Then take the partial of the x component with respect to x, the partial of the y component with respect to y, and the partial of the z component with respect to z. Then add the resulting 3 equations. This will be bulletproof.

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