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Fick's Laws Of Diffusion
Fick's laws of diffusion describe diffusion and were first posited by Adolf Fick in 1855 on the basis of largely experimental results. They can be used to solve for the diffusion coefficient, . Fick's first law can be used to derive his second law which in turn is identical to the diffusion equation. ''Fick's first law'': Movement of particles from high to low concentration (diffusive flux) is directly proportional to the particle's concentration gradient. ''Fick's second law'': Prediction of change in concentration gradient with time due to diffusion. A diffusion process that obeys Fick's laws is called normal or Fickian diffusion; otherwise, it is called anomalous diffusion or non-Fickian diffusion. History In 1855, physiologist Adolf Fick first reported* * his now well-known laws governing the transport of mass through diffusive means. Fick's work was inspired by the earlier experiments of Thomas Graham, which fell short of proposing the fundamental laws for which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cardiac Output
In cardiac physiology, cardiac output (CO), also known as heart output and often denoted by the symbols Q, \dot Q, or \dot Q_ , edited by Catherine E. Williamson, Phillip Bennett is the volumetric flow rate of the heart's pumping output: that is, the volume of blood being pumped by a single Ventricle (heart), ventricle of the heart, per unit time (usually measured per minute). Cardiac output (CO) is the product of the heart rate (HR), i.e. the number of heartbeats per minute (bpm), and the stroke volume (SV), which is the volume of blood pumped from the left ventricle per beat; thus giving the formula: :CO = HR \times SV Values for cardiac output are usually denoted as L/min. For a healthy individual weighing 70 kg, the cardiac output at rest averages about 5 L/min; assuming a heart rate of 70 beats/min, the stroke volume would be approximately 70 mL. Because cardiac output is related to the quantity of blood delivered to various parts of the body, it is an important com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gradient
In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p gives the direction and the rate of fastest increase. The gradient transforms like a vector under change of basis of the space of variables of f. If the gradient of a function is non-zero at a point p, the direction of the gradient is the direction in which the function increases most quickly from p, and the magnitude of the gradient is the rate of increase in that direction, the greatest absolute directional derivative. Further, a point where the gradient is the zero vector is known as a stationary point. The gradient thus plays a fundamental role in optimization theory, where it is used to minimize a function by gradient descent. In coordinate-free terms, the gradient of a function f(\mathbf) may be defined by: df=\nabla f \cdot d\mathbf where df is the total infinitesimal change in f for a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
References
A reference is a relationship between Object (philosophy), objects in which one object designates, or acts as a means by which to connect to or link to, another object. The first object in this relation is said to ''refer to'' the second object. It is called a ''name'' for the second object. The next object, the one to which the first object refers, is called the ''referent'' of the first object. A name is usually a phrase or expression, or some other Symbol, symbolic representation. Its referent may be anything – a material object, a person, an event, an activity, or an abstract concept. References can take on many forms, including: a thought, a sensory perception that is Hearing (sense), audible (onomatopoeia), visual perception, visual (text), olfaction, olfactory, or tactile, emotions, emotional state, relationship with other, spacetime coordinates, symbolic system, symbolic or alpha-numeric grid, alpha-numeric, a physical object, or an energy projection. In some cases, meth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heat Kernel
In the mathematical study of heat conduction and diffusion, a heat kernel is the fundamental solution to the heat equation on a specified domain with appropriate boundary conditions. It is also one of the main tools in the study of the spectrum of the Laplace operator, and is thus of some auxiliary importance throughout mathematical physics. The heat kernel represents the evolution of temperature in a region whose boundary is held fixed at a particular temperature (typically zero), such that an initial unit of heat energy is placed at a point at time . Definition ] The most well-known heat kernel is the heat kernel of -dimensional Euclidean space , which has the form of a time-varying Gaussian function, K(t,x,y) = \frac \exp\left(-\frac\right), which is defined for all x,y\in\mathbb^d and t > 0. This solves the heat equation \left\{ \begin{aligned} & \frac{\partial K}{\partial t}(t,x,y) = \Delta_x K(t,x,y)\\ & \lim_{t \to 0} K(t,x,y) = \delta(x-y) = \delta_x(y) \end{aligned} ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Solution
In mathematics, a fundamental solution for a linear partial differential operator is a formulation in the language of distribution theory of the older idea of a Green's function (although unlike Green's functions, fundamental solutions do not address boundary conditions). In terms of the Dirac delta "function" , a fundamental solution is a solution of the inhomogeneous equation Here is ''a priori'' only assumed to be a distribution. This concept has long been utilized for the Laplacian in two and three dimensions. It was investigated for all dimensions for the Laplacian by Marcel Riesz. The existence of a fundamental solution for any operator with constant coefficients — the most important case, directly linked to the possibility of using convolution to solve an arbitrary right hand side — was shown by Bernard Malgrange and Leon Ehrenpreis, and a proof is available in Joel Smoller (1994). In the context of functional analysis, fundamental solutions are usually ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heat Equation
