Basica the equation of Bernoulli is the integral of the equation of Euler:

rho (dv/dt)=-grad p

with rho the density, v the velocity and p the pressure which is valid for a frictionless flow. One can include an external force field derived from a potential into a modified pressure. dv/dt=partial v/partial t+(v.nabla)v is the Lagrangian time particle acceleration including the convective term (v.nabla)v. For an irrotational flow rot v=0 and (v.nabla)=grad (|v|^2)/2. Furthermore as rot v=0 we can define a scalar potential phi such that v=grad phi. If now we assume an homentropic flow (uniform specific entropy s) or grad s=0 we can write on the basis the first law of thermodynamics as:

0=de+p d(1/rho)=dh-(dp)/rho

or 

(grad p)/rho=grad h

where e is the specific internal energy and h the specific enthalpy.

The equation of Euler can be rewritten as:

grad( (partial phi/partial t)+h+(|v|^2)/2)=0

integration yields the unsteady compressible Bernoulli equation:

(partial phi/partial t)+h+(|v|^2)/2= f(t)

where f(t) is determined by the boundary conditions.

When the flow is not-homentrope but still isentropic and irrotational (rot v=0), one can integrate for steady flows along a streamline defined as the path such that v.dx=0.

The the steady compressible equation of Bernoulli:

h+|v|^2/2=h_0

is valid along the streamline.

Obviously the equation of Bernoulli involves the kinetic energy of the flow. Thermodynamics is in principle involved with statics and cannot be used to derive this equation. The generalised energy conservation law for thermodynamic systems, including the kinetic energy is not a Thermodynamic law but a general conservation principle. Also this integral equation does not specify that the flow is irrotational or that we follow a particle along a streamline. Hence even from this general statement we cannot derive the various forms of the equation of Bernoulli without additional assumptions which are not results from the thermodynamics.

My conclusion is that the Equation of Bernoulli in compressible form involves thermodynamics but cannot be deduced from thermodynamics only.

Regards, Mico

You can find these arguments in books on Gas Dynamics such as:

J. Oswareck, Fundamentals of Gas Dynamics.

the answer is yes.

please refer to the book of Pattrick H. Oosthuizen, and William E. Carscallen,"Compressible fluid flow", McGraw-Hill,  international eidition 1997, section 2.7

Read my book: Mecánica Y Termodinámica de Sistemas Materiales Continuos.

Answer is no. Bernoulli is a mechanical law of conservation of mechanical energy under no dissipation or viscous stress on a streamline or tube. Thermodynamics involves heat flow additionally and dissipation over closed or open volumes. A streamtube is an open volume, nothing else.

There are two posibility to derive the Bernoulli equation :

The first involves integration of the Euler equation whereou assume the flow to be incompressible.

The second method is to apply conservation of energy along a streamline without taking into account i: heat transfer, compressibility and viscosity.

What the first principle of thermodynamic would say is that the work over the system increases the internal energy. That's all.

There is a general deduction of Bernoulli Theorem for compresible and viscous flows.

See my book in researchgate or directly:

https://www.academia.edu/11946074/Teorema_de_Bernoulli

A more general form can be found, in the following course with Special cases of Bernouilli's Theorem

http://www.whoi.edu/cms/files/12.800_Chapter_10_%2706_25345.pdf

First law of Thermodynamics can be simplified easily into Bernoulli's equation, for ideal flows, with no heat transfer.  

The deduction from

http://www.whoi.edu/cms/files/12.800_Chapter_10_%2706_25345.pdf

The before deduction is from first principle of thermodynamic for open volumes in differential form. Obviously includes the conservation of mechanical energy. But it is a detour to reach to a result is reached more directly from the equation of motion.

You can derive Bernoulli's equation from the first law of thermodynamics for open system (control volume) by including the kinetic and potential energy changes terms. You will reach Bernoulli's equation by assuming the system is internally reversible, and the work is equal to zero.

Of course, the equation of motion is a simpler route to it.

