| 2571. |
- Almqvist, Andreas, et al.
(författare)
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On lower-dimensional models in lubrication, Part A : Common misinterpretations and incorrect usage of the Reynolds equation
- 2021
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Ingår i: Proceedings of the Institution of mechanical engineers. Part J, journal of engineering tribology. - : Sage Publications. - 1350-6501 .- 2041-305X. ; 235:8, s. 1692-1702
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Tidskriftsartikel (refereegranskat)abstract
- Most of the problems in lubrication are studied within the context of Reynolds’ equation, which can be derived by writing the incompressible Navier-Stokes equation in a dimensionless form and neglecting terms which are small under the assumption that the lubricant film is very thin. Unfortunately, the Reynolds equation is often used even though the basic assumptions under which it is derived are not satisfied. One example is in the mathematical modelling of elastohydrodynamic lubrication (EHL). In the EHL regime, the pressure is so high that the viscosity changes by several orders of magnitude. This is taken into account by just replacing the constant viscosity in either the incompressible Navier-Stokes equation or the Reynolds equation by a viscosity-pressure relation. However, there are no available rigorous arguments which justify such an assumption. The main purpose of this two-part work is to investigate if such arguments exist or not. In Part A, we formulate a generalised form of the Navier-Stokes equation for piezo-viscous incompressible fluids. By dimensional analysis of this equation we, thereafter, show that it is not possible to obtain the Reynolds equation, where the constant viscosity is replaced with a viscosity-pressure relation, by just neglecting terms which are small under the assumption that the lubricant film is very thin. The reason is that the lone assumption that the fluid film is very thin is not enough to neglect the terms, in the generalised Navier-Stokes equation, which are related to the body forces and the inertia. However, we analysed the coefficients in front of these (remaining) terms and provided arguments for when they may be neglected. In Part B, we present an alternative method to derive a lower-dimensional model, which is based on asymptotic analysis of the generalised Navier-Stokes equation as the film thickness goes to zero.
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| 2572. |
- Almqvist, Andreas, et al.
(författare)
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On lower-dimensional models in lubrication, Part B : Derivation of a Reynolds type of equation for incompressible piezo-viscous fluids
- 2021
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Ingår i: Proceedings of the Institution of mechanical engineers. Part J, journal of engineering tribology. - : Sage Publications. - 1350-6501 .- 2041-305X. ; 235:8, s. 1703-1718
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Tidskriftsartikel (refereegranskat)abstract
- The Reynolds equation is a lower-dimensional model for the pressure in a fluid confined between two adjacent surfaces that move relative to each other. It was originally derived under the assumption that the fluid is incompressible and has constant viscosity. In the existing literature, the lower-dimensional Reynolds equation is often employed as a model for the thin films, which lubricates interfaces in various machine components. For example, in the modelling of elastohydrodynamic lubrication (EHL) in gears and bearings, the pressure dependence of the viscosity is often considered by just replacing the constant viscosity in the Reynolds equation with a given viscosity-pressure relation. The arguments to justify this are heuristic, and in many cases, it is taken for granted that you can do so. This motivated us to make an attempt to formulate and present a rigorous derivation of a lower-dimensional model for the pressure when the fluid has pressure-dependent viscosity. The results of our study are presented in two parts. In Part A, we showed that for incompressible and piezo-viscous fluids it is not possible to obtain a lower-dimensional model for the pressure by just assuming that the film thickness is thin, as it is for incompressible fluids with constant viscosity. Here, in Part B, we present a method for deriving lower-dimensional models of thin-film flow, where the fluid has a pressure-dependent viscosity. The main idea is to rescale the generalised Navier-Stokes equation, which we obtained in Part A based on theory for implicit constitutive relations, so that we can pass to the limit as the film thickness goes to zero. If the scaling is correct, then the limit problem can be used as the dimensionally reduced model for the flow and it is possible to derive a type of Reynolds equation for the pressure.
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| 2573. |
- Almqvist, Andreas, et al.
(författare)
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On lower-dimensional models of thin film flow, Part C: Derivation of a Reynolds type of equation for fluids with temperature and pressure dependent viscosity
- 2023
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Ingår i: Proceedings of the Institution of mechanical engineers. Part J, journal of engineering tribology. - : Sage. - 1350-6501 .- 2041-305X. ; 237:3, s. 514-526
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Tidskriftsartikel (refereegranskat)abstract
- This paper constitutes the third part of a series of works on lower-dimensional models in lubrication. In Part A, it was shown that implicit constitutive theory must be used in the modelling of incompressible fluids with pressure-dependent viscosity and that it is not possible to obtain a lower-dimensional model for the pressure just by letting the film thickness go to zero, as in the proof of the classical Reynolds equation. In Part B, a new method for deriving lower-dimensional models of thin-film flow of fluids with pressure-dependent viscosity was presented. Here, in Part C, we also incorporate the energy equation so as to include fluids with both temperature and pressure dependent viscosity. By asymptotic analysis of this system, as the film thickness goes to zero, we derive a simplified model of the flow. We also carry out an asymptotic analysis of the boundary condition, in the case where the normal stress is specified on one part of the boundary and the velocity on the remaining part.
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| 2574. |
- Altmann, Robert, et al.