In mathematics and physics (more specifically thermodynamics), the heat equation is a parabolic partial differential equation. The theory of the heat equation was first developed by Joseph Fourier in 1822 for the purpose of modeling how a quantity such as heat diffuses through a given region. Since then, the heat equation and its variants have been found to be fundamental in many parts of both pure and applied mathematics. Definition Given an open subset of and a subinterval of , one says that a function is a solution of the heat equation if : \frac = \frac + \cdots + \frac, where denotes a general point of the domain. It is typical to refer to as time and as spatial variables, even in abstract contexts where these phrases fail to have their intuitive meaning. The collection of spatial variables is often referred to simply as . For any given value of , the right-hand side of the equation is the Laplace operator, Laplacian of the function . As such, the heat equation is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laplacian
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the nabla operator), or \Delta. In a Cartesian coordinate system, the Laplacian is given by the sum of second partial derivatives of the function with respect to each independent variable. In other coordinate systems, such as cylindrical and spherical coordinates, the Laplacian also has a useful form. Informally, the Laplacian of a function at a point measures by how much the average value of over small spheres or balls centered at deviates from . The Laplace operator is named after the French mathematician Pierre-Simon de Laplace (1749–1827), who first applied the operator to the study of celestial mechanics: the Laplacian of the gravitational potential due to a given mass density distribution is a constant multiple of that de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Differential Equation
In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to how is thought of as an unknown number solving, e.g., an algebraic equation like . However, it is usually impossible to write down explicit formulae for solutions of partial differential equations. There is correspondingly a vast amount of modern mathematical and scientific research on methods to numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such as existence, uniqueness, regularity and stability. Among the many open questions are the existence ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kinetic Theory Of Gases
The kinetic theory of gases is a simple classical model of the thermodynamic behavior of gases. Its introduction allowed many principal concepts of thermodynamics to be established. It treats a gas as composed of numerous particles, too small to be seen with a microscope, in constant, random motion. These particles are now known to be the atoms or molecules of the gas. The kinetic theory of gases uses their collisions with each other and with the walls of their container to explain the relationship between the macroscopic properties of gases, such as volume, pressure, and temperature, as well as transport properties such as viscosity, thermal conductivity and mass diffusivity. The basic version of the model describes an ideal gas. It treats the collisions as perfectly elastic and as the only interaction between the particles, which are additionally assumed to be much smaller than their average distance apart. Due to the time reversibility of microscopic dynamics ( microsco ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fugacity
In thermodynamics, the fugacity of a real gas is an effective partial pressure which replaces the mechanical partial pressure in an accurate computation of chemical equilibrium. It is equal to the pressure of an ideal gas which has the same temperature and molar Gibbs free energy as the real gas. Fugacities are determined experimentally or estimated from various models such as a Van der Waals gas that are closer to reality than an ideal gas. The real gas pressure and fugacity are related through the dimensionless fugacity coefficient \varphi = \frac. For an ideal gas, fugacity and pressure are equal, and so . Taken at the same temperature and pressure, the difference between the molar Gibbs free energies of a real gas and the corresponding ideal gas is equal to . The fugacity is closely related to the thermodynamic activity. For a gas, the activity is simply the fugacity divided by a reference pressure to give a dimensionless quantity. This reference pressure is called the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Universal Gas Constant
The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature, temperature increment per amount of substance, rather than energy per temperature increment per ''particle''. The constant is also a combination of the constants from Boyle's law, Charles's law, Avogadro's law, and Gay-Lussac's law. It is a physical constant that is featured in many fundamental equations in the physical sciences, such as the ideal gas law, the Arrhenius equation, and the Nernst equation. The gas constant is the constant of proportionality that relates the energy scale in physics to the temperature scale and the scale used for amount of substance. Thus, the value of the gas constant ultimately derives from historical decisions and accidents in the setting of units of energy, temperature and amount of substance. The Boltzmann constan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chemical Potential
In thermodynamics, the chemical potential of a Chemical specie, species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potential of a species in a mixture is defined as the rate of change of Thermodynamic free energy, free energy of a thermodynamic system with respect to the change in the number of atoms or molecules of the species that are added to the system. Thus, it is the partial derivative of the free energy with respect to the amount of the species, all other species' concentrations in the mixture remaining constant. When both temperature and pressure are held constant, and the number of particles is expressed in moles, the chemical potential is the partial Molar concentration, molar Gibbs free energy. At chemical equilibrium or in phase equilibrium, the total sum of the product of chemical potentials and stoichiometric coefficients is zero, as the free en ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
