Continuity, momentum and flow

Abstract

This document provides a foundational introduction to aerodynamics through the lens of inviscid flow theory and the fundamental conservation equations governing fluid motion. Beginning with an accessible analogy of humans immersed in air like fish in water, it establishes aerodynamics as the mathematical study of life in motion—from flight and speed to everyday phenomena.

The text introduces the distinction between viscous and inviscid flow regimes, emphasizing that inviscid analysis (neglecting internal friction forces) offers tractable mathematical models that closely approximate real fluid behavior under key assumptions: irrotational flow, incompressible density, and energy conservation.

Three fundamental conservation equations are presented as the mathematical foundation of aerodynamics: the continuity equation (mass conservation, showing that for incompressible flow ∇·u = 0), the momentum equation (connecting force to flow changes, culminating in the Euler equations for inviscid flow), and the energy equation (balancing total energy within a control volume). Each equation is developed in both differential and integral forms, with the material derivative introduced to track changes following fluid particles.

The document bridges conceptual understanding with mathematical rigor, showing how these “three best friends” work together to describe the movement of air and enable the engineering marvels of modern flight, sports, and transportation.

Key words

Aerodynamics

The branch of fluid dynamics that studies the motion of air and other gases, and their interaction with solid objects. It encompasses the mathematical and physical principles governing flight, speed, and fluid flow phenomena.

Inviscid flow

An idealized flow regime where internal friction forces (viscosity) are neglected. While all real fluids have viscosity, inviscid analysis provides mathematically tractable solutions that closely approximate actual flow behavior in many practical situations.

Viscous flow

Flow regime where internal friction forces within the fluid are considered. These forces add complexity to the analysis but are essential for understanding phenomena like boundary layers and drag.

Conservation equations

The fundamental mathematical laws governing fluid motion: continuity (mass conservation), momentum conservation, and energy conservation. Together, these three equations form the complete description of fluid flow.

Continuity equation

Mathematical expression of mass conservation stating that mass cannot be created or destroyed. For incompressible flow, it simplifies to ∇·u = 0, meaning the fluid neither spreads out nor converges at any point.

Momentum equation

Expression of Newton’s second law for fluids, relating the rate of change of momentum to applied forces. In inviscid flow, this becomes the Euler equation: ρ Du/Dt = -∇p + ρg.

Energy equation

Conservation law stating that the total energy (kinetic plus potential) within a control volume remains constant, accounting for energy entering/leaving through boundaries and work done on the system.

Material (substantial derivative)

Operator D/Dt = ∂/∂t + (u·∇) that describes the total rate of change experienced by a fluid particle, accounting for both temporal changes and convective transport as the particle moves through space.

Incompressible flow

Flow where density remains constant (ρ = constant). This assumption simplifies the governing equations significantly and applies well to low-speed aerodynamics where Mach number < 0.3.

Divergence

Mathematical operator measuring how much a vector field spreads out or converges at a point. In the continuity equation, ∇·u = 0 means the flow is neither expanding nor compressing.

Euler equations

The momentum equations for inviscid flow, obtained from the Navier-Stokes equations by setting viscosity to zero: ρ Du/Dt = -∇p + ρg.

Navier-Stokes equations

The complete momentum equations for viscous fluid flow, accounting for pressure gradients, body forces, and viscous stresses. They represent one of the most powerful (and complex) descriptions of fluid motion.

Irrotational flow

Flow without rotation or swirling motion, where fluid particles do not spin about their own axes (though they may follow curved paths). Mathematically: ∇×u = 0.

Let’s talk about aerodynamics, the beautiful branch of physics, the beautiful branch of fluid dynamics that focuses on the study of the movement of air (or other gasses). So we must stop here for a little moment and really think about what this is. I want you to picture a little orange fish in the sea, it is swimming at the bottom of the ocean, it is going happily, it is moving through the water; obviously it is surrounded by water. Now, in the same way the little fish us surrounded by water, we are little humans that are surrounded by air, this beautiful component of life that not only allows us to breathe and to live, but also to move.

