A shallow-liquid theory in magnetohydrodynamics

1960 ◽  
Vol 7 (1) ◽  
pp. 81-107 ◽  
Author(s):  
L. E. Fraenkel
Keyword(s):  
Magnetic Field ◽  
Mechanical Energy ◽  
Fluid Velocity ◽  
Linear Equations ◽  
Qualitative Difference ◽  
Meridional Plane ◽  
Plane Flow ◽  
Non Linear ◽  
Moderate Magnetic Field ◽  
Approximate Equations

The non-linear and linear ‘shallow-water’ theories, which describe long gravity waves on the free surface of an inviscid liquid, are extended to the case of an electrically conducting liquid on a horizontal bottom, in the presence of a vertical magnetic field. The dish holding the liquid, and the medium outside it, are assumed to be non-conducting. The approximate equations are based on a small ratio of depth to wavelength, on the properties of mercury, and on a moderate magnetic field strength. These equations have a ‘magneto-hydraulic’ character, for in the shallow liquid layer the horizontal fluid velocity and current density are independent of the vertical co-ordinate.Some explicit solutions of the linear equations are obtained for plane flows and for axi-symmetric flows in which the velocity vector lies in a vertical, meridional plane. The amplitudes of waves in a dish, and the amplitudes behind wave fronts progressing into undisturbed liquid, are found to be exponentially damped, the mechanical energy associated with a disturbance being dissipated by Joule heating.The approximate non-linear equations for plane flow are studied by means of characteristic variables, and it appears that, because of the magnetic damping effect, there is less qualitative difference between solutions of the non-linear and linear approximate equations at large times than is the case when the magnetic field is absent. In particular, the characteristic curves depart only a finite distance from their ‘undisturbed positions’.

scholarly journals Inclined magnetic field effects on Marangoni flow of Carreau liquid

2020 ◽  
Vol 24 (2 Part B) ◽  
pp. 1131-1141
Author(s):  
Rahila Naz ◽  
Fazle Mabood ◽  
Tasawar Hayat
Keyword(s):  
Magnetic Field ◽  
Nusselt Number ◽  
Radiation Effects ◽  
Linear Equations ◽  
Convection Flow ◽  
Magnetic Field Effects ◽  
Convective Boundary ◽  
Non Linear ◽  
Order Scheme ◽  
Fifth Order

Marangoni convection flow of Carreau liquid by an inclined porous surface is addressed. Magnetic field is taken inclined. Non-linear thermal radiation effects are incorporated considering the Rosseland?s approximation. Runge-Kutta-Fehl?berg fourth fifth order scheme is utilized to solve the non-linear equations subject to non-linear convective boundary conditions. Non-linear expression of Nusselt number is derived. Concrete graphical description is present out for flow velocity, temperature and Nusselt number. Numerical treatment of non-linear Nusselt number is performed and analyzed.

Non-linear dependence of the dielectric properties of chick embryo myoblast membranes exposed to a sinusoidal 50 Hz magnetic field

1991 ◽  
Vol 60 (6) ◽  
pp. 877-890 ◽  
Author(s):  
Martino Grandolfo ◽  
Maria Santini ◽  
Paolo Vecchia ◽  
Adalberto Bonincontro ◽  
Cesare Cametti ◽  
...  
Keyword(s):  
Magnetic Field ◽  
Dielectric Properties ◽  
Chick Embryo ◽  
Linear Dependence ◽  
Non Linear

scholarly journals A master exceptional field theory

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Guillaume Bossard ◽  
Axel Kleinschmidt ◽  
Ergin Sezgin
Keyword(s):  
Field Theory ◽  
Gauge Invariance ◽  
Sigma Model ◽  
Linear Equations ◽  
Field Theories ◽  
Gauge Invariant ◽  
Non Linear ◽  
Non Linear Equations

Abstract We construct a pseudo-Lagrangian that is invariant under rigid E11 and transforms as a density under E11 generalised diffeomorphisms. The gauge-invariance requires the use of a section condition studied in previous work on E11 exceptional field theory and the inclusion of constrained fields that transform in an indecomposable E11-representation together with the E11 coset fields. We show that, in combination with gauge-invariant and E11-invariant duality equations, this pseudo-Lagrangian reduces to the bosonic sector of non-linear eleven-dimensional supergravity for one choice of solution to the section condi- tion. For another choice, we reobtain the E8 exceptional field theory and conjecture that our pseudo-Lagrangian and duality equations produce all exceptional field theories with maximal supersymmetry in any dimension. We also describe how the theory entails non-linear equations for higher dual fields, including the dual graviton in eleven dimensions. Furthermore, we speculate on the relation to the E10 sigma model.

scholarly journals Steady Motion of Ice Sheets

1980 ◽  
Vol 25 (92) ◽  
pp. 229-246 ◽  
Author(s):  
L. W. Morland ◽  
I. R. Johnson
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Power Law ◽  
Perturbation Expansion ◽  
Ice Sheet ◽  
Leading Order ◽  
Plane Flow ◽  
General Law ◽  
Non Linear ◽  
Linear Differential

