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Burgers Vortex
In fluid dynamics, the Burgers vortex or Burgers–Rott vortex is an exact solution to the Navier–Stokes equations governing viscous flow, named after Jan Burgers and Nicholas Rott. The Burgers vortex describes a stationary, self-similarity, self-similar flow. An inward, radial flow, tends to concentrate vorticity in a narrow column around the symmetry axis. At the same time, viscosity, viscous diffusion tends to spread the vorticity. The stationary Burgers vortex arises when the two effects balance. The Burgers vortex, apart from serving as an illustration of the vortex stretching mechanism, may describe such flows as tornados, where the vorticity is provided by continuous convection-driven vortex stretching. Flow field The flow for the Burgers vortex is described in cylindrical (r,\theta,z) coordinates. Assuming axial symmetry (no \theta-dependence), the flow field associated with the axisymmetric stagnation point flow is considered: :v_r= -\alpha r, :v_z=2\alpha z, :v_\thet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fluid Dynamics
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) and hydrodynamics (the study of liquids in motion). Fluid dynamics has a wide range of applications, including calculating forces and moments on aircraft, determining the mass flow rate of petroleum through pipelines, predicting weather patterns, understanding nebulae in interstellar space and modelling fission weapon detonation. Fluid dynamics offers a systematic structure—which underlies these practical disciplines—that embraces empirical and semi-empirical laws derived from flow measurement and used to solve practical problems. The solution to a fluid dynamics problem typically involves the calculation of various properties of the fluid, such as flow velocity, pressure, density, and temperature, as functions of space and ti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gamma Function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except the non-positive integers. For every positive integer , \Gamma(n) = (n-1)!\,. Derived by Daniel Bernoulli, for complex numbers with a positive real part, the gamma function is defined via a convergent improper integral: \Gamma(z) = \int_0^\infty t^ e^\,dt, \ \qquad \Re(z) > 0\,. The gamma function then is defined as the analytic continuation of this integral function to a meromorphic function that is holomorphic in the whole complex plane except zero and the negative integers, where the function has simple poles. The gamma function has no zeroes, so the reciprocal gamma function is an entire function. In fact, the gamma function corresponds to the Mellin transform of the negative exponential function: \Gamma(z) = \mathcal M ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Potential Flow
In fluid dynamics, potential flow (or ideal flow) describes the velocity field as the gradient of a scalar function: the velocity potential. As a result, a potential flow is characterized by an irrotational velocity field, which is a valid approximation for several applications. The irrotationality of a potential flow is due to the curl of the gradient of a scalar always being equal to zero. In the case of an incompressible flow the velocity potential satisfies Laplace's equation, and potential theory is applicable. However, potential flows also have been used to describe compressible flows. The potential flow approach occurs in the modeling of both stationary as well as nonstationary flows. Applications of potential flow are for instance: the outer flow field for aerofoils, water waves, electroosmotic flow, and groundwater flow. For flows (or parts thereof) with strong vorticity effects, the potential flow approximation is not applicable. Characteristics and applications ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Incomplete Gamma Function
In mathematics, the upper and lower incomplete gamma functions are types of special functions which arise as solutions to various mathematical problems such as certain integrals. Their respective names stem from their integral definitions, which are defined similarly to the gamma function but with different or "incomplete" integral limits. The gamma function is defined as an integral from zero to infinity. This contrasts with the lower incomplete gamma function, which is defined as an integral from zero to a variable upper limit. Similarly, the upper incomplete gamma function is defined as an integral from a variable lower limit to infinity. Definition The upper incomplete gamma function is defined as: \Gamma(s,x) = \int_x^ t^\,e^\, dt , whereas the lower incomplete gamma function is defined as: \gamma(s,x) = \int_0^x t^\,e^\, dt . In both cases is a complex parameter, such that the real part of is positive. Properties By integration by parts we find the recurrence rela ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Keith Moffatt
