Fluid mechanics

Fluid mechanics is the branch of physics that studies the mechanics of fluids (liquids, gases, and plasmas) and the forces on them. Fluid mechanics has a wide range of applications, including for mechanical engineering, chemical engineering, geophysics, astrophysics, and biology. Fluid mechanics can be divided into fluid statics, the study of fluids at rest; and fluid dynamics, the study of the effect of forces on fluid motion. It is a branch of continuum mechanics, a subject which models matter without using the information that it is made out of atoms; that is, it models matter from a macroscopic viewpoint rather than from microscopic. Fluid mechanics, especially fluid dynamics, is an active field of research with many problems that are partly or wholly unsolved. Fluid mechanics can be mathematically complex, and can best be solved by numerical methods, typically using computers. A modern discipline, called computational fluid dynamics (CFD), is devoted to this approach to solving fluid mechanics problems. Particle image velocimetry, an experimental method for visualizing and analyzing fluid flow, also takes advantage of the highly visual nature of fluid flow.

Brief history

The study of fluid mechanics goes back at least to the days of ancient Greece, when Archimedes investigated fluid statics and buoyancy and formulated his famous law known now as the Archimedes' principle, which was published in his work On Floating Bodies – generally considered to be the first major work on fluid mechanics. Rapid advancement in fluid mechanics began with Leonardo da Vinci (observations and experiments), Evangelista Torricelli (invented the barometer), Isaac Newton (investigated viscosity) and Blaise Pascal (researched hydrostatics, formulated Pascal's law), and was continued by Daniel Bernoulli with the introduction of mathematical fluid dynamics in Hydrodynamica (1738).

Inviscid flow was further analyzed by various mathematicians (Leonhard Euler, Jean le Rond d'Alembert, Joseph Louis Lagrange, Pierre-Simon Laplace, Siméon Denis Poisson) and viscous flow was explored by a multitude of engineers including Jean Léonard Marie Poiseuille and Gotthilf Hagen. Further mathematical justification was provided by Claude-Louis Navier and George Gabriel Stokes in the Navier–Stokes equations, and boundary layers were investigated (Ludwig Prandtl, Theodore von Kármán), while various scientists such as Osborne Reynolds, Andrey Kolmogorov, and Geoffrey Ingram Taylor advanced the understanding of fluid viscosity and turbulence.

Main branches

Fluid statics

Fluid statics or hydrostatics is the branch of fluid mechanics that studies fluids at rest. It embraces the study of the conditions under which fluids are at rest in stable equilibrium; and is contrasted with fluid dynamics, the study of fluids in motion.

Hydrostatics is fundamental to hydraulics, the engineering of equipment for storing, transporting and using fluids. It is also relevant to geophysics and astrophysics (for example, in understanding plate tectonics and the anomalies of the Earth's gravitational field), to meteorology, to medicine (in the context of blood pressure), and many other fields.

Hydrostatics offers physical explanations for many phenomena of everyday life, such as why atmospheric pressure changes with altitude, why wood and oil float on water, and why the surface of water is always flat and horizontal whatever the shape of its container.

Fluid dynamics

Fluid dynamics is a subdiscipline of fluid mechanics that deals with fluid flow—the natural science of fluids (liquids and gases) in motion. It has several subdisciplines itself, including aerodynamics (the study of air and other gases in motion) and hydrodynamics (the study of liquids in motion). Fluid dynamics has a wide range of applications, including calculating forces and moments on aircraft, determining the mass flow rate of petroleum through pipelines, predicting weather patterns, understanding nebulae in interstellar space and modelling fission weapon detonation. Some of its principles are even used in traffic engineering, where traffic is treated as a continuous fluid, and crowd dynamics.

Fluid dynamics offers a systematic structure—which underlies these practical disciplines—that embraces empirical and semi-empirical laws derived from flow measurement and used to solve practical problems. The solution to a fluid dynamics problem typically involves calculating various properties of the fluid, such as velocity, pressure, density, and temperature, as functions of space and time.

Relationship to continuum mechanics

Fluid mechanics is a subdiscipline of continuum mechanics, as illustrated in the following table.

