Kajal Work: March 2011

Sunday, March 27, 2011

Viscosity

Viscosity is a measure of the resistance of a fluid which is being deformed by either shear stress or tensile stress.
In everyday terms (and for fluids only), viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity. Put simply, the less viscous the fluid is, the greater its ease of movement .

Shear stress in fluids

Any real fluids (liquids and gases included) moving along solid boundary will incur a shear stress on that boundary.
The speed of the fluid at the boundary (relative to the boundary) is zero, but at some height from the boundary the flow speed must equal that of the fluid.
The region between these two points is aptly named the boundary layer. For all Newtonian fluids in laminar flow the shear stress is proportional to the strain rate in the fluid where the viscosity is the constant of proportionality.
However for Non Newtonian fluids, this is no longer the case as for these fluids the viscosity is not constant.
The shear stress is imparted onto the boundary as a result of this loss of velocity. The shear stress, for a Newtonian fluid, at a surface element parallel to a flat plate, at the point y, is given by:

$\tau (y) = \mu \frac{\partial u}{\partial y}~~,$

where

μ

is the dynamic viscosity of the fluid;

u

is the velocity of the fluid along the boundary;

y

is the height above the boundary.

A shear stress, $\tau\,$ is applied to the top of the square while the bottom is held in place. This stress results in a strain, or deformation, changing the square into a parallelogram.

Specifically, the wall shear stress is defined as:

$\tau_\mathrm{w} \equiv \tau(y=0)= \mu \left.\frac{\partial u}{\partial y}\right|_{y = 0}~~.$

In case of wind, the shear stress at the boundary is called wind stress.

Viscosity:

Laminar shear of fluid between two plates. Friction between the fluid and the moving boundaries causes the fluid to shear. The force required for this action is a measure of the fluid's viscosity. This type of flow is known as a Couette flow.
In general, in any flow, layers move at different velocities and the fluid's viscosity arises from the shear stress between the layers that ultimately opposes any applied force.
The relationship between the shear stress and the velocity gradient can be obtained by considering two plates closely spaced at a distance y, and separated by a homogeneous substance.
Assuming that the plates are very large, with a large area A, such that edge effects may be ignored, and that the lower plate is fixed, let a force F be applied to the upper plate. If this force causes the substance between the plates to undergo shear flow with a velocity gradient u(as opposed to just shearing elastically until the shear stress in the substance balances the applied force), the substance is called a fluid.
The applied force is proportional to the area and velocity gradient in the fluid and inversely proportional to the distance between the plates. Combining these three relations results in the equation:

$F=\mu A \frac{u}{y}$ ,

where μ is the proportionality factor called viscosity.

This equation can be expressed in terms of shear stress

$\tau=\frac{F}{A}$ .

Thus as expressed in differential form by Isaac Newton for straight, parallel and uniform flow, the shear stress between layers is proportional to the velocity gradient in the direction perpendicular to the layers:

$\tau=\mu \frac{\partial u}{\partial y}$

Hence, through this method, the relation between the shear stress and the velocity gradient can be obtained.
Note that the rate of shear deformation is $\frac{u} {y}$ which can be also written as a shear velocity, $\frac{du} {dy}$ .
James Clerk Maxwell called viscosity fugitive elasticity because of the analogy that elastic deformation opposes shear stress in solids, while in viscous fluids, shear stress is opposed by rate of deformation.

Pressure

Pressure (the symbol: P) is the force per unit area applied in a direction perpendicular to the surface of an object.

Pressure is an effect which occurs when a force is applied on a surface. Pressure is the amount of force acting on a unit area. The symbol of pressure is P.

Formula

$P = \frac{F}{A}\ \mbox{or}\ P = \frac{dF_n}{dA}$ Mathematically:

where:

P is the pressure,

F is the normal force,

A is the area.

Pressure is a scalar quantity. It relates the vector surface element (a vector normal to the surface) with the normal force acting on it. The pressure is the scalar proportionality constant that relates the two normal vectors:

$d\mathbf{F}_n=-P\,d\mathbf{A} = -P\,\mathbf{n}\,dA$

The minus sign comes from the fact that the force is considered towards the surface element, while the normal vector points outwards.

The SI unit for pressure is the pascal (Pa), equal to one newton per square meter (N/m² or kg·m⁻¹·s⁻²).

This special name for the unit was added in 1971; before that, pressure in SI was expressed simply as N/m²

Although pressure itself is a scalar, we can define a pressure force to be equal to the pressure (force/area) times the surface area in a direction perpendicular to the surface. The pressure force is a vector quantity.
Pressure forces have some unique qualities as compared to gravitational or mechanical forces. In the figure shown above on the right, we have a red gas that is confined in a box.
A mechanical force is applied to the top of the box. The pressure force within the box opposes the applied force according to Newton's third law of motion. The scalar pressure equals the external force divided by the area of the top of the box. Inside the gas, the pressure acts in all directions. So the pressure pushes on the bottom of the box and on the sides.
This is different from simple solid mechanics. If the red gas were a solid, there would be no forces applied to the sides of the box; the applied force would be simply transmitted to the bottom. But in a gas, because the molecules are free to move about and collide with one another, a force applied in the vertical direction causes forces in the horizontal direction.

Fluid pressure

Fluid pressure is the pressure at some point within a fluid, such as water or air.

Fluid pressure occurs in one of two situations:

an open condition, such as the ocean, a swimming pool, or the atmosphere; or
a closed condition, such as a water line or a gas line.

Pressure in open conditions usually can be approximated as the pressure in "static" or non-moving conditions (even in the ocean where there are waves and currents), because the motions create only negligible changes in the pressure. Such conditions conform with principles of fluid statics.
The pressure at any given point of a non-moving (static) fluid is called the hydrostatic pressure.

Closed bodies of fluid are either "static," when the fluid is not moving, or "dynamic," when the fluid can move as in either a pipe or by compressing an air gap in a closed container. The pressure in closed conditions conforms with the principles of fluid dynamics.

Liquid pressure or pressure at depth

Used with liquid columns of constant density or at a depth within a substance (ex. pressure at 20km depth in the earth).

P = g d h

Where:

P is Pressure

g is gravity at the surface of overlaying material

d is density of liquid or overlaying material

h is height of liquid or depth within material

Kinetic Theory

The temperature of a gas is a measure of the mean kinetic energy of the gas. The molecules are in constant random motion, and there is an energy (mass x square of the velocity) associated with that motion. The higher the temperature, the greater the motion.

In a solid, the location of the molecules relative to each other remains almost constant. But in a gas, the molecules can move around and interact with each other and with their surroundings in different ways. As mentioned above, there is always a random component of molecular motion. The entire fluid can be made to move as well in an ordered motion (flow).

The ordered motion is superimposed, or added to, the normal random motion of the molecules. At the molecular level, there is no distinction between the random component and the ordered component.

We measure the pressure produced by the random component as the static pressure. The pressure produced by the ordered motion is called dynamic pressure. And Bernoulli's equation tells us that the sum of the static and dynamic pressure is the total pressure which we can also measure.

Temperature

We've heard the TV meteorologist give the daily value of the temperature of theatmosphere (15 degrees Celsius, for example).
We know that a hot object has a high temperature, and a cold object has a low temperature. And we know that the temperature of an object changes when we heat the object or cool it.
Scientists, however, must be more precise than simply describing an object as "hot" or "cold." An entire branch of physics, called thermodynamics, is devoted to studying the temperature of objects and the transfer of heat between objects of different temperatures.
let you study how temperature varies with height through the atmosphere.

Turning to the large scale, the temperature of a gas is something that we can determine qualitatively with our senses. We can sense that one gas is hotter than another gas and therefore has a higher temperature. But to determine the temperature quantitatively, to assign a number, we must use some principles from thermodynamics:

The first principle is the observation that the temperature of an object can affect some physical property of the object, such as the length of a solid, or the gas pressure in a closed vessel, or the electrical resistance of a wire. You can explore the effects of temperature on the pressure of a gas at the animated gas lab.
The second principle is the definition of thermodynamic equilibrium between two objects. Two objects are in thermodynamic equilibrium when they have the same temperature.
And the final principle is the observation that if two objects of different temperatures are brought into contact with one another, they will eventually establish a thermodynamic equilibrium. The word "eventually" is important. Insulating materials reach equilibrium after a very long time, while conducting materials reach equilibrium very quickly.

