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Encyclopedia > Fluid mechanics
Continuum mechanics Conservation of mass
Conservation of momentum
Navier–Stokes equations
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## Relationship to continuum mechanics GA_googleFillSlot("encyclopedia_square");

Fluid mechanics is a subdiscipline of continuum mechanics, as illustrated in the following table. Continuum mechanics is a branch of physics (specifically mechanics) that deals with continuous matter, including both solids and fluids (i. ...

 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: which describes materials that return to their rest shape after an applied stress. Plasticity: which describes materials that permanently deform after a large enough applied stress. Rheology: the study of materials with both solid and fluid characteristics Fluid mechanics: the study of the physics of continuous materials which take the shape of their container. Non-Newtonian fluids Newtonian fluids

In a mechanical view, a fluid is a substance that does not support tangential stress; that is why a fluid at rest has the shape of its containing vessel. A fluid at rest has no shear stress. Continuum mechanics is a branch of physics (specifically mechanics) that deals with continuous matter, including both solids and fluids (i. ... Solid mechanics is the branch of physics and mathematics that concern the behavior of solid matter under external actions (e. ... Elasticity is a branch of physics which studies the properties of elastic materials. ... Stress is a measure of force per unit area within a body. ... For other uses, see Plasticity. ... Rheology is the study of the deformation and flow of matter under the influence of an applied stress. ... A non-Newtonian fluid is a fluid in which the viscosity changes with the applied strain rate. ... A Newtonian fluid (named for Isaac Newton) is a fluid that flows like waterâ€”its shear stress is linearly proportional to the velocity gradient in the direction perpendicular to the plane of shear. ...

## 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 hold true. For example, consider an incompressible fluid in three dimensions. The assumption that mass is conserved means that for any fixed closed surface (such as a sphere) the rate of mass passing from outside to inside the surface must be the same as rate of mass passing the other way. (Alternatively, the mass inside remains constant, as does the mass outside). This can be turned into an integral equation over the surface. In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign. ...

Fluid mechanics assumes that every fluid obeys the following:

Further, it is often useful (and realistic) to assume a fluid is incompressible - that is, the density of the fluid does not change. Liquids can often be modelled as incompressible fluids, whereas gases cannot. The law of conservation of mass/matter, also known as law of mass/matter conservation (or the Lomonosov-Lavoisier law), states that the mass of a closed system of substances will remain constant, regardless of the processes acting inside the system. ... In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves. ... In fluid mechanics, an incompressible fluid is a fluid whose density (often represented by the Greek letter ρ) is constant: it is the same throughout the field and it does not change through time. ...

Similarly, it can sometimes be assumed that the viscosity of the fluid is zero (the fluid is inviscid). Gases can often be assumed to be 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). For other uses, see Viscosity (disambiguation). ... PIPE can refer to PIPE (explosive) PIPE Networks Private Investment in Public Equity (PIPE) Physical Interface for PCI Express (PIPE) For other meanings, see also pipe. ... In fluid dynamics, the no-slip condition states that fluids stick to surfaces past which they flow. ...

### The 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. In science, a molecule is the smallest particle of a pure chemical substance that still retains its chemical composition and properties. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ...

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. (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 unity are best evaluated using statistical mechanics for reliable solutions. Statistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... The Knudsen number (Kn) is the ratio of the molecular mean free path length to a representative physical length scale. ... For sound waves in an enclosure, the mean free path is the average distance the wave travels between reflections off of the enclosures walls. ... The concept of scale is applicable if a system is represented proportionally by another system. ... One redirects here. ...

## Navier-Stokes equations

Main article: Navier-Stokes equations The Navier-Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, are a set of equations which describe the motion of fluid substances such as liquids and gases. ...

The Navier-Stokes equations are differential equations which describe the motion of a fluid. Such equations establish relations among the rates of change 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. In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ...

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 other uses, see Calculus (disambiguation). ... In fluid mechanics, the Reynolds number may be described as the ratio of inertial forces (vsÏ) to viscous forces (Î¼/L) and, consequently, it quantifies the relative importance of these two types of forces for given flow conditions. ...

For more complex situations, such as global weather systems like El Niño or lift in a wing, 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. A computer simulation of high velocity air flow around the Space Shuttle during re-entry. ...

### General form of the equation

The general form of the Navier-Stokes equations for the conservation of momentum is: $rhofrac{Dmathbf{v}}{D t} = nablacdotmathbb{P} + rhomathbf{f}$

where

• $rho$ is the fluid density,
• $frac{D}{D t}$ is the substantive derivative (also called the material derivative),
• $mathbf{v}$ is the velocity vector,
• $mathbf{f}$ is the body force vector, and
• $mathbb{P}$ is a tensor that represents the surface forces applied on a fluid particle (the comoving stress tensor).

