|
Squire's Theorem
In fluid dynamics, Squire's theorem states that of all the perturbations that may be applied to a shear flow (i.e. a velocity field of the form \mathbf = (U(z), 0, 0)), the perturbations which are least stable are two-dimensional, i.e. of the form \mathbf' = (u'(x,z,t),0,w'(x,z,t)), rather than the three-dimensional disturbances. This applies to incompressible flows which are governed by the Navier–Stokes equations. The theorem is named after Herbert Squire, who proved the theorem in 1933.Squire, H. B. (1933). On the stability for three-dimensional disturbances of viscous fluid flow between parallel walls. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 142(847), 621-628. Squire's theorem allows many simplifications to be made in stability theory. If we want to decide whether a flow is unstable or not, it suffices to look at two-dimensional perturbations. These are governed by the Orr–Sommerfeld equation for vis ... [...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 time. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perturbation Theory
In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle step that breaks the problem into "solvable" and "perturbative" parts. In perturbation theory, the solution is expressed as a power series in a small parameter The first term is the known solution to the solvable problem. Successive terms in the series at higher powers of \varepsilon usually become smaller. An approximate 'perturbation solution' is obtained by truncating the series, usually by keeping only the first two terms, the solution to the known problem and the 'first order' perturbation correction. Perturbation theory is used in a wide range of fields, and reaches its most sophisticated and advanced forms in quantum field theory. Perturbation theory (quantum mechanics) describes the use of this method in quantum mechanics. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shear Flow
The term shear flow is used in solid mechanics as well as in fluid dynamics. The expression ''shear flow'' is used to indicate: * a shear stress over a distance in a thin-walled structure (in solid mechanics);Higdon, Ohlsen, Stiles and Weese (1960), ''Mechanics of Materials'', article 4-9 (2nd edition), John Wiley & Sons, Inc., New York. Library of Congress CCN 66-25222 * the flow ''induced'' by a force (in a fluid). In solid mechanics For thin-walled profiles, such as that through a beam or semi-monocoque structure, the shear stress distribution through the thickness can be neglected. Furthermore, there is no shear stress in the direction normal to the wall, only parallel. In these instances, it can be useful to express internal shear stress as shear flow, which is found as the shear stress multiplied by the thickness of the section. An equivalent definition for shear flow is the shear force ''V'' per unit length of the perimeter around a thin-walled section. Shear flow has the di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Velocity Field
In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the flow velocity vector is the flow speed and is a scalar. It is also called velocity field; when evaluated along a line, it is called a velocity profile (as in, e.g., law of the wall). Definition The flow velocity ''u'' of a fluid is a vector field : \mathbf=\mathbf(\mathbf,t), which gives the velocity of an '' element of fluid'' at a position \mathbf\, and time t.\, The flow speed ''q'' is the length of the flow velocity vector :q = \, \mathbf \, and is a scalar field. Uses The flow velocity of a fluid effectively describes everything about the motion of a fluid. Many physical properties of a fluid can be expressed mathematically in terms of the flow velocity. Some common examples follow: Steady flow The flow of a fluid is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Philip Drazin
Philip Gerald Drazin (25 May 1934 – 10 January 2002) was a British mathematician and a leading international expert in fluid dynamics. He completed his PhD at the University of Cambridge under G. I. Taylor in 1958. He was awarded the Smith's Prize in 1957. After leaving Cambridge, he spent two years at MIT before moving to the University of Bristol, where he stayed and became a Professor until retiring in 1999. After retiring, he lectured at the University of Oxford and the University of Bath until his death in 2002. Drazin worked on hydrodynamic stability and the transition to turbulence. His 1974 paper ''On a model of instability of a slowly-varying flow'' introduced the concept of a global mode solution to a system of partial differential equations such as the Navier-Stokes equations. He also worked on solitons. In 1998 he was awarded the Symons Gold Medal of the Royal Meteorological Society. References External links Philip Gerald Drazinat the Mathematics Geneal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compressibility
In thermodynamics and fluid mechanics, the compressibility (also known as the coefficient of compressibility or, if the temperature is held constant, the isothermal compressibility) is a measure of the instantaneous relative volume change of a fluid or solid as a response to a pressure (or mean stress) change. In its simple form, the compressibility \kappa (denoted in some fields) may be expressed as :\beta =-\frac\frac, where is volume and is pressure. The choice to define compressibility as the negative of the fraction makes compressibility positive in the (usual) case that an increase in pressure induces a reduction in volume. The reciprocal of compressibility at fixed temperature is called the isothermal bulk modulus. Definition The specification above is incomplete, because for any object or system the magnitude of the compressibility depends strongly on whether the process is isentropic or isothermal. Accordingly, isothermal compressibility is defined: :\beta_T=-\frac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Navier–Stokes Equations