The natural state of life is movement, everything is always moving: the earth by itself, we as humans, but also more objects like the leaves of a tree, our hair when we go out for a walk in that green park. Aerodynamics is not just the study of air, it is not just the study of movement, but it is also the study of life.

For me it is crazy how we have “normalized’” the word “fly.” Just think about it, your friend tells you: “hey, tomorrow I will fly to Spain!” I am sorry, but what did you just say? You are going to fly? Like a bird? What we have always dreamed?

What is it about flying that makes us feel so free? How beautiful it is to see a bird to take flight, or to see just how it is gliding in the blue sky. What is it about speed that makes us feel so alive? Why do we even watch Formula One? What is that component of the car that makes it go so fast?. What is it about the beauty of soccer or american football or golf? How can the athletes be so precise?

Aerodynamics may be seen as a topic that is not that big, that is kind or constrained to just airplanes, but in reality it is open to everything. Open to the design of life.

Now, the most beautiful part of aerodynamics is that one that is visual, that part that you can feel with your senses; however, how can we become masters of air? Well, we have to speak its language, and the language of aerodynamics is mathematics and we didn’t learn to speak by creating complete sentences, but little by little, word by word.

Following this same line of thought, we can begin by mentioning that yes, the language of aerodynamics is mathematics and well, it really speaks it really good, actually, it is very difficult to understand its “accent”. Have you ever talked with a person that has an accent difficult to understand? Well, something similar happens with aerodynamics, its equation are so complicated and long that in order to understand it mathematicians and physicist, very intelligently, created two branches for its studies denominated as: viscous[1] and inviscid flow. In the first one, the internal forces of the fluid, like friction, are taken into account, these forces add complexity to the analysis. The second term refers to fluids in which the internal forces are neglected, and despite that all fluids in reality are viscous, the analysis of inviscid flows can yield results that would be extremely similar as for viscous fluids and with easier mathematics. So then, when learning the language of aerodynamics, we begin focusing on this second type of fluids.

Inviscid fluids

When we are analyzing in the inviscid regime, we say that we are analyzing ideal fluids (which mean non real, but ideal), so here we make some assumptions like:

No rotation (the fluid will not start swirling, like it is seen when we flush the toilet or in a tornado).
No change in density (imagine we put air inside an invisible box, the box is full and well closed, so it won’t let go air neither new air will enter).
No change in energy.

Conservation equations

Following these rules for inviscid fluids, now we can start talking about the mathematics that describe its movement. For this we will invite these three best friends that belong to a rock band called “conservative equations,” and so they are:

Continuity of mass equation.
Momentum conservation equation.
Energy conservation equation.

Now, it is time to start introducing them and to talk about the things they can do.

Continuity equation

I remember once in my physics subject how my professor used to tell us: “learn concepts, not just the equations,” therefore I am going to tell you that the continuity equation tells you if the density at a point changes, it must because more mass is flowing into the point than out or viceversa. In other words:

(How density changes at a certain point) + (Divergence of mass flux) = 0

Or in its mathematical form it would then be:

$\frac{\partial \rho}{\partial t}+\vec \nabla \cdot (\rho u)=0$

However, can your remember (and if not, just go 5 sentences above) how we are neglecting the change in density? So with this in mind, the continuity equation would then say that the divergence of velocity is equal to zero (the fluid does not spread out nor converges):

$\vec \nabla \cdot u=0$

Convective form

Now, if we are analyzing a fluid that is flowing, we then evaluate with something called as the “material (or substantial) derivative” that explains the net rate of change that a point experiences (change in time and divergence), and it looks like:

$\frac{D\rho}{Dt}=\frac{\partial}{\partial t}+(u\cdot \vec \nabla)$

So then, with this material derivative, the continuity equation would look like:

$\frac{D\rho}{Dt}+\rho \vec \nabla \cdot u=0$

And now, I think that we all have listened to that very famous law of physics “mass cannot be created or destroyed”, well that phrase can be seen in the next equation, in which we see how the sum of the rate of change of mass over the volume plus the net mass flux out of the surface surrounding the volume is equal to zero:

$\int_{v} \frac{\partial \rho}{\partial t} dv+\oint_{s}\rho u \cdot n dA=0$

Momentum equation

Now let’s move to the next member of this band, the momentum equation. This equation explains how momentum changes in a small region as time passes; in the same way, the change in momentum is equal to the forces applied [2].

