What is Fluid Mechanics?

The area of physics known as fluid mechanics is focused on the forces acting on and within fluids (liquids, gases, and plasmas).

It has uses in many fields, including biology, geophysics, oceanography, meteorology, astrophysics, and mechanical, aeronautical, civil, chemical, and biomedical engineering.

Fluid dynamics is the study of how forces affect fluid motion, and fluid statics is the study of fluids at rest.?

It is a subfield of continuum mechanics, a field that models matter from a macroscopic perspective rather than a microscopic one and does so without employing the knowledge that matter is composed of atoms.

Research in fluid mechanics, especially fluid dynamics, is lively and frequently involves challenging maths.

Numerous issues are partially or completely unsolvable and are best handled by numerical approaches, which are often carried out on computers.

This method is covered in a contemporary discipline called computational fluid dynamics (CFD).

The highly visual character of fluid flow is also used to advantage by particle image velocimetry, an experimental technique for visualizing and assessing fluid flow.

History

It is generally agreed that Archimedes' work On Floating Bodies, which is regarded as the first significant work on fluid mechanics, is where the study of fluid mechanics first began.

Archimedes investigated fluid statics and buoyancy and developed his famous law, now known as the Archimedes' principle.

With the observations and experiments of Leonardo da Vinci, Evangelista Torricelli, Isaac Newton, Blaise Pascal, and Blaise Pascal's formulation of Pascal's law in Hydrodynamica (1739), fluid mechanics advanced rapidly.

Daniel Bernoulli continued this trend by introducing mathematical fluid dynamics.

Numerous mathematicians, such as Jean le Rond d'Alembert, Joseph Louis Lagrange, Pierre-Simon Laplace, and Siméon Denis Poisson, went on to further analyse inviscid flow, while numerous engineers, such as Jean Léonard Marie Poiseuille and Gotthilf Hagen, investigated viscous flow.

The Navier-Stokes equations were further mathematically justified by Claude-Louis Navier and George Gabriel Stokes, boundary layers were studied by Ludwig Prandtl and Theodore von Kármán, and the understanding of fluid viscosity and turbulence was advanced by several scientists including Osborne Reynolds, Andrey Kolmogorov, and Geoffrey Ingram Taylor.

Main branches

Fluids Statics

The area of fluid mechanics that analyses fluids at rest is known as fluid statics or hydrostatics. In contrast to fluid dynamics, which is the study of fluids in motion, it encompasses the study of the circumstances in which fluids are at rest in stable equilibrium.

Many everyday occurrences can be physically explained by hydrostatics, including why air pressure varies with altitude, why wood and oil float, and why water's surface is always level regardless of the shape of its container.

Hydraulics, the engineering of machinery for storing, moving, and utilizing fluids, depends on hydrostatics. It is also significant to many other subjects, including meteorology, medicine (in the context of blood pressure), and some parts of geophysics and astronomy (for instance, in understanding plate tectonics and anomalies in the Earth's gravitational field).

Fluid Dynamics

Fluid flow, or the study of liquids and gases in motion, is the subject of the branch of fluid mechanics known as fluid dynamics. Underpinning these applied sciences, fluid dynamics provides a systematic framework that encompasses empirical and semi-empirical laws generated from flow measurement and applied to practical issues.

A fluid dynamics problem is often solved by computing the fluid's many parameters as functions of space and time, including velocity, pressure, density, and temperature. Its own subdisciplines include aerodynamics, which is the study of gases in motion like air and water in motion, and hydrodynamics, which is the study of liquids in motion.

Many different things can be calculated using fluid dynamics, including forces and movements on an aeroplane, the mass flow rate of oil through pipelines, forecasting changing weather patterns, comprehending nebulae in interstellar space, and modelling explosions.

Both crowd dynamics and traffic engineering make use of some fluid-dynamical concepts.

Assumptions

Mathematical equations can be used to express the presumptions behind a fluid mechanics analysis of a physical system. Fundamentally, it is considered that every fluid mechanical system complies with:

• Mass conservation
• power-saving measures
• Maintaining momentum
• The continuum hypothesis

For instance, if mass is assumed to be conserved, then the rate of change of the mass contained in any fixed control volume (for instance, a spherical volume) enclosed by a control surface is equal to the rate at which mass is passing through the surface from outside to inside, less the rate at which mass is passing from inside to outside.

This can be written as an integral form equation over the control volume. Fluid characteristics, which are average values of molecular properties, can change continuously from one volume element to another. In applications like supersonic speed flows or molecule fluxes at the nanoscale, the continuum hypothesis can produce erroneous results.

Statistical mechanics can be used to address those issues where the continuum hypothesis falls short. The Knudsen number, defined as the ratio of the molecular mean free path to the characteristic length scale, is assessed to determine whether the continuum hypothesis holds true.

