The study of fluid dynamics might seem like a narrow topic in physics. And why would you want to spend so much time just looking at the motion of something so mundane? But this way of thinking misunderstands the nature of the study of fluids and ignores the many different applications of fluid dynamics. As well as being useful for understanding things like ocean currents, fluid dynamics has applications in areas like plate tectonics, stellar evolution, blood circulation and meteorology.

The key concepts are also crucial for engineering and design, and mastery of fluid dynamics opens doors to working with things like aerospace engineering, wind turbines, air conditioning systems, rocket engines and pipe networks.

The first step to unlocking the understanding you need to work on projects like these, though, is to understand the basics of fluid dynamics, the terms physicists use when talking about it and the most important equations governing it.

The meaning of fluid dynamics can be understood if you break down the individual words in the phrase. There is a close relationship between fluid dynamics, fluid mechanics and aerodynamics. Aerodynamics, on the other hand, deals exclusively with gases, while fluid dynamics covers both gases and liquids.

The key focus of fluid dynamics is how fluids flowand so understanding the basics is crucial for any student.

### Statics For Dummies Cheat Sheet

However, the key points are intuitively simple: Fluids flow downhill and as a result of pressure differences. The continuity equation is a fairly complicated-looking expression but it really just conveys a very simple point: Matter is conserved during fluid flow. So the amount of fluid flowing past point 1 must match the point flowing past point 2, in other words, the mass flow rate is constant.

The equation makes it easy to see specifically what this means:. The remaining piece of the puzzle, the density, ensures that this is balanced against the amount of compression of the fluid at different points. This is so that if a gas is compressed between point 1 and point 2, the greater amount of matter per unit volume at point 2 is accounted for in the equation. The equation explicitly matches the rate of flow of matter at two different points on its journey.

It essentially states that the energy density i. In symbols:. If you know the initial values and at least some details of the pipe and fluid after the chosen point, you can find out the remaining value by re-arranging the equation.

It assumes that both points lie on a streamline, that the flow is steady, that there is no friction and that the fluid has a constant density. These are restrictive limitations on the formula, and if you were being strictly accurate, no moving fluids would meet these requirements. It can help to think about it as the opposite to turbulent flow, where there are fluctuations, vortices and other irregular behavior. In this steady flow, the important quantities like velocity and pressure used to characterize the flow remain constant, and the fluid flow can be thought of as taking place in layers.

For example, on a horizontal surface, the flow could be modeled as a series of parallel, horizontal layers of water, or through a tube it could be thought of as a series of increasingly small concentric cylinders. Some examples of laminar flow should help you understand what it is, and one everyday example is the water emerging from the bottom of a tap. At first, it dribbles, but if you open the tap a little more, you get a smooth, perfect stream of water out of it — this is laminar flow — and at higher levels still it becomes turbulent.

The smoke emerging from the tip of a cigarette also shows laminar flow, a smooth stream at first, but then becomes turbulent as it gets farther away from the tip. Laminar flow is more common when the fluid is moving slowly, when it has high viscosity or when it only has a small amount of space to flow through. This was demonstrated in a famous experiment by Osborne Reynolds known for the Reynolds number, which will be discussed more in the next sectionin which he injected dye into a fluid flow through a glass tube.

When the flow was slower, the dye moved in a straight line path, at higher speeds it moves to a transitional pattern, while at much higher speeds it becomes turbulent.

Turbulent flow is the chaotic flow motion that tends to happen at higher speeds, where the fluid has a larger space to flow through and where the viscosity is low. This is characterized by vortices, eddies and wakes, which makes it very difficult to predict the precise motions in the flow because of the chaotic behavior. In turbulent flow, the speed and direction i.

The reason for this is that laminar flow really only happens under special circumstances.Fluid statics or hydrostatics is the branch of fluid mechanics that studies " fluids at rest and the pressure in a fluid or exerted by a fluid on an immersed body". It encompasses the study of the conditions under which fluids are at rest in stable equilibrium as opposed to fluid dynamicsthe study of fluids in motion.