Basica the equation of Bernoulli is the integral of the equation of Euler:

rho (dv/dt)=-grad p

with rho the density, v the velocity and p the pressure which is valid for a frictionless flow. One can include an external force field derived from a potential into a modified pressure. dv/dt=partial v/partial t+(v.nabla)v is the Lagrangian time particle acceleration including the convective term (v.nabla)v. For an irrotational flow rot v=0 and (v.nabla)=grad (|v|^2)/2. Furthermore as rot v=0 we can define a scalar potential phi such that v=grad phi. If now we assume an homentropic flow (uniform specific entropy s) or grad s=0 we can write on the basis the first law of thermodynamics as:

0=de+p d(1/rho)=dh-(dp)/rho

or 

(grad p)/rho=grad h

where e is the specific internal energy and h the specific enthalpy.

The equation of Euler can be rewritten as:

grad( (partial phi/partial t)+h+(|v|^2)/2)=0

integration yields the unsteady compressible Bernoulli equation:

(partial phi/partial t)+h+(|v|^2)/2= f(t)

where f(t) is determined by the boundary conditions.

When the flow is not-homentrope but still isentropic and irrotational (rot v=0), one can integrate for steady flows along a streamline defined as the path such that v.dx=0.

The the steady compressible equation of Bernoulli:

h+|v|^2/2=h_0

is valid along the streamline.

Obviously the equation of Bernoulli involves the kinetic energy of the flow. Thermodynamics is in principle involved with statics and cannot be used to derive this equation. The generalised energy conservation law for thermodynamic systems, including the kinetic energy is not a Thermodynamic law but a general conservation principle. Also this integral equation does not specify that the flow is irrotational or that we follow a particle along a streamline. Hence even from this general statement we cannot derive the various forms of the equation of Bernoulli without additional assumptions which are not results from the thermodynamics.

My conclusion is that the Equation of Bernoulli in compressible form involves thermodynamics but cannot be deduced from thermodynamics only.

Regards, Mico

You can find these arguments in books on Gas Dynamics such as:

J. Oswareck, Fundamentals of Gas Dynamics.

First law of thermodynamics only states conservation of energy for an isolated system. Bernoulli's equation is a conservation mass and energy. The conservation of mass is in some form assumed for incompressible fluids but is necessary in the derivation for compressible fluids.

Dear Ravindra Aglave,

Indeed when applying the first law of thermodynamics to a fluid particle, one does assume a certain material element can be isolated and treated as a closed thermodynamical system (without mass exchange). This corresponds to the continuum approximation (see discussion in the book of Landau and Lifchitz, Theoretical Physics Vol VI, Fluid Mechanics).

Regards, mico Hirschberg

there is an elephant in the room: pressure-energy (pV). In the paper on the Joule-Thompson experiment (throttle)  [see this site] it is tried to address that issue, for a flow process where Bernoulli does not apply. However pressure energy will make the step from the first law to the Bernoulli-eqn much smaller. You only have to assume that friction is small, hence Q can be neglected. 

When I teach Fluid Dynamics I teach Bernoulli's eqn as a substitute for the energy equation.   This is because some problems require ALL the physics and require using: conservation of mass, conservation of momentum, and Bernoulli.  But using conservation of mass, conservation of energy, AND Bernoulli (for momentum) - never works.  (Because Bernoulli is always redundant with the conservation of energy eqn.)   

It is indeed tricky.   Confusion is especially prevalent because of the classic (and formally correct - but ultimately confusing) derivation of Bernoulli from the momentum equation (So people think it is equivalent to momentum eqn - which it is not).   You can certainly derive the kinetic energy equation from the momentum equation (as they do) - and the simplest form of Bernoulli is just the KE equation (energy).  

BUT ...  The total energy equation contains KE in its total energy.  So it also contains Bernoulli in its guts. In fact - the more general derivation is from this total energy.

When Bernoulli is derived from the energy equation you can see exactly why there are certain restrictions on its use (and how to generalize it from the simplified case).