The continuum hypothesis can be used to assess issues with Knudsen numbers under 0.1, however statistical mechanics can be used to identify the fluid motion for issues with larger Knudsen numbers.

viscous and inviscid fluids

The viscosity of an inviscid fluid is zero, or v=0. A mathematical treatment is made easier by the idealization of an inviscid flow. Superfluidity is the only state where purely inviscid flows have been observed.

If not, fluids are typically viscous, a characteristic that is frequently crucial in boundary layers close to solid surfaces, where the flow must fulfil the solid's no-slip requirement. By assuming that the fluid outside of boundary layers is inviscid and then mapping its solution onto that for a thin laminar boundary layer, it is possible in some circumstances to treat the mathematics of a fluid mechanical system.

The fluid velocity between the free fluid and the fluid in the porous media might differ for fluid flow over a porous boundary (this is connected to the Beavers and Joseph condition). Additionally, it is helpful to consider gas to be incompressible at low subsonic speeds, meaning that the density of the gas remains constant even when the speed and static pressure fluctuate.

Fluids that are Newtonian versus non-Newtonian

A fluid that has a shear stress that is directly proportional to the velocity gradient in the direction perpendicular to the shear plane is known as a Newtonian fluid (named after Isaac Newton).

According to this definition, fluids continue to flow despite external forces acting on them.

For instance, water exhibits fluid qualities regardless matter how much it is agitated or combined, making it a Newtonian fluid. The drag of a small object moving slowly through a fluid is proportional to the force given to the object, which is a slightly less exact description. Consider friction. Under typical Earthly conditions, significant fluids like water and most gases behave roughly as a Newtonian fluid.?

A non-Newtonian fluid, however, can leave a "hole" behind when stirred. This will progressively fill up over time; things like pudding, Oobleck, or sand (although sand isn't precisely a fluid) exhibit this behavior. In contrast, swirling a non-Newtonian fluid can result in a drop in viscosity, giving the fluid the appearance of being "thinner" (this is the case with non-drip paints).

As something that does not adhere to a specific property, non-Newtonian fluids come in a variety of forms. For instance, most fluids with long molecular chains can behave in a non-Newtonian way.

Basic fluid characteristics

Because they are made up of discrete molecules, fluids are not precisely continuous media in the sense that all of Euler's and Bernoulli's predecessors have assumed. But because the molecules are so tiny and there are so many of them per milliliter-except in gases at very low pressures-it is unnecessary to think of them as distinct entities.

The great majority of fluids, including air and water, are isotropic. However, there exist a small number of liquids, known as liquid crystals, in which the molecules are packed together in such a way that the properties of the medium are locally anisotropic.

According to fluid mechanics, the state of an isotropic fluid can be completely described by defining its mean mass per unit volume, or density, its temperature, and its velocity, or v, at every point in space.

It is not directly relevant how these macroscopic properties relate to the positions and velocities of individual molecules.

Perhaps some clarification is necessary regarding the distinction between gases and liquids, even though this distinction is simpler to understand than to articulate. Gases have a tendency to expand to cover any available volume because the molecules in them are sufficiently spaced apart to move practically independently of one another.

The molecules in liquids are in contact, and the short-range attraction interactions between them keep them together. They move too quickly to form the ordered arrays that are distinctive of solids, but not too quickly that they can fly apart.

As a result, unlike samples of gas, samples of liquid can exist as drops or jets with free surfaces or they can sit in beakers with only gravity acting as a constraint. As molecules gradually gain enough speed to escape over the free surface and are not replaced, such samples may eventually evaporate. However, the duration of liquid droplets and jets is typically sufficient to neglect evaporation.

A brick held between two hands can be used to show the differences between the two types of stress that may exist in any solid or fluid medium.

The holder applies pressure to the brick if he moves his hands towards one another; if he moves one hand towards his body and the other away from it, he applies a shear stress to the brick. A solid substance, like a brick, can bear both sorts of loads, but fluids must always surrender to shear forces, regardless of how little they may be.

The rate at which they proceed is set by the viscosity of the fluid. The rate at which they proceed is set by the viscosity of the fluid. This feature, which will be discussed in more detail later, measures the friction that develops when adjacent fluid layers slide over one another.

Pascal's law states that the pressure (i.e., force per unit area) acting perpendicular to all planes in a fluid is the same regardless of the direction of those planes because shear stresses in a fluid at rest and in equilibrium are everywhere zero.

There is only one local pressure (p) value that is consistent with the provided values for and T for an isotropic fluid in equilibrium. The equation of state for the fluid, also known as the relationship between these three quantities, connects them together.

