# Fluid Mechanics

## The Fundamental Introduction of Fluid Mechanics

**Abstract**

The branch of
physics mechanical engineering that is connected to mechanics of fluids (liquids,
gases, and plasmas) and their related forces; It called as ** Fluid
Mechanics**. Fluid Mechanics has applications in a wide range of use of
engineering (especially mechanical engineering) and scientific fields. It can
be divided to

*fluid statics*and

*fluid dynamics*. It is a branch of continuum mechanics that models matter without using any data about atoms and molecules. In other word, it modeling matter from a macroscopic viewpoint rather than from microscopic.

Fluid mechanics is an active field of research (especially fluid dynamics) that has a very complex modeling in mathematical equations. Since many problems in fluid mechanics are unsolved (partly or wholly), they are best addressed by numerical methods. To achieve a solution for its problems, we use a modern discipline which called computational fluid dynamics (CFD). CFD is a devoted to this approach.

*** Fluid Statics**

Investigations of fluid mechanics in the rest state refers to fluid statics. Fluid statics or hydrostatics is the branch of fluid mechanics that studies fluids at rest. The rest state in this situation of fluid mechanics means stable equilibrium. There are some questions in our life that fluid statics answer them, such as why atmospheric pressure changes with altitude, why wood and oil float on water, and why surface of water is always level and horizontal whatever the shape of its container. The engineering of equipment for storing, transporting and using fluids called hydraulics that hydrostatics is fundamental to it.

*** Fluid Dynamics**

Fluid flow is
the science of liquids and gases in motion. Fluid dynamics is a branch or in
better word subdiscipline of fluid mechanics that deals with fluid flow. There
is a systematic structure in fluid dynamics – which underlines these practical
disciplines – that used to solve, analysis, and simulation practical problems.
Also, empirical and semi-empirical laws that rived from flow measurement used
to fluid dynamics analysis. The solution to a fluid dynamics problem typically
involves calculating various properties of the fluid, such as velocity,
pressure, density, and temperature, as function of space and time. Fluid
dynamics has several subdisciplines by itself, including aerodynamics and
hydrodynamics. Aerodynamics is the study of air and other gases in motion and
hydrodynamics is the study of liquids in motion. There is a wide range of applications
of fluid dynamics, including calculating forces and movements on air craft,
determining the mass flow rate of petroleum through pipelines, predicting
evolving weather patterns, understanding nebulae in interstellar space and
modeling explosions.

*** Assumptions**

There are some terms in mathematical equations to simplifying the governing equations of fluid flow as the assumptions inherent to fluid mechanical treatment of physical system. these assumptions can express in terms of mechanical systems. Fundamentally, every fluid mechanical system is assumed to obey:

- Conservation of mass
- Conservation of energy
- Conservation of momentum
- The continuum assumption

The assumption
of conservation of mass means that for any fixed control volume – enclosed by a
control surface – difference between two fluxes is equal to the rate of change
of the mass contained in that volume. These fluxes men the rate at which mass
is passing through the surface from *outside to inside*, minus the rate at
which mass is passing from *inside to outside*.

*‘’ This
can be expressed as an equation in integral form over the control volume ‘’*

An idealization of continuum mechanics for fluid treatments is the continuum assumption. Under this assumption fluids can be treated as continuous, even though, macroscopic scale. Under the continuum assumption, macroscopic (observed/measurable) properties such as density, pressure, temperature, and balk velocity are taken to be well-defined at “infinitesimal’’ volume elements – small in comparison to the characteristic length scale of the system, but large in comparison to molecular length scale. For fluid properties, there are average values of the molecular properties which are used, because fluid properties can vary continuously from one volume element to another.

In some of flows such as supersonic speed flows, or molecular flows on Nano scale in applications, the continuum hypothesis can lead to inaccurate results. These flows can be solved using statistical mechanics. Thus, there is a demand to existing of a criterion. To determine whether or not the continuum hypothesis applies, the Knudsen number, defined as the ratio of the molecular free path to the characteristic length scale, is evaluated. To use of continuum hypothesis, Knudsen number must be below 0.1, but molecular approach (statistical mechanics) can be applied for all ranges of Knudsen numbers.

*** Navier-Stokes
Equation**

The differential
equations that describe the force balance at a given point within a fluid,
called Navier-Stokes equations. For a given physical problem, solutions of the
Navier-Stokes equations must be sought with the help of calculus. In practical
terms only the simplest cases can be solved exactly in this way. In fact, these
cases are not terms which exist in real problems, because these cases generally
involve non-turbulent, steady flow in which Reynolds number is small. In real
problems, there are more complexity. In these cases, especially those involving
turbulence, complex geometry, non-stable flows such as global weather systems,
aerodynamics, hydrodynamics, and many more, solutions of the Navier-Stokes
equations can currently only be found with the numerical computations. This
branch of science is called computational fluid dynamics or ** CFD**.