Take the total energy equation and make a control volume that is a long thin curved cylinder that follows a streamline from point A to point B.   (Restriction 1 for Bernoulli is along a streamline).    Since the cylinder follows the streamline the flow is always along (and not through the curved cylinder side walls).  So the CV balance for the advection and pressure work terms only involves the two end caps.  Make the end caps small enough that everything is constant on them when doing the integrals.  (Also you will need mass conservation in your tube - also just involves caps).

Restriction 2 for Bernoulli is steady.  That gets rid of the change of total energy inside the control volume.  So any volume info goes away.

Restriction 3 is no losses (small friction and heat fluxes).  That makes any fluxes on the contorted curved surface that are not equal to zero (because of perpendicular velocity) equal to zero (because of this restriction 3).

Now look at what is left.  Just a balance between the two end caps (which are points A and B on your Bernoulli streamline).    Also note that this full version of Bernoulli also contains the internal energy.  This is the 'full' temperature varying variable density Bernoulli equation.  

The one in books has the unnecessary restriction 4 (constant temperature) to eliminate the internal energy.    Also this shows you that Bernoulli works fine for variable density situations (like littoral ocean (with salt) and atmosphere), BUT only as    P + rho*vel*vel/2 + .....     And NOT as  P/rho + vel*vel/2 + .....   This later version of Bernoulli (the one derived from the momentum eqn) only works for constant density.     

Hint:   If you want to start from Thermo book to derive Bernoulli.  Start in the chapter on control volumes (which is missing in half the books).   

Compare Energy eq.(14) and (18) pag.146, with Bernoulli eq.(7) and (8) p.188

(see comments after equations) of my book:

https://www.academia.edu/11942882/Mec%C3%A1nica_y_Termodin%C3%A1mica_de_Sistemas_Materiales_Continuos

Then make your conclusions.

For me the transient term in Bernoulli and the pressure work for Energy are the principal differences between those equations. Energy and Momentum have similarities but no more.

BUT only as P + rho*vel*vel/2 + ..... And NOT as P/rho + vel*vel/2 + .....

Sorry this statement is exactly the wrong way around.  When you divide by m_dot on both sides the density is in the denominator of P (and nowhere else).  

I think that it is possible. In the attached file you can see some stuff from the course that I teach in my University. You have here all the necessary information in order to derive the General Bernoulli equation from the conservation principles formulated in the integral way, and using the Reynolds Theorem Transport... but..., sorry!,... the slides are in spanish.... ; )

Some references (in spanish) about this topic are: ( you can easily obtain the english versions,... but I do know the exact place where the subject is developed in each),

White (5ta edic.) cap. III, pp 148:159-166:186,

Çengel Cimbala, (1ra edic.) cap. V, pp. 180:226, cap VI, pp. 223:259.

Fox, (2da edic.), cap IV, pp. 113:199.

Franzini, (9na edic.), cap VI, pp. 117:141, cap V, pp. 85:108

Potter (3ra edic.), cap III, pp. 95:103, cap IV , pp. 114:161

Dear Migel Coussirat,

In fact you are deriving the general integral law for energy conservation and applying it to a pipe flow where you assume a uniform flow velocity. A generalisation of this for non-uniform flow velocity profiles is provided in the book of Bird et al. (Transport Phenomena). Calling this the equation of Bernoulli might be confusing. This is a question of definition of what we call "The equation of Bernouli" and "The integral energy conservation". I feel that neglecting friction and heat transfer is an essential part of what is commonly used to define the equation of Bernoulli as an integral of the equation of motion for an isentropic potential flow. A simple example of the difference between these two equations is the shock relations. Across an adiabatic shock the adiabatic equation of energy conservation states that the "total enthalpy" (reservoir enthalpy) is conserved. Bernoulli is not valid here because there because the flow is not isentropic. There is entropy production due to viscous dissipation and heat transfer. The same occurs when considering a throttling valve in a cooling engine.

