CFD for Beginners: Master Navier–Stokes, Continuity & Fluid

In this lecture, we derive the differential form of the continuity equation, one of the fundamental governing equations of Computational Fluid Dynamics.

Starting from conservation of mass, we analyze a small fluid element and examine how mass enters and leaves through its faces.

You will learn how mass accumulation and net mass flux are balanced to obtain the continuity equation in differential form.

We explain the physical meaning of each term and show how the equation represents mass conservation at every point in the flow field.

Special attention is given to incompressible flow, where the equation simplifies into an important and widely used form.

This lecture is presented with clear visual animations so that the derivation becomes intuitive, structured, and easy to remember.

In this lecture, we derive the integral form of the continuity equation using the principle of conservation of mass.

We apply control volume analysis to understand how mass can accumulate inside a region while fluid simultaneously enters and leaves through its boundaries.

You will learn how the rate of mass increase within the control volume balances the net mass flow across the control surface.

The physical meaning of each term is explained clearly, making it easier to understand how real engineering systems such as pipes, nozzles, tanks, and ducts are analyzed.

This integral form is one of the most important starting points in fluid mechanics and Computational Fluid Dynamics before converting equations into differential form.

The lecture is taught step by step with visual animations to make the concepts intuitive and easy to follow.

In this lecture, we begin one of the most important governing equations in Computational Fluid Dynamics: the conservation of momentum equation.

We start with an intuitive introduction to momentum balance and understand how Newton’s Second Law is applied to a fluid element.

Then we derive the pressure force term by analyzing how pressure acts on opposite faces of a differential control volume.

You will learn how pressure variations in space create a net force on the fluid and how this contribution appears in the momentum equation.

The physical meaning of pressure gradients and their role in driving fluid motion are explained clearly.

This lecture builds the foundation for the complete momentum equation, where pressure, viscous forces, body forces, and acceleration terms are combined.

The concepts are taught step by step with detailed visual animations to make the derivation intuitive and easy to understand.

In this lecture, we continue the derivation of the momentum equation by developing the viscous force term and the body force term.

We begin with viscous forces by analyzing stresses acting on the faces of a small fluid element. You will learn how shear and normal viscous stresses create net forces due to spatial variation, leading to the viscous contribution in the momentum equation.

Next, we study body forces, which act throughout the fluid volume rather than only on surfaces. Common examples such as gravity are used to understand how volumetric forces are incorporated into the governing equation.

The physical meaning of each term is explained clearly so that students understand not only the mathematics, but also the flow physics behind the derivation.

This lecture is an important step toward assembling the complete momentum equation used in Computational Fluid Dynamics.

The topic is taught visually with detailed animations to make difficult concepts intuitive and easy to remember.

In this lecture, we break down and understand the individual terms of the Navier-Stokes equation, one of the most important equations in Computational Fluid Dynamics.

After completing the derivation, we now focus on the physical meaning of each term and how it influences fluid motion.

You will learn the significance of transient acceleration, convective acceleration, pressure forces, viscous diffusion, and body forces such as gravity.

The lecture explains how these terms represent inertia, momentum transport, resistance due to viscosity, and external forces acting on the fluid.

Special attention is given to building intuition so that the equation no longer appears as a collection of symbols, but as a complete balance of forces and accelerations.

Understanding these terms is essential before studying simplifications, numerical methods, turbulence models, and commercial CFD software.

The concepts are explained visually with animations to make the Navier-Stokes equation clear, practical, and easy to remember.

In this lecture, we begin the study of the conservation of energy equation, one of the fundamental governing equations of Computational Fluid Dynamics.

We start with the physical meaning of energy conservation in fluid flow and understand how thermal energy changes within a moving fluid element.

You will learn the different forms of energy present in fluids, including internal energy, kinetic energy, and potential energy, and why internal energy is commonly used in practical energy equation formulations.

The lecture also introduces the major mechanisms through which energy is transferred or generated, such as heat conduction, pressure work, viscous work, and volumetric source terms.

Special attention is given to building intuition so that students understand why the energy equation is needed in addition to mass and momentum conservation.

This lecture creates the foundation for the detailed derivation of the complete energy equation in later lessons.

The concepts are explained clearly with visual animations to make this important topic intuitive and easy to understand.

In this lecture, we begin the detailed derivation of the energy equation by focusing on internal energy and heat added through conduction.

We first develop the rate of change of internal energy for a moving fluid element using the total derivative, showing how thermal energy changes with time and motion.

Next, we derive the heat conduction term by analyzing a differential control volume and examining heat transfer through its faces.

Using Taylor series expansion and Fourier’s law of heat conduction, we obtain the compact conduction form used in the governing energy equation.

The lecture also explains the physical meaning of temperature gradients and how heat naturally flows from hotter regions to colder regions.

This topic is essential for understanding thermal transport in cooling systems, heat exchangers, boundary layers, combustion, and many engineering applications.

The derivation is taught step by step with clear animations so that students can understand both the mathematics and the physics behind the equation.

In this lecture, we continue the derivation of the energy equation by developing the pressure work term.

You will learn how pressure forces acting on the surfaces of a differential fluid element can transfer mechanical energy to or from the fluid.

By analyzing opposite faces of the control volume and applying Taylor series expansion, we derive the net work done by pressure forces in differential form.

The lecture explains how this term represents compression, expansion, and energy transfer caused by pressure gradients within flowing fluids.

Special attention is given to the physical interpretation of pressure work and its importance in compressible flows, nozzles, turbines, compressors, engines, and many engineering systems.

This lecture is an important step toward assembling the complete conservation of energy equation used in Computational Fluid Dynamics.

The derivation is taught visually with detailed animations so that both the mathematics and physics become clear and intuitive.

In this lecture, we continue the derivation of the energy equation by developing the viscous work term.

You will learn how viscous stresses acting on the surfaces of a differential fluid element can transfer mechanical energy within the fluid.

We begin by understanding the physical origin of viscosity as internal friction caused by relative motion between neighboring fluid layers.

Then, by analyzing stresses on the control volume faces and applying differential expansion, we derive the net viscous work contribution used in the energy equation.

The lecture explains how this term becomes important in flows with strong shear, lubrication systems, boundary layers, mixing regions, polymer flows, and highly viscous fluids.

Special attention is given to the physical meaning of viscous work as the transfer of mechanical energy due to internal fluid friction.

This lecture is an essential step toward completing the full conservation of energy equation used in Computational Fluid Dynamics.

The topic is taught with clear visual animations so that students can understand both the derivation and the real flow physics behind the equation.

In this lecture, we complete the derivation of the conservation of energy equation by introducing the source term and reviewing the physical meaning of every term in the final equation.

You will learn how volumetric energy sources or sinks act within a fluid element, representing internal energy generation or removal throughout the control volume.

Common examples such as combustion, electrical heating, radiation absorption, chemical reactions, and internal cooling are discussed to build practical understanding.

After deriving the source contribution, we consolidate the complete energy equation and explain the significance of each term, including internal energy change, heat conduction, pressure work, viscous work, and source effects.

This lecture helps students move beyond mathematical symbols and understand the real physical processes represented in the governing equation.

It also serves as an important summary of the complete energy equation before advancing to numerical solution methods and applied CFD topics.

The concepts are explained step by step with visual animations to make the final equation clear, intuitive, and easy to remember.