Equação de Cauchy e o papel de uma equação constitutiva

Hey everyone! How's it going? In today's lesson, we'll arrive at one of the most important results in all of classical Fluid Mechanics: 👉 Cauchy's Equation. It represents the principle of conservation of linear momentum applied to an infinitesimal fluid particle and forms the basis upon which virtually all the equations we use to model flows are built. Over the last few lessons, we've carefully developed the necessary mathematical tools to get here: Eulerian description, material derivative, Reynolds Transport Theorem, continuity equation, and the transition between integral and differential formulations. Now we'll finally see how these pieces fit together to produce one of the most fundamental equations in Continuous Media Mechanics. 🔍 In this lesson, you will understand: How the principle of linear momentum balance applies to a fluid particle; The physical meaning of the terms present in Cauchy's Equation; The interpretation of body forces and surface forces; How the Cauchy equation can be written in its conservative form; How to obtain the more well-known form involving the material derivative; The relationship between both representations; Why different numerical methods often prefer different forms of the same equation; The conceptual meaning of constitutive equations; What we understand by the principle of material indifference. 💡 The central point of this lesson One of the great achievements of Continuous Media Mechanics was realizing that extremely general physical laws can be expressed through local differential equations. The Cauchy equation represents exactly this idea. It is not a specific equation for fluids, nor for solids, nor for particular materials. It expresses only a universal physical principle: 👉 the rate of change of linear momentum must be equal to the resultant of the forces acting on the material element. All the richness of different materials emerges later, when we introduce constitutive models capable of describing how each material responds to the deformations to which it is subjected. 📘 What's Next? From this point in the course, we definitively begin to enter the territory of hydrodynamics. In the following classes, we will see how the introduction of specific constitutive hypotheses allows us to obtain: Euler's equations; Navier-Stokes equations; Newtonian fluid models; Physical interpretation of viscosity; Mathematical structure of the equations that underpin modern CFD. In other words, we are crossing the bridge that connects the Mechanics of Continuous Media to Fluid Mechanics as it is used in contemporary Engineering and scientific research. This class is brought to you by L2C – Solutions in Scientific Computing. 🚀 L2C Educational Initiatives If this type of content sparks your interest and you want to learn, in a structured way, how to transform mathematical models into algorithms and algorithms into tools capable of solving real Engineering problems, learn about L2C's initiatives. 📘 Course: From Calculus to Computer Simulation Registration for the 3rd class is now open. Over four months we will build a complete journey connecting: ✔ Numerical methods ✔ Scientific programming ✔ Mathematical modeling ✔ Computational simulation ✔ Real-world applications in Engineering and Applied Sciences 📅 Start date: August 19, 2026 📅 End date: December 16, 2026 🗓️ Live meetings on Wednesdays ⏰ From 7 PM to 10 PM 🌐 Learn more: www.l2c.dev.br/lp 📺 Watch the course launch live stream: https://youtube.com/live/gXvueRTRRwo 🌪️ Coming soon: Fundamentals of CFD with Applications in OpenFOAM 🌐 www.l2c.dev.br/lp2 📗 Calculus Manual Numerical – Fundamentals with Applications 🌐 www.l2c.dev.br/lp3 📺 Want to follow this journey? The Fluid Mechanics 2 course is completely free, and a new lesson is published weekly here on the Ciência e Brisa channel. Subscribe to the channel, activate notifications, and follow the upcoming episodes. And, if you want access to the summary slides containing the main concepts presented in each lesson, also follow the L2C page on LinkedIn. See you in class! Prof. Rafael Gabler Gontijo 🌐 www.rafaelgabler.com.br 🌐 www.l2c.dev.br