Regards, Mico

Dear J. Blair Perot ,

Of course the equation of Bernoulli derived as an integral of the equation of motion is a single scalar equation and cannot be equivalent to the original 3 components of the equation of motion. This is not a restriction of the equation of Bernoulli but a fact we should remember why using this equation in practice.

Yours sincerely,

Mico

I believe we all are on a futile discussion if we are not talking on the same assuptions.

First: What Bernoulli. Compressible (there is a versión for compressible flow) or incompressible. The density is divided, maybe different at any pont of the stream. For compressible or/and transient flow Bernoulli has terms with line integral. Depends on the following road. Not on entrance and exit.

Second: What energy. If it is stationary, mechanical energy transport involves mass of entrance  and exit. The density is multiplied. It is the pressure of Bernoulli equation a energy or equivalently a work of pressure forces. In the energy first principle is a work.

Third: What time condition. Transient (there is a version of Bernoulli for transient flow) or steady. For steady state, energy depends only on entrance and exit. Do not depends on the line or volume of integration. A different density in each point or area. At entrance and exit.

Fourth: What space condition. Integral (there is a version of Bernoulli for stream tubes) or differential on a streamline. Maybe another line? The curvature of the surface of flow (entrance or exit) is important or not?. Only one entrace and one exit?

For me they continue being different based on the aforementioned subtlety.

On Reynolds Transport Theorems. They are three. See for example:

https://www.academia.edu/11947082/Reynolds_Transport_theorems_and_Conservation_Principles_as_Special_Applications_of_Leibniz_Rule

Does the partial time derivative means something outside the integral volume integration over the control volume? It is a rigid volume always? The change of the metric over a surface affect what part of the theorem?

In my opinion, considering the key in the total energy conservation equation, conversion of kinetic into heat energy is self-contained, no matter of entropy balance.Thus, when you use properly the thermodinamic of reversible systems you can get the same conclusion of Bernouille principle.

As Perot wrote, using momentum equation is often not enlighting (at least for students...) on the real meaning of the Bernouille integral.

There is a discussion related to your question in our undergraduate text “Transport Phenomena: A Unified Approach” by Brodkey and Hershey.  The discussion of the integral energy balance starts on page 286 and goes to page 305.  There are many worked examples within this range.  However, the text gets directly into the topic in the last paragraph on page 291 and refers back to the thermodynamic  approach that started on page 286.

 The eq.(7.52) in p.291 E=Q=W_s=0 of the recommended book is not Bernoulli for streamtube eq.(20) or eq.(27) of my paper under two analysis alpha_v or \alpha_c (without neglecting anything, but may select Tau_w=0 for non-viscous fluid).

https://www.academia.edu/11946074/Teorema_de_Bernoulli

The eq.(7.52) has two additional terms (including for incompressible flows) and has the means of  v^3 ( valid comparison is with alpha_c).

Dear Avraham,

Yes, I agree with you, and for the set of reasons that you gave, I try to distinguish in my slides, from Bernoulli without friction, to a Generalised Bernoulli Equation. Also, of course, that you can obtain a relation for generalised Bernoulli equation with entropy by using the Reynolds Transport Theorem... and of course the 'correct' definition of what is the real Bernoulli equation depends strongly on the viewpoint that someone is studying the problem, as Andres said. I think that is no a futile discussion... it's funny!, it is very instructive for me to see opinions of other people about this broadly discussed topic.

You can also derive Bernoulli's equation, by using an Eulerian variational principle,

in which the basic Lagrangian $L$ is the kinetic minus the internal energy density of the fluid. The Lagrangian is modified by using Lagnrange multipliers, which ensure

mass conservation, and entropy advection with the flow, as well as the so-called Lin constraint is satisfied (this is a version of Kelvin's theorem). The Lagrange multipliers are called Clebsch variables. This method is explained in the paper by Zakharov and Kuznetsov (1997), Hamiltonian formalism for nonlinear waves, Uspekhi, Vol. 40, 1087-116. The Lagrange multiplier $\phi$ that ensures mass conservation turns out to be the velocity potential in Bernouuli's equation. Bernoulli's equation does not by itself describe flows with vorticity.

