In physics, **fluid dynamics** is a branch of fluid mechanics that studies the behavior of fluids (i.e., liquids and gases) in motion, as opposed to fluid statics; solving a fluid dynamic problem generally involves solving (analytically or numerically) complex differential equations to calculate various properties of the fluid including velocity, pressure, density, or temperature, as a function of position in space and time.

The study of fluids in motion, or fluid dynamics, makes up the more significant part of fluid mechanics. Branches of fluid dynamics include hydrodynamics (the study of liquids in motion) and aerodynamics (the study of gases in motion) as well as vortex dynamics, gas dynamics, computational fluid dynamics (CFD), convection heat transfer, flows of turbomachinery, acoustics, bio-fluids, physical oceanography, atmospheric dynamics, wind engineering, and the dynamics of two-phase flows. The modern design of aircraft, spacecraft, automobiles, ships, land and marine structures, power and propulsion systems, or heat exchangers relies on a clear understanding of the relevant fluid mechanics.

A **fluid flow** may be described in two different ways: the Lagrangian approach (named after the French mathematician Joseph Louis Lagrange), and the Eulerian approach (named after Leonhard Euler, a famous Swiss mathematician).

In the Lagrangian approach, one particle is chosen and is followed as it moves through space with time. The line traced out by that one particle is called a particle pathline.

A Eulerian approach is used to obtain a clearer idea of the airflow at one particular instant. One can look at a “photograph” of the flow of, for instance, surface ocean currents at a particular fixed time. The entire flow field is easily visualized. The lines comprising this flow field are called streamlines (see streamlining). Thus, a pathline refers to the trace of a single particle in time and space, whereas a streamline presents the line of motion of many particles at a fixed time. The question of whether particle pathlines and streamlines are ever the same is considered next.

Of primary importance in understanding fluid movements about an object is the concept of a steady flow. On a windy day, a person calls the wind steady if, from where she stands, it blows continuously from the same direction at a constant speed. If, however, the speed or direction changes, the wind is “gusty” or unsteady. Similarly, the flow of a fluid (both liquid and air) about an object is steady if its velocity (speed and direction) at each point in the flow remains constant – this does not necessarily require that the velocity be the same at all points in the fluid. This means that for unsteady flows, particle pathlines (the Lagrangian point of view) and streamlines (the Eulerian approach) are not equivalent. For a steady flow, however, a particle pathline and streamline are equivalent, and the Lagrangian point of view is the same as the Eulerian approach for flow visualization.

As well as steady or unsteady, fluid flow can be rotational or irrotational. If the elements of fluid at each point in the flow have no net angular (spin) velocity about the points, the fluid flow is said to be irrotational. One can imagine a small paddle wheel immersed in a moving fluid. If the wheel translates (or moves) without rotating, the motion is irrotational. If the wheel rotates in a flow, the flow is rotational.

According to a theorem of Hermann von Helmholtz, a German physicist who contributed much to theoretical aerodynamics, assuming zero viscosity, if a fluid flow is initially irrotational, it remains irrotational. In real life, viscosity effects are limited to a small region near the surface of the airfoil and in its wake. Most of the flow may still be treated as irrotational. A simplifying argument often used to aid in understanding basic ideas about fluid flow is that of one-dimensional fluid flow. Flows may be considered one-dimensional where the flow parameters (for example density, velocity, temperature, pressure) vary as a function of one spatial variable (for example, length) and variations in the other two spatial dimensions (i.e., y and z) are negligible by comparison.

Simplifying assumptions about fluids are made: the first is that fluid is considered to be inviscid (no viscosity); the second is that it is incompressible. Further, the flow is considered steady and one-dimensional. Fluids with these characteristics are said to be ideal fluids or perfect fluids. Once solutions of problems relating to the lift and drag of ideal fluids, or the inviscid flow, have been made, a solution of the viscous flow in the thin boundary layer allows the effects of skin friction drag to be calculated.