In classical fluid dynamics, the Navier-Stokes equations for incompressible viscous fluids and its special (limiting) case the Euler equations for inviscid fluids are sets of non-linear partial differential equations that describes the spatiotemporal evolution of a fluid (gas). Both equations are derived from conservative principles and they model the behavior of some macroscopic variables namely: mass density, velocity and temperature.
The evolution of a fluid (gas) can also be described by the exact dynamics of the individual particles that constitutes the fluid (gas) in terms of Newton equations. However, this is complicated in the sense that in order to compute the time evolution of the fluid, one will have to solve a system of 6N first order differential equations with 6N unknowns constituting the position and velocity vectors. A perquisite for this computation is the knowledge of 6N initial
…show more content…
The Boltzmann equation provides a connection between the Newton equations and the spatiotemporal evolution of the macroscopic properties of a gas. In other words, the Boltzmann equation lies in between the two cases described above.
The Boltzmann equation is a nonlinear integro-differential equation and it describes the evolution of the density of particles (molecules) in a monatomic rarefied gas. It utilizes the fact that free streaming and collisions are the mechanisms responsible for the increase or decrease of particles in a given domain (small volume) of a fluid.
The range of application of the original Boltzmann equation has grown way beyond just dealing with a rarefied gas with one constituent and as a result, several generalizations of the Boltzmann equation have been developed and this includes the reactive Boltzmann
As the temperature increases, the movements of molecules also increase. This is the kinetic theory. When the temperature is increased the particles gain more energy and therefore move around faster. This gives the particles more of a chance with other particles and with more force.
so they collide more frequent as they have more energy. Therefore the reaction will speed up, a decreased temperature will have less energy it will move slower, collide less often slowing the reaction down. Varying the catalyst If I increase the catalyst , the particles will move a lot faster as the catalyst speeds up the rate of reaction, because they are moving
Matter is assumed to be composed of an enormous number of very tiny particles which are indestructible. Gas is a state of matter. These tiny particles are separated by relatively large distances, which interact elastically. This large space between the particles make it easy to compress a gas. Which gives low mass to volume ratio. Particles must be in continual motion. These particles are very fast (usually about 500 meters per second). The molecules in a gaseous state have enough kinetic energy to be essentially independent of each other.
is related to the ability of the gas to do expansion work. Heat capacity at constant volume, Cv can be described using the equipartition theory, which states that each mode of motion will contribute to a molecule or atom's energy.
5. In a gas increasing the pressure means molecules are more squashed up together, so there will be more collisions. My Investigation. I am going to investigate the concentration variable. I have chosen this because in my opinion it will be the easiest one to measure.
Introduction to Aerodynamics Aerodynamics is the study of the motion of fluids in the gas state and bodies in motion relative to the fluid/air. In other words, the study of aerodynamics is the study of fluid dynamics specifically relating to air or the gas state of matter. When an object travels through fluid/air there are two types of flow characteristics that happen, laminar and turbulent. Laminar flow is a smooth, steady flow over a smooth surface and it has little disturbance. Intuition would lead to the belief that this type of air flow would be desirable.
A differential equation is defined as an equation which relates an unknown function to one or more derivatives. When solved and transformed into its original equation in the form f(x), an exact value can be found at any given point. While some differential equations can be solved, it is important to realize that very few differential equations that come from "real world" problems can be solved explicitly, and often it is necessary to resort to numerical integration for their solutions. For the exploration I will be using an example in which a differential equation is used in the real world, specifically involving Newton's Law of Cooling. To approximate values at various points of the original equation (Which will be able to be found analytically for means of having the exact values to compare to the approximations. For purposes of the exploration, however, we will assume that the differential equation cannot be solved and we must thus resort to numerical methods), Euler's method will be used and compared with other methods to evaluate how accurate each one is when compared to the true value that is being found. Euler's method, being the earliest discovered approach to approximating solutions for differential equations, is an easy, yet rather inaccurate method when compared to more newly discovered methods that differ in their solving processes. I aim to start with Euler's method, and go on to using other methods in order of increasing accuracy for the same example.
If there is a large particle with a large surface area, and many small particles, the smaller particles have a higher chance of colliding with the larger particles. However, if there are small particles, and small particles of another compound, then the reaction rate would be slower, because the particles wouldn’t collide as easily as they would with particles of a bigger size. The third factor that affects collisions is the temperature. If there is a higher temperature, then the particles are able to move freer and faster, than they would if the temperature was lower. This means that the reaction rate would be faster, because the collisions of the particles are more frequent.
Monty, J.P., Jones M.B. and Ooi, A., 436-352 Thermofluids 3 – Compressible Flow. Lecture series distributed by the Dept. on Mech. & Manuf. Engineering, University of Melbourne, 2005.
On a more scientific note I am interested in mechanics of fluids. This interest was enforced last year when I had the opportunity to attend a lecture on fluid mechanics at P&G. At the conference I greatly expanded my knowledge regarding the physical aspect of fluids and their properties. In last year's AS course we have met a topic in this field. I will be applying ideas and knowledge gathered from last year for this investigation.
of a gas, liquid, or other substance-are excited so that more of them are at
In conclusion, the field of thermodynamics is a vast and diverse with applications that touch different aspects of the human lives. It is unfathomable to think where the world would be today without the revolutionary discoveries of Sadi Carnot and all the other early discoverers. Steam engines, power plants, air conditioning, locomotives and all the other comforts we enjoy today thanks to the applications of thermodynamics would not have been invented or have been as efficient as they are today. It is likely that the Carnot model will continue to be considered in constructing engines and with adequate research maybe the most efficient and ideal engine will be created.
The study of the relation between internal energy, heat, and work is the basic foundation in thermodynamics. How they interact can be applied to mechanics and experiments. For example, if you add heat to a piston, the gas contained inside will begin to expand and cause displacement, doing work. Gases are heavily studied in thermodynamics, because the internal energy is easier to account for. Gases only have kinetic energy because the potential energy is negligible since the far apart molecules cannot interact with each other. The four main types of thermodynamic processes- isovolumetric, isothermal, adiabatic, and isobaric-all involve the relation between work, heat, and internal energy on gases.
Today, Mathematical Physics has gone far. Due to the rapid advancement and the presence of modern technology like computers, direct numerical method using computers to formulate mathematical models become more and more essential. Using new technologies, the process involved in the formulation of mathematical models becomes simpler and inexpensive.