Hydrostatics are categorized as a part of the fluid statics, which is the study of all fluids, incompressible or not, at rest. Hydrostatics is fundamental to hydraulicsthe engineering of equipment for storing, transporting and using fluids.

It is also relevant to geophysics and astrophysics for example, in understanding plate tectonics and the anomalies of the Earth's gravitational fieldto meteorologyto medicine in the context of blood pressureand many other fields. Hydrostatics offers physical explanations for many phenomena of everyday life, such as why atmospheric pressure changes with altitudewhy wood and oil float on water, and why the surface of still water is always level.

Some principles of hydrostatics have been known in an empirical and intuitive sense since antiquity, by the builders of boats, cisternsaqueducts and fountains. Archimedes is credited with the discovery of Archimedes' Principlewhich relates the buoyancy force on an object that is submerged in a fluid to the weight of fluid displaced by the object. The Roman engineer Vitruvius warned readers about lead pipes bursting under hydrostatic pressure.

The concept of pressure and the way it is transmitted by fluids was formulated by the French mathematician and philosopher Blaise Pascal in The "fair cup" or Pythagorean cupwhich dates from about the 6th century BC, is a hydraulic technology whose invention is credited to the Greek mathematician and geometer Pythagoras. It was used as a learning tool. The cup consists of a line carved into the interior of the cup, and a small vertical pipe in the center of the cup that leads to the bottom.

The height of this pipe is the same as the line carved into the interior of the cup. The cup may be filled to the line without any fluid passing into the pipe in the center of the cup. However, when the amount of fluid exceeds this fill line, fluid will overflow into the pipe in the center of the cup. Due to the drag that molecules exert on one another, the cup will be emptied. Heron's fountain is a device invented by Heron of Alexandria that consists of a jet of fluid being fed by a reservoir of fluid.

The fountain is constructed in such a way that the height of the jet exceeds the height of the fluid in the reservoir, apparently in violation of principles of hydrostatic pressure. The device consisted of an opening and two containers arranged one above the other. The intermediate pot, which was sealed, was filled with fluid, and several cannula a small tube for transferring fluid between vessels connecting the various vessels. Trapped air inside the vessels induces a jet of water out of a nozzle, emptying all water from the intermediate reservoir.

Pascal made contributions to developments in both hydrostatics and hydrodynamics. Pascal's Law is a fundamental principle of fluid mechanics that states that any pressure applied to the surface of a fluid is transmitted uniformly throughout the fluid in all directions, in such a way that initial variations in pressure are not changed.

Due to the fundamental nature of fluids, a fluid cannot remain at rest under the presence of a shear stress. However, fluids can exert pressure normal to any contacting surface.Fluid statics or hydrostatics is the branch of fluid mechanics that studies " fluids at rest and the pressure in a fluid or exerted by a fluid on an immersed body". It encompasses the study of the conditions under which fluids are at rest in stable equilibrium as opposed to fluid dynamicsthe study of fluids in motion.

Hydrostatics are categorized as a part of the fluid statics, which is the study of all fluids, incompressible or not, at rest. Hydrostatics is fundamental to hydraulicsthe engineering of equipment for storing, transporting and using fluids. It is also relevant to geophysics and astrophysics for example, in understanding plate tectonics and the anomalies of the Earth's gravitational fieldto meteorologyto medicine in the context of blood pressureand many other fields. Hydrostatics offers physical explanations for many phenomena of everyday life, such as why atmospheric pressure changes with altitudewhy wood and oil float on water, and why the surface of still water is always level.

Some principles of hydrostatics have been known in an empirical and intuitive sense since antiquity, by the builders of boats, cisternsaqueducts and fountains. Archimedes is credited with the discovery of Archimedes' Principlewhich relates the buoyancy force on an object that is submerged in a fluid to the weight of fluid displaced by the object. The Roman engineer Vitruvius warned readers about lead pipes bursting under hydrostatic pressure.

