| 1. |
- Almqvist, Andreas, et al.
(författare)
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New insights on lubrication theory for compressible fluids
- 2019
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Ingår i: International Journal of Engineering Science. - : Elsevier. - 0020-7225 .- 1879-2197. ; 145
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Tidskriftsartikel (refereegranskat)abstract
- The fact that the film is thin is in lubrication theory utilised to simplify the full Navier–Stokes system of equations. For incompressible and iso-viscous fluids, it turns out that the inertial terms are small enough to be neglected. However, for a compressible fluid, we show that the influence of inertia depends on the (constitutive) density-pressure relationship and may not always be neglected. We consider a class of iso-viscous fluids obeying a power-law type of compressibility, which in particular includes both incompressible fluids and ideal gases. We show by scaling and asymptotic analysis, that the degree of compressibility determines whether the terms governing inertia may or may not be neglected. For instance, for an ideal gas, the inertial terms remain regardless of the film height-to-length ratio. However, by means of a specific modified Reynolds number that we define we show that the magnitudes of the inertial terms rarely are large enough to be influential. In addition, we consider fluids obeying the well-known Dowson and Higginson density-pressure relationship and show that the inertial terms can be neglected, which allows for obtaining a Reynolds type of equation. Finally, some numerical examples are presented in order to illustrate our theoretical results.
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| 2. |
- Almqvist, Andreas, et al.
(författare)
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On flow of power-law fluids between adjacent surfaces : Why is it possible to derive a Reynolds-type equation for pressure-driven flow, but not for shear-driven flow?
- 2023
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Ingår i: Applications in Engineering Science. - : Elsevier. - 2666-4968. ; 15
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Tidskriftsartikel (refereegranskat)abstract
- Flows of incompressible Navier–Stokes (Newtonian) fluids between adjacent surfaces are encountered in numerous practical applications, such as seal leakage and bearing lubrication. In seals, the flow is primarily pressure-driven, whereas, in bearings, the dominating driving force is due to shear. The governing Navier–Stokes system of equations can be significantly simplified due to the small distance between the surfaces compared to their size. From the simplified system, it is possible to derive a single lower-dimensional equation, known as the Reynolds equation, which describes the pressure field. Once the pressure field is computed, it can be used to determine the velocity field. This computational algorithm is much simpler to implement than a direct numerical solution of the Navier–Stokes equations and is therefore widely employed by engineers. The primary objective of this article is to investigate the possibility of deriving a type of Reynolds equation also for non-Newtonian fluids, using the balance of linear momentum. By considering power-law fluids we demonstrate that it is not possible for shear-driven flows, whereas it is feasible for pressure-driven flows. Additionally, we demonstrate that in the full 3D model, a normal stress boundary condition at the inlet/outlet implies a Dirichlet condition for the pressure in the Reynolds equation associated with pressure-driven flow. Furthermore, we establish that a Dirichlet condition for the velocity at the inlet/outlet in the 3D model results in a Neumann condition for the pressure in the Reynolds equation.
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| 3. |
- 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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| 4. |
- 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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| 5. |
- 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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| 6. |
- Burtseva, Evgeniya, 1988-
(författare)
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Boundedness of some linear operators in various function spaces
- 2020
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This PhD thesis is devoted to boundedness of some classical linear operators in various function spaces. We prove boundedness of weighted Hardy type operators and the weighted Riesz potential in Morrey—Orlicz spaces. Furthermore, we consider central Morrey—Orlicz spaces and prove boundedness of the Riesz potential in these spaces. We also present results concerning boundedness of Hardy type operators in Hölder type spaces. The thesis consists of four papers (Papers A—D), two complementary appendices (A1, B1) and an introduction.The introduction is divided into three parts. In the first part we give main definitions and properties of Morrey spaces, Orlicz spaces and Morrey—Orlicz spaces. In the second part we consider boundedness of the Riesz potential and Hardy type operators in various Banach ideal spaces. These operators have lately been studied in Lebesgue spaces, Morrey spaces and Orlicz spaces by many authors. We briefly describe this development and thereafter we present how these results have been extended to Morrey—Orlicz spaces (Paper A) and central Morrey—Orlicz spaces (Paper B). Finally, in the third part, we introduce Hölder type spaces and present our main results from Paper C and Paper D, which concern boundedness of Hardy type operators in Hölder type spaces. In Paper A we prove boundedness of the Riesz fractional integral operator between distinct Morrey—Orlicz spaces, which is a generalization of the Adams type result. Moreover, we investigate boundedness of some weighted Hardy type operators and weighted Riesz fractional integral operator between distinct Morrey—Orlicz spaces. The Appendix A1 contains detailed calculations of some examples, which illustrate one of our main results presented in Paper A.In Paper B we prove strong and weak boundedness of the Riesz potential in central Morrey—Orlicz spaces. We also give some examples, which illustrate the main theorem. Detailed calculations connected to one of the examples are described in the Appendix B1. In Paper C we consider n-dimensional Hardy type operators and prove that these operators are bounded in Hölder spaces. In Paper D we develop the results from paper C and derive necessary and sufficient conditions for the boundedness of n-dimensional weighted Hardy type operators in Hölder type spaces. We also obtain necessary and sufficient conditions for the boundedness of weighted Hardy operators in Hölder spaces on compactification of Rn.