The answer is NO, provided we are considering the Classical Bernouilli Equation. This equation arises from the projection of the vetorial product (rot v X v) over the streamlines after assuming that the mass density depends only on the pressure. This issue is not available in the general energy balance.

Can Bernoulli's equation properly be derived from the first principle of thermodynamics? I can state  definitely that the Bernoulli's equation can  be derived properly  (I made this in 2007) from the generalized formulation (presented in my book of 2007, and in my articles of 2014, which are avalable on this website) of the first principle ()low of thermodynamics! 

I would also suggest reading the derivation from energy relation as reported in the book of Chorin & Marsden. I always appreciated it exactly because start from thermodinamics

Rogerio:

  Bernoulli's equation is about potential flow, with no vorticity. I may have given the wrong impression in my comments.

 Gary

Dear Gary,

But, even in a context of nonzero vorticity, Bernoulli equation holds along a streamline. This fact arises from mathematical arguments (perhaps with low importance for engineering purposes). Nevertheless I do not see how this condition may be ensured from the local energy balance...

Rogério

Dear Rogerio:

 As I understand it, the full velocity can be split up into a  potential flow piece,

described by the gradient of the velocity potential, plus other pieces associated with

fluid spin generated by entropy gradients, plus another fluid spin piece which is due 

to initial fluid spin, and not associated with entropy gradients. In Bernoulli's equation

(as derived from the ZK variational principle) one obtains $d\phi/dt=(1/2) u^2-w-\Phi(x)$  where $u$ is the full fluid speed and $d/dt$ is the Lagrangian time derivative following the flow, $w$ is the enthalpy and $\Phi(x)$ is the gravitational potential.   There are other equations governing the fluid spin potentials

 The full fluid velocity in this approach comes from varying the fluid velocity in

the variational principle, which gives

 ${\bf u}=\nabla\phi-(\beta/\rho)\nabla S-(\lambda/\rho)\nabla\mu$

The $\beta$ term is associated with fluid spin generated by entropy gradients

(sometimes referred to as the baroclinic effect in tornadoes), whereas  $\mu$

is associated with Kelvin's theorem, and initial vorticity not due to entropy gradients

  The energy conservation, momentum conservation and vorticity related conservation equations come out of the variational principle by using  either Noether's first theorem or Noether's second theorem, using either Lie point symmetries, or fluid relabeling symmetries.

   Best regards, Gary Webb

As a matter of fact Bernoulli's equation may be obtained from the NS equations themselves. To clear it out, consider the incompressible case where the flow equations are

(1) div(u)=0

(2) D/Dt(u) = -grad(p) + nu lap(u),

where the operator D/Dt is the substantial derivative, u is the eulerian velocity field, p is the modified pressure field and nu is the kinematical viscosity of the the (non-generalised) newtonian fluid, which means nu is a constant. Equation (2) may also be written as 

(3) grad(p+1/2u.u) = u x (curl(u)) -[d/dt(u) + nu curl(curl(u))],

where the operator d/dt means partial differentiation with time t. Taking the dot product of (3) with the velocity direction (unitary) field eu = u/||u|| yields the important result

(4) d/deu(p + 1/2u.u) = -[d/dt(u) + nu curl(curl(u))].eu

since u, curl(u) and eu are linearly dependents. 

So for STEADY FLOWS and "CURLLESS VORTICITY FLOWS" (I am not even sure how these are called, so please correct me for an more appropriate terminology), the "mechanical energy" p+1/2u.u is conserved along a streamline of the flow, i.e. the left hand side of equation (4) simply vanishes. The last requirement may be avoided when considering an inviscid fluid, or a frictionless flow, and the supposed energy is conserved everywhere in the flow domain.