The concept of pressure and the way it is transmitted by fluids was formulated by the French mathematician and philosopher Blaise Pascal in The "fair cup" or Pythagorean cupwhich dates from about the 6th century BC, is a hydraulic technology whose invention is credited to the Greek mathematician and geometer Pythagoras.

It was used as a learning tool. The cup consists of a line carved into the interior of the cup, and a small vertical pipe in the center of the cup that leads to the bottom. The height of this pipe is the same as the line carved into the interior of the cup.

The cup may be filled to the line without any fluid passing into the pipe in the center of the cup. However, when the amount of fluid exceeds this fill line, fluid will overflow into the pipe in the center of the cup. Due to the drag that molecules exert on one another, the cup will be emptied. Heron's fountain is a device invented by Heron of Alexandria that consists of a jet of fluid being fed by a reservoir of fluid. The fountain is constructed in such a way that the height of the jet exceeds the height of the fluid in the reservoir, apparently in violation of principles of hydrostatic pressure.

The device consisted of an opening and two containers arranged one above the other. The intermediate pot, which was sealed, was filled with fluid, and several cannula a small tube for transferring fluid between vessels connecting the various vessels. Trapped air inside the vessels induces a jet of water out of a nozzle, emptying all water from the intermediate reservoir. Pascal made contributions to developments in both hydrostatics and hydrodynamics.

### List of equations in fluid mechanics

Pascal's Law is a fundamental principle of fluid mechanics that states that any pressure applied to the surface of a fluid is transmitted uniformly throughout the fluid in all directions, in such a way that initial variations in pressure are not changed.

Due to the fundamental nature of fluids, a fluid cannot remain at rest under the presence of a shear stress. However, fluids can exert pressure normal to any contacting surface. If a point in the fluid is thought of as an infinitesimally small cube, then it follows from the principles of equilibrium that the pressure on every side of this unit of fluid must be equal.

If this were not the case, the fluid would move in the direction of the resulting force. Thus, the pressure on a fluid at rest is isotropic ; i. This characteristic allows fluids to transmit force through the length of pipes or tubes; i.

This principle was first formulated, in a slightly extended form, by Blaise Pascal, and is now called Pascal's law. In a fluid at rest, all frictional and inertial stresses vanish and the state of stress of the system is called hydrostatic. For a barotropic fluid in a conservative force field like a gravitational force field, pressure exerted by a fluid at equilibrium becomes a function of force exerted by gravity.

The hydrostatic pressure can be determined from a control volume analysis of an infinitesimally small cube of fluid. For water and other liquids, this integral can be simplified significantly for many practical applications, based on the following two assumptions: Since many liquids can be considered incompressiblea reasonable good estimation can be made from assuming a constant density throughout the liquid.

The same assumption cannot be made within a gaseous environment.Variation of Pressure in a Stationary Fluid. Variation of Pressure along Horizontal in Accelerating Fluids. Simulation for Hydraulic Lift Works Principle. A fluid is a substance, such as liquid or gas that has no rigidity like solids.

Liquids are distinguished from gases by the presence of a surface. As fluids have no rigidity, they fail to support a shear stress. When a fluid is subjected to a shear stress, the layers of the fluid slide relative to each other. This characteristic gives the fluid, an ability to flow or change shapes. An object with less density than the fluid will float upward until it reaches the surface of the fluid.

At that position, only part of the object is submerged. Fluids obey Newton's laws of motion. Any infinitesimal volume of fluid accelerates according to net force acting on it.

Under static condition, the only force component that needs to be considered is one that acts normal or perpendicular to the surface of the fluid. The magnitude of normal force per unit surface area is called pressure.

Pressure is a scalar quantity and has no directional properties. Consider an elemental area dA inside a fluid, the fluid on one side of area dA presses the fluid on the other side and vice-versa. We define the pressure p a that point as the normal force per unit area, i. If the pressure is same at all the points of a finite plane surface with area A, then.

