Fluid Thread Breakup
Fluid thread breakup is the process by which a single mass of fluid breaks into several smaller fluid masses. The process is characterized by the elongation of the fluid mass forming thin, thread-like regions between larger nodules of fluid. The thread-like regions continue to thin until they break, forming individual droplets of fluid. Thread breakup occurs where two fluids or a fluid in a vacuum form a free surface with surface energy. If more surface area is present than the minimum required to contain the volume of fluid, the system has an excess of surface energy. A system not at the minimum energy state will attempt to rearrange so as to move toward the lower energy state, leading to the breakup of the fluid into smaller masses to minimize the system surface energy by reducing the surface area. The exact outcome of the thread breakup process is dependent on the surface tension, viscosity, density, and diameter of the thread undergoing breakup. History The examination of drople ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Surface Energy
In surface science, surface energy (also interfacial free energy or surface free energy) quantifies the disruption of intermolecular bonds that occurs when a surface is created. In solid-state physics, surfaces must be intrinsically less energetically favorable than the bulk of the material (that is, the atoms on the surface must have more energy than the atoms in the bulk), otherwise there would be a driving force for surfaces to be created, removing the bulk of the material by sublimation. The surface energy may therefore be defined as the excess energy at the surface of a material compared to the bulk, or it is the work required to build an area of a particular surface. Another way to view the surface energy is to relate it to the work required to cut a bulk sample, creating two surfaces. There is "excess energy" as a result of the now-incomplete, unrealized bonding between the two created surfaces. Cutting a solid body into pieces disrupts its bonds and increases the surfac ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Green's Function
In mathematics, a Green's function (or Green function) is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means that if L is a linear differential operator, then * the Green's function G is the solution of the equation where \delta is Dirac's delta function; * the solution of the initial-value problem L y = f is the convolution Through the superposition principle, given a linear ordinary differential equation (ODE), one can first solve for each , and realizing that, since the source is a sum of delta functions, the solution is a sum of Green's functions as well, by linearity of . Green's functions are named after the British mathematician George Green, who first developed the concept in the 1820s. In the modern study of linear partial differential equations, Green's functions are studied largely from the point of view of fundamental solutions instead. Under many ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Bessel Function
Bessel functions, named after Friedrich Bessel who was the first to systematically study them in 1824, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary complex number \alpha, which represents the ''order'' of the Bessel function. Although \alpha and -\alpha produce the same differential equation, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly smooth functions of \alpha. The most important cases are when \alpha is an integer or half-integer. Bessel functions for integer \alpha are also known as cylinder functions or the cylindrical harmonics because they appear in the solution to Laplace's equation in cylindrical coordinates. Spherical Bessel functions with half-integer \alpha are obtained when solving the Helmholtz equation in spherical coordinates. Applications Bessel's equation arises when finding separa ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Plateau–Rayleigh Instability
In fluid dynamics, the Plateau–Rayleigh instability, often just called the Rayleigh instability, explains why and how a falling stream of fluid breaks up into smaller packets with the same total volume but less surface area per droplet. It is related to the Rayleigh–Taylor instability and is part of a greater branch of fluid dynamics concerned with fluid thread breakup. This fluid instability is exploited in the design of a particular type of Ink jet#Continuous inkjet, ink jet technology whereby a jet of liquid is perturbed into a steady stream of droplets. The driving force of the Plateau–Rayleigh instability is that liquids, by virtue of their surface tensions, tend to minimize their surface area. A considerable amount of work has been done recently on the final pinching profile by attacking it with self-similarity, self-similar solutions. History The Plateau–Rayleigh instability is named for Joseph Plateau and John Strutt, 3rd Baron Rayleigh, Lord Rayleigh. In 187 ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Wavenumber
In the physical sciences, the wavenumber (or wave number), also known as repetency, is the spatial frequency of a wave. Ordinary wavenumber is defined as the number of wave cycles divided by length; it is a physical quantity with dimension of reciprocal length, expressed in SI units of cycles per metre or reciprocal metre (m−1). Angular wavenumber, defined as the wave phase divided by time, is a quantity with dimension of angle per length and SI units of radians per metre. They are analogous to temporal frequency, respectively the '' ordinary frequency'', defined as the number of wave cycles divided by time (in cycles per second or reciprocal seconds), and the ''angular frequency'', defined as the phase angle divided by time (in radians per second). In multidimensional systems, the wavenumber is the magnitude of the '' wave vector''. The space of wave vectors is called ''reciprocal space''. Wave numbers and wave vectors play an essential role in optics and the physics ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Ohnesorge Number
