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I'm writing this question here because it is a mathematics issue not a physics problem in particular.

Here I was trying to write an extended version of Euler-Bernoulli (EB) equation for compressible and viscous flow when I realised it is actually the equation of conservation of linear momentum (Navier-Stokes NS) in Lagrangian form alongside the stream. As it has been mentioned in the book "Transport Phenomena" by Bird second edition page 86, to drive EB from NS we should multiply both sides of the NS equation by $\vec{v}\cdot$ and then change the $\vec{v}\cdot\vec{

abla}$ parameters to $\left|\vec{v} \right|\frac{\partial}{\partial s}$. So let's start with the most general form of linear momentum equation for a steady-state flow:

$$\rho \check{v} \check{

abla}^T \check{v} =\check{

abla} \check{\sigma}\tag{1}$$

$$\rho \left(\boldsymbol{v} \cdot \boldsymbol{

abla}\right) \boldsymbol{v} = \rho \boldsymbol{v} \cdot \left( \boldsymbol{

abla} \boldsymbol{v} \right) =\boldsymbol{

abla} \cdot \boldsymbol{\sigma}\tag{1 tensor form}$$

Where $\check{\sigma}$ is the matrix representation of Cauchy stress tensor $\boldsymbol{\sigma}$

$$\check{\sigma}= \check{\tau} -P\check{I}\tag{2}$$

Assuming Newtonian fluid

$$\check{\tau}=\eta\left( \check{

abla}^T \check{v}+ \left( \check{

abla}^T \check{v} \right)^T \right) +\lambda \left(\check{

abla}\check{v}^T\right) \check{I}\tag{3}$$

$$\boldsymbol{\tau}=\eta\left( \boldsymbol{

abla} \boldsymbol{v}+ \left( \boldsymbol{

abla} \boldsymbol{v} \right)^T \right) +\lambda \left(\boldsymbol{

abla}\cdot\boldsymbol{v}\right) \boldsymbol{I}\tag{3 tensor form}$$

Assuming the viscosity is constant:

$$\rho \check{v} \check{

abla}^T \check{v} = \eta\left( \check{

abla}\check{

abla}^T \check{v}+ \check{

abla}\left( \check{

abla}^T \check{v} \right)^T \right) + \check{

abla} \left(\lambda\check{

abla}\check{v}^T-P\right) \tag{4}$$

Now if I have calculated correctly

$$\check{

abla}\left(\check{

abla}^T \check{v}\right)^T=\check{

abla}\left(\check{

abla} \check{v}^T\right)\tag{5}$$

$$\boldsymbol{

abla}\cdot\left(\boldsymbol{

abla} \boldsymbol{v}\right)^T=\boldsymbol{

abla}\left(\boldsymbol{

abla} \cdot \boldsymbol{v}\right)\tag{5 tensor form}$$

and we can write the equation in vector form:

$$ \rho \vec{v} \cdot \left(\vec{

abla}\otimes \vec{v}\right) = \eta\left( \left( \vec{

abla} \cdot \vec{

abla} \right) \vec{v}+ \vec{

abla}\left( \vec{

abla} \cdot \vec{v} \right) \right) + \vec{

abla} \left(\lambda\vec{

abla}\cdot\vec{v}-P\right) \tag{6}$$

Expanding the left side as discussed here leads to:

$$ \vec{v} \cdot \left(\vec{

abla}\otimes \vec{v}\right)= \left(\vec{v} \cdot \vec{

abla}\right) \vec{v} = \frac{1}{2}\vec{

abla}\left(\vec{v} \cdot\vec{v} \right)-\vec{v} \times \left( \vec{

abla} \times\vec{v} \right) \tag{7}$$

Multiplying the sides of the equation with $\vec{v}\cdot$ and if my presumptions of

$$\vec{v}\left(\vec{

abla}\cdot\vec{

abla} \right)\vec{v}=\left(\vec{v}\cdot\vec{

abla}\right)\left(\vec{

abla}\cdot\vec{v}\right)\tag{8}$$

and

$$\vec{v}\cdot\left( \vec{v} \times \left( \vec{

abla} \times\vec{v} \right)\right)=0\tag{9}$$

are correct we can write:

$$ \frac{1}{2}\rho\frac{\partial}{\partial s} \left(\vec{v} \cdot\vec{v} \right) = \frac{\partial}{\partial s} \left(\left(2\eta+\lambda\right)\left(\vec{

abla}\cdot\vec{v}\right)-P\right) \tag{10}$$

Which is mesmerizingly elegant and for incompressible flow $\vec{

abla}\cdot\vec{v}=0$ or inviscid fluids $\eta,\lambda=0$ it reduces to EB equation. However the issue is that the standard EB equation is valid for inviscid and incompressible and and irrotational flow, not just inviscid or incompressible. There must be terms for inviscid-compressible and viscous-incompressible which does not exist in my result. Presumably I have done some calculation mistakes.

I would appreciate if you could help me know where are my mistakes and how I can solve them.