Hydrostatics

It is well known that the atmosphere's pressure, or about 10^5 newtons per square meter, is caused by the weight of air above the surface of the Earth. This pressure decreases as one climbs higher, and conversely, increases as one dives deeper into a lake (or other similar body of water).

Separate manometers

Differential manometers are devices for comparing pressures, and the most basic differential manometer is a U-tube filled with liquid, as seen in Figure.

The difference in height, h, between the two menisci is measured after the two pressures of interest, p1 and p2, are conveyed to the two ends of the liquid column through an inert gas, whose density is minimal in compared to the liquid density.

Simply said, a manometer with p2 set to zero or as close to zero as is practical is a barometer for measuring the atmospheric pressure in absolute terms. Evangelista Torricelli, an Italian physicist and mathematician, created the first barometer in the 17th century, and it is still in use today. It can be flipped after being filled with liquid and turned with the sealed end facing down.

If the liquid column is long enough, a temporary negative pressure may form there during inversion; however, cavitation typically takes place there and the column displaces the sealed end of the tube, as depicted in the image.

Torricelli believed there to be a vacuum between the two, but if the barometer has been filled without taking meticulous measures to guarantee that all dissolved or adsorbed gases, which would otherwise accumulate in this space, have first been eliminated, it may be very far from that condition.

The liquid's vapor is always contained in the Torricellian vacuum, even when no contaminating gas is present.

This vapor exerts pressure that may be slight but is never completely zero. Traditionally, mercury has been employed as the liquid in a Torricelli barometer since it has a low vapor pressure and a high density.

h is only around 760 millimeters due to the high density; if water were utilized, h would need to be about 10 meters.

The pressure within the upper container, p2, is ambient because it is exposed to the atmosphere.

The pressure p1 in the bottom container needs to be increased by ρgh to counteract this and the weight of the liquid column in between. Equilibrium is obviously unattainable if the bottom container is also exposed to the atmosphere; as a result, the liquid flows downward due to the liquid column's weight.

The syphon only works if the column is continuous; if a bubble of gas builds up in the tube or if cavitation takes place, it stops working. Cavitation thus restricts the level differences across which syphons can be utilised as well as the depth of wells from which water can be pumped using suction alone (to a maximum of around 10 meters).

Archimedes Principle

Consider a d-sided cube with its top and bottom faces horizontally submerged in a liquid. Since pressure is force per unit area and the area of a cube face is d2, the resulting upthrust on the cube is gd3.

The pressure on the bottom face will be greater than on the top by ρgd^3. This is a straightforward illustration of the so-called Archimedes' principle, according to which the upward thrust that a body experiences while submerged or floating is always equal to the weight of the fluid that the body displaces.

There is no need to demonstrate this with a thorough analysis of the pressure differential between the top and bottom, as Archimedes must have realized. Regardless of the shape of the body, it is certainly true. It is clear because the system would still be in balance if the solid body could be removed and the resulting cavity could somehow be filled with extra fluid.

However, the extra fluid would then be experiencing the upthrust that the solid body had been previously experiencing, and it would not be in equilibrium unless this were just enough to balance its weight.

Archimedes' challenge was to determine whether King Hieron II's crown was made of pure gold or gold that had been diluted with silver using what is now known as a nondestructive test.

He realized that the density of the pure metal and the alloy would be different, and that he could calculate the density of the crown by weighing it to establish its mass and measuring its volume separately.

He may have been inspired (while taking a bath) by the realization that any object's volume may be determined by drowning it in liquid in a container resembling a measuring cylinder (i.e., one with vertical sides that have been appropriately calibrated) and measuring the displacement of the liquid surface.

If so, he immediately realized that the concept that bears his name can serve as the foundation for a more sophisticated and precise way of calculating density.

In this approach, the object is weighed twice: once while floating in a vacuum (usually, air would do), and once while completely submerged in a liquid of density.

Liquid surface tension

The most important hydrostatic phenomenon where surface tension of liquids is involved is probably capillarity. Think about what occurs when a capillary tube, which has a small bore, is dipped into a liquid.

The liquid surface within the tube produces a concave meniscus, which is a nearly spherical surface with the same radius, r, as the interior of the tube, if the liquid "wets" the tube (with zero contact angle).

Hydrodynamics

The Bernoulli's law

Fluids at rest have previously been the main emphasis. This section discusses fluids that are moving steadily, with the fluid velocity at each location in space remaining constant across time.

Any steady flow pattern in this sense can be represented by a series of streamlines, which represent the trajectories of hypothetical particles transported by the fluid and suspended in it.

The fluid is in motion but the streamlines are stationary in constant flow. The fluid velocity is very rapid where the streamlines congregate; conversely, the fluid becomes comparatively sluggish where they spread out.