Henry Keith Moffatt, FRS FRSE (born 12 April 1935) is a Scottish mathematician with research interests in the field of fluid dynamics, particularly magnetohydrodynamics and the theory of turbulence. He was Professor of Mathematical Physics at the University of Cambridge from 1980 to 2002. Early life and education Moffatt was born on 12 April 1935 to Emmeline Marchant and Frederick Henry Moffatt''.'' He was schooled at George Watson's College, Edinburgh, going on to study Mathematical Sciences at the University of Edinburgh, graduating in 1957. He then went to Trinity College, Cambridge, where he studied mathematics and, 1959, he was a Wrangler. In 1960, he was awarded a Smith's Prize while preparing his PhD. He received his PhD in 1962, the title of his dissertation was ''Magnetohydrodynamic Turbulence.'' Career After completing his PhD, Moffatt joined the staff of the Mathematics Faculty in Cambridge as an Assistant Lecturer and became a Fellow of Trinity College. He was ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Philip Saffman
Philip Geoffrey Saffman FRS (19 March 1931 – 17 August 2008) was a mathematician and the Theodore von Kármán Professor of Applied Mathematics and Aeronautics at the California Institute of Technology.. Education and early life Saffman was born to a Jewish family in Leeds, England, and educated at Roundhay Grammar School and Trinity College, Cambridge which he entered aged 15. He received his Bachelor of Arts degree in 1953, studied for Part III of the Cambridge Mathematical Tripos in 1954 and was awarded his PhD in 1956 for research supervised by George Batchelor. Career and research Saffman started his academic career as a lecturer at the University of Cambridge, then joined King's College London as a Reader. Saffman joined the Caltech faculty in 1964 and was named the Theodore von Kármán Professor in 1995. According to Dan Meiron, Saffman "really was one of the leading figures in fluid mechanics," and he influenced almost every subfield of that discipline. He is kno ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reynolds Number
In fluid mechanics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between inertial and viscous forces. At low Reynolds numbers, flows tend to be dominated by laminar (sheet-like) flow, while at high Reynolds numbers flows tend to be 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 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 to turbulent flow and is used in the scaling of similar but different-sized flow situations, such as between an aircraft model in a wind tunnel and the full-size v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kelvin–Helmholtz Instability
The Kelvin–Helmholtz instability (after Lord Kelvin and Hermann von Helmholtz) is a fluid instability that occurs when there is velocity shear in a single continuous fluid or a velocity difference across the interface between two fluids. Kelvin-Helmholtz instabilities are visible in the atmospheres of planets and moons, such as in cloud formations on Earth or the Red Spot on Jupiter, and the atmospheres of the Sun and other stars. Theory overview and mathematical concepts Fluid dynamics predicts the onset of instability and transition to turbulent flow within fluids of different densities moving at different speeds. If surface tension is ignored, two fluids in parallel motion with different velocities and densities yield an interface that is unstable to short-wavelength perturbations for all speeds. However, surface tension is able to stabilize the short wavelength instability up to a threshold velocity. If the density and velocity vary continuously in space (with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vortex Sheet
A vortex sheet is a term used in fluid mechanics for a surface across which there is a discontinuity in fluid velocity, such as in slippage of one layer of fluid over another. While the tangential components of the flow velocity are discontinuous across the vortex sheet, the normal component of the flow velocity is continuous. The discontinuity in the tangential velocity means the flow has infinite vorticity on a vortex sheet. At high Reynolds numbers, vortex sheets tend to be unstable. In particular, they may exhibit Kelvin–Helmholtz instability The Kelvin–Helmholtz instability (after Lord Kelvin and Hermann von Helmholtz) is a fluid instability that occurs when there is velocity shear in a single continuous fluid or a velocity difference across the interface between two fluids. K .... The formulation of the vortex sheet equation of motion is given in terms of a complex coordinate z = x + iy. The sheet is described parametrically by z(s,t) where s is the arclength b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
