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Journal articles on the topic 'Equations Navier-Stokes incompressibles'

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Published: 24 July 2022

Last updated: 8 August 2022

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1

Yang, JianWei, and Shu Wang. "Convergence of compressible Navier-Stokes-Maxwell equations to incompressible Navier-Stokes equations." Science China Mathematics 57, no. 10 (February 28, 2014): 2153–62. http://dx.doi.org/10.1007/s11425-014-4792-4.

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2

Cutland, Nigel J., and Brendan Enright. "Stochastic nonhomogeneous incompressible Navier–Stokes equations." Journal of Differential Equations 228, no. 1 (September 2006): 140–70. http://dx.doi.org/10.1016/j.jde.2006.04.009.

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3

Vyskrebtsov, V. G. "Integration of Navier-Stokes equations." Izvestiya MGTU MAMI 8, no. 2-4 (July 20, 2014): 23–31. http://dx.doi.org/10.17816/2074-0530-67399.

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The author considers the integration of the motion equations of a Newtonian fluid (Navier-Stokes equations) in vector form, taking into consideration a separation of vector Navier-Stokes equation on the two equations containing separately linear and quadratic terms. On this basis, the paper demonstrates the possibility of integration of separated motion equations of an incompressible viscous fluid, which is determined in a greater extent by the characteristics of flow: boundary conditions, axisymmetric, nonaxisymmetric flow and others.

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4

Gustafsson, Bertil, and Hans Stoor. "Navier–Stokes Equations for Almost Incompressible Flow." SIAM Journal on Numerical Analysis 28, no. 6 (December 1991): 1523–47. http://dx.doi.org/10.1137/0728078.

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5

Ercan, Ali, and M. Levent Kavvas. "Self-similarity in incompressible Navier-Stokes equations." Chaos: An Interdisciplinary Journal of Nonlinear Science 25, no. 12 (December 2015): 123126. http://dx.doi.org/10.1063/1.4938762.

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6

Danchin, Raphaël, and Piotr Bogusław Mucha. "The Incompressible Navier‐Stokes Equations in Vacuum." Communications on Pure and Applied Mathematics 72, no. 7 (December 21, 2018): 1351–85. http://dx.doi.org/10.1002/cpa.21806.

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7

Hanek, Martin, Jakub Šístek, and Pavel Burda. "Multilevel BDDC for Incompressible Navier--Stokes Equations." SIAM Journal on Scientific Computing 42, no. 6 (January 2020): C359—C383. http://dx.doi.org/10.1137/19m1276479.

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8

Soh, W. Y., and John W. Goodrich. "Unsteady solution of incompressible Navier-Stokes equations." Journal of Computational Physics 79, no. 1 (November 1988): 113–34. http://dx.doi.org/10.1016/0021-9991(88)90007-1.

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9

Henriksen, Martin Ofstad, and Jens Holmen. "Algebraic Splitting for Incompressible Navier–Stokes Equations." Journal of Computational Physics 175, no. 2 (January 2002): 438–53. http://dx.doi.org/10.1006/jcph.2001.6907.

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10

Nazarov, Serdar, Muhammetberdi Rakhimov, and Gurbanyaz Khekimov. "Linearization of the Navier-Stokes equations." E3S Web of Conferences 216 (2020): 01060. http://dx.doi.org/10.1051/e3sconf/202021601060.

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This paper studies mathematical models of the heat transfer process of a viscous incompressible fluid. Optimal control methods are used to solve the problem of optimal modeling. Questions of linearization of the Navier-Stokes equation for a plane fluid flow are considered. The optimal modes (optimal functional dependencies) of the pump and heating device are found depending on the fluid flow rate.

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11

Meng, Zhi-Jun, Yao-Ming Zhou, and Dong-Mu Mei. "On three-dimensional incompressible Navier-Stokes fluid on cantor sets in spherical Cantor type co-ordinate system." Thermal Science 20, suppl. 3 (2016): 853–58. http://dx.doi.org/10.2298/tsci16s3853m.

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This paper addresses the systems of the incompressible Navier-Stokes equations on Cantor sets without the external force involving the fractal heat-conduction problem vial local fractional derivative. The spherical Cantor type co-ordinate method is used to transfer the incompressible Navier-Stokes equation from the Cantorian co-ordinate system into the spherical Cantor type co-ordinate system.

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12

Adanhounme, V., A. Adomou, and F. P. Codo. "Analytical Solutions for Navier-Stokes Equations in the Cylindrical Coordinates." Bulletin of Society for Mathematical Services and Standards 2 (June 2012): 16–23. http://dx.doi.org/10.18052/www.scipress.com/bsmass.2.16.

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We consider the problem of convective heat transport in the incompressible fluid flow and the motion of the fluid in the cylinder which is described by the Navier-Stokes equations with the heat equation.The exact solutions of the Navier-Stokes equations, the temperature field and the vorticity vector are obtained.

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13

Fortin, Michel. "Finite element solution of the Navier—Stokes equations." Acta Numerica 2 (January 1993): 239–84. http://dx.doi.org/10.1017/s0962492900002373.

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Viscous incompressible flows are of considerable interest for applications. Let us mention, for example, the design of hydraulic turbines or rheologically complex flows appearing in many processes involving plastics or molten metals. Their simulation raises a number of difficulties, some of which are likely to remain while others are now resolved. Among the latter are those related to incompressibility which are also present in the simulation of incompressible or nearly incompressible elastic materials. Among the still unresolved are those associated with high Reynolds numbers which are also met in compressible flows. They involve the formation of boundary layers and turbulence, an ever present phenomenon in fluid mechanics, implying that we have to simulate unsteady, highly unstable phenomena.

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14

Henshaw, Willam D., Heinz-Otto Kreiss, and Luis G. Reyna. "Estimates of the local minimum scale for the incompressible navier-stokes equations navier-stokes equations." Numerical Functional Analysis and Optimization 16, no. 3-4 (January 1995): 315–44. http://dx.doi.org/10.1080/01630569508816621.

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15

Geng, Fan, Shu Wang, and Yongxin Wang. "The Regularity Criteria and the A Priori Estimate on the 3D Incompressible Navier-Stokes Equations in Orthogonal Curvilinear Coordinate Systems." Journal of Function Spaces 2020 (October 13, 2020): 1–9. http://dx.doi.org/10.1155/2020/2816183.

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The paper considers the regularity problem on three-dimensional incompressible Navier-Stokes equations in general orthogonal curvilinear coordinate systems. We establish one regularity criteria of the weak solutions involving only in a vorticity component ω 3 and one a priori estimate on the solution that H 3 u 3 L ∞ 0 , T ; L p ℝ 3 is bounded for 1 ≤ p ≤ ∞ to three-dimensional incompressible Navier-Stokes equations in orthogonal curvilinear coordinate systems. These extent greatly the corresponding results on axisymmetric cylindrical flow.

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16

Goswami, Deepjyoti, and Pedro D. Damázio. "A Two-Grid Finite Element Method for Time-Dependent Incompressible Navier-Stokes Equations with Non-Smooth Initial Data." Numerical Mathematics: Theory, Methods and Applications 8, no. 4 (November 2015): 549–81. http://dx.doi.org/10.4208/nmtma.2015.m1414.

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AbstractWe analyze here, a two-grid finite element method for the two dimensional time-dependent incompressible Navier-Stokes equations with non-smooth initial data. It involves solving the non-linear Navier-Stokes problem on a coarse grid of size H and solving a Stokes problem on a fine grid of size h, h « H. This method gives optimal convergence for velocity in H1-norm and for pressure in L2-norm. The analysis mainly focuses on the loss of regularity of the solution at t = 0 of the Navier-Stokes equations.

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17

Rempfer, Dietmar. "On Boundary Conditions for Incompressible Navier-Stokes Problems." Applied Mechanics Reviews 59, no. 3 (May 1, 2006): 107–25. http://dx.doi.org/10.1115/1.2177683.