However as stated before, from a pure thermodynamics starting point it seems not likely to derive Bernoulli's equation. The main reason for that, as far as I can tell, seem to be linked to the material itself, in this case the fluid. Different 'versions' of this conservation assertion are given in the literature, e.g. for incompressible and compressible FLOWS of NEWTONIAN FLUIDS. Since a conservation statement is trying to be achieved the nature of the stresses in the fluid is vital for counting for irreversibilities, or this case how to avoid them. Perhaps you would also be interested in deriving conservation statement alike for more sophisticated materials such as non-newtonian fluids. Either way, I hope it eased some aspects of it.

Best regards with your research 

Dear Fernando Soares,

What you call "CURLLESS VORTICITY FLOWS" is a potential flow in which the velocity field can be written as the gradient of a scalar potential Phi. Almost by definition rot v=0 implies v=grad Phi and v=grad Phi implies rot v=0. Indeed when the flow is not frictionless it cannot be a potential flow. If it is frictionless this depends on the initial/boundary conditions.

Regards, Mico

Dear Mr. Hirschberg,

I believe I might have not come as clear as I wanted to. What I mean by "CURLLESS VORTICITY FLOWS" are flows in which the vorticity field is conservative, rather than the velocity itself. By what is stated in (4), considering also steady flows (d/dt = 0), since the vorticity = curl(u) = grad(Psi), the friction due to fluid stresses is curl(grad(Psi)) = 0, for incompressible flows. This means a flow may be rotational and yet conserve the quantity p + 1/2u.u (throughout a streamline).

I simply did not remember how such flows are called. For instance when the velocity field is "curlless" is said to irrotational, but when considering the vorticity field which is irrotational I simply do not  know what it is called.

Anyway thank you for the reply and best regards.

Fernando Soares.

First, I assume you mean to discuss the **steady** form of the Bernoulli equation, as taught in introductory fluids courses and many introductory physics courses. The unsteady version applies under the same conditions as the steady version, except of course for the unsteady term.

Second, I assume you wish to avoid going through some form of the momentum equation, such as N-S or Euler. The closest version I'm familiar with is the path followed in Chapters 2 and 3 of Zucker and Biblarz, "Fundamentals of Gas Dynamics", 2/e, Wiley, 2002. However, this path to Bernoulli clearly shows that you cannot get there without also imposing an equation of state (which means incompressible flow in this case) and invoking the 2nd Law of Thermodynamics (2LTD) to show that entropy generation must not occur if the usual form of the Bernoulli equation is to be obtained.

Requiring a specific equation of state and 2LTD seems to invalidate the claim that it can be obtained through 1LTD only.

Finally, I'm confused by the original statement that the Bernoulli equation is a product of mechanics only when obtained from the Euler equations by integration. The derivation requires an equation of state to connect pressure and density. The assumption of constant density is a thermodynamics assumption about the pressure which, for example, relates to our model of the flow work in the process.

Yes, Bernoulli's Equation can be derived from the first law of thermodynamics and has been many times. Just by looking at the usual Bernoulli terms it should be obvious that it's an energy per unit volume equation, that is (pressure + kinetic + potential) energy/volume is constant along a streamline or everywhere for an irrotational flow.  Here's one derivation from the 1st Law….https://www.av8n.com/physics/bernoulli.htm

Without going into detail here, I can offer you as a reference "Transport Phenomena: A Unified Approach", which goes into the problem in same detail.  Sec. 7.1.4 in pages 286-305 and some additional comments on page 323, could be the answer you need.

I derived the Bernoulli's Equation from the Simonenko's generalized formulation of the first law of thermodynamics (see my article "The practically confirmed validity of the .... " published in American Journal of Earth Sciences, 2015). The classical Gibbs' formulation is generalized  by  the Simonenko's generalized formulation of the first law of thermodynamics (see my articles of 2014-2015).

Pressure is not an energy. But is an isotropic force making a work.

Bernoulli includes transient and compressible flows, and even viscous efects in the term of loss.

First law is valid only in quasi static state and it is an integral formulation (even in open systems). Bernoulli is a diferential formulation, inclusively in the stream tube formulation (transient).

Derivation Bernoulli from thermodynamics first law without heat, is like to derive hydrostatics equation from Navier-Stokes equation without motion (viscosity and compressibility).

