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My understanding of Euler-Bernoulli (EB) equation is that it is the energy balance equation in Lagrangian form for an inviscid, incompressible and steady-state flow . From different sources I see different equations attempting towards extending the equation for compressible or viscous flows. For example the wikipedia entry for the matter offers:

$$v dv+\frac{dP}{\rho}=0 \tag{1}$$

along the stream for compressible flow (assuming there is no body force i.e. gravity) and I have also seen:

$$v dv+ dh=0 \tag{2}$$

also along the stream for viscous flows (which paradoxically does not include viscosity).

If my understanding of the EB equation is correct then we must be able to drive them from Eulerian form of continuum equation of energy balance neglecting conduction and radiation:

$$ \left\{ \begin{matrix} \rho \left( \check{v} \check{\nabla}^T \right) e=\check{\sigma} : \check{\nabla}^T \check{v}\\ \left( \check{v} \check{\nabla}^T \right) \left( h+\frac{v^2}{2} \right)=0 \end{matrix}\right. \tag{3}$$

Where

$$ \check{\sigma}= \check{\tau} -P\check{I} \tag{4}$$

and

$$\check{\tau}=\eta\left( \check{\nabla}^T \check{v}+ \left( \check{\nabla}^T \check{v} \right)^T \right) +\lambda \left(\check{\nabla}\check{v}^T\right) \check{I} \tag{5}$$

For Newtonian fluids.

So my questions are:

  1. Am I right about extended EB equations being Lagrangian form of the energy balance equation for steady state flow?
  2. If no then how these two distinct set of equations are related? the EB equation and continuum equations for conservation of mass (i.e. continuity), momentum (i.e. Navier-Stokes) and energy.
  3. If yes how we can drive the extended EB equation for compressible viscous flow from eq 3 and what is the correct form?
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    $\begingroup$ The answers to everything you're asking is in Transport Phenomena (of which I know you have a copy). By the way, what is cte? $\endgroup$
    – Chet Miller
    Commented Mar 8, 2018 at 16:52
  • $\begingroup$ Hi @ChesterMiller nice to have you here. true. I have access to many books but finding the right information in these books is a hassle. and sometimes they use wired notations which I do not understand. That's why I'm asking. Maybe you can help me find the right section of the book driving an extended version of of EB equations from continuum eqs? $\endgroup$
    – Foad
    Commented Mar 8, 2018 at 16:58
  • $\begingroup$ I don't think the Bernoulli equation for compressible flow is valid if viscosity forces are important. In 1D, how do you describe viscous forces? $\endgroup$
    – Ján Lalinský
    Commented Mar 29, 2019 at 0:37

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I think the equation(s) you are looking for are in table 11.4-1 of BSL. You will be dealing with basically two types of energy balance: overall energy balance and mechanical energy balance. The overall energy balance is basically the 1st law of thermodynamics, and the mechanical energy balance is basically the equation of motion dotted with the velocity vector. These equations each contain some common information and some separate information, and can be coupled to yield other forms of energy equation. The coupled form (or forms) you choose depends on which one works best for the problem you are solving at that time.

Your question about enthalpy relates to all this. Eqn. 1 is a steady state simplified version of the equation of motion dotted with the velocity vector. Eqn. 2 is a simplified version of the overall energy balance for steady state approximately adiabatic reversible flow. For the latter, $dh=TdS+vdP=vdP=\frac{dP}{\rho}$

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  • $\begingroup$ Thanks. It seems the correct way to go is to use the $\vec{v}\cdot$ of linear momentum equation. Page 86 BSL. So far I have been able to reduce to $$\rho \check{v}\check{v}^T\check{\nabla}\check{v}^T =\eta\check{v} \left(\check{\nabla} \check{\nabla}^T \check{v}+ \check{\nabla} \left( \check{\nabla} \check{v}^T \right) \right)^T + \check{v}\check{\nabla}^T \left(\lambda\check{\nabla}\check{v}^T-P\right) $$ $\endgroup$
    – Foad
    Commented Mar 9, 2018 at 12:41
  • $\begingroup$ and then replace $\check{v}\check{\nabla}^T$ with $\left|\vec{v}\right|\frac{\partial}{\partial s}$ $\endgroup$
    – Foad
    Commented Mar 9, 2018 at 12:47
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    $\begingroup$ It depends on the specific problem you are trying to solve. If temperature changes are not important, then the mechanical energy equation, as you indicated, is probably the way to go. But, if it's compressible flow in a nozzle, for example, temperature changes are important. $\endgroup$
    – Chet Miller
    Commented Mar 9, 2018 at 12:48
  • $\begingroup$ FYI the final goal is to explain this $\endgroup$
    – Foad
    Commented Mar 9, 2018 at 12:50
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    $\begingroup$ OK. This is certainly a case where the mechanical energy balance alone won't be adequate. The presence of the $\kappa$s in the equations seems to be an indication that the deformation is being treated as adiabatic and reversible. But it would involve real work to slog through piecing together where all this came from (beyond what I would be inclined to do). $\endgroup$
    – Chet Miller
    Commented Mar 9, 2018 at 12:58

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