Questions tagged [fluid-dynamics]

When to Use This Tag

Use fluid-dynamics when asking questions about the response of a fluid to externally applied forces, which results in fluid motion. When asking questions about the response of fluid to external forces which does not result in fluid motion, use the tag fluid-statics. Most questions with these tags will be subsumed in the field of continuum-mechanics, in which we are concerned with the large-scale behavior of fluids that may be modeled as continuous substances. When dealing with nanoscale fluid flow, or in cases where the fluid particles may be highly rarefied (such as the extremely low-density gas in the upper atmosphere), a classical continuum description may be insufficient, and it may be necessary to invoke statistical-mechanics to model the behavior of the fluid. Note that questions regarding only the material properties of a fluid and not regarding its large-scale motion may also be more appropriately tagged under statistical-mechanics or even material-science.

Introduction

From a dynamics point of view, a fluid (i.e., a liquid, gas, or plasma) is distinguished from a solid by its inability to support shear stress. When a constant shear stress is applied to a solid, it experiences some definite displacement which is referred to as strain. A fluid, on the other hand, does not experience such a finite displacement but rather deforms continuously at some well-defined strain rate. The dynamic behavior of a solid substance such as a metal beam may therefore be described by specifying a (possibly time-dependent) displacement for each point in the substance. For a fluid, we specify instead the velocity at each point. This is the main mathematical difference between fluid mechanics and solid-mechanics.

Equations of Motion

A fluid flow is said to be fully characterized when the kinematic history of the fluid (i.e., the time and space history of its velocity field) is known, in addition to the time history of the internal pressure distribution. As in other areas of classical-mechanics, this dynamic behavior is governed by the transport equations of mass, momentum, and energy. These may be expressed under quite general hypotheses as, respectively:

$$ \frac{\partial\rho}{\partial t} + \frac{\partial\rho u_i}{\partial x_i} = 0 $$

$$ \frac{\partial\rho u_i}{\partial t} + \frac{\partial\rho u_i u_j}{\partial x_j} = \frac{\partial\sigma_{ij}}{\partial x_j} $$

$$ \frac{\partial}{\partial t}\left( \rho e + \rho\frac{u_i^2}{2} \right) + \frac{\partial}{\partial x_i}\left( \rho e u_i + \rho u_i \frac{u_i^2}{2} \right) = -\frac{\partial q_i}{\partial x_i} + \frac{\partial \sigma_{ij}u_j}{\partial x_j} $$

where $ e $ is the internal energy of the fluid which may be related to the thermodynamic temperature, $ \rho $ is the density, $ q_i $ denotes the heat flux vector, and $ \sigma_{ij} $ denotes the Cartesian stress tensor. All the foregoing equations have been written in Cartesian tensor notation and employ the Einstein summation convention for compactness. According to this convention, a repeated subscript indicates summation over that index. The stress tensor may be decomposed into a pressure term and a viscous term, viz

$$ \sigma_{ij} = -p\delta_{ij} + \tau_{ij} $$

The quantities $ q_i $ and $ \tau_{ij} $, as well as all others not in terms of primitive variables, must be modeled in some way to close the formulation. In many situations, the heat flux may be approximated by Fourier's law, and the viscous stress tensor may be assumed to be Newtonian (see questions marked newtonian-fluid).

In the case that the flow is incompressible and the fluid is Newtonian, we encounter one of the most important special cases of the above equations. The mass conservation equation (also known as the continuity equation) reduces to

$$ \frac{\partial u_i}{\partial x_i} = 0 $$

which indicates that there is no local fluid dilatation, or in other words no local change in specific volume of a fluid element. The momentum equation reduces to the well-known Navier-Stokes equation:

$$ \frac{\partial u_i}{\partial t} + u_j\frac{\partial u_i}{\partial x_j} = -\frac{1}{\rho}\frac{\partial p}{\partial x_i} + \nu\frac{\partial^2u_i}{\partial x_j x_j} $$

where $ \nu = \mu/\rho $ is the kinematic viscosity of the fluid. In the incompressible limit, the thermal energy equation may be removed from the formulation, because the fluid mechanics and heat transfer problems have become one-way coupled. That is, the fluid velocity and pressure fields may affect the temperature field, but the temperature field has no significant effect on the pressure and velocity fields.

Prerequisites to Learn Fluid Dynamics:

Phys: Elementary thermodynamics and rigid body dynamics.

Math: Vector calculus (e.g. Green's theorem, Divergence theorem), elementary tensor calculus, Ordinary Differential Equations (ODEs), Partial Differential Equations (PDEs).