(författare)
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Numerical homogenization beyond scale separation
- 2021
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Ingår i: Acta Numerica. - : Cambridge University Press (CUP). - 0962-4929 .- 1474-0508. ; 30, s. 1-86
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Tidskriftsartikel (refereegranskat)abstract
- Numerical homogenization is a methodology for the computational solution of multiscale partial differential equations. It aims at reducing complex large-scale problems to simplified numerical models valid on some target scale of interest, thereby accounting for the impact of features on smaller scales that are otherwise not resolved. While constructive approaches in the mathematical theory of homogenization are restricted to problems with a clear scale separation, modern numerical homogenization methods can accurately handle problems with a continuum of scales. This paper reviews such approaches embedded in a historical context and provides a unified variational framework for their design and numerical analysis. Apart from prototypical elliptic model problems, the class of partial differential equations covered here includes wave scattering in heterogeneous media and serves as a template for more general multi-physics problems.
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| 2575. |
- Ames, E., et al.
(författare)
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Dynamics of gravitational collapse in the axisymmetric Einstein-Vlasov system
- 2021
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Ingår i: Classical and Quantum Gravity. - : IOP Publishing. - 0264-9381 .- 1361-6382. ; 38:10
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Tidskriftsartikel (refereegranskat)abstract
- We numerically investigate the dynamics near black hole formation of solutions to the Einstein-Vlasov system in axisymmetry. Our results are obtained using a particle-in-cell and finite difference code based on the (2 + 1) + 1 formulation of the Einstein field equations in axisymmetry. Solutions are launched from non-stationary initial data and exhibit type I critical behaviour. In particular, we find lifetime scaling in solutions containing black holes, and support that the critical solutions are stationary. Our results contain examples of solutions that form black holes, perform damped oscillations, and appear to disperse. We prove that complete dispersal of the solution implies that it has nonpositive binding energy.
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| 2576. |
- Ananth, S., et al.
(författare)
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E 8 in N= 8 supergravity in four dimensions
- 2018
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Ingår i: Journal of High Energy Physics. - 1029-8479 .- 1126-6708. ; 2018:1
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Tidskriftsartikel (refereegranskat)abstract
- We argue that N= 8 supergravity in four dimensions exhibits an exceptional E 8(8) symmetry, enhanced from the known E 7(7) invariance. Our procedure to demonstrate this involves dimensional reduction of the N= 8 theory to d = 3, a field redefinition to render the E 8(8) invariance manifest, followed by dimensional oxidation back to d = 4.
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| 2577. |
- Andersson, Kristoffer, et al.
(författare)
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Convergence of a robust deep FBSDE method for stochastic control
- 2023
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Ingår i: SIAM Journal of Scientific Computing. - 1064-8275 .- 1095-7197. ; 45:1, s. A226-A255
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Tidskriftsartikel (refereegranskat)abstract
- In this paper, we propose a deep learning based numerical scheme for strongly coupled FBSDEs, stemming from stochastic control. It is a modification of the deep BSDE method in which the initial value to the backward equation is not a free parameter, and with a new loss function being the weighted sum of the cost of the control problem, and a variance term which coincides with the mean squared error in the terminal condition. We show by a numerical example that a direct extension of the classical deep BSDE method to FBSDEs, fails for a simple linear-quadratic control problem, and motivate why the new method works. Under regularity and boundedness assumptions on the exact controls of time continuous and time discrete control problems, we provide an error analysis for our method. We show empirically that the method converges for three different problems, one being the one that failed for a direct extension of the deep BSDE method.
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| 2578. |
- Andreasson, Håkan, 1966, et al.
(författare)
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Existence of axially symmetric static solutions of the Einstein-Vlasov system
- 2011
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Ingår i: Commun. Math. Phys.. - : Springer Science and Business Media LLC. - 0010-3616 .- 1432-0916. ; 308, s. 23-47
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Tidskriftsartikel (refereegranskat)abstract
- We prove the existence of static, asymptotically flat non-vacuum spacetimes with axial symmetry where the matter is modeled as a collisionless gas. The axially symmetric solutions of the resulting Einstein-Vlasov system are obtained via the implicit function theorem by perturbing off a suitable spherically symmetric steady state of the Vlasov-Poisson system.
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| 2579. |
- Andreasson, Håkan, 1966
(författare)
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Existence of Steady States of the Massless Einstein-Vlasov System Surrounding a Schwarzschild Black Hole
- 2021
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Ingår i: Annales Henri Poincare. - : Springer Science and Business Media LLC. - 1424-0637 .- 1424-0661. ; 22
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Tidskriftsartikel (refereegranskat)abstract
- We show that there exist steady states of the spherically symmetric massless Einstein-Vlasov system which surround a Schwarzschild black hole. The steady states are (thick) shells with finite mass and compact support. Furthermore we prove that an arbitrary number of shells, necessarily well separated, can surround the black hole. To our knowledge this is the first result of static self-gravitating solutions to any massless Einstein-matter system which surround a black hole. We also include a numerical investigation about the properties of the shells.
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| 2580. |
- Andreasson, Håkan, 1966, et al.
(författare)
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Formation of trapped surfaces for the spherically symmetric Einstein-Vlasov system
- 2010
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Ingår i: Journal of Hyperbolic Differential Equations. - Göteborg : Chalmers University of Technology. - 0219-8916. ; 7:4, s. 707-731
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Tidskriftsartikel (refereegranskat)abstract
- We consider the spherically symmetric, asymptotically flat, non-vacuum Einstein equations, using as matter model a collisionless gas as described by the Vlasov equation. We find explicit conditions on the initial data which guarantee the formation of a trapped surface in the evolution which in particular implies that weak cosmic censorship holds for these data. We also analyze the evolution of solutions after a trapped surface has formed and we show that the event horizon is future complete. Furthermore we find that the apparent horizon and the event horizon do not coincide. This behavior is analogous to what is found in certain Vaidya spacetimes. The analysis is carried out in Eddington-Finkelstein coordinates.
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