With these three thermodynamic principles, we can construct a device for measuring temperature, a thermometer, which assigns a number to the temperature of an object. When the thermometer is brought into contact with another object, it quickly establishes a thermodynamic equilibrium. By measuring the thermodynamic effect on some physical property of the thermometer at some fixed conditions, like the boiling point and freezing point of water, we can establish a scale for assigning temperature values.

The Celsius scale, designated with a C, uses the freezing point of pure water as the zero point and the boiling point as 100 degrees with a linear scale in between these extremes.

The Fahrenheit scale, designated with an F, is a lot more confusing. It originally used the freezing point of sea water as the zero point and the freezing point of pure water as 30 degrees, which made the temperature of a healthy person equal to 96 degrees.

On this scale, the boiling point of pure water was 212 degrees. So Fahrenheit adjusted the scale to make the boiling point of pure water 212 and the freezing point of pure water 32, which gave 180 degrees between the two reference points.

180 degrees was chosen because it is evenly divisible by 2, 3, 4, 5, and 6. On the new temperature scale, the temperature of a healthy person is 98.6 degrees F. Because there are 100 degrees C and 180 degrees F between the same reference condition:

1 degree C = 1 degree F * 100 / 180

= 1 degree F * 5 / 9

Since the scales start at different zero points, we can convert from the temperature on the Fahrenheit scale (TF) to the temperature on the Celsius scale (TC) by using this equation:

TF = 32 + (9 / 5) * TC

Of course, you can have temperatures below the freezing point of water and these are assigned negative numbers. When scientists began to study the coldest possible temperature, they determined an absolute zero at which molecular kinetic energy is a minimum (but not strictly zero!).

They found this value to be at -273.16 degrees C. Using this point as the new zero point we can define another temperature scale called the absolute temperature. If we keep the size of a single degree to be the same as the Celsius scale, we get a temperature scale which has been named after Lord Kelvin and designated with a K. Then:

K = C + 273.16

There is a similar absolute temperature corresponding to the Fahrenheit degree. It is named after the scientist Rankine and designated with an R.

R = F + 459.69

Absolute temperatures are used in the equation of state, the derivation of the state variables enthalpy, and entropy, and determining the speed of sound.

Density

Density is a scalar quantity; it has a magnitude but no direction associated with it.
An important property of any fluid is its density. Density is defined as the mass of an object divided by its volume, and most of our experiences with density involve solids.
We know that some objects are heavier than other objects, even though they are the same size. A brick and a loaf of bread are about the same size, but a brick is heavier--it is more dense.
Among metals, aluminum is less dense than iron. That's why airplanes and rockets and some automobile parts are made from aluminum. For the same volume of material, one metal weighs less than another if it has a lower density.
Since density is defined to be the mass divided by the volume, density depends directly on the size of the container in which a fixed mass of gas is confined.

As a simple example, consider Case #1 on our figure. We have 26 molecules of a mythical gas. Each molecule has a mass of 20 grams (.02 kilograms), so the mass of this gas is .52 kg. We have confined this gas in a rectangular tube that is 1 meter on each side and 2 meters high. We are viewing the tube from the front, so the dimension into the slide is 1 meter for all the cases considered. The volume of the tube is 2 cubic meters, so the density is .26 kg/cubic meter.
This corresponds to air density at about 13 kilometers altitude. If the size of our container were decreased to 1 meter on all sides, as in Case #3, and we kept the same number of molecules, that density would increase to .52 kg/cubic meter. Notice that we have the same amount of material; it is just contained in a smaller volume. How we decrease the volume is very important for the final value of pressure and temperature. (This example REALLY works only for a very large number of molecules moving at random. Case #2 is just an illustration.)
Another way to obtain the same density for a smaller volume is to remove molecules from the container. In Case #4, the container is the same size as in Case #3, but the number of molecules (the mass) has been decreased to only 13 molecules. The density is .26 kg/cubic meter, which is the same density seen in the blue box of Case #2 and throughout Case #1. A careful study of these four cases will help you understand the meaning of gas density.

Note: For Density Problems, use the formula D=M/V
For problems 1-4, use the mass and volume given to find the density of the column of air.

(1) Mass = 10 kg & Volume = 8m³
Density =_________
(2) Mass = 15 kg & Volume = 10m³

Density = _________
(3) Mass = 5 kg & Volume = 2m x 2m x 2m (Hint: You first need to calculate the volume!)