Continuum mechanics The study of the physics of continuous materials	Solid mechanics The study of the physics of continuous materials with a defined rest shape.	Elasticity Describes materials that return to their rest shape after applied stresses are removed.
		Plasticity Describes materials that permanently deform after a sufficient applied stress.	Rheology The study of materials with both solid and fluid characteristics.
	Fluid mechanics The study of the physics of continuous materials which deform when subjected to a force.	Non-Newtonian fluids do not undergo strain rates proportional to the applied shear stress.
		Newtonian fluids undergo strain rates proportional to the applied shear stress.

In a mechanical view, a fluid is a substance that does not support shear stress; that is why a fluid at rest has the shape of its containing vessel. A fluid at rest has no shear stress.

Assumptions

Balance for some integrated fluid quantity in a control volume enclosed by a control surface.

Like any mathematical model of the real world, fluid mechanics makes some basic assumptions about the materials being studied. These assumptions are turned into equations that must be satisfied if the assumptions are to be held true.

For example, consider a fluid in three dimensions. The assumption that mass is conserved means that for any fixed control volume (for example a sphere) – enclosed by a control surface – the rate of change of the mass contained is equal to the rate at which mass is passing from outside to inside through the surface, minus the rate at which mass is passing the other way, from inside to outside. (A special case would be when the mass inside and the mass outside remain constant). This can be turned into an equation in integral form over the control volume.^[1]

Fluid mechanics assumes that every fluid obeys the following:

• Conservation of mass
• Conservation of energy
• Conservation of momentum
• The continuum hypothesis, detailed below.

Further, it is useful at low subsonic speeds to assume that a gas is incompressible — that is, the density of the gas does not change even though the speed and static pressure change.

Similarly, it can sometimes be assumed that the viscosity of the fluid is zero (the fluid is inviscid). If a fluid is viscous, and its flow contained in some way (e.g. in a pipe), then the flow at the boundary must have zero velocity. For a viscous fluid, if the boundary is not porous, the shear forces between the fluid and the boundary results also in a zero velocity for the fluid at the boundary. This is called the no-slip condition. For a porous media otherwise, in the frontier of the containing vessel, the slip condition is not zero velocity, and the fluid has a discontinuous velocity field between the free fluid and the fluid in the porous media (this is related to the Beavers and Joseph condition).

Continuum hypothesis

Fluids are composed of molecules that collide with one another and solid objects. The continuum assumption, however, considers fluids to be continuous. That is, properties such as density, pressure, temperature, and velocity are taken to be well-defined at "infinitely" small points, defining a REV (Reference Element of Volume), at the geometric order of the distance between two adjacent molecules of fluid. Properties are assumed to vary continuously from one point to another, and are averaged values in the REV. The fact that the fluid is made up of discrete molecules is ignored.

The continuum hypothesis is basically an approximation, in the same way planets are approximated by point particles when dealing with celestial mechanics, and therefore results in approximate solutions. Consequently, assumption of the continuum hypothesis can lead to results which are not of desired accuracy. However, under the right circumstances, the continuum hypothesis produces extremely accurate results.

Those problems for which the continuum hypothesis does not allow solutions of desired accuracy are solved using statistical mechanics. To determine whether or not to use conventional fluid dynamics or statistical mechanics, the Knudsen number is evaluated for the problem. The Knudsen number is defined as the ratio of the molecular mean free path length to a certain representative physical length scale. This length scale could be, for example, the radius of a body in a fluid. (More simply, the Knudsen number is how many times its own diameter a particle will travel on average before hitting another particle). Problems with Knudsen numbers at or above one are best evaluated using statistical mechanics for reliable solutions.

Navier–Stokes equations

The Navier–Stokes equations (named after Claude-Louis Navier and George Gabriel Stokes) are the set of equations that describe the motion of fluid substances such as liquids and gases. These equations state that changes in momentum (force) of fluid particles depend only on the external pressure and internal viscous forces (similar to friction) acting on the fluid. Thus, the Navier–Stokes equations describe the balance of forces acting at any given region of the fluid.

The Navier–Stokes equations are differential equations which describe the motion of a fluid. Such equations establish relations among the rates of change of the variables of interest. For example, the Navier–Stokes equations for an ideal fluid with zero viscosity states that acceleration (the rate of change of velocity) is proportional to the derivative of internal pressure.