*** Inviscid and Viscous
Fluids**

In practice, an inviscid fluid is an idealization, one that facilities mathematical treatment. Because Inviscid flow has no viscosity. Purely inviscid flows are only known to be realized in the case of super fluidity. Otherwise, fluids are generally viscous. The property that is called viscosity, that is often most important within a boundary layer near a solid surface, where the flow must match onto the no-slip condition at the solid. The most common application of inviscid flow is study of fluid flow outside the boundary layer. In some cases, the mathematics of a fluid mechanical system can be treated by assuming that the fluid outside the boundary layers is inviscid, and then matching its solution onto that for a thin laminar boundary layer. There is a difference for boundary layer over a porous media solid. For fluid flow over a porous boundary, the fluid velocity can be discontinuous between the free fluid and the fluid in the porous media (this is related to the Beavers and Joseph condition). Further, it is useful at low subsonic speeds to assume that a gas is incompressible – that is, the density of the gas does not change even though the speed and static pressure change.

*** Newtonian versus
non-Newtonian Fluids**

In a general classification for fluids in order to behavior of them when it is exposed in shear stress, is the Newtonian or non-Newtonian fluids. A Newtonian fluid is defined to be a fluid whose shear stress is linearly proportional to the velocity gradient in the direction perpendicular to the plane of shear. This definition means regardless of the forces acting on a fluid, it continues to flow. For better lucidity, for example, water is a Newtonian fluid, because it continues to display fluid properties no matter how much it is stirred or mixed. Important fluids, like water as well as most gases, behave – to good approximation – as a Newtonian fluid under normal conditions on Earth. A slightly less rigorous definition is that the drag of small object being moved slowly through the fluid is proportional to the force applied to the object.

By contrast, stirring a non-Newtonian fluid can leave a “hole” behind. This will gradually fill up over time – this behavior is seen in materials such as pudding, oobleck, or sand (although sand isn’t strictly a fluid). Alternatively, stirring a non-Newtonian fluid can cause the viscosity to decrease, so the fluids appear “thinner” (this is seen in non-drip paints). There are many types of non-Newtonian fluids, as they are defined to be something that fails to obey a particular property. For example, most fluids with long molecular chains can react in a non-Newtonian manner.

*** Pathlines and
Streamlines**

What is pathline? The path which a fluid element traces out in space is called a pathline.

How about streamlines? For steady non-fluctuating flows where a path line is followed continuously by a number of fluid elements, the pathline is called streamline. A streamline is the imaginary line whose tangent gives the velocity of flow at all times; if the flow is steady, however in an unsteady flow, the streamline constantly changing and thus the tangent gives the velocity of an element at an instant of time. A common practice in analysis is tacking some of the walls of control volume to be along streamlines. Since there is no flow perpendicular to streamlines, only the flow across the other boundaries need be considered.

*** Introduction of
Control Volume Analysis**

A fluid dynamic
system can be analyzed using a ** control volume** which is an
imaginary surface enclosing a volume of interest. A control volume is a volume
in space that can be fixed or moving, and also it can be rigid or deformable.
Thus, we will have to write the most general case of the laws of mechanics to
deal with control volume (C.V).

The first equation we can write is the conservation of mass over time. Consider a system where mass flow is given by dm/dt, where m is the mass of the system. We have,

For steady flow, we have dm/dt=0, thus

And for incompressible flow, we have:

If consider a flow through a tube, we have, for steady flow,

And for
incompressible steady flow, A_{1}V_{1} = A_{2}V_{2}
.

Law of conservation of momentum as applied to a control volume state that

Where **V**
is the velocity vector and n is the unit vector normal to the control surface
at that point. Law conservation of energy (First Law of Thermodynamics):

Where e is the energy per unit mass.

*** Bernoulli’s Equation**

Bernoulli’s equation considers frictionless flow along a streamline. For steady, incompressible flow along a streamline, we have:

In fact, this equation is just the law of conservation of energy without the heat transfer and work. It may seem that Bernoulli’s equation can only be applied in a very limited set of situations, as it requires ideal conditions. However, since the equation applies to streamlines, we can consider a streamline still the area of interest where it is satisfied, and it might still give good results, i.e., you do not need a control volume for the actual analysis (although one is used in the derivation of the equation).

*** Energy in terms of
Head**

Bernoulli’s equation can be recast as:

This constant can be called head of the water, and is a representation of the amount of work that can be extracted from it. For example, for a water in a dam, at the inlet of the penstock, the pressure is low (atmospheric) while the velocity is high. The value of head calculated above remains constant (ignoring frictionless losses).

*** Mechanical Energy
Balance Equation**

Another variation of the Bernoulli’s equation is the mechanical energy balance equation. The mechanical energy balance equation is an equation for primary analysis of practical problems. The mechanical energy balance equation is useful when needing to consider things such as work or losses due to friction, or if there are differences between the outlet and inlet (such as pressure, velocity, and height).

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