Introduction

Bernoulli’s equation is one of the most useful equations in the field of fluid mechanics, owing to its simplicity and broad applicability. That applicability does have limits, however, which has led to it being one of the most misunderstood and also misused equations in fluid mechanics. The goal of this post is to present Bernoulli’s equation, its derivation based on energy conservation, and interpretations based on both momentum and energy conservation, and its limitations. The bulk of this article will be written at a level appropriate for an undergraduate STEM student who is familiar with calculus and differential equations.

The control volume and streamtubes

In fluid mechanics, it is helpful to utilize the concept of a control volume (CV): a fixed volume in space through which fluid flows. For a given steady flow field, if we imagine a line drawn by a particle moving through the flow, this line will trace out a streamline. (Note: strictly speaking, this is called a pathline, but for a steady flow, pathlines and streamlines are identical.) Given that the fluid velocity is everywhere tangent to the streamline, no flow can cross such lines. Therefore, it is convenient to draw a curve through the flow field and follow the streamlines along that curve some distance downstream. This procedure traces out a streamtube which is commonly and conveniently employed as a control volume since flow will only enter and exit the streamtube through the inlet and outlet curves. Additionally, we can draw the inlet and outlet curves such that the local flow is normal to the inlet and outlet surfaces, simplifying the mathematics considerably.

Conservation of energy and Bernoulli’s equation

The above image is a typical streamtube created by drawing a curve, ##C_1##, and following the streamlines to some end curve, ##C_2##.

We typically concern ourselves with the conservation of three quantities in a control volume: mass, momentum, and energy. It is beyond the scope of this article to discuss all of the forms that these laws may take, but for now, we will apply the Reynolds transport theorem to a steady flow through a streamtube as it pertains to energy.

Conservation of energy in a steady flow states that the rate of external work done on the CV plus shaft work added to the CV plus the rate of heat conducted into the CV plus the rate of internal heat generation equals the integrated net outward energy flux across the control surface (CS). If we assume the flow is inviscid, shaft work is zero, the heat conduction is zero, and the heat generation is zero, then we have

[tex]

0 = \oint\limits_{CS}\rho\left(e+\dfrac{U^2}{2} + \dfrac{p}{\rho} + gz\right)\vec{U}\cdot\hat{n}\;dA,

[/tex]

where ##\rho## is density, ##e## is internal energy, ##\vec{U}## is the velocity, ##p## is the static (thermodynamic) pressure, ##g## is the acceleration due to gravity, ##z## is the height above the reference surface for gravitational potential energy, ##\hat{n}## is the outward surface normal vector of the CS, and ##dA## is a differential area on the CS. This is simply a mathematical statement that the net flux of mechanical energy out of our control volume is zero under these conditions.

So, if we note that we have defined the streamtube inlet and outlet as being everywhere normal to the velocity vector and denote the inlet surface bounded by ##C_1## as ##S_1## and the outlet surface bounded by ##C_2## as ##S_2##, then we have

[tex]

0 = -\oint\limits_{S_1}\rho_1\left(e_1+\dfrac{1}{2} U_1^2 + \dfrac{p_1}{\rho_1} + gz_1\right)U_1\;dA + \oint\limits_{S_2}\rho_2\left(e_2+\dfrac{1}{2} U_2^2 + \dfrac{p_2}{\rho_2} + gz_2\right)U_2\;dA,

[/tex]

If we then take the limit of this streamtube as ##A_1## and ##A_2## (the areas of ##S_1## and ##S_2## respectively) go to zero such that the integrands are constant over the integral (i.e. as the streamtube approaches being simply a streamline), we find the relationship

[tex]

0 = -\left(e_1+\dfrac{1}{2} U_1^2 + \dfrac{p_1}{\rho_1} + gz_1\right)\rho_1 U_1 A_1 + \left(e_2+\dfrac{1}{2} U_2^2 + \dfrac{p_2}{\rho_2} + gz_2\right)\rho_2 U_2 A_2.