This means that solutions of the Navier–Stokes equations for a given physical problem must be sought with the help of calculus. In practical terms only the simplest cases can be solved exactly in this way. These cases generally involve non-turbulent, steady flow (flow does not change with time) in which the Reynolds number is small.

For more complex situations, involving turbulence, such as global weather systems, aerodynamics, hydrodynamics and many more, solutions of the Navier–Stokes equations can currently only be found with the help of computers. This branch of science is called computational fluid dynamics.

General form of the equation

The general form of the Cauchy momentum equation is:

$\rho\frac{D\mathbf{u}}{D t} = \nabla\cdot\boldsymbol{\sigma} + \rho\mathbf{f}$

where

• $\rho\$ is the fluid density,
• $\frac{D}{D t}$ is the substantive derivative (also called the material derivative),
• $\mathbf{u}$ is the flow velocity vector,
• $\mathbf{f}$ is the specific body force vector, and
• $\boldsymbol{\sigma}$ is the stress tensor.

Unless the fluid is made up of spinning degrees of freedom like vortices, $\boldsymbol{\sigma}$ is a symmetric tensor. In Navier Stokes equations the stress tensor can be decomposed as

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

where $-p \delta_{ij}$ is a static isotropic stress state (that would exist if the fluid were at rest), and $\tau_{ij}$ is the deviatoric stress tensor, corresponding to the part of the stress due to the fluid motion. Generally, the scalar $p$ can be taken as the thermodynamic pressure, whereas $\tau_{ij}$ is called the viscous stress tensor. Furthermore, the diagonal components of tensor $\mathbf{\tau}$ are called normal stresses and the off-diagonal components are called shear stresses.

The vectorial Cauchy equation above can be written then as

$\rho\frac{D\mathbf{u}}{D t} = -\nabla p + \nabla\cdot\mathbf{\tau} + \rho\mathbf{f}$

This is actually a set of three equations, one per dimension. By themselves, these equations are not sufficient to produce a solution. However, adding other conservation laws and appropriate boundary conditions to the system of equations produces a solvable set of equations. The conservation of mass provides another equation relating the density and the flow velocity:

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

On the other hand, the identification of $p$ with the thermodynamic pressure is usually possible (unless the fluid is not in thermodynamic equilibrium; such situation is however rare [e.g. shock waves]). Therefore, a thermodynamic equation of state must be used to connect the pressure with the density and another state property, such as temperature or enthalpy. This in turn brings another unknown to the problem so that an equation for conservation of thermal energy must also be solved along with momentum and mass conservations.

In the case of an incompressible fluid there is no relationship between the pressure and the density. The Navier–Stokes equations and mass conservation are then sufficient to determine the solution to a fluid mechanics problem. Actually, the absolute pressure in an incompressible fluid is indeterminate, and only its gradient is relevant for the equations of motion. Taking the divergence of the Navier–Stokes equation and using the mass conservation equation to simplify the result gives a Poisson equation for the pressure.

Additionally, in order to close the system of equations a constitutive equation relating the viscous stress tensor to the velocity field must be introduced. This constitutive model, which depends on the nature of the fluid, is the basis for the distinction between Newtonian and non-Newtonian fluids.

Newtonian versus non-Newtonian fluids

A Newtonian fluid (named after Isaac Newton) is defined to be a fluid whose shear stress is linearly proportional to the velocity gradient in the direction perpendicular to the plane of shear. This definition means regardless of the forces acting on a fluid, it continues to flow. For example, water is a Newtonian fluid, because it continues to display fluid properties no matter how much it is stirred or mixed. A slightly less rigorous definition is that the drag of a small object being moved slowly through the fluid is proportional to the force applied to the object. (Compare friction). Important fluids, like water as well as most gases, behave – to good approximation – as a Newtonian fluid under normal conditions on Earth.^[2]

By contrast, stirring a non-Newtonian fluid can leave a "hole" behind. This will gradually fill up over time – this behaviour is seen in materials such as pudding, oobleck, or sand (although sand isn't strictly a fluid). Alternatively, stirring a non-Newtonian fluid can cause the viscosity to decrease, so the fluid appears "thinner" (this is seen in non-drip paints). There are many types of non-Newtonian fluids, as they are defined to be something that fails to obey a particular property – for example, most fluids with long molecular chains can react in a non-Newtonian manner.^[2]

