Fluid mechanics is the branch of physics that studies fluids and the forces on them. Fluid
mechanics can be divided into fluid statics, the study of fluids at rest; fluid kinematics,
the study of fluids in motion; 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 a microscopic viewpoint. Fluid mechanics, especially fluid dynamics,
is an active field of research with many unsolved or partly solved problems. Fluid mechanics
can be mathematically complex, and can best be solved by numerical methods, typically
using computers. A modern discipline, called computational fluid dynamics, 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, Evangelista Torricelli, Isaac Newton and Blaise Pascal,
and was continued by Daniel Bernoulli with the introduction of mathematical fluid dynamics
in Hydrodynamica. Inviscid flow was further analyzed by various
mathematicians 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, while various scientists such as Osborne Reynolds, Andrey
Kolmogorov, and Geoffrey Ingram Taylor advanced the understanding of fluid viscosity and turbulence.
Relationship to continuum mechanics Fluid mechanics is a subdiscipline of continuum
mechanics, as illustrated in the following table.
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
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 – 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.. This can be turned into an equation
in integral form over the control volume. 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 often useful to assume a fluid
is incompressible – that is, the density of the fluid does not change.
Similarly, it can sometimes be assumed that the viscosity of the fluid is zero. Gases
can often be assumed to be inviscid. If a fluid is viscous, and its flow contained in
some way, 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. 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, 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. That said, 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.. Problems with
Knudsen numbers at or above one are best evaluated using statistical mechanics for reliable solutions.
Navier–Stokes equations
The Navier–Stokes equations are the set of equations that describe the motion of fluid
substances such as liquids and gases. These equations state that changes in momentum of
fluid particles depend only on the external pressure and internal viscous forces 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 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 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 is a field of sciences by its own called
computational fluid dynamics. General form of the equation
The general form of the Navier–Stokes equations for the conservation of momentum is:
where is the fluid density,
is the substantive derivative, is the velocity vector,
is the specific body force vector, and is a tensor that represents the surface forces
applied on a fluid particle. Unless the fluid is made up of spinning degrees
of freedom like vortices, is a symmetric tensor. Usually the stress tensor is decomposed as
where is a static isotropic stress state, and is the desviatoric stress tensor, corresponding
to the part of the stress due to the fluid motion. Generally, the scalar can be taken
as the thermodynamic pressure, whereas is called the viscous stress tensor. Furthermore,
the diagonal components of tensor are called normal stresses and the off-diagonal components
are called shear stresses. The vectorial Navier–Stokes equation above
can be written then as
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 fluid
velocity:
On the other hand, the identification of with the thermodynamic pressure is usually possible.
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 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.. Important fluids,
like water as well as most gases, behave — to good approximation — as a Newtonian fluid
under normal conditions on Earth. 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. Alternatively, stirring a non-Newtonian
fluid can cause the viscosity to decrease, so the fluid appears "thinner". 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. 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
where is the shear stress exerted by the fluid
is the fluid viscosity – a constant of proportionality 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 is
where is the shear stress on the face of a fluid
element in the direction is the velocity in the direction
is the direction coordinate. If the fluid is not incompressible the general
form for the viscous stress in a Newtonian fluid is
where is the second viscosity coefficient. 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, viscoelatic.
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 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. When the viscosity is negleted, the term containing the viscous
stress tensor 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
References Batchelor, George K., An Introduction to Fluid
Dynamics, Cambridge University Press, ISBN 0-521-66396-2 Falkovich, Gregory, Fluid Mechanics, Cambridge
University Press, ISBN 978-1-107-00575-4 Kundu, Pijush K.; Cohen, Ira
M., Fluid Mechanics, Academic Press, ISBN 978-0-12-373735-9 Currie, I. G., Fundamental Mechanics of Fluids,
McGraw-Hill, Inc., ISBN 0-07-015000-1 Massey, B.; Ward-Smith, J., Mechanics of Fluids,
Taylor & Francis, ISBN 978-0-415-36206-1 White, Frank M., Fluid Mechanics, McGraw–Hill,
ISBN 0-07-240217-2 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