Understanding Couette Flow requires a basic understanding of Fluid Dynamics which is fundamentally based upon the concept of Viscosity.
The viscosity of a fluid is a measure of its resistance to gradual deformation by shear stress or tensile stress.
For liquids, it corresponds to the informal concept of “thickness”.
For example, honey has a much higher viscosity than water.
Viscosity is a property arising from friction between neighboring particles in a fluid that are moving at different velocities.
When the fluid is forced through a tube, the particles which comprise the fluid generally move faster near the tube’s axis and more slowly near its walls: therefore some stress, (such as a pressure difference between the two ends of the tube), is needed to overcome the friction between particle layers and keep the fluid moving.
For the same velocity pattern, the stress required is proportional to the fluid’s viscosity.
A fluid that has no resistance to shear stress is known as an ideal fluid or inviscid fluid.
Zero viscosity is observed only at very low temperatures, in superfluids.
Otherwise, all fluids have positive viscosity, and are technically said to be viscous or viscid.
In common parlance, however, a liquid is said to be viscous if its viscosity is substantially greater than water’s, and may be described as mobile if the viscosity is noticeably less than water’s.
If the viscosity is very high, for instance in pitch, the fluid will appear to be a solid in the short term.
The dynamic viscosity of a fluid expresses its resistance to shearing flows when adjacent layers move parallel to each other with different speeds.
Laminar shear of fluid between two plates.
Friction between the fluid and the moving boundaries causes the fluid to shear.
The force required for this action is a measure of the fluid’s viscosity.
The SI physical unit of dynamic viscosity is the pascal-second (Pa•s).
The SI physical unit of dynamic viscosity is the pascal-second (Pa•s).
If a fluid with a viscosity of one Pa•s is placed between two plates, and one plate is pushed sideways with a shear stress of one pascal, it moves a distance equal to the thickness of the layer between the plates in one second.
Water at 20 °C has a viscosity of 0.001002 Pa•s, while a typical motor oil could have a viscosity of about 0.250 Pa•s.
The dynamic viscosities of liquids are typically several orders of magnitude higher than dynamic viscosities of gases.
The viscosity of a liquid tends to decrease with increasing temperature.
The viscosity of a gas tends to increase with increasing temperature.
However, the viscosity of a fluid is not the whole story when inertial forces are involved.
When a fluid is flowing [over a surface or around an obstacle] or an object is moving through a fluid then velocity, shape and surface area also become important.
The terminal velocity of an object is the velocity of the object when the sum of the drag force (Fd) and buoyancy equals the downward force of gravity (FG) acting on the object.
Since the net force on the object is zero, the object has zero acceleration.
In fluid dynamics, an object is moving at its terminal velocity if its speed is constant due to the restraining force exerted by the fluid through which it is moving.
As the speed of an object increases, the drag force acting on the object, resultant of the substance (e.g., air or water) it is passing through, increases.
At some speed, the drag or force of resistance will equal the gravitational pull on the object (buoyancy is considered below).
At this point the object ceases to accelerate and continues falling at a constant speed called terminal velocity (also called settling velocity).
An object moving downward with greater than terminal velocity (for example because it was thrown downwards or it fell from a thinner part of the atmosphere or it changed shape) will slow down until it reaches terminal velocity.
Drag depends on the projected area, and this is why objects with a large projected area relative to mass, such as parachutes, have a lower terminal velocity than objects with a small projected area relative to mass, such as bullets.
Based on wind resistance, for example, the terminal velocity of a skydiver in a belly-to-earth (i.e., face down) free-fall position is about 195 km/h (122 mph or 54 m/s).
This velocity is the asymptotic limiting value of the acceleration process, because the effective forces on the body balance each other more and more closely as the terminal velocity is approached.
In this example, a speed of 50% of terminal velocity is reached after only about 3 seconds, while it takes 8 seconds to reach 90%, 15 seconds to reach 99% and so on.
Higher speeds can be attained if the skydiver pulls in his or her limbs (see also freeflying).
In this case, the terminal velocity increases to about 320 km/h (200 mph or 90 m/s), which is almost the terminal velocity of the Peregrine Falcon diving down on its prey.
In Fluid Mechanics “different flow regimes” are characterised by a dimensionless quantity called the Reynolds Number which represents the ratio of inertial forces to viscous forces.
In fluid mechanics, the Reynolds number (Re) is a dimensionless quantity that is used to help predict similar flow patterns in different fluid flow situations.
The concept was introduced by George Gabriel Stokes in 1851, but the Reynolds number is named after Osborne Reynolds (1842–1912), who popularized its use in 1883.
The Reynolds number is defined as the ratio of inertial forces to viscous forces and consequently quantifies the relative importance of these two types of forces for given flow conditions.