Equations for a Newtonian fluid

The constant of proportionality between the viscous stress tensor and the velocity gradient is known as the viscosity. A simple equation to describe incompressible Newtonian fluid behaviour is

$\tau=-\mu\frac{dv}{dy}$

where

$\tau$ is the shear stress exerted by the fluid ("drag")

$\mu$ is the fluid viscosity – a constant of proportionality

$\frac{dv}{dy}$ is the velocity gradient perpendicular to the direction of shear.

For a Newtonian fluid, the viscosity, by definition, depends only on temperature and pressure, not on the forces acting upon it. If the fluid is incompressible the equation governing the viscous stress (in Cartesian coordinates) is

$\tau_{ij}=\mu\left(\frac{\partial v_i}{\partial x_j}+\frac{\partial v_j}{\partial x_i} \right)$

where

$\tau_{ij}$ is the shear stress on the $i^{th}$ face of a fluid element in the $j^{th}$ direction

$v_i$ is the velocity in the $i^{th}$ direction

$x_j$ is the $j^{th}$ direction coordinate.

If the fluid is not incompressible the general form for the viscous stress in a Newtonian fluid is

$\tau_{ij} = \mu \left( \frac{\partial v_i}{\partial x_j} + \frac{\partial v_j}{\partial x_i} - \frac{2}{3} \delta_{ij} \nabla \cdot \mathbf{v} \right) + \kappa \delta_{ij} \nabla \cdot \mathbf{v}$

where $\kappa$ is the second viscosity coefficient (or bulk viscosity). If a fluid does not obey this relation, it is termed a non-Newtonian fluid, of which there are several types. Non-Newtonian fluids can be either plastic, Bingham plastic, pseudoplastic, dilatant, thixotropic, rheopectic, viscoelastic.

In some applications another rough broad division among fluids is made: ideal and non-ideal fluids. An Ideal fluid is non-viscous and offers no resistance whatsoever to a shearing force. An ideal fluid really does not exist, but in some calculations, the assumption is justifiable. One example of this is the flow far from solid surfaces. In many cases the viscous effects are concentrated near the solid boundaries (such as in boundary layers) while in regions of the flow field far away from the boundaries the viscous effects can be neglected and the fluid there is treated as it were inviscid (ideal flow). When the viscosity is negleted, the term containing the viscous stress tensor $\mathbf{\tau}$ in the Navier–Stokes equation vanishes. The equation reduced in this form is called the Euler equation.

See also

• Aerodynamics
• Applied mechanics
• Bernoulli's principle
• Communicating vessels
• Secondary flow
• Different types of boundary conditions in fluid dynamics

Notes

1. Batchelor (1967), p. 74.
2. Batchelor (1967), p. 145.

References

• Batchelor, George K. (1967), An Introduction to Fluid Dynamics, Cambridge University Press, ISBN 0-521-66396-2

Further reading

• Falkovich, Gregory (2011), Fluid Mechanics (A short course for physicists), Cambridge University Press, ISBN 978-1-107-00575-4
• Kundu, Pijush K.; Cohen, Ira M. (2008), Fluid Mechanics (4th revised ed.), Academic Press, ISBN 978-0-12-373735-9
• Currie, I. G. (1974), Fundamental Mechanics of Fluids, McGraw-Hill, Inc., ISBN 0-07-015000-1
• Massey, B.; Ward-Smith, J. (2005), Mechanics of Fluids (8th ed.), Taylor & Francis, ISBN 978-0-415-36206-1
• White, Frank M. (2003), Fluid Mechanics, McGraw–Hill, ISBN 0-07-240217-2
• Nazarenko, Sergey (2014), Fluid Dynamics via Examples and Solutions, CRC Press (Taylor & Francis group), ISBN 978-1-43-988882-7

External links

• Free Fluid Mechanics books
• Annual Review of Fluid Mechanics
• CFDWiki – the Computational Fluid Dynamics reference wiki.
• Educational Particle Image Velocimetry – resources and demonstrations