The pressure inside a fluid can be calculated by considering the following. If we analyse the force acting on surface CD we have to take into account the weight of the fluid column just above CD other parts of the liquid cannot exert any force because liquid cannot exert shear stress.

If the pressure at the surface p 0 is changed, an equal change in pressure is felt at all depths i. T his phenomenon will be described while discussing Pascal's law. Fluids exert force on all surfaces in contact with them. They exert thrust in a direction normal to the surface in contact. The understand this, let us consider a surface in contact with a fluid.

Let a be the area of the surface in contact. Hence, the thrust on a surface depends on the volume of fluid above it. This suggests that if a body is immersed in a fluid the thrust on the lower surface will be more than that on the upper surface. It I the pressure of the earth's atmosphere. Fluid force acts perpendicular to any surface in the fluid, not matter how that surface is oriented. Hence, pressure has no intrinsic direction of its own, i.

It is a scalar. The excess pressure above atmospheric pressure is called gauge pressure, and total pressure is called absolute pressure. The pressure increases with depth y. The shaded element accelerates to the right.

This animations shows how a hydraulic lift works -in principle. This is an example of Pascal's Principle. The small force and small area on the pump side equals the large force and large area on the other side.Physics is filled with equations and formulas that deal with angular motion, Carnot engines, fluids, forces, moments of inertia, linear motion, simple harmonic motion, thermodynamics, and work and energy.

Equations of angular motion are relevant wherever you have rotational motions around an axis. You must use radians to measure the angle. Also, if you know that the distance from the axis is r, then you can work out the linear distance traveled, svelocity, vcentripetal acceleration, a cand force, F c. A heat engine takes heat, Q hfrom a high temperature source at temperature T h and moves it to a low temperature sink temperature T c at a rate Q c and, in the process, does mechanical work, W.

This process can be reversed such that work can be performed to move the heat in the opposite direction — a heat pump. The amount of work performed in proportion to the amount of heat extracted from the heat source is the efficiency of the engine. A Carnot engine is reversible and has the maximum possible efficiency, given by the following equations. The equivalent of efficiency for a heat pump is the coefficient of performance. A force, Fover an area, Agives rise to a pressure, P.

The pressure of a fluid at a depth of h depends on the density and the gravitational constant, g. Objects immersed in a fluid causing a mass of weight, W water displacedgive rise to an upward directed buoyancy force, F buoyancy. Because of the conservation of mass, the volume flow rate of a fluid moving with velocity, vthrough a cross-sectional area, Ais constant.

A mass, maccelerates at a rate, adue to a force, Facting. Two masses, m 1 and m 2separated by a distance, rattract each other with a gravitational force, given by the following equations, in proportion to the gravitational constant G :.

The moments of inertia for various shapes are shown here:. Rectangle rotating around an axis along one edge where the other edge is of length r :. Rectangle rotating around an axis parallel to one edge and passing through the center, where the length of the other edge is r :.

When an object at position x moves with velocity, vand acceleration, a, resulting in displacement, seach of these components is related by the following equations:. One example of such a force is provided by a spring with spring constant, k. The position, xvelocity, vand acceleration, a, of an object undergoing simple harmonic motion can be expressed as sines and cosines.

The random vibrational and rotational motions of the molecules that make up an object of substance have energy; this energy is called thermal energy. When an object receives an amount of heat, its temperature, Trises. You can use these formulas to convert from one temperature scale to another:.The most remarkable thing about this expression is what it does not include.

The fluid pressure at a given depth does not depend upon the total mass or total volume of the liquid. The above pressure expression is easy to see for the straight, unobstructed column, but not obvious for the cases of different geometry which are shown. Because of the ease of visualizing a column height of a known liquid, it has become common practice to state all kinds of pressures in column height units, like mmHg or cm H 2 O, etc.

Pressures are often measured by manometers in terms of a liquid column height. Note that this static fluid pressure is dependent on density and depth only; it is independent of total mass, weight, volume, etc. Static Fluid Pressure The pressure exerted by a static fluid depends only upon the depth of the fluid, the density of the fluid, and the acceleration of gravity.

