Fluid Statics And Dynamics Pdf

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Niklas Terki

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Aug 3, 2024, 5:02:03 PM8/3/24

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Our experience suggests that most students from geosciences, natural sciences, and other backgrounds are not required to take a fluid mechanics class, and most engineering students who took a fluid mechanics class earlier may still need to reinforce their physical understanding of the fluid mechanics concepts. Thereby, this chapter is intended to provide readers with some basics of fluid mechanics (particularly, in terms of physics) that are essential to understand and appreciate flow through geologic media.

This chapter first introduces readers to the basic continuum assumption (i.e., volume averaging concept). Then, definitions of fluid properties and their units follow. Forces of static fluids and the relationship between forces and energies are discussed subsequently. The chapter then examines fluid dynamics, in which the fixed and moving coordinate systems are introduced, the time derivatives associated with these two coordinate systems are brought forth, and fluid acceleration is investigated. Using force balance and acceleration concepts, we derive the Bernoulli equation for flow through pipe; we elucidate its physical meaning in term of conservation of energy, which leads to the concept of total, pressure, velocity, and potential heads. Head loss due to viscous forces ignored by the Bernoulli equation then explains the limitation of the equation for flow through real-world systems.

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Fluid statics or hydrostatics is the branch of fluid mechanics that studies fluids at hydrostatic equilibrium[1] and "the pressure in a fluid or exerted by a fluid on an immersed body".[2]

It encompasses the study of the conditions under which fluids are at rest in stable equilibrium as opposed to fluid dynamics, the study of fluids in motion. Hydrostatics is a subcategory of fluid statics, which is the study of all fluids, both compressible or incompressible, at rest.

Hydrostatics is fundamental to hydraulics, the 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 field), to meteorology, to medicine (in the context of blood pressure), and many other fields.

Hydrostatics offers physical explanations for many phenomena of everyday life, such as why atmospheric pressure changes with altitude, why wood and oil float on water, and why the surface of still water is always flat, level and horizontal.

Some principles of hydrostatics have been known in an empirical and intuitive sense since antiquity, by the builders of boats, cisterns, aqueducts and fountains. Archimedes is credited with the discovery of Archimedes' Principle, which 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.[3]

The "fair cup" or Pythagorean cup, which 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.[citation needed]

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.[citation needed]

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. 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.e., it acts with equal magnitude in all directions. This characteristic allows fluids to transmit force through the length of pipes or tubes; i.e., a force applied to a fluid in a pipe is transmitted, via the fluid, to the other end of the pipe. This principle was first formulated, in a slightly extended form, by Blaise Pascal, and is now called Pascal's law.[citation needed]

So in this case the pressure difference is the opposite of the difference of the scalar potential associated to the body force.In the other particular case of a body force of constant direction along z:

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 incompressible, a reasonable good estimation can be made from assuming a constant density throughout the liquid. The same assumption cannot be made within a gaseous environment. Also, since the height Δ z \displaystyle \Delta z of the fluid column between z and z0 is often reasonably small compared to the radius of the Earth, one can neglect the variation of g. Under these circumstances, one can transport out of the integral the density and the gravity acceleration and the law is simplified into the formula

where Δ z \displaystyle \Delta z is the total height of the liquid column above the test area to the surface, and p0 is the atmospheric pressure, i.e., the pressure calculated from the remaining integral over the air column from the liquid surface to infinity. This can easily be visualized using a pressure prism.

If there are multiple types of molecules in the gas, the partial pressure of each type will be given by this equation. Under most conditions, the distribution of each species of gas is independent of the other species.

Any body of arbitrary shape which is immersed, partly or fully, in a fluid will experience the action of a net force in the opposite direction of the local pressure gradient. If this pressure gradient arises from gravity, the net force is in the vertical direction opposite that of the gravitational force. This vertical force is termed buoyancy or buoyant force and is equal in magnitude, but opposite in direction, to the weight of the displaced fluid. Mathematically,

Liquids can have free surfaces at which they interface with gases, or with a vacuum. In general, the lack of the ability to sustain a shear stress entails that free surfaces rapidly adjust towards an equilibrium. However, on small length scales, there is an important balancing force from surface tension.

When liquids are constrained in vessels whose dimensions are small, compared to the relevant length scales, surface tension effects become important leading to the formation of a meniscus through capillary action. This capillary action has profound consequences for biological systems as it is part of one of the two driving mechanisms of the flow of water in plant xylem, the transpirational pull.

Without surface tension, drops would not be able to form. The dimensions and stability of drops are determined by surface tension. The drop's surface tension is directly proportional to the cohesion property of the fluid.

The Armfield F1-29 is designed to demonstrate the properties of Newtonian fluids and their behaviour under hydrostatic conditions (fluid at rest). This enables students to develop an understanding and knowledge of a wide range of fundamental principles and techniques, before studying fluids in motion. These include the use of fluids in manometers to measure pressure and pressure differences in gases and liquids.

The apparatus is constructed from PVC and clear acrylic, and consists of a vertical reservoir containing water that is connected to a series of vertical manometer tubes. These tubes can be used individually or in combination for the different demonstrations of hydrostatic principles and manometry. One tube includes changes in cross section to demonstrate that the level of a free surface is not affected by the size or the shape of the tube. The right-hand manometer tube is separate from the other tubes and incorporates a pivot and indexing mechanism at the base that enables this tube to be inclined at fixed angles of 5, 30, 60 and 90 (vertical).