See this (excuse if it is in spanish):

https://www.academia.edu/11946074/Teorema_de_Bernoulli

More in detail:

https://www.academia.edu/11942882/Mec%C3%A1nica_y_Termodin%C3%A1mica_de_Sistemas_Materiales_Continuos

Chapter IX, section 1,4, pp.189-199

Well, pressure can be seen as energy...it is a simple matter to work with the total energy equation for a volume of fluid and derive (pressure produces reversible mechanical work in a non-zero velocity field) the Bernouilli integral. Note that the total energy equation for a volume of fluid is nothing else that the well known thermodynamic law dE/dt= Q-W

Irving H. Shames book, Mechanics of Fluids actually devotes a chapter to this called "BERNOULLI'S EQUATION FROM THE 1ST LAW OF THERMODYNAMICS." It is assumed that the heat transfer is zero and there is no change in internal energy. The opposite, where you derive the first law from the Bernoulli Equation is not true.

When I first started teaching thermodynamics at the Naval Academy in 1958, "Flow Work" was used to derive the steady state energy equation, which could be further reduced to a form of the steady state Bernoulli equation.  I later learned in graduate school that this was just an over simplified way to avoid accounting for energy storage in an open system. I was embarrassed that I had been teaching "bum dope" and vowed to not oversimplify when constructing mathematical models of real systems unless one also  explained the consequences of that oversimplification. See attached derivation section 4.2 of the Equations of Motion for CFD which starts with Cauchy's equations.

Because the formulas can't be shown correctly, you can see my attachment.

The answer is yes. The Bernoulli's equation can be derived from the Energy Conservation Law.

There are three assumptions used for Bernoulli's equation:

(1) ignoring the thermal conduction, which means that doing work is the only way to change the energy.

(2) ignoring the shear stresses, which means that the stresses equal hydrostatic pressure and they are isotropic;

(3) the flow is constant, which means that the flow line and the trace line coincide. So the Bernoulli's equation can be described by two positions at a flow line or by two moments of a fluid micelle’s moving.

There are three kinds of energy in the question:

(1) kinetic energy;

(2) gravity potential;

(3) internal energy;

When a micelle moves, the total energy will be changed by the work and the thermal conduction. Because of the assumption 1, only the work exists. Two kinds of force are applied to the micelle, including the gravity force and the pressure. As mentioned above, the gravity potential is used, which stands for the effect of the gravity force, so only the work done by the pressure is considered. So the energy and the work can be associated as follows:

    (1)

In the above formula, TE means the total energy, and w means the rate of work, and t means the time.

The total energy per mass unit can be expressed as:

    (2)

Three components of the above formula are the three kinds of energy mentioned above, each of which is a value per mass unit.

The rate of work in a volume Ω and a surface S can be expressed as:

    (3)

In the above formula, p means the pressure, and v means the velocity. The derivation uses the Gauss formula.

According to the formula (3), the rate of work per volume unit is:

    (4-1)

So the rate of work per mass unit is:

    (4-2)

Some formula transformation is conducted:

    (5)

    (6)

    (7)

Notice that the physical meaning of  is the relative rate of volume, so the formula (7) is transformed rightly. Then the formula (5) is transformed as follows:

    (8)

Then the formula (1) is transformed:

    (9)

Another form is:

    (10)

Then the Bernoulli's equation is derived:

    (11)

Its first three components are the total energy, and the forth one stand for the potential work done by the pressure. The derivation of the equation uses only the Energy Conservation Law, and no fluid mechanics but some knowledge of the pressure is needed.

I agree, Bernoulli is within the total energy conservation (integral) equation.

That topic is developed since the fundamental book of Batchelor (Sec. 3.5). Other books reported such development, too.

Markus,

See the following link.

https://www.researchgate.net/project/Bernoulli-principle-and-the-field-derivative

I don't think so if Bernoulli equation can be derived from the principle of energy conservation. Actually, the Bernoulli equation is valid for a stream line....