[/tex]

Next, we note that conservation of mass implies that the mass flow rate into the CV equals the mass flow rate out of the CV, so ##\rho_1 U_1 A_1 = \rho_2 U_2 A_2##. This gives us

[tex]

0 = -\left(e_1+\dfrac{1}{2} U_1^2 + \dfrac{p_1}{\rho_1} + gz_1\right) + \left(e_2+\dfrac{1}{2} U_2^2 + \dfrac{p_2}{\rho_2} + gz_2\right).

[/tex]

Finally, we make one final assumption that the flow is incompressible. This implies that the density is constant and that the volume of a fluid element does not change, which, when combined with the lack of heat transfer, implies the the internal energy is constant. Therefore, we are left with

[tex]

\dfrac{1}{2} U_1^2 + \dfrac{p_1}{\rho} + gz_1 = \dfrac{1}{2} U_2^2 + \dfrac{p_2}{\rho} + gz_2.

[/tex]

This is one form of the Bernoulli equation. In effect, it amounts to a statement of conservation of energy. It is perhaps more intuitive to write the version that has been multiplied by density,

[tex]

\dfrac{1}{2} \rho U_1^2 + p_1 + \rho gz_1 = \dfrac{1}{2} \rho U_2^2 + p_2 + \rho gz_2.

[/tex]

In this form, it should be very clear to see that the ##\rho U^2/2## term represents kinetic energy per unit volume (often called dynamic pressure), the ##\rho g z## term represents the gravitational potential energy, and the static pressure, ##p##, represents the remaining pool of total energy, in this case stored by moving molecules in the form of pressure.

Limitations of the Bernoulli equation

The limitations of the Bernoulli equation arise due to the assumption used to derive it. In fact, this is the reason for the choice to derive it based on energy conservation rather than momentum, as the assumptions are more obvious. These limitation are

Steady flow Incompressible flow Inviscid flow No shaft work, heat conduction, or heat generation Flow along a streamline*

The final assumption has an asterisk because this is a “weak” requirement. If we rewrite the Bernoulli equation,

[tex]

\dfrac{1}{2} \rho U^2 + p + \rho gz = B = const,

[/tex]

where we will call ##B## the Bernoulli constant. We can then note that ##B## is constant under the preceding assumptions along a streamline. However, if an entire region of streamlines originates from the same reservoir with the same ##B##, then ##B## is constant throughout the entire region and Bernoulli’s equation can be applied more broadly. As it turns out, this is often the case in practical situations and the final requirement above can be dropped.

Static, dynamic, and stagnation pressure

Often, changes in elevation (##z##) are negligible in a given flow situation. In that case, Bernoulli’s equation can be expressed as

[tex]

p_0 = p + \dfrac{1}{2}\rho U^2,

[/tex]

where ##p_0## is called the total or stagnation pressure and replaces the Bernoulli constant under these conditions. It represents the pressure that would be achieved if the flow was brought isentropically (adiabatically and reversibly) to zero velocity, thus the “stagnation” nomenclature. At any point in a flow, it is the static pressure ##p## that is actually “felt” by any submerged surface. Stagnation pressure is only felt at a stagnation point, and dynamic pressure is never felt directly.

Interpretation as a force balance

As a sanity check, we can perform a thought experiment based on a force balance (momentum conservation). If the velocity increases from point 1 to point 2, then that acceleration must be the result of a force in the same direction. Bernoulli’s equation tells us that the velocity increase corresponds to a decrease in pressure. This means that a fluid element passing through this pressure gradient will experience a higher pressure on its side in the direction of point 1 than on its side in the direction of point 2, and will have a net force accelerating it in the 2 direction. This is consistent with the simple thought experiment. In fact, this is the basis of the equation’s derivation based on Euler’s equation.

Conclusion

The Bernoulli equation is a powerful and deceptively simple relationship. The simplicity comes with a cost, however, in terms of a relatively strict set of limitations. With a bit of care in its use, as outlined above, it can be applied to a broad range of fluid dynamics problems, either directly or with slight modification, ranging from pipe flows to airplanes.