# Driving Euler-Bernoulli equation for a compressible viscous flow from continuum conservation equations of an steady-state flow

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?
• 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? – Chet Miller Mar 8 '18 at 16:52
• 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? – Foad Mar 8 '18 at 16:58
• 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? – Ján Lalinský Mar 29 '19 at 0:37

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, $dh=TdS+vdP=vdP=\frac{dP}{\rho}$
• 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 $$\rho \check{v}\check{v}^T\check{\nabla}\check{v}^T =\eta\check{v} \left(\check{\nabla} \check{\nabla}^T \check{v}+ \check{\nabla} \left( \check{\nabla} \check{v}^T \right) \right)^T + \check{v}\check{\nabla}^T \left(\lambda\check{\nabla}\check{v}^T-P\right)$$ – Foad Mar 9 '18 at 12:41
• and then replace $\check{v}\check{\nabla}^T$ with $\left|\vec{v}\right|\frac{\partial}{\partial s}$ – Foad Mar 9 '18 at 12:47
• 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). – Chet Miller Mar 9 '18 at 12:58