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| 7. |
- Burtseva, Evgeniya, 1988-, et al.
(författare)
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Boundedness of the Riesz potential in central Morrey-Orlicz spaces
- 2022
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Ingår i: Positivity (Dordrecht). - : Springer Nature. - 1385-1292 .- 1572-9281. ; 26:1
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Tidskriftsartikel (refereegranskat)abstract
- Boundedness of the maximal operator and the Calderón–Zygmund singular integral operators in central Morrey–Orlicz spaces were proved in papers (Maligranda et al. in Colloq Math 138:165–181, 2015; Maligranda et al. in Tohoku Math J 72:235–259, 2020) by the second and third authors. The weak-type estimates have also been proven. Here we show boundedness of the Riesz potential in central Morrey–Orlicz spaces and the corresponding weak-type version.
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| 8. |
- Burtseva, Evgeniya, 1988-, et al.
(författare)
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Multi–dimensional Hardy type inequalities in Hölder spaces
- 2018
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Ingår i: Journal of Mathematical Inequalities. - Zagreb : Element D.O.O.. - 1846-579X .- 1848-9575. ; 12:3, s. 719-729
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Tidskriftsartikel (refereegranskat)abstract
- Most Hardy type inequalities concern boundedness of the Hardy type operators in Lebesgue spaces. In this paper we prove some new multi-dimensional Hardy type inequalities in Hölder spaces.
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| 9. |
- Burtseva, Evgeniya, 1988-, et al.
(författare)
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Necessary and sufficient conditions for the boundedness of weighted Hardy operators in Hölder spaces
- 2018
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Ingår i: Mathematische Nachrichten. - : John Wiley & Sons. - 0025-584X .- 1522-2616. ; 291:11-12, s. 1655-1665
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Tidskriftsartikel (refereegranskat)abstract
- We study one‐ and multi‐dimensional weighted Hardy operators on functions with Hölder‐type behavior. As a main result, we obtain necessary and sufficient conditions on the power weight under which both the left and right hand sided Hardy operators map, roughly speaking, functions with the Hölder behavior only at the singular point to functions differentiable for and bounded after multiplication by a power weight. As a consequence, this implies necessary and sufficient conditions for the boundedness in Hölder spaces due to the corresponding imbeddings. In the multi‐dimensional case we provide, in fact, stronger Hardy inequalities via spherical means. We also separately consider the case of functions with Hölder‐type behavior at infinity (Hölder spaces on the compactified ).
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| 10. |
- Burtseva, Evgeniya, 1988-
(författare)
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Operators and Inequalities in various Function Spaces and their Applications
- 2016
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This Licentiate thesis is devoted to the study of mapping properties of different operators (Hardy type, singular and potential) between various function spaces.The main body of the thesis consists of five papers and an introduction, which puts these papers into a more general frame.In paper A we prove the boundedness of the Riesz Fractional Integration Operator from a Generalized Morrey Space to a certain Orlicz-Morrey Space, which covers the Adams resultfor Morrey Spaces. We also give a generalization to the case of Weighted Riesz Fractional Integration Operators for a class of weights.In paper B we study the boundedness of the Cauchy Singular Integral Operator on curves in complex plane in Generalized Morrey Spaces. We also consider the weighted case with radial weights. We apply these results to the study of Fredholm properties of Singular Integral Operators in Weighted Generalized Morrey Spaces.In paper C we prove the boundedness of the Potential Operator in Weighted Generalized Morrey Spaces in terms of Matuszewska-Orlicz indices of weights and apply this result to the Hemholtz equation with a free term in such a space. We also give a short overview of some typical situations when Potential type Operators arise when solving PDEs.In paper D some new inequalities of Hardy type are proved. More exactly, the boundedness of multidimensional Weighted Hardy Operators in Hölder Spaces are proved in cases with and without compactification.In paper E the mapping properties are studied for Hardy type and Generalized Potential type Operators in Weighted Morrey type Spaces.
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