Unless the fluid is made up of spinning degrees of freedom like vortices, $mathbb{P}$ is a symmetric tensor. In general, (in three dimensions) $mathbb{P}$ has the form: In mathematics and continuum mechanics, including fluid dynamics, the substantive derivative (sometimes the Lagrangian derivative, material derivative or advective derivative), written , is the rate of change of some property of a small parcel of fluid. ... In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ... This article is in need of attention from an expert on the subject. ... $mathbb{P} = begin{pmatrix} sigma_{xx} & tau_{xy} & tau_{xz} tau_{yx} & sigma_{yy} & tau_{yz} tau_{zx} & tau_{zy} & sigma_{zz} end{pmatrix}$

where

• $sigma$ are normal stresses, and
• $tau$ are tangential stresses (shear stresses).

The above is actually a set of three equations, one per dimension. By themselves, these aren't sufficient to produce a solution. However, adding conservation of mass and appropriate boundary conditions to the system of equations produces a solvable set of equations.

## Newtonian vs. 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 through the fluid is proportional to the force applied to the object. (Compare friction). Sir Isaac Newton FRS (4 January 1643 â€“ 31 March 1727) [ OS: 25 December 1642 â€“ 20 March 1727] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist. ... This box:      A fluid is defined as a substance that continually deforms (flows) under an applied shear stress regardless of how small the applied stress. ... Shear stress is a stress state where the stress is parallel or tangential to a face of the material, as opposed to normal stress when the stress is perpendicular to the face. ... This article is about velocity in physics. ... For other uses, see Gradient (disambiguation). ... Fig. ... An object moving through a gas or liquid experiences a force in direction opposite to its motion. ... For other uses, see Friction (disambiguation). ...

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. A non-Newtonian fluid is a fluid in which the viscosity changes with the applied strain rate. ... Oobleck was originally popularized as a fictional form of green precipitation described by Dr. Seuss in Bartholomew and the Oobleck. ... For other uses, see Sand (disambiguation). ... For other uses, see Paint (disambiguation). ...

### Equations for a Newtonian fluid

Main article: Newtonian fluid A Newtonian fluid (named for Isaac Newton) is a fluid that flows like waterâ€”its shear stress is linearly proportional to the velocity gradient in the direction perpendicular to the plane of shear. ...

The constant of proportionality between the shear stress and the velocity gradient is known as the viscosity. A simple equation to describe Newtonian fluid behaviour is For other uses, see Viscosity (disambiguation). ... $tau=-mufrac{dv}{dx}$

where

τ is the shear stress exerted by the fluid ("drag")
μ is the fluid viscosity - a constant of proportionality $frac{dv}{dx}$ 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 and viscosity is constant across the fluid, the equation governing the shear stress (in Cartesian coordinates) is An object moving through a gas or liquid experiences a force in direction opposite to its motion. ... For other uses, see Temperature (disambiguation). ... This article is about pressure in the physical sciences. ... In fluid mechanics, an incompressible fluid is a fluid whose density (often represented by the Greek letter ρ) is constant: it is the same throughout the field and it does not change through time. ... Fig. ... $tau_{ij}=muleft(frac{partial v_i}{partial x_j}+frac{partial v_j}{partial x_i} right)$

where

τij is the shear stress on the ith face of a fluid element in the jth direction
vi is the velocity in the ith direction
xj is the jth direction coordinate

If a fluid does not obey this relation, it is termed a non-Newtonian fluid, of which there are several types. A non-Newtonian fluid is a fluid in which the viscosity changes with the applied strain rate. ... Wikibooks has more on the topic of
Fluid mechanics

Image File history File links Portal. ... Image File history File links Wikibooks-logo-en. ... Wikibooks logo Wikibooks, previously called Wikimedia Free Textbook Project and Wikimedia-Textbooks, is a wiki for the creation of books. ... Applied mechanics, also known as theoretical and applied mechanics, is a branch of the physical sciences and the practical application of mechanics. ... In fluid dynamics, a secondary flow is a relatively minor flow superimposed on the primary flow, where the primary flow usually matches very closely the flow pattern predicted using simple analytical techniques and assuming the fluid is inviscid. ... Bernoullis Principle states that for an ideal fluid (low speed air is a good approximation), with no work being performed on the fluid, an increase in velocity occurs simultaneously with decrease in pressure or a change in the fluids gravitational potential energy. ... Results from FactBites:

 Fluid mechanics - definition of Fluid mechanics in Encyclopedia (302 words) Fluid mechanics is the study of the macroscopic physical behaviour of fluids. Fluids are specifically liquids and gases though some other materials and systems can be described in a similar way. Fluid mechanics offers a mathematical structure that underlies these practical discipines which often also embrace empirical and semi-empirical laws, derived from flow measurement, to solve practical problems.
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