Index fluid pressure calculation Pressure concepts. Fluid Pressure Calculation Discussion. Fluid column height in the relationship is often used for the measurement of pressure.

After entering the relevant data, any one of the highlighted quantities below can be calculated by clicking on it. Index fluid pressure discussion Pressure concepts.These balance equations arise from applying Isaac Newton's second law to fluid motiontogether with the assumption that the stress in the fluid is the sum of a diffusing viscous term proportional to the gradient of velocity and a pressure term—hence describing viscous flow.

The main difference between them and the simpler Euler equations for inviscid flow is that Navier—Stokes equations also factor in the Froude limit no external field and are not conservation equationsbut rather a dissipative systemin the sense that they cannot be put into the quasilinear homogeneous form:. The Navier—Stokes equations are useful because they describe the physics of many phenomena of scientific and engineering interest.

They may be used to model the weather, ocean currentswater flow in a pipe and air flow around a wing. The Navier—Stokes equations, in their full and simplified forms, help with the design of aircraft and cars, the study of blood flow, the design of power stations, the analysis of pollution, and many other things. Coupled with Maxwell's equationsthey can be used to model and study magnetohydrodynamics. The Navier—Stokes equations are also of great interest in a purely mathematical sense.

Despite their wide range of practical uses, it has not yet been proven whether solutions always exist in three dimensions and, if they do exist, whether they are smooth — i. These are called the Navier—Stokes existence and smoothness problems.

The solution of the equations is a flow velocity. It is a vector field - to every point in a fluid, at any moment in a time interval, it gives a vector whose direction and magnitude are those of the velocity of the fluid at that point in space and at that moment in time. It is usually studied in three spatial dimensions and one time dimension, although the two spatial dimensional case is often useful as a model, and higher-dimensional analogues are of both pure and applied mathematical interest.

Once the velocity field is calculated, other quantities of interest such as pressure or temperature may be found using dynamical equations and relations. This is different from what one normally sees in classical mechanicswhere solutions are typically trajectories of position of a particle or deflection of a continuum. Studying velocity instead of position makes more sense for a fluid; however for visualization purposes one can compute various trajectories.

## List of equations in fluid mechanics

In particular, the streamlines of a vector field, interpreted as flow velocity, are the paths along which a massless fluid particle would travel. These paths are the integral curves whose derivative at each point is equal to the vector field, and they can represent visually the behavior of the vector field at a point in time.

The Navier—Stokes momentum equation can be derived as a particular form of the Cauchy momentum equationwhose general convective form is. In this form, it is apparent that in the assumption of an inviscid fluid -no deviatoric stress- Cauchy equations reduce to the Euler equations. This is often written: [3].

The left side of the equation describes acceleration, and may be composed of time-dependent and convective components also the effects of non-inertial coordinates if present. The right side of the equation is in effect a summation of hydrostatic effects, the divergence of deviatoric stress and body forces such as gravity. All non-relativistic balance equations, such as the Navier—Stokes equations, can be derived by beginning with the Cauchy equations and specifying the stress tensor through a constitutive relation.

By expressing the deviatoric shear stress tensor in terms of viscosity and the fluid velocity gradient, and assuming constant viscosity, the above Cauchy equations will lead to the Navier—Stokes equations below. A significant feature of the Cauchy equation and consequently all other continuum equations including Euler and Navier—Stokes is the presence of convective acceleration: the effect of acceleration of a flow with respect to space.

While individual fluid particles indeed experience time-dependent acceleration, the convective acceleration of the flow field is a spatial effect, one example being fluid speeding up in a nozzle. The compressible momentum Navier—Stokes equation results from the following assumptions on the Cauchy stress tensor: [4].

Since the trace of the rate-of-strain tensor in three dimensions is:. So by alternatively decomposing the stress tensor into isotropic and deviatoric parts, as usual in fluid dynamics: [5].

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