Collisional alpha transport in a weakly rippled magnetic field

2019 ◽  
Vol 85 (2) ◽  
Author(s):  
Peter J. Catto
Keyword(s):  
Magnetic Field ◽  
Boundary Layer ◽  
Pitch Angle ◽  
High Field ◽  
Fusion Reactor ◽  
Alpha Particles ◽  
Boundary Layer Problem ◽  
Self Consistent ◽  
Consistent Manner ◽  
Layer Problem

To properly treat the collisional transport of alpha particles due to a weakly rippled tokamak magnetic field the tangential magnetic drift due to its gradient (the $\unicode[STIX]{x1D735}B$ drift) and pitch angle scatter must be retained. Their combination gives rise to a narrow boundary layer in which collisions are able to match the finite trapped response to the ripple to the vanishing passing response of the alphas. Away from this boundary layer collisions are ineffective. There the $\unicode[STIX]{x1D735}B$ drift of the alphas balances the small radial drift of the trapped alphas caused by the ripple. A narrow collisional boundary layer is necessary since this balance does not allow the perturbed trapped alpha distribution function to vanish at the trapped–passing boundary. The solution of this boundary layer problem allows the alpha transport fluxes to be evaluated in a self-consistent manner to obtain meaningful constraints on the ripple allowable in a tokamak fusion reactor. A key result of the analysis is that collisional alpha losses are insensitive to the ripple near the equatorial plane on the outboard side where the ripple is high. As the high field side ripple is normally very small, collisional $\sqrt{\unicode[STIX]{x1D708}}$ ripple transport is unlikely to be a serious issue.

MHD Marangoni boundary layer problem for hybrid nanofluids with thermal radiation

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Emad H. Aly ◽  
Abdelhalim Ebaid
Keyword(s):  
Magnetic Field ◽  
Boundary Layer ◽  
Copper Nanoparticles ◽  
Volume Fraction ◽  
Content Type ◽  
Boundary Layer Problem ◽  
Nanoparticles Volume Fraction ◽  
Hybrid Nanofluids ◽  
Marangoni Boundary Layer ◽  
Layer Problem

Purpose The purpose of this paper is to study flow of the Marangoni boundary layer pasta surface embedded in a porous medium saturated by a hybrid nanofluid in the presence of a magnetic field and thermal radiation. Design/methodology/approach The governing model was converted into ordinary differential equations applying proper similarity transformations. Therefore, Laplace transform was used to exactly solve the resulted equations. Hence, the influence of the velocity profile and temperature distribution was investigated under impacts of the involved parameters. Findings In the case of regular fluid, i.e. the solid volume fractions are zeros, the current results are in a very good agreement with those in the literature. It was found that the velocity decreases (increases) on increasing the parameters of copper-nanoparticles volume fraction, magnetic field and suction (permeability and injection). Further, the temperature increases (decreases) with an increase of the copper-nanoparticles volume fraction, magnetic field, injection and radiation (permeability and suction). Originality/value The current results of the Marangoni boundary layer problem for hybrid nanofluids are new, original and extend the previous problems investigated by many authors for the case of regular/nano fluids.

Numerical Study of Boundary Layers With Reverse Wedge Flows Over a Semi-Infinite Flat Plate

2009 ◽  
Vol 77 (2) ◽  
Author(s):  
R. Ahmad ◽  
K. Naeem ◽  
Waqar Ahmed Khan
Keyword(s):  
Boundary Layer ◽  
Flat Plate ◽  
Numerical Study ◽  
Boundary Layer Equation ◽  
Wedge Angle ◽  
Adverse Pressure Gradient ◽  
Problem Solution ◽  
Weighted Residual Method ◽  
Boundary Layer Problem ◽  
Layer Problem

This paper presents the classical approximation scheme to investigate the velocity profile associated with the Falkner–Skan boundary-layer problem. Solution of the boundary-layer equation is obtained for a model problem in which the flow field contains a substantial region of strongly reversed flow. The problem investigates the flow of a viscous liquid past a semi-infinite flat plate against an adverse pressure gradient. Optimized results for the dimensionless velocity profiles of reverse wedge flow are presented graphically for different values of wedge angle parameter β taken from 0≤β≤2.5. Weighted residual method (WRM) is used for determining the solution of nonlinear boundary-layer problem. Finally, for β=0 the results of WRM are compared with the results of homotopy perturbation method.

On a Boundary Layer Problem

2002 ◽  
Vol 108 (4) ◽  
pp. 369-398 ◽  
Author(s):  
R. Wong ◽  
Heping Yang
Keyword(s):  
Boundary Layer ◽  
Boundary Layer Problem ◽  
Layer Problem

The boundary layer over an impulsively started flat plate

1969 ◽  
Vol 310 (1502) ◽  
pp. 401-414 ◽  
Keyword(s):  
Boundary Layer ◽  
Leading Edge ◽  
Surface Shear Stress ◽  
Unsteady State ◽  
Similarity Condition ◽  
Surface Shear ◽  
Boundary Layer Problem ◽  
Independent Variables ◽  
Selection Of ◽  
Layer Problem

A numerical solution has been obtained for the development of the flow from the initial unsteady state described by Rayleigh to the ultimate steady state described by Blasius. The usual formulation of the problem in two independent variables is dropped, and three independent variables, in space and time, are reverted to. The boundary-layer problem is unconventional in that the boundary conditions are not completely known. Instead, it is known that the solution should satisfy a similarity condition, and use is made of this to obtain a solution by iteration. A finite-difference technique of a mixed, explicit-implicit, type is employed. The iteration converges rapidly. It is terminated where the maximum errors are estimated to be about 0.04%. A selection of the results for the velocity profiles and the surface shear stress is presented. One striking feature is the rapidity of the transition from the Rayleigh to the Blasius state. The change is practically complete, at a given station on the plate, by the time the plate has moved a distance equal to four times the distance from the station to the leading edge of the plate.

Helical Magnetic Cavities: Kinetic Model and Comparison with MMS Observations

2021 ◽  
Author(s):  
Jinghuan Li ◽  
Xuzhi Zhou ◽  
Fan Yang ◽  
Anton V. Artemyev ◽  
Qiugang Zong
Keyword(s):  
Magnetic Field ◽  
Kinetic Model ◽  
Pitch Angle ◽  
Maxwell Equations ◽  
Equilibrium Solution ◽  
Space Plasma ◽  
Azimuthal Magnetic Field ◽  
Particle Distributions ◽  
Multi Scale ◽  
Self Consistent

<p>Magnetic cavities are sudden depressions of magnetic field strength widely observed in the space plasma environments, which are often accompanied by plasma density and pressure enhancement. To describe these cavities, a self-consistent kinetic model has been proposed as an equilibrium solution to the Vlasov-Maxwell equations. However, observations from the Magnetospheric Multi-Scale (MMS) constellation have shown the existence of helical magnetic cavities characterized by the presence of azimuthal magnetic field, which could not be reconstructed by the aforementioned model. Here, we take into account another invariant of motion, the canonical axial momentum, to construct the particle distributions and accordingly modify the equilibrium model. The reconstructed magnetic cavity shows excellent agreement with the MMS1 observations not only in the electromagnetic field and plasma moment profiles but also in electron pitch-angle distributions. With the same set of parameters, the model also predicts signatures of the neighboring MMS3 spacecraft, matching its observations satisfactorily.</p>

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