Lagrangian viewpoint

The starting point for obtaining quantitative descriptions of flow phenomena is Newton s second law, which states that the vector sum of forces acting on a body equals the rate of change of momentum of the body. This force balance can be made in many different ways. It may be appHed over a body of finite size or over each infinitesimal portion of the body. It may be utilized in a coordinate system moving with the body (the so-called Lagrangian viewpoint) or in a fixed coordinate system (the Eulerian viewpoint). Described herein is derivation of the equations of motion from the Eulerian viewpoint using the Cartesian coordinate system. The equations in other coordinate systems are described in standard references (1,2). 

The term macromixing refers to the overall mixing performance in a reactor. It is usually described by the residence time distribution (RTD). Originally introduced by Danckwerts (1958), this concept is based on a macroscopic lumped population balance. A fluid element is followed from the time at which it enters the reactor (Lagrangian viewpoint - observer moves with the fluid). The probability that the fluid element will leave the reactor after a residence time t is expressed as the RTD function. This function characterises the scale of mixedness in a reactor. 

To overcome some of the limitations just mentioned that are associated with purely empirical models, simulations that include various aspects of the inhaled aerosol dynamics have been developed. The simplest of these belong to a class of models we refer to as Lagrangian dynamical models (LDMs), meaning that the model simulates some of the dynamical behavior of the aerosol in a frame of reference that travels with the aerosol (i.e., a Lagrangian viewpoint ). 

For mathematical convenience, boundary conditions and initial conditions must be prescribed. For the simple marine propeller problem, a Lagrangian viewpoint was adopted. The frame of reference was attached to the propeller so that the propeller was fixed but the vessel was rotating. The boundary condition was then a zero velocity on the impeller, while the vessel wall rotated at -Qimpdier- The free surface was considered to be fiat, therefore the normal velocity was zero and a shear-free condition was assumed. It should be noted that in the Lagrangian viewpoint, the frame of reference is in rotation. The fluid is therefore subjected to a constant acceleration and the momentum conservation equation [Eq. (6)] must be modified to account for centrifugal forces and Coriolis forces.An advantage is, however, that the flow can be solved numerically at steady state provided the flow is fully periodic, which limits the computational efforts significantly. 

In the case of the coaxial mixer, the rotation kinematics is much more complex since the two sets of agitators counter-rotate at different speeds. For the sake of simplicity, we decided to simulate the flow using the frame of reference of the anchor. In this Lagrangian viewpoint, the anchor is fixed but the vessel wall rotates at —Qanchor and the turbine rotates at anchor + turbine- such a situation. Contrary to the simple propeller problem, the resolution of the flow equations is time-dependent as the position of the central agitator changes with time. 

This is messy problem analytically. It is fairly easy if you take the viewpoint of someone riding on the jump (the lagrangian viewpoint) and solve by trial and error for the jump velocity that satisfies the hydraulic jump equation in the moving frame of reference. 

There are two ways to look at a cloud of diffusing molecules the Eulerian viewpoint or the Lagrangian viewpoint. In the Eulerian viewpoint, the diffusion equations would be derived from a consideration of concentrations and flux at a fixed point in space. Such quantities are easily observable. It is possible to arrive at the same results fi om the Lagrangian viewpoint, which focuses on the history of the random movements of the diffusing material. Statistical properties of the random motions would have to be mathematically described to produce useful results. 

In the Lagrangian viewpoint, one keeps track of a fluid particle, which, at time t, is located at a position x, i — 1, 2, 3) in Cartesian coordinates. The trajectory of the particle would be given by x, (r), where x, is a dependent variable and t is an independent variable. The velocity of the fluid particle is therefore given by... 

Riding along with a fluid packet is a Lagrangian notion. However, in the limit of dt - 0, the distance traveled dx vanishes. In this limit, (i.e., at a point in time and space) the Eulerian viewpoint is achieved. The relationship between the Lagrangian and Eulerian representations is established in terms of Eq. 2.52, recognizing the equivalence of the displacement rate in the flow direction and the flow velocity. In the Eulerian framework the... 

In an engineering view the ensemble of system points moving through phase space behaves much like a fluid in a multidimensional space, and there are numerous similarities between our imagination of the ensemble and the well known notions of fluid dynamics [35]. Then, the substantial derivative in fluid dynamics corresponds to a derivative of the density as we follow the motion of a particular differential volume of the ensemble in time. The material derivative is thus similar to the Lagrangian picture in fluid d3mamics in which individual particles are followed in time. The partial derivative is defined at fixed (q,p). It can be interpreted as if we consider a particular fixed control volume in phase space and measure the time variation of the density as the ensemble of system points flows by us. The partial derivative at a fixed point in phase space thus resembles the Eulerian viewpoint in fluid dynamics. 

Although it is difficult to induce turbulence (so-called Eulerian chaos) in microchannels, the mixing performance obtained in low Reynolds number flow regimes can be enhanced via the chaotic advection mechanism (or so-called Lagrangian chaos). Chaotic advection occurs in regular, smooth (from a Eulerian viewpoint)... 

There are two viewpoints that one may take to describe the fluid motion the Lagrangian and the Eulerian viewpoints. 

For steady-state flows (here we consider the steady-state in the Eulerian viewpoint. In a steady Eulerian velocity field (which is not necessarily Lagrangian steady), 0M,/0f = 0. For the analysis of polymer melt flow occurring at a very low Reynolds number, the inertia term puj dujldxj will usually be neglected. The body force effect is usually negligible too. This reduces the equations of motion to... 

From an analytical viewpoint, the flow fields in laminar devices are highly dependent on geometry, and individual drops experience varied deformation paths of long time scale that are difficult to analyze. Even if Lagrangian tracking of deformation and breakup history of many drops were possible, it would be difficult to apply this information to real-life systems. Therefore, most studies... 

A course in classical mechanics is an essential requirement of any first degree course in physics. In this volume Dr Brian Cowan provides a clear, concise and self-contained introduction to the subject and covers all the material needed by a student taking such a course. The author treats the material from a modern viewpoint, culminating in a final chapter showing how the Lagrangian and Hamiltonian formulations lend themselves particularly well to the more modem areas of physics such as quantum mechanics. Worked examples are included in. the text and there are exercises, with answers, for the student. 

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