In fluid mechanics, a fluid is said to be in hydrostatic equilibrium or hydrostatic balance when it is at rest, or when the flow velocity at each point is constant over time. This occurs when external forces such as gravity are balanced by a pressure-gradient force. For instance, the pressure-gradient force prevents gravity from collapsing Earth's atmosphere into a thin, dense shell, whereas gravity prevents the pressure gradient force from diffusing the atmosphere into space.
The Grashof number (Gr) is a dimensionless number in fluid dynamics and heat transfer which approximates the ratio of the buoyancy to viscous force acting on a fluid. It frequently arises in the study of situations involving natural convection and is analogous to the Reynolds number. It's believed to be named after Franz Grashof. Though this grouping of terms had already been in use, it wasn't named until around 1921, 28 years after Franz Grashof's death. It's not very clear why the grouping was named after him.
In 1851, George Gabriel Stokes derived an expression, now known as Stokes' law, for the frictional force – also called drag force – exerted on spherical objects with very small Reynolds numbers in a viscous fluid. Stokes' law is derived by solving the Stokes flow limit for small Reynolds numbers of the Navier–Stokes equations.
In viscous fluid dynamics, the Archimedes number (Ar), named after the ancient Greek scientist Archimedes is used to determine the motion of fluids due to density differences. It is a dimensionless number, the ratio of gravitational forces to viscous forces and has the form:
Darcy's law is an equation that describes the flow of a fluid through a porous medium. The law was formulated by Henry Darcy based on results of experiments on the flow of water through beds of sand, forming the basis of hydrogeology, a branch of earth sciences.
Stokes flow, also named creeping flow or creeping motion, is a type of fluid flow where advective inertial forces are small compared with viscous forces. The Reynolds number is low, i.e. . This is a typical situation in flows where the fluid velocities are very slow, the viscosities are very large, or the length-scales of the flow are very small. Creeping flow was first studied to understand lubrication. In nature this type of flow occurs in the swimming of microorganisms and sperm and the flow of lava. In technology, it occurs in paint, MEMS devices, and in the flow of viscous polymers generally.
In physics, a perfect fluid is a fluid that can be completely characterized by its rest frame mass density and isotropic pressure p.
In physics the Einstein relation is a previously unexpected connection revealed independently by William Sutherland in 1904, Albert Einstein in 1905, and by Marian Smoluchowski in 1906 in their papers on Brownian motion. The more general form of the equation is
The Ergun equation, derived by the Turkish chemical engineer Sabri Ergun in 1952, expresses the friction factor in a packed column as a function of the modified Reynolds number.
In fluid dynamics, the Morton number (Mo) is a dimensionless number used together with the Eötvös number or Bond number to characterize the shape of bubbles or drops moving in a surrounding fluid or continuous phase, c. It is named after Rose Morton, who described it with W. L. Haberman in 1953.
The intent of this article is to highlight the important points of the derivation of the Navier–Stokes equations as well as its application and formulation for different families of fluids.
Volume viscosity becomes important only for such effects where fluid compressibility is essential. Volume viscosity is mainly related to the vibrational energy of the molecules. It is zero for monatomic gases at low density, but can be large for fluids with larger molecules. The volume viscosity is important in describing sound attenuation, and the absorption of sound energy into the fluid depends on the sound frequency i.e. the rate of fluid compression and expansion. Volume viscosity is also important in describing the fluid dynamics of liquids containing gas bubbles. For an incompressible liquid the volume viscosity is superfluous, and does not appear in the equation of motion.
In nonideal fluid dynamics, the Hagen–Poiseuille equation, also known as the Hagen–Poiseuille law, Poiseuille law or Poiseuille equation, is a physical law that gives the pressure drop in an incompressible and Newtonian fluid in laminar flow flowing through a long cylindrical pipe of constant cross section. It can be successfully applied to air flow in lung alveoli, or the flow through a drinking straw or through a hypodermic needle. It was experimentally derived independently by Jean Léonard Marie Poiseuille in 1838 and Gotthilf Heinrich Ludwig Hagen, and published by Poiseuille in 1840–41 and 1846.
In fluid dynamics, hydrodynamic stability is the field which analyses the stability and the onset of instability of fluid flows. The study of hydrodynamic stability aims to find out if a given flow is stable or unstable, and if so, how these instabilities will cause the development of turbulence. The foundations of hydrodynamic stability, both theoretical and experimental, were laid most notably by Helmholtz, Kelvin, Rayleigh and Reynolds during the nineteenth century. These foundations have given many useful tools to study hydrodynamic stability. These include Reynolds number, the Euler equations, and the Navier–Stokes equations. When studying flow stability it is useful to understand more simplistic systems, e.g. incompressible and inviscid fluids which can then be developed further onto more complex flows. Since the 1980s, more computational methods are being used to model and analyse the more complex flows.
The Reynolds number is an important dimensionless quantity in fluid mechanics used to help predict flow patterns in different fluid flow situations. At low Reynolds numbers, flows tend to be dominated by laminar (sheet-like) flow, while at high Reynolds numbers turbulence results from differences in the fluid's speed and direction, which may sometimes intersect or even move counter to the overall direction of the flow. These eddy currents begin to churn the flow, using up energy in the process, which for liquids increases the chances of cavitation. The Reynolds number has wide applications, ranging from liquid flow in a pipe to the passage of air over an aircraft wing. It is used to predict the transition from laminar to turbulent flow, and is used in the scaling of similar but different-sized flow situations, such as between an aircraft model in a wind tunnel and the full size version. The predictions of the onset of turbulence and the ability to calculate scaling effects can be used to help predict fluid behaviour on a larger scale, such as in local or global air or water movement and thereby the associated meteorological and climatological effects.
In fluid mechanics, dynamic similarity is the phenomenon that when there are two geometrically similar vessels with the same boundary conditions and the same Reynolds and Womersley numbers, then the fluid flows will be identical. This can be seen from inspection of the underlying Navier-Stokes equation, with geometrically similar bodies, equal Reynolds and Womersley Numbers the functions of velocity (u’,v’,w’) and pressure (P’) for any variation of flow.
In fluid mechanics, the Rayleigh–Plesset equation or Besant–Rayleigh–Plesset equation is an ordinary differential equation which governs the dynamics of a spherical bubble in an infinite body of incompressible fluid. Its general form is usually written as
The Reynolds Equation is a partial differential equation governing the pressure distribution of thin viscous fluid films in Lubrication theory. It should not be confused with Osborne Reynolds' other namesakes, Reynolds number and Reynolds-averaged Navier–Stokes equations. It was first derived by Osborne Reynolds in 1886. The classical Reynolds Equation can be used to describe the pressure distribution in nearly any type of fluid film bearing; a bearing type in which the bounding bodies are fully separated by a thin layer of liquid or gas.
