Fluid mechanics
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Fluid mechanics is the branch of physics concerned with the mechanics of fluids (liquids, gases, and plasmas) and the forces on them.[1]: 3 It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical, and biomedical engineering, as well as geophysics, oceanography, meteorology, astrophysics, and biology.
It can be divided into
Fluid mechanics, especially fluid dynamics, is an active field of research, typically mathematically complex. Many problems are partly or wholly unsolved and are best addressed by
History
The study of fluid mechanics goes back at least to the days of
Inviscid flow was further analyzed by various mathematicians (
Main branches
Fluid statics
Fluid dynamics
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. |
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 fluid 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
The assumptions inherent to a fluid mechanical treatment of a physical system can be expressed in terms of mathematical equations. Fundamentally, every fluid mechanical system is assumed to obey:
- Conservation of mass
- Conservation of energy
- Conservation of momentum
- The continuum assumption
For example, the assumption that mass is conserved means that for any fixed
The continuum assumption is an idealization of
The Navier–Stokes equations (named after
- .
These differential equations are the analogues for deformable materials to Newton's equations of motion for particles – the Navier–Stokes equations describe changes in momentum (force) in response to pressure and viscosity, parameterized by the
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 cases, especially those 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.[17][18][19][20][21]
Inviscid and viscous fluids
An inviscid fluid has no viscosity, . In practice, an inviscid flow is an
For fluid flow over a porous boundary, the fluid velocity can be discontinuous between the free fluid and the fluid in the porous media (this is related to the Beavers and Joseph condition). Further, it is useful at low
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 gasses, behave—to good approximation—as a Newtonian fluid under normal conditions on Earth.[11]: 145
By contrast, stirring a
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 behavior is
where
- is the shear stress exerted by the fluid ("drag"),
- is the fluid viscosity—a constant of proportionality, and
- is the velocity gradient perpendicular to the direction of shear.
For a Newtonian fluid, the viscosity, by definition, depends only on
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 (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 neglected, 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
- Transport phenomena
- Aerodynamics
- Applied mechanics
- Bernoulli's principle
- Communicating vessels
- Computational fluid dynamics
- Compressor map
- Secondary flow
- Different types of boundary conditions in fluid dynamics
References
- ^ ISBN 978-0-07-352934-9.
- ISBN 978-0080982434.
- ^ Mariam Rozhanskaya and I. S. Levinova (1996), "Statics", p. 642,
- ^ Batchelor, C. K., & Batchelor, G. K. (2000). An introduction to fluid dynamics. Cambridge University Press.
- ^ Bertin, J. J., & Smith, M. L. (1998). Aerodynamics for engineers (Vol. 5). Upper Saddle River, NJ: Prentice Hall.
- ^ Anderson Jr, J. D. (2010). Fundamentals of aerodynamics. Tata McGraw-Hill Education.
- ^ Houghton, E. L., & Carpenter, P. W. (2003). Aerodynamics for engineering students. Elsevier.
- ^ Milne-Thomson, L. M. (1973). Theoretical aerodynamics. Courier Corporation.
- ^ Milne-Thomson, L. M. (1996). Theoretical hydrodynamics. Courier Corporation.
- ^ Birkhoff, G. (2015). Hydrodynamics. Princeton University Press.
- ^ ISBN 0-521-66396-2.
- ISBN 978-1-4822-9297-8.
- ^ Constantin, P., & Foias, C. (1988). Navier-stokes equations. University of Chicago Press.
- ^ Temam, R. (2001). Navier-Stokes equations: theory and numerical analysis (Vol. 343). American Mathematical Society.
- ^ Foias, C., Manley, O., Rosa, R., & Temam, R. (2001). Navier-Stokes equations and turbulence (Vol. 83). Cambridge University Press.
- ^ Girault, V., & Raviart, P. A. (2012). Finite element methods for Navier-Stokes equations: theory and algorithms (Vol. 5). Springer Science & Business Media.
- ^ Anderson, J. D., & Wendt, J. (1995). Computational fluid dynamics (Vol. 206). New York: McGraw-Hill.
- ^ Chung, T. J. (2010). Computational fluid dynamics. Cambridge University Press.
- ^ Blazek, J. (2015). Computational fluid dynamics: principles and applications. Butterworth-Heinemann.
- ^ Wesseling, P. (2009). Principles of computational fluid dynamics (Vol. 29). Springer Science & Business Media.
- ^ Anderson, D., Tannehill, J. C., & Pletcher, R. H. (2016). Computational fluid mechanics and heat transfer. Taylor & Francis.
- ISBN 978-0124059351.
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, ISBN 0-07-015000-1
- Massey, B.; Ward-Smith, J. (2005), Mechanics of Fluids (8th ed.), Taylor & Francis, ISBN 978-0-415-36206-1
- 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. Archived 2009-01-19 at the Wayback Machine.
- CFDWiki – the Computational Fluid Dynamics reference wiki.
- Educational Particle Image Velocimetry – resources and demonstrations