So now, in the conservative way, the momentum equation is the rate of change of density + the flux of momentum (momentum carried in or out) = body forces + divergence of the stress tensor (compresses or stretches).

$\frac{\partial\rho}{\partial t}+\vec \nabla \cdot (\rho uu)=\rho g+\vec \nabla \cdot \overline{\overline{\tau}}$

Convective form

Its convective form would be that the material change of momentum is equal to the forces:

$\rho\frac{Du}{Dt}=\rho g+\vec \nabla \cdot \overline{\overline{\tau}}$

And now, if we integrate this, or better said, if we make the sum of the momentum change in the volume of the fluid + the momentum flux on the surface = body forces acting on the volume + forces acting on the pressure:

$\int_V \frac{\partial \rho u}{\partial t}dV+\oint_S \rho uu\cdot ndA=\int_V \rho g dV+\oint_Sf(x,n)dA$

And now you are starting to see how everything is really contributing, how it all is shaping the movement or air and the movement of living things. Now, we can show up just a little bit about one of the most powerful equations in fluid dynamics, that is the Navier-Stokes equation (for momentum), in particular the next equation shows how the fluid changes at a point based on physical effects like forces, pressure and viscosity. Look at it in this way, if you want to predict how the speed and direction of a point will change, you need to know the pushes and pulls and all the in’s and out’s flows of momentum.

$\rho(\frac{\partial u}{\partial t}i+u\cdot \frac{\partial n}{\partial x}j)=\frac{\partial p}{\partial x_j}+\rho g j+\frac{\partial}{\partial x_i}[(\mu\frac{\partial u_i}{\partial x_i}+\frac{\partial u_j}{\partial x_j})+(\mu-\frac{2}{3}\mu)\frac{\partial u_m}{\partial x_m}\delta_{ij}]$

However, remember the assumptions hat we established almost at the beginning: inviscid; so then, we can rewrite this equation, not in the Navier-Stokes way, but in the Euler form in which viscosity is not taken into account (an equation for incompressible flows):

$\rho\frac{Du}{Dt}=-\vec \nabla p+\rho g$

Energy equation

Basically this is the total particles moving in our system, each single molecule has kinetic and potential energy and the sum of this must be considered as constant inside a certain volume. So then, this equation is balance between the rate of change of total energy and all the ways energy can enter or leave the control volume.

Now, directly ignoring viscosity we would have the equation:

$\frac{\partial[\rho (e+\frac{v^2}{2})]}{\partial t}+\vec \nabla \cdot [e(e+\frac{v^2}{2})\vec u+p\vec u]=Work$

So then, its integral in the convective form would be that the integral of the material derivative of energy + the divergence of flux is equal to the work produced.

$\int \frac{De}{Dt}+\vec \nabla \cdot \vec up=Work$

Conclusion

Aerodynamics is a subject that not only studies the movement of air (and other gasses), but also the movement of life, and this is nowadays super relevant in our life as it shapes the way of how we move, obtain energy and even enjoy the world.

To start understanding the language of aerodynamics, you first need to learn about the 2 ways to analyze its mathematics like the viscous and inviscid form, being the second one the most used in general to allow the comprehension of it; then you introduce yourself to the 3 best friends, the 3 equations that govern motion that are: continuity, momentum and energy.

Footnotes

[1] It is important to say here that the term “viscosity” refers to the resistance to movement. For example, which one do you think has higher viscosity (more resistance to movement), honey or water? Just think about what would happen if you let them fall.

[2] You can go more deep with this concept here.

Continuity, momentum and flow

Abstract

Key words

Inviscid fluids

Conservation equations

Conclusion

Footnotes

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