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## Fluid mechanics
## Additional recommended knowledge
## Relationship to continuum mechanicsFluid 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 tangential stress; that is why a fluid in rest has the shape of their containing vessel.And fluid in rest have zero shear stress ## AssumptionsLike 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 Fluid mechanics assumes that every fluid obeys the following: - Conservation of mass
- Conservation of momentum
- The
*continuum hypothesis*, detailed below.
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. Similarly, it can sometimes be assumed that the viscosity of the fluid is zero (the fluid is ## The continuum hypothesisFluids 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. 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. ## Navier-Stokes equations
The 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. 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 less. 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. ## General form of the equationThe general form of the Navier-Stokes equations for the conservation of momentum is: where - ρ is the fluid density,
- is the substantive derivative (also called the material derivative)
- is the velocity vector,
- is the body force vector, and
- 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, is a symmetric tensor. In general, (in three dimensions) has the form: where - σ are normal stresses, and
- τ 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 fluidsA 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). Alternativley, 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. ## Equations for a Newtonian fluid
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 where - τ is the shear stress exerted by the fluid ("drag")
- μ 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 and viscosity is constant across the fluid, the equation governing the shear stress (in Cartesian coordinates) is where - τ
_{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 a fluid does not obey this relation, it is termed a non-Newtonian fluid, of which there are several types. ## See also
- Applied mechanics
- Secondary flow
## References- White, Frank M. (2003).
*Fluid Mechanics.*McGraw-Hill.__ISBN 0072402172__ - Cramer, Mark. "The Gallery of Fluid Mechanics"
Categories: Continuum mechanics | Fluid mechanics |
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This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Fluid_mechanics". A list of authors is available in Wikipedia. |