The Ohnesorge number (Oh) is a dimensionless number that relates the viscous forces to inertial and surface tension forces. The number was defined by Wolfgang von Ohnesorge in his 1936 doctoral thesis. It is defined as: : \mathrm = \frac = \frac \sim \frac Where * ''μ'' is the dynamic viscosity of the liquid * ''ρ'' is the density of the liquid * ''σ'' is the surface tension * ''L'' is the characteristic length scale (typically drop diameter) * Re is the Reynolds number * We is the Weber number Applications The Ohnesorge number for a 3 mm diameter rain drop is typically ~0.002. Larger Ohnesorge numbers indicate a greater influence of the viscosity. This is often used to relate to free surface fluid dynamics such as dispersion of liquids in gases and in spray technology. English translation: In inkjet printing, liquids whose Ohnesorge number are in the range 0.1 < ''Oh'' < 1.0 are jettable (1 [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Weber Number
The Weber number (We) is a dimensionless number in fluid mechanics that is often useful in analysing fluid flows where there is an interface between two different fluids, especially for multiphase flows with strongly curved surfaces. It is named after Moritz Weber (1871–1951). It can be thought of as a measure of the relative importance of the fluid's inertia compared to its surface tension. The quantity is useful in analyzing thin film flows and the formation of droplets and bubbles. Mathematical expression The Weber number may be written as: :\mathrm = \frac = \frac = \frac where * \rho is the density of the fluid ( kg/ m3). * v is its velocity (m/ s). * l is its characteristic length, typically the droplet diameter (m). * \sigma is the surface tension ( N/m). * \rho\,v^2 is the inertial or dynamic pressure scale. * \sigma/l is the Laplace pressure scale. The above is the force perspective to define the Weber number. We can also define it using energy perspec ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Reynolds Number
In fluid dynamics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between Inertia, inertial and viscous forces. At low Reynolds numbers, flows tend to be dominated by laminar flow, laminar (sheet-like) flow, while at high Reynolds numbers, flows tend to be turbulence, turbulent. The turbulence results from differences in the fluid's speed and direction, which may sometimes intersect or even move counter to the overall direction of the flow (Eddy (fluid dynamics), eddy currents). These eddy currents begin to churn the flow, using up energy in the process, which for liquids increases the chances of cavitation. The Reynolds number has wide applications, ranging from liquid flow in a pipe to the passage of air over an aircraft wing. It is used to predict the transition from laminar–turbulent transition, laminar to turbulent flow and is used in the scaling of similar but different-sized fl ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Asymptotic Expansion
In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to a given function as the argument of the function tends towards a particular, often infinite, point. Investigations by revealed that the divergent part of an asymptotic expansion is latently meaningful, i.e. contains information about the exact value of the expanded function. The theory of asymptotic series was created by Poincaré (and independently by Stieltjes) in 1886. The most common type of asymptotic expansion is a power series in either positive or negative powers. Methods of generating such expansions include the Euler–Maclaurin summation formula and integral transforms such as the Laplace and Mellin transforms. Repeated integration by parts will often lead to an asymptotic expansion. Since a '' convergent'' Taylor s ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Self-similarity
In mathematics, a self-similar object is exactly or approximately similar to a part of itself (i.e., the whole has the same shape as one or more of the parts). Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many scales. Self-similarity is a typical property of fractals. Scale invariance is an exact form of self-similarity where at any magnification there is a smaller piece of the object that is similar to the whole. For instance, a side of the Koch snowflake is both symmetrical and scale-invariant; it can be continually magnified 3x without changing shape. The non-trivial similarity evident in fractals is distinguished by their fine structure, or detail on arbitrarily small scales. As a counterexample, whereas any portion of a straight line may resemble the whole, further detail is not revealed. Peitgen ''et al.'' explain the concept as such: Since mathematically, a fractal may sho ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Fourier Transform
In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the transform is a complex-valued function of frequency. The term ''Fourier transform'' refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made, the output of the operation is sometimes called the frequency domain representation of the original function. The Fourier transform is analogous to decomposing the sound of a musical chord into the intensities of its constituent pitches. Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa, a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statist ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |