Same Solution Between Momentum Balance Equations and Mesoscale Eddies

2021 ◽  
pp. 179-194
Author(s):  
Shu-Tang Liu ◽  
Yu-Pin Wang ◽  
Zhi-Min Bi ◽  
Yin Wang
Keyword(s):  
Mesoscale Eddies ◽  
Momentum Balance ◽  
Balance Equations

Dynamical modelling of an anaerobic digestion fluidized bed reactor

1997 ◽  
Vol 36 (5) ◽  
pp. 285-292 ◽  
Author(s):  
Benjamin Bonnet ◽  
Denis Dochain ◽  
Jean-Philippe Steyer
Keyword(s):  
Fluidized Bed ◽  
Solid Phase ◽  
Biofilm Reactor ◽  
Momentum Balance ◽  
Balance Equations ◽  
Biological Variables ◽  
Dynamical Simulation ◽  
Model Derivation ◽  
The Stability ◽  
Bed Expansion

One of the main difficulties in modelling a Fluidized Bed Biofilm Reactor (FBBR) is to take into account hydraulic phenomena (such as bed expansion) and its interactions with the biological variables. In this paper, we shall present a dynamical model of the process, analyse the stability of the hydrodynamics and illustrate its performances in simulation. A key feature of the model is that it combines mass balance of the process components with momentum balance equations in order to emphasise the different hydrodynamics of the liquid phase and of the solid phase, and the interactions between both phases. The model derivation finally leads to a set of partial differential equations (PDE). This model is intended to be used as a basis for the derivation of controllers and for dynamical simulation.

scholarly journals THE INFLUENCE OF “INJECTED” AND “THERMAL” MAGNONS ON A SPIN WAVE CURRENT AND DRAG EFFECT IN HYBRID STRUCTURES

2018 ◽  
Vol 185 ◽  
pp. 01022
Author(s):  
Igor Lyapilin ◽  
Mikhail Okorokov
Keyword(s):  
Spin Wave ◽  
Metal Structure ◽  
Relaxation Mechanism ◽  
Seebeck Effect ◽  
Hybrid Structures ◽  
Momentum Balance ◽  
Drag Effect ◽  
Balance Equations ◽  
Spin Seebeck Effect ◽  
Magnetic Insulator

The formation of the two: injected and thermally excited, different in energies magnon subsystems and the influence of its interaction with phonons and between on drag effect under spin Seebeck effect conditions in the magnetic insulator part of the metal/ferromagnetic insulator/metal structure is studied. The analysis of the macroscopic momentum balance equations of the systems of interest conducted for different ratios of the drift velocities of the magnon and phonon currents show that the injected magnons relaxation on the thermal ones is possible to be dominant over its relaxation on phonons. This interaction will be the defining in the forming of the temperature dependence of the spin-wave current under spin Seebeck effect conditions, and inelastic part of the magnon-magnon interaction is the dominant spin relaxation mechanism.

Continuum Theories for the Viscoelasticity of Flexible Homogeneous Polymeric Liquids

2007 ◽  
Author(s):  
Chang Dae Han
Keyword(s):  
Constitutive Equation ◽  
Flow Field ◽  
Polymer Blend ◽  
Constitutive Equations ◽  
Viscoelastic Fluids ◽  
Phenomenological Approach ◽  
The State ◽  
Momentum Balance ◽  
Molecular Approach ◽  
Balance Equations

There are two primary reasons for seeking a precise mathematical description of the constitutive equations for viscoelastic fluids, which relate the state of stress to the state of deformation or deformation history. The first reason is that the constitutive equations are needed to predict the rheological behavior of viscoelastic fluids for a given flow field. The second reason is that constitutive equations are needed to solve the equations of motion (momentum balance equations), energy balance equations, and/or mass balance equations in order to describe the velocity, stress, temperature, and/or concentration profiles in a given flow field that is often encountered in polymer processing operations. There are two approaches to developing constitutive equations for viscoelastic fluids: one is a continuum (phenomenological) approach and the other is a molecular approach. Depending upon the chemical structure of a polymer (e.g., flexible homopolymer, rigid rodlike polymer, microphase-separated block copolymer, segmented multicomponent polymers, highly filled polymer, miscible polymer blend, immiscible polymer blend), one may take a different approach to the formulation of the constitutive equation. In this chapter we present some representative constitutive equations for flexible, homogeneous viscoelastic liquids that have been formulated on the basis of the phenomenological approach. In the next chapter we present the molecular approach to the formulation of constitutive equations for flexible, homogeneous viscoelastic fluids. In the formulation of the constitutive equations using a phenomenological approach, emphasis is placed on the relationship between the components of stress and the components of the rate of deformation (or strain) or deformation (or strain) history, such that the responses of a fluid to a specified flow field or stress can adequately be described. The parameters appearing in a constitutive equation are supposed to represent the characteristics of the fluid under consideration. More often than not, the parameters appearing in a phenomenological constitutive equation are determined by curve fitting to experimental results. Thus phenomenological constitutive equations shed little light on the effect of the molecular parameters of the fluid under investigation to the rheological responses of the fluid.

Do Eddies Play a Role in the Momentum Balance of the Leeuwin Current?

2005 ◽  
Vol 35 (6) ◽  
pp. 964-975 ◽  
Author(s):  
Ming Feng ◽  
Susan Wijffels ◽  
Stuart Godfrey ◽  
Gary Meyers
Keyword(s):  
Kinetic Energy ◽  
Pressure Gradient ◽  
Reynolds Stress ◽  
Wind Stress ◽  
Mesoscale Eddies ◽  
Eddy Kinetic Energy ◽  
Momentum Balance ◽  
Gradient Force ◽  
Leeuwin Current ◽  
Eastern Boundary

Abstract The Leeuwin Current is a poleward-flowing eastern boundary current off the western Australian coast, and alongshore momentum balance in the current has been hypothesized to comprise a southward pressure gradient force balanced by northward wind and bottom stresses. This alongshore momentum balance is revisited using a high-resolution upper-ocean climatology to determine the alongshore pressure gradient and altimeter and mooring observations to derive an eddy-induced Reynolds stress. Results show that north of the Abrolhos Islands (situated near the shelf break between 28.2° and 29.3°S), the alongshore momentum balance is between the pressure gradient and wind stress. South of the Abrolhos Islands, the Leeuwin Current is highly unstable and strong eddy kinetic energy is observed offshore of the current axis. The alongshore momentum balance on the offshore side of the current reveals an increased alongshore pressure gradient, weakened alongshore wind stress, and a significant Reynolds stress exerted by mesoscale eddies. The eddy Reynolds stress has a −0.5 Sv (Sv ≡ 106 m3 s−1) correction to the Indonesian Throughflow transport estimate from Godfrey’s island rule. The mesoscale eddies draw energy from the mean current through mixed barotropic and baroclinic instability, and the pressure gradient work overcomes the negative wind work to supply energy for the instability process. Hence the anomalous large-scale pressure gradient in the eastern Indian Ocean drives the strongest eddy kinetic energy level among all the midlatitude eastern boundary currents.

Derivation of radial electric fields using kinetic theory in tokamak

2013 ◽  
Vol 79 (5) ◽  
pp. 513-517
Author(s):  
K. NOORI ◽  
P. KHORSHID ◽  
M. AFSARI
Keyword(s):  
Kinetic Theory ◽  
Electric Field ◽  
Electric Fields ◽  
Driving Forces ◽  
Radial Electric Field ◽  
Radial Pressure ◽  
Momentum Balance ◽  
Balance Equations ◽  
Radial Pressure Gradient ◽  
Poloidal Rotation

AbstractIn the current study, radial electric field with fluid equations has been calculated. The calculation started with kinetic theory, Boltzmann and momentum balance equations were derived, the negligible terms compared with others were eliminated, and the radial electric field expression in steady state was derived. As mentioned in previous researches, this expression includes all types of particles such as electrons, ions, and neutrals. The consequence of this solution reveals that three major driving forces contribute in radial electric field: radial pressure gradient, poloidal rotation, and toroidal rotation; rotational terms mean Lorentz force. Therefore, radial electric field and plasma rotation are connected through the radial momentum balance.

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