AbstractSteady plane flow under gravity of a symmetric ice sheet resting on a horizontal rigid bed, subject to surface accumulation and ablation, basal drainage, and basal sliding according to a shear-traction-velocity power law, is treated. The surface accumulation is taken to depend on height, and the drainage and sliding coefficient also depend on the height of overlying ice. The ice is described as a general non-linearly viscous incompressible fluid, with illustrations presented for Glen’s power law, the polynomial law of Colbeck and Evans, and a Newtonian fluid. Uniform temperature is assumed so that effects of a realistic temperature distribution on the ice response are not taken into account. In dimensionless variables a small paramter ν occurs, but the ν = 0 solution corresponds to an unbounded sheet of uniform depth. To obtain a bounded sheet, a horizontal coordinate scaling by a small factor ε(ν) is required, so that the aspect ratio ε of a steady ice sheet is determined by the ice properties, accumulation magnitude, and the magnitude of the central thickness. A perturbation expansion in ε gives simple leading-order terms for the stress and velocity components, and generates a first order non-linear differential equation for the free-surface slope, which is then integrated to determine the profile. The non-linear differential equation can be solved explicitly for a linear sliding law in the Newtonian case. For the general law it is shown that the leading-order approximation is valid both at the margin and in the central zone provided that the power and coefficient in the sliding law satisfy certain restrictions.

scholarly journals Survival of molecular gas in a stellar feedback-driven outflow witnessed with the MUSE TIMER project and ALMA

2019 ◽  
Vol 488 (3) ◽  
pp. 3904-3928 ◽  
Author(s):  
Ryan Leaman ◽  
Francesca Fragkoudi ◽  
Miguel Querejeta ◽  
Gigi Y C Leung ◽  
Dimitri A Gadotti ◽  
...  
Keyword(s):  
Magnetic Field ◽  
Star Formation ◽  
Mechanical Energy ◽  
Molecular Gas ◽  
Ionized Gas ◽  
Star Forming ◽  
Stellar Feedback ◽  
Dominant Component ◽  
The Galaxy ◽  
Field Lines

ABSTRACT Stellar feedback plays a significant role in modulating star formation, redistributing metals, and shaping the baryonic and dark structure of galaxies – however, the efficiency of its energy deposition to the interstellar medium is challenging to constrain observationally. Here we leverage HST and ALMA imaging of a molecular gas and dust shell ($M_{\mathrm{ H}_2} \sim 2\times 10^{5}\, {\rm M}_{\odot }$) in an outflow from the nuclear star-forming ring of the galaxy NGC 3351, to serve as a boundary condition for a dynamical and energetic analysis of the outflowing ionized gas seen in our MUSE TIMER survey. We use starburst99 models and prescriptions for feedback from simulations to demonstrate that the observed star formation energetics can reproduce the ionized and molecular gas dynamics – provided a dominant component of the momentum injection comes from direct photon pressure from young stars, on top of supernovae, photoionization heating, and stellar winds. The mechanical energy budget from these sources is comparable to low luminosity active galactic neuclei, suggesting that stellar feedback can be a relevant driver of bulk gas motions in galaxy centres – although here ≲10−3 of the ionized gas mass is escaping the galaxy. We test several scenarios for the survival/formation of the cold gas in the outflow, including in situ condensation and cooling. Interestingly, the geometry of the molecular gas shell, observed magnetic field strengths and emission line diagnostics are consistent with a scenario where magnetic field lines aided survival of the dusty ISM as it was initially launched (with mass-loading factor ≲1) from the ring by stellar feedback. This system’s unique feedback-driven morphology can hopefully serve as a useful litmus test for feedback prescriptions in magnetohydrodynamical galaxy simulations.

Vibration and Stability Behaviour of Sandwiched Viscoelastic Pipes Conveying a Non-Newtonian Fluid

2010 ◽  
Author(s):  
Vincent O. S. Olunloyo ◽  
Charles A. Osheku ◽  
Sidikat I. Kuye
Keyword(s):  
Newtonian Fluid ◽  
Fluid Velocity ◽  
Linear Equations ◽  
Steel Pipes ◽  
Flow Parameters ◽  
Internal Fluid ◽  
Reduction Mechanisms ◽  
Internal Fluid Flow ◽  
Flow Induced Vibrations ◽  
Flow Variables

Internal fluid flow parameters in conjunction with elastomechanical properties of conveyance systems have significantly modulated flow induced vibrations in pipeline and riser systems. Recent advances on the mechanics of sandwich elastic systems as effective vibration and noise reduction mechanisms have simulated the possibility of replacing traditional steel pipes with sandwich pipes in deepwater environment. The dynamic behaviour and stability of sandwich elastic pipes conveying a non-Newtonian fluid are investigated in this paper. For this problem, a set of generalised non-linear equations governing the vibration of sandwich pipes held together in pressurised environment and conveying a non-Newtonian fluid is presented. By linearizing the governing partial differential equation matching the problem physics, under slight perturbation of the internal fluid velocity and other flow variables closed form analytical results for the system dual natural frequencies and stability under external excitation are computed for field designs and applications. Results show that for a given length of pipe, beyond the critical velocity, instability increases with the velocity of conveyance.

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