Density = _________
(4) Mass = 35 kg & Volume = 3m x 3m x 3m

Density = _________

Now answer the following questions.

(5) Looking at your answers above, which density will produce the greatest amount of lift(Raise)?

(6) Which will produce the least amount of lift?

(7) Given your answers to Problems 5 & 6 above, briefly state what effect the density of the air has on the lift of an airplane.

Saturday, March 26, 2011

Mass Flow Rate

The conservation of mass is a fundamental concept of physics. Within some problem domain, the amount of mass remains constant --mass is neither created nor destroyed.
The mass of any object is simply the volume that the object occupies times the density of the object. For a fluid (a liquid or a gas) the density, volume, and shape of the object can all change within the domain with time. And mass can move through the domain.

On the figure, we show a flow of gas through a constricted tube. There is no accumulation(Build up/growth) or destruction of mass through the tube; the same amount of mass leaves the tube as enters the tube.
At any plane perpendicular to the center line of the tube, the same amount of mass passes through. We call the amount of mass passing through a plane the mass flow rate.
The conservation of mass (continuity) tells us that the mass flow rate through a tube is a constant. We can determine the value of the mass flow rate from the flow conditions.

If the fluid initially passes through an area A at velocity V, we can define a volume of mass to be swept out in some amount of time t. The volume v is:

v = A * V * tA units check gives area x length/time x time = area x length = volume. The mass m contained in this volume is simply density r times the volume.

m = r * A * V * tTo determine the mass flow rate mdot, we divide the mass by the time. The resulting definition of mass flow rate is shown on the slide in red.

mdot = r * A * V

Speed of Sound

Air is a gas, and a very important property of any gas is the speed of sound through the gas.
Why are we interested in the speed of sound? The speed of "sound" is actually the speed of transmission of a small disturbance through a medium.
Sound itself is a sensation created in the human brain in response to sensory inputs from the inner ear.
Disturbances are transmitted through a gas as a result of collisions(accident/crash) between the randomly moving molecules in the gas.
The conditions in the gas are the same before and after the disturbance passes through. Because the speed of transmission depends on molecular collisions, the speed of sound depends on the state of the gas.

The speed of sound is a constant within a given gas and the value of the constant depends on the type of gas (air, pure oxygen, carbon dioxide, etc.) and the temperature of the gas. An analysis based on conservation of mass and momentum(force) shows that the speed of sound a is equal to the square root of the ratio of specific heats g times the gas constant R times the temperature T.

a = sqrt [g * R * T]

Notice that the temperature must be specified on an absolute scale (Kelvin or Rankine). The dependence on the type of gas is included in the gas constant R. which equals the universal gas constant divided by the molecular weight of the gas, and the ratio of specific heats.

Conservation of Mass

Solid Mechanics

The conservation of mass is a fundamental concept of physics along with the conservation of energy and the conservation of momentum.
Within some problem domain, the amount of mass remains constant--mass is neither created nor destroyed. This seems quite obvious, as long as we are not talking about black holes or very exotic physics problems.
The mass of any object can be determined by multiplying the volume of the object by the density of the object.
When we move a solid object, as shown at the top of the slide, the object retains its shape, density, and volume. The mass of the object, therefore, remains a constant between state "a" and state "b."

Fluid Statics

In the center of the figure, we consider an amount of a static fluid , liquid or gas. If we change the fluid from some state "a" to another state "b" and allow it to come to rest, we find that, unlike a solid, a fluid may change its shape.
The amount of fluid, however, remains the same. We can calculate the amount of fluid by multiplying the density times the volume. Since the mass remains constant, the product of the density and volume also remains constant. (If the density remains constant, the volume also remains constant.) The shape can change, but the mass remains the same.

Fluid Dynamics

Finally, at the bottom of the slide, we consider the changes for a fluid that is moving through our domain. There is no accumulation(Build up) or depletion (Running Down/ Reduction) of mass, so mass is conserved within the domain. Since the fluid is moving, defining the amount of mass gets a little tricky(complicated).
Let's consider an amount of fluid that passes through point "a" of our domain in some amount of time t. If the fluid passes through an area A at velocity V, we can define the volume Vol to be:

Vol = A * V * t

A units check gives area x length/time x time = area x length = volume. Thus the mass at point "a" ma is simply density r times the volume at "a".

ma = (r * A * V * t)a

If we compare the flow through another point in the domain, point "b," for the same amount of time t, we find the mass at "b"mb to be the density times the velocity times the area times the time at "b":

mb = (r * A * V * t)b

From the conservation of mass, these two masses are the same and since the times are the same, we can eliminate the time dependence.