In physics, the Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician George Gabriel Stokes. They were developed over several decades of progressively building the theories, from 1822 (Navier) to 1842–1850 (Stokes). The Navier–Stokes equations mathematically express conservation of momentum and conservation of mass for Newtonian fluids. They are sometimes accompanied by an equation of state relating pressure, temperature and density. They arise from applying Isaac Newton's second law to fluid motion, together with the assumption that the stress in the fluid is the sum of a diffusing viscous term (proportional to the gradient of velocity) and a pressure term—hence describing ''viscous flow''. The difference between them and the closely related Euler equations is that Navier–Stokes equations take ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Herbert Squire
Herbert Brian Squire FRS (13 July 1909 – 22 November 1961), was a British aerospace engineer and Zaharoff Professor of Aviation at Imperial College London. Biography Born on 13 July 1909, Squire was educated at Bedford School and at Balliol College, Oxford, where he read mathematics. After research at the University of Oxford, and at the University of Göttingen between 1932 and 1933, he became a scientific officer at the Royal Aircraft Establishment. In 1946 he was appointed as chairman of the Helicopter Committee of the Aeronautics Research Council and, in 1947, he was appointed as principal scientific officer at the Royal Aircraft Establishment, working on jet propulsion. Between 1952 and 1961 he was Zaharoff Professor of Aviation at Imperial College London. He was elected as a Fellow of the Royal Society The Royal Society, formally The Royal Society of London for Improving Natural Knowledge, is a learned society and the United Kingdom's national academy of scien ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stability Theory
In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions. The heat equation, for example, is a stable partial differential equation because small perturbations of initial data lead to small variations in temperature at a later time as a result of the maximum principle. In partial differential equations one may measure the distances between functions using Lp norms or the sup norm, while in differential geometry one may measure the distance between spaces using the Gromov–Hausdorff distance. In dynamical systems, an orbit is called ''Lyapunov stable'' if the forward orbit of any point is in a small enough neighborhood or it stays in a small (but perhaps, larger) neighborhood. Various criteria have been developed to prove stability or instability of an orbit. Under favorable circumstances, the question may be reduced to a well-studied problem involvi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orr–Sommerfeld Equation
The Orr–Sommerfeld equation, in fluid dynamics, is an eigenvalue equation describing the linear two-dimensional modes of disturbance to a viscous parallel flow. The solution to the Navier–Stokes equations for a parallel, laminar flow can become unstable if certain conditions on the flow are satisfied, and the Orr–Sommerfeld equation determines precisely what the conditions for hydrodynamic stability are. The equation is named after William McFadden Orr and Arnold Sommerfeld, who derived it at the beginning of the 20th century. Formulation The equation is derived by solving a linearized version of the Navier–Stokes equation for the perturbation velocity field :\mathbf = \left(U(z)+u'(x,z,t), 0 ,w'(x,z,t)\right), where (U(z), 0, 0) is the unperturbed or basic flow. The perturbation velocity has the wave-like solution \mathbf' \propto \exp(i \alpha (x - c t)) (real part understood). Using this knowledge, and the streamfunction representation for the flow, the following ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Viscosity
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity quantifies the internal frictional force between adjacent layers of fluid that are in relative motion. For instance, when a viscous fluid is forced through a tube, it flows more quickly near the tube's axis than near its walls. Experiments show that some stress (such as a pressure difference between the two ends of the tube) is needed to sustain the flow. This is because a force is required to overcome the friction between the layers of the fluid which are in relative motion. For a tube with a constant rate of flow, the strength of the compensating force is proportional to the fluid's viscosity. In general, viscosity depends on a fluid's state, such as its temperature, pressure, and rate of deformation. However, the dependence on some of these properties is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rayleigh's Equation (fluid Dynamics)
In fluid dynamics, Rayleigh's equation or Rayleigh stability equation is a linear ordinary differential equation to study the hydrodynamic stability of a parallel, incompressible and inviscid shear flow. The equation is: :(U-c) (\varphi'' - k^2 \varphi) - U'' \varphi=0, with U(z) the flow velocity of the steady base flow whose stability is to be studied and z is the cross-stream direction (i.e. perpendicular to the flow direction). Further \varphi(z) is the complex valued amplitude of the infinitesimal streamfunction perturbations applied to the base flow, k is the wavenumber of the perturbations and c is the phase speed with which the perturbations propagate in the flow direction. The prime denotes differentiation with respect to z. Background The equation is named after Lord Rayleigh, who introduced it in 1880. The Orr–Sommerfeld equation – introduced later, for the study of stability of parallel viscous flow – reduces to Rayleigh's equation when the viscosity is zero. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
