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Fluid mechanics is the study of fluids and the forces on them. (Fluids include liquids, gases, and plasmas.) Fluid mechanics can be divided into fluid kinematics, the study of fluid motion, and fluid dynamics, the study of the effect of forces on fluid motion, which can further be divided into fluid statics, the study of fluids at rest, and fluid kinetics, the study of fluids in motion. Fluid mechanics is very important to engineers when observing flow in pipes, viscous effects of fluids, and the forces that act on a fluid. As a student, I am suppose to demonstrate an adequate understanding of many properties involved fluid mechanics. Some learning outcomes that must be accomplished by taking this class are: * Demonstrate understanding of fluid mechanics fundamentals, fluid and flow properties such as compressibility, viscosity, buoyancy, hydrostatic pressure and forces on surfaces * Apply Bernoulli equation to solve problems in fluid mechanics * Solve fluid mechanics problem using control volume analysis using conservation of mass, energy equation and irreversible flow * Use differential analysis of fluid flow, potential flow theory, viscous flow, Navier Stokes equations to solve problems * Perform modeling and similitude using Buckingham Pi theorem, correlation of experimental data. * Analyze flow in pipes to determine laminar and turbulent flow behaviors. * Apply energy and momentum equations to determine performance of centrifugal pumps, fans and turbine for proper selection of turbo machines.

Compressibility of fluids is how easily a particular fluid can change it’s volume (and thus the density) when there is a change in pressure. Viscosity is a measure of the resistance of a fluid which is being deformed by either shear stress or tensile stress. Buoyancy is an upward acting force exerted by a fluid , that opposes an object's weight. If the object is either less dense than the liquid or is shaped appropriately, the force can keep the object afloat. Hydrostatic pressure is the pressure exerted by a fluid at equilibrium due to the force of gravity. A fluid in this condition is known as a hydrostatic fluid. In fluid dynamics, Bernoulli's principle states that for an inviscid flow, an increase in the speed of the fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy. Bernoulli's principle can be applied to various types of fluid flow, resulting in what is loosely denoted as Bernoulli's equation. In fact, there are different forms of the Bernoulli equation for different types of flow. The simple form of Bernoulli's principle is valid for incompressible flows (e.g. most liquid flows) and also for compressible flows moving at low Mach numbers. More advanced forms may in some cases be applied to compressible flows at higher Mach numbers (see the derivations of the Bernoulli equation). Bernoulli’s equation is as follows:

A system is a collection of matter of fixed identity which may move, flow and interact with its surroundings. A control volume is a volume in space through which fluid may flow.

A system may consist of a relatively large amount of mass (earth’s atmosphere) or an infinitesimal size (single fluid particle). The conservation of mass principle for a system is simply stated as “time rate of change of the system mass = 0” . The total stored energy per unit mass e consists of internal energy per unit mass u, kinetic energy per unit mass V2/2 and potential energy per unit mass, gz, denoted in the energy equation:

The Navier–Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of fluid substances. These equations arise from applying Newton's second law to fluid motion, together with the assumption that the fluid stress is the sum of a diffusing viscous term (proportional to the gradient of velocity), plus a pressure term. The Navier–Stokes equations are based on the assumption that the fluid, at the scale of interest, is a continuum, in other words is not made up of discrete particles but rather a continuous substance. Another necessary assumption is that all the fields of interest like pressure, velocity, density, temperature and so on are differentiable, weakly at least.

The equations are derived from the basic principles of conservation of mass, momentum, and energy. For that matter, sometimes it is necessary to consider a finite arbitrary volume, called a control volume, over which these principles can be applied. This finite volume is denoted by Ω and its bounding surface . The control volume can remain fixed in space or can move with the fluid.

The Vaschy-Buckingham π theorem is a key theorem in dimensional analysis. The theorem loosely states that if we have a physically meaningful equation involving a certain number, n, of physical variables, and these variables are expressible in terms of k independent fundamental physical quantities, then the original expression is equivalent to an equation involving a set of p = n − k dimensionless parameters constructed from the original variables: it is a scheme for nondimensionalization. This provides a method for computing sets of dimensionless parameters from the given variables, even if the form of the equation is still unknown. However, the choice of dimensionless parameters is not unique: Vaschy-Buckingham's theorem only provides a way of generating sets of dimensionless parameters, and will not choose the most 'physically meaningful'.

Laminar flow, sometimes known as streamline flow, occurs when a fluid flows in parallel layers, with no disruption between the layers. At low velocities the fluid tends to flow without lateral mixing, and adjacent layers slide past one another like playing cards. There are no cross currents perpendicular to the direction of flow, nor eddies or swirls of fluids. In laminar flow the motion of the particles of fluid is very orderly with all particles moving in straight lines parallel to the pipe walls. In fluid dynamics, laminar flow is a flow regime characterized by high momentum diffusion and low momentum convection.

