Inviscid flow

Inviscid flow is the flow of an inviscid fluid, in which the viscosity of the fluid is equal to zero. The Reynolds number of inviscid flow approaches infinity, as the viscosity approaches zero. Though there are limited examples of invisicd fluids, known as superfluids, inviscid flow has many applications in fluid dynamics. When viscous forces are neglected, such as the case of inviscid flow, the Navier-Stokes equation can be simplified to a form known as the Euler equation. This simplified equation is applicable to inviscid flow as well as flow with low viscosity and a Reynolds number much greater than one. Using the Euler equation, many fluid dynamics problems involving low viscosity are easily solved, however, the assumed negligible viscosity is no longer valid in the region of fluid near a solid boundary.

Reynolds number

The Reynolds number (Re) is a dimensionless group, having no distinct units, that is commonly used in fluid dynamics and engineering. Originally described by George Gabriel Stokes in 1850, it became popularized by Osborne Reynolds, who the concept was specifically named after in 1908 by Arnold Sommerfeld. The Reynolds number is calculated as:

$Re={l_{c}v\rho \over \mu }$

Symbol	Description	Units
		SI	Standard
$l_{c}$	characteristic length	m	ft
$v$	fluid velocity	m/s	ft/s
$\rho$	fluid density	kg/m^3	slug/ft^3
$\mu$	fluid viscosity	Pa*s	lb*s/ft^2

Examining the dimensions of the individual parameters, it becomes apparent that the Reynolds number is unit-less. The value represents the ratio of inertial forces to viscous forces in a fluid, and is useful in determining the relative importance of viscosity. In inviscid flow, since the viscous forces are zero, the Reynolds number approaches infinity. When viscous forces are negligible, the Reynolds number is much greater than one. In such cases (Re>>1), assuming inviscid flow can be useful in simplifying many fluid dynamics problems.

Euler equations

Inviscid flow around a wing

In a 1757 publication, Leonhard Euler described a set of equations governing inviscid flow:

$\rho {Dv \over Dt}=-\nabla p+\rho g$

Symbol	Description	Units
		SI	Standard
${D \over Dt}$	material derivative
$\nabla$	del operator
$p$	pressure	Pa	psi
$g$	is the acceleration due to gravity	m/s^2	ft/s^2

Assuming inviscid flow, or negligible viscosity, allows the use of the Euler equation, and can apply to many examples of flow in which viscous forces are insignificant. Some examples include flow around an airplane wing, upstream flow around bridge supports in a river, and ocean currents.

Navier Stokes equations

In 1845, George Gabriel Stokes published another important set of equations, today known as the Navier-Stokes equations. Claude-Louis Navier developed the equations first using molecular theory, which was further confirmed by Stokes using continuum theory. The Navier Stokes equations describe the motion of fluids:

$\rho {Dv \over Dt}=-\nabla p+\mu \nabla ^{2}v+\rho g$

When the flow is inviscid, or the viscosity can be assumed to be negligible, the Navier Stokes equation simplifies to the Euler equation: This simplification is much easier to solve, and can apply to many types of flow in which viscosity is negligible. Some examples include flow around an airplane wing, upstream flow around bridge supports in a river, and ocean currents.

Flow developing over a solid surface

Solid boundaries

It is important to note, that negligible viscosity can no longer be assumed near solid boundaries, such as the case of the airplane wing. In turbulent flow regimes (Re >> 1), viscosity can typically be neglected, however this is only valid at distances far from solid interfaces. When considering flow in the vicinity of a solid surface, such as flow through a pipe or around a wing, it is convenient to categorize four distinct regions of flow near the surface:

• Main turbulent stream: Furthest from the surface, viscosity can be neglected.
• Inertial sub-layer: The start of the main turbulent stream, viscosity has only minor importance.
• Buffer layer: The transformation between inertial and viscous layers.
• Viscous sub-layer: Closest to the surface, here viscosity is important.

Although these distinctions can be a useful tool in illustrating the significance of viscous forces near solid interfaces, it is important to note that these regions are fairly arbitrary. Assuming inviscid flow can be a useful tool in solving many fluid dynamics problems, however, this assumption requires careful consideration of the fluid sub layers when solid boundaries are involved.

Superfluids

Superfluid Helium

To date, Helium is the only fluid to exhibit superfluidity, that has been discovered. Helium becomes a superfluid once it is cooled to below 2.2k, this point is known as the lambda point. Temperatures above the lambda point, Helium exists as a liquid exhibiting normal fluid dynamic behavior. Once it is cooled to below 2.2k it begins to exhibit quantum behavior. For example, at the lambda point there is a sharp increase in heat capacity, as it is continued to be cooled, the heat capacity begins to decrease with temperature. In addition, the thermal conductivity is very large, contributing to the excellent coolant properties of superfluid Helium.

Applications

Spectrometers are kept at a very low temperature using Helium as the coolant. This allows for minimal background flux in far-infrared readings. Some of the designs for the spectrometers may be simple, but even the frame is at its warmest less than 20 Kelvin. These devices are not commonly used as it is very expensive to use superfluid helium over other coolants.

Superfluid Helium has a very high thermal conductivity which makes it very useful for cooling superconductors. Superconductors such as the ones used at the LHC (Large Hadron Collidor) are cooled to temperatures approximately 1.9 Kelvin. This temperature allows the niobium-titanium magnets to reach a superconductor state. Without the use of the superfluid Helium this temperature would not be possible. Cooling to these temperatures, with this fluid, is a very expensive system and there are few compared to other cooling systems.

Another application of the superfluid Helium is its uses in understanding quantum mechanics. Using lasers to look at small droplets allows scientists to view behavior’s that may not normally be viewable. This is due to all the helium in each droplet being at the same quantum state. This application does not have any practical uses by itself, but it gives a look into better understand quantum mechanics which will have later applications.

See also

• Viscosity
• Fluid dynamics
• Potential flow, a special case of inviscid flow
• Stokes flow, in which the viscous forces are much greater than inertial forces.
• Couette flow