Lagrangian potential functionExtended lagrange's equationsNavier-Stokes equationViscous compressible flowIn classical fluid mechanics, variational principles have been applied to derive the Navier-Stokes equations with relative success. The common procedure uses indirect and semi-direct methods with nonstandard...
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When atoms combine to form molecules, they create chemical bonds which can be described by a potential energy function. An accurate description of the bonding requires quantum mechanics, as we discuss in Chapters 6 and 8, but many of the features can be understood with the “classical” picture...
where f(t) is an arbitrary function of time. This equation is a first integral of the equations of potential flow. The function f(t) in equation (9.3) can be put equal to zero without loss of generality, because the potential is not uniquely defined: since the velocity is the space de...
19. Fluid Mechanics(170) Density (43) Intro to Pressure (46) Pascal's Law & Hydraulic Lift (6) Pressure Gauge: Barometer (6) Pressure Gauge: Manometer (1) Pressure Gauge: U-shaped Tube (0) Buoyancy & Buoyant Force (36) Ideal vs Real Fluids ...
Using principles from optimal control theory and continuum mechanics, we can derive a system of PDE to characterize Nash equilibrium. Introducing a value function $$\begin{aligned} u(t,x) = \inf _{\gamma (t) = x}{\left\{ \int _t^T {\Big ( L{\Big ( \gamma (\tau ),{\dot{\...
Newtonian potential function 牛顿位势函数 Newtonian attraction 牛顿引力 Newtonian fluid 牛顿流体 Newtonian material 牛顿材料 Newtonian aberration 牛顿象差 Newtonian cooling 牛顿定理冷却,按牛顿定理冷却 Newtonian flow 牛顿流动 Newtonian liquid 牛顿液体 Newtonian mechanics 牛顿力学 相似...
Whenever you have a potential V(x) as a function of x, the force F(x) is F(x) = -dV(x)/dx Then V(x) = -∫F(x)dx = -∫k(x - L)dx = -k(x - L)2/2 + c, where c is a constant. To find c, you would need to state a known value of V0 at some reference ...
5. The work required to detach one fluid from another fluid with a surface of l02 is estimated as (22)2σrmin2=∑i∈A,j∈BP(|ri−rj|) Kondo et al. [82] also proposed a new potential function (Eq. (23)), which is much smoother than Shirakawa’s function. (23)P(r)=C3...
The velocity potential function and the determination of the velocity components from this scalar function are described. A description of the reduction of the equations of motion for ‘ideal’ (irrotational, incompressible and inviscid) flow to a single equation, viz., the Laplace equation, is ...