When a fluid is flowing through a closed channel such as a pipe or between two flat plates, either two types of flow may occur depending on the velocity of the fluid: laminar flow or turbulent flow. Laminar flow is the opposite of turbulent flow which occurs at higher velocities where eddies or small packets of fluid particles form leading to lateral mixing. In nonscientific terms laminar flow is "smooth", while turbulent flow is "rough."

The type of flow occurring in a fluid in a channel is important in fluid dynamics problems. The dimensionless Reynolds number is an important parameter in the equations that describe whether flow conditions lead to laminar or turbulent flow. In the case of flow through a straight pipe with a circular cross-section, Reynolds numbers of less than 2100 are generally considered to be of a laminar type; however, the Reynolds number upon which laminar flows become turbulent is dependent upon the flow geometry. When the Reynolds number is much less than 1, Creeping motion or Stokes flow occurs. This is an extreme case of laminar flow where viscous (friction) effects are much greater than inertial forces. The common application of laminar flow would be in the smooth flow of a viscous liquid through a tube or pipe. In that case, the velocity of flow varies from zero at the walls to a maximum along the centerline of the vessel. The flow profile of laminar flow in a tube can be calculated by dividing the flow into thin cylindrical elements and applying the viscous force to them.

Turbomachinery, in mechanical engineering, describes machines that transfer energy between a rotor and a fluid, including both turbines and compressors. While a turbine transfers energy from a fluid to a rotor, a compressor transfers energy from a rotor to a fluid. The two types of machines are governed by the same basic relationships including Newton's second Law of Motion and Euler's energy equation for compressible fluids. Centrifugal pumps are also turbomachines that transfer energy from a rotor to a fluid, usually a liquid, while turbines and compressors usually work with a gas.

In general, two kinds of turbomachines are encountered in practice. These are open and closed turbomachines. Open machines such as propellers, windmills, and unshrouded fans act on an infinite extent of fluid, whereas, closed machines operate on a finite quantity of fluid as it passes through a housing or casing.

Turbomachines are also categorized according to the type of flow. When the flow is parallel to the axis of rotation, they are called axial flow machines, and when flow is perpendicular to the axis of rotation, they are referred to as radial (or centrifugal) flow machines. There is also a third category, called mixed flow machines, where both radial and axial flow velocity components are present.

Turbomachines may be further classified into two additional categories: those that absorb energy to increase the fluid pressure, i.e. pumps, fans, and compressors, and those that produce energy such as turbines by expanding flow to lower pressures. Of particular interest are applications which contain pumps, fans, compressors and turbines. These components are essential in almost all mechanical equipment systems, such as power and refrigeration cycles.

Energy is supplied to the rotating shaft and transferred to the fluid by the blades in pumps. Energy is transferred from the fluid to the blades and transferred to shaft as shaft power (turbines). Many turbo machines contain housing that surrounds the rotating blade or rotor. The rotor and housing forms an internal passageway through which fluid flows.

These are some of the vast principles involved in fluid mechanics. The main topics I covered in this paper are compressibility of fluids, buoyancy, viscocity, hydrostatic pressure, Bernoulli’s principle, Laminar and turbulent flow, Buckingham Pi theorem, energy equation, and the Navier-Stokes equations. There are many more topics involved with fluid mechanics. These are are the main topics that are covered in the fluid mechanics course. As for the student learning outcomes, I believe that I have a decent understanding of these topics and how they apply to fluid mechanics as well as engineering.

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...This is page i Printer: Opaque this A Mathematical Introduction to Fluid Mechanics Alexandre Chorin Department of Mathematics University of California, Berkeley Berkeley, California 94720-3840, USA Jerrold E. Marsden Control and Dynamical Systems, 107-81 California Institute of Technology Pasadena, California 91125, USA ii iii A Mathematical Introduction to Fluid Mechanics iv Library of Congress Cataloging in Publication Data Chorin, Alexandre A Mathematical Introduction to Fluid Mechanics, Third Edition (Texts in Applied Mathematics) Bibliography: in frontmatter Includes. 1. Fluid dynamics (Mathematics) 2. Dynamics (Mathematics) I. Marsden, Jerrold E. II. Title. III. Series. ISBN 0-387 97300-1 American Mathematics Society (MOS) Subject Classiﬁcation (1980): 76-01, 76C05, 76D05, 76N05, 76N15 Copyright 1992 by Springer-Verlag Publishing Company, Inc. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher, Springer-Verlag Publishing Company, Inc., 175 Fifth Avenue, New York, N.Y. 10010. Typesetting and illustrations prepared by June Meyermann, Gregory Kubota, and Wendy McKay The cover illustration shows a computer simulation of a shock diﬀraction by a pair of cylinders, by John Bell, Phillip Colella, William Crutchﬁeld, Richard Pember, and Michael......

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