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Bernoulli's principle is a key concept in fluid dynamics that relates pressure, density, speed and height. Bernoulli's principle states that an increase in the speed of a parcel of fluid occurs simultaneously with a decrease in either the pressure or the height above a datum. [1]:
Bernoulli equations are special because they are nonlinear differential equations with known exact solutions. A notable special case of the Bernoulli equation is the logistic differential equation . Transformation to a linear differential equation
Dynamic pressure is one of the terms of Bernoulli's equation, which can be derived from the conservation of energy for a fluid in motion. [1] At a stagnation point the dynamic pressure is equal to the difference between the stagnation pressure and the static pressure, so the dynamic pressure in a flow field can be measured at a stagnation point ...
In mathematics, the Bernoulli numbers B n are a sequence of rational numbers which occur frequently in analysis.The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, in Faulhaber's formula for the sum of m-th powers of the first n positive integers, in the Euler–Maclaurin formula, and in expressions for certain ...
Bernoulli's equation is foundational to the dynamics of incompressible fluids. In many fluid flow situations of interest, changes in elevation are insignificant and can be ignored. With this simplification, Bernoulli's equation for incompressible flows can be expressed as [2] [3] [4] + =, where:
Using Bernoulli's equation, the pressure coefficient can be further simplified for potential flows (inviscid, and steady): [3] | = = where: is the flow speed at the point at which pressure coefficient is being evaluated
Eq.2b is a fundamental equation for most of discrete models. The equation can be solved by recurrence and iteration method for a manifold. It is clear that Eq.2a is limiting case of Eq.2b when ∆X → 0. Eq.2a is simplified to Eq.1 Bernoulli equation without the potential energy term when β=1 whilst Eq.2 is simplified to Kee's model [6] when β=0
The Bernoulli equation applicable to incompressible flow shows that the stagnation pressure is equal to the dynamic pressure and static pressure combined. [1]: § 3.5 In compressible flows, stagnation pressure is also equal to total pressure as well, provided that the fluid entering the stagnation point is brought to rest isentropically.