(r * A * V)a = (r * A * V)b
or r * A * V = constant

The conservation of mass gives us an easy way to determine the velocity of flow in a tube if the density is constant.
If we can determine (or set) the velocity at some known area, the equation tells us the value of velocity for any other area.
In our animation, the area of "b" is one half the area of "a." Therefore, the velocity at "b" must be twice the velocity at "a." If we desire a certain velocity in a tube, we can determine the area necessary to obtain that velocity.
This information is used in the design of wind tunnels. The quantity density times area times velocity has the dimensions of mass/time and is called the mass flow rate. This quantity is an important parameter in determining the thrust (drive/move forward) produced by a propulsion system.
As the speed of the flow approaches the speed of sound the density of the flow is no longer a constant and we must then use a compressible form of the mass flow rate equation.
The conservation of mass equation also occurs in a differential form as part of the Navier-Stokes equations of fluid flow.

Fluids Pressure & Depth

A fluid is a substance that flows easily. Gases and liquids are fluids, although sometimes the dividing line between liquids and solids is not always clear. Because of their ability to flow, fluids can exert buoyant forces, multiply forces in a hydraulic systems, allow aircraft to fly and ships to float.

The topic that this page will explore will be pressure and depth. If a fluid is within a container then the depth of an object placed in that fluid can be measured. The deeper the object is placed in the fluid, the more pressure it experiences. This is because is the weight of the fluid above it. The more dense the fluid above it, the more pressure is exerted on the object that is submerged, due to the weight of the fluid.

The formula that gives the P pressure on an object submerged in a fluid is:

P = r * g * h

where

r (rho) is the density of the fluid, ( Density= mass/volume)
g is the acceleration of gravity
h is the height of the fluid above the object

If the container is open to the atmosphere above, the added pressure must be included if one is to find the total pressure on an object.

The total pressure is the same as absolute pressure on pressure gauges(measure) readings, while the gauge(measure) pressure is the same as the fluid pressure alone, not including atmospheric pressure.

P_total = P_atmosphere + P_fluid

P_total = P_atmosphere + ( r * g * h )

A Pascal is the unit of pressure in the metric system. It represents 1 newton/m²

Example:
Find the pressure on a scuba diver when she is 12 meters below the surface of the ocean. Assume standard atmospheric conditions.
Solution:
The density of sea water is 1.03 X 10³ kg/m³ and the atmospheric pressure is 1.01 x 10⁵ N/m².

P_fluid = r g h = (1.03 x10 ³ kg/m³) (9.8 m/s²) (12 m)

= 1.21 x 10⁵ Newtons/m²P_total = P_atmosphere + P_fluid = (1.01 x 10⁵) + (1.21 x 10⁵ ) Pa

= 2.22 x 10² kPa (kilo Pascals)

Exercises :

What is the pressure experienced at a point on the bottom of a swimming pool 9 meters in depth? The density of water is 1.00 x 10³ kg/m³. (Answer=189kp)
The interior of a submarine located at a depth of 45 meters is maintained at normal atmospheric conditions. Find the total force exerted on a 20 cm by 20 cm square window. Use the density of sea water given above.
(Answer=1.81 x 10.4 newtons)
How many atmospheres is a depth of 100 meters of ocean water?
(Answer= 11 atm)
If the weight density of pure water is 62 pounds/ft³, find the weight of water in a swimming pool whose dimensions are 20 ft by 10 ft by 6 feet.
(Answer=74,400 pounds)
An airplane in level flight whose mass is 20,000 kg has a wing area of 60 m². What is the pressure difference between the upper and lower surfaces of its wing? Express your answer in atmospheres.
(Answer=.032 atm)

What is Fluid?

Fluid: a substance that flows; any liquid or gas .

Sentences:

Fluids like tea and juice become round like balls when they are floating on the space station.

Water is a fluid.