Reynolds numbers frequently arise when performing scaling of fluid dynamics problems, and as such can be used to determine dynamic similitude between two different cases of fluid flow.
They are also used to characterize different flow regimes within a similar fluid, such as laminar or turbulent flow:
laminar flow occurs at low Reynolds numbers, where viscous forces are dominant, and is characterized by smooth, constant fluid motion;
turbulent flow occurs at high Reynolds numbers and is dominated by inertial forces, which tend to produce chaotic eddies, vortices and other flow instabilities.
In practice, matching the Reynolds number is not on its own sufficient to guarantee similitude.
Fluid flow is generally chaotic, and very small changes to shape and surface roughness can result in very different flows.
Nevertheless, Reynolds numbers are a very important guide and are widely used.
The Reynolds Number can be calculated using customised formulae for several different situations where a fluid is in relative motion to a surface e.g. flow in pipe, flow in a wide duct, flow in an open channel and flow around airfoils.
The Reynolds Number for an object in a fluid [called the Particle Reynolds Number and often denoted as Rep] is important when considering the nature of the surrounding flow, whether or not vortex shedding will occur, and its fall velocity.
However, before digging any deeper into Fluid Dynamics it is necessary to understand that Fluid Dynamics is not a bastion of Settled Science because the science encompasses: natural complexity, exceptions, approximations, special cases and controversy.
The viscosity of some fluids may depend on other factors.
A magnetorheological fluid, for example, becomes thicker when subjected to a magnetic field, possibly to the point of behaving like a solid.
Newton’s law of viscosity is a constitutive equation (like Hooke’s law, Fick’s law, Ohm’s law): it is not a fundamental law of nature but an approximation that holds in some materials and fails in others.
A fluid that behaves according to Newton’s law, with a viscosity μ that is independent of the stress, is said to be Newtonian. Gases, water and many common liquids can be considered Newtonian in ordinary conditions and contexts. There are many non-Newtonian fluids that significantly deviate from that law in some way or other.
some authors have claimed that amorphous solids, such as glass and many polymers, are actually liquids with a very high viscosity (e.g.~greater than 1012 Pa•s).
Although these expressions are each exact, in order to calculate the viscosity of a dense fluid using these relations currently requires the use of molecular dynamics computer simulations.
Valid for temperatures between 0 < T < 555 K with an error due to pressure less than 10% below 3.45 MPa.
The Chapman-Enskog equation may be used to estimate viscosity for a dilute gas
The viscosity of the blend of two or more liquids can be estimated using the Refutas equation.
In the study of turbulence in fluids, a common practical strategy for calculation is to ignore the small-scale vortices (or eddies) in the motion and to calculate a large-scale motion with an eddy viscosity that characterizes the transport and dissipation of energy in the smaller-scale flow (see large eddy simulation).
In liquids, the additional forces between molecules become important.
This leads to an additional contribution to the shear stress though the exact mechanics of this are still controversial.
For fluids of variable density such as compressible gases or fluids of variable viscosity such as non-Newtonian fluids, special rules apply.
The velocity may also be a matter of convention in some circumstances, notably stirred vessels.
In practice, matching the Reynolds number is not on its own sufficient to guarantee similitude. Fluid flow is generally chaotic, and very small changes to shape and surface roughness can result in very different flows. Nevertheless, Reynolds numbers are a very important guide and are widely used.
The development of Fluid Dynamics has [in many cases] been driven by engineers looking for practical solutions to real world problems which [in some cases] can still only be resolved via modelling and experimentation.
Fluid dynamics offers a systematic structure – which underlies these practical disciplines – that embraces empirical and semi-empirical laws derived from flow measurement and used to solve practical problems.
The solution to a fluid dynamics problem typically involves calculating various properties of the fluid, such as velocity, pressure, density, and temperature, as functions of space and time.
It is believed that turbulent flows can be described well through the use of the Navier–Stokes equations. Direct numerical simulation (DNS), based on the Navier–Stokes equations, makes it possible to simulate turbulent flows at moderate Reynolds numbers. Restrictions depend on the power of the computer used and the efficiency of the solution algorithm. The results of DNS have been found to agree well with experimental data for some flows.
Most flows of interest have Reynolds numbers much too high for DNS to be a viable option, given the state of computational power for the next few decades.
Any flight vehicle large enough to carry a human (L > 3 m), moving faster than 72 km/h (20 m/s) is well beyond the limit of DNS simulation (Re = 4 million).
Transport aircraft wings (such as on an Airbus A300 or Boeing 747) have Reynolds numbers of 40 million (based on the wing chord).
In order to solve these real-life flow problems, turbulence models will be a necessity for the foreseeable future.