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We revisit the issue of finding proper boundary conditions for the field equations describing incompressible flow problems, for quantities like pressure or vorticity, which often do not have immediately obvious “physical” boundary conditions. Most of the issues are discussed for the example of a primitive-variables formulation of the incompressible Navier-Stokes equations in the form of momentum equations plus the pressure Poisson equation. However, analogous problems also exist in other formulations, some of which are briefly reviewed as well. This review article cites 95 references.

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18

Ju, Qiangchang, Fucai Li, and Shu Wang. "Convergence of the Navier–Stokes–Poisson system to the incompressible Navier–Stokes equations." Journal of Mathematical Physics 49, no. 7 (July 2008): 073515. http://dx.doi.org/10.1063/1.2956495.

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19

Hamouda, Makram, and Roger Temam. "Boundary Layers for the Navier–Stokes Equations. The Case of a Characteristic Boundary." gmj 15, no. 3 (September 2008): 517–30. http://dx.doi.org/10.1515/gmj.2008.517.

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Abstract We prove the existence of a strong corrector for the linearized incompressible Navier–Stokes solution on a domain with characteristic boundary. This case is different from the noncharacteristic case considered in [Hamouda and Temam, Some singular perturbation problems related to the Navier–Stokes equations: Springer Verlag, 2006] and somehow physically more relevant. More precisely, we show that the linearized Navier–Stokes solutions behave like the Euler solutions except in a thin region, close to the boundary, where a certain heat equation solution is added (the corrector). Here, the Navier–Stokes equations are considered in an infinite channel of but our results still hold for more general bounded domains.

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20

Nasu, Shoichi, and Mutsuto Kawahara. "An Analysis of Compressible Viscous Flows Around a Body Using Finite Element Method." Advanced Materials Research 403-408 (November 2011): 461–65. http://dx.doi.org/10.4028/www.scientific.net/amr.403-408.461.

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The objective of this paper is an analysis of a body in a compressible viscous flow using the finite element method. Generally, when the fluid flow is analyzed, an incompressible viscous flow is often applied. However fluids have compressibility in actual phenomena. Therefore, the compressibility should be concerned in Computational Fluid Dynamics [CFD]. In this study, two kind of equation is applied to basic equations. One is compressible Navier-stokes equation, the other is incompressible Navier-stokes equation considering density variation. These analysis results of both equations are compared.

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21

Zdanski, Paulo S. B., M. A. Ortega, and Nide G. C. R. Fico. "NUMERICAL SIMULATION OF THE INCOMPRESSIBLE NAVIER–STOKES EQUATIONS." Numerical Heat Transfer, Part B: Fundamentals 46, no. 6 (December 2004): 549–79. http://dx.doi.org/10.1080/104077990503663.

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22

Zhao, Wenyan, and Zhibo Zheng. "On the Incompressible Navier-Stokes Equations with Damping." Applied Mathematics 04, no. 04 (2013): 652–58. http://dx.doi.org/10.4236/am.2013.44089.

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23

Edwards, W. S., L. S. Tuckerman, R. A. Friesner, and D. C. Sorensen. "Krylov Methods for the Incompressible Navier-Stokes Equations." Journal of Computational Physics 110, no. 1 (January 1994): 82–102. http://dx.doi.org/10.1006/jcph.1994.1007.

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24

Cai, Zhiqiang, and Yanqiu Wang. "Pseudostress-velocity formulation for incompressible Navier-Stokes equations." International Journal for Numerical Methods in Fluids 63, no. 3 (May 30, 2010): 341–56. http://dx.doi.org/10.1002/fld.2077.

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25

Sheu, Tony W. H., and R. K. Lin. "Newton linearization of the incompressible Navier–Stokes equations." International Journal for Numerical Methods in Fluids 44, no. 3 (January 12, 2004): 297–312. http://dx.doi.org/10.1002/fld.639.

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26

Carpio, Ana. "Large-Time Behavior in Incompressible Navier–Stokes Equations." SIAM Journal on Mathematical Analysis 27, no. 2 (March 1996): 449–75. http://dx.doi.org/10.1137/s0036141093256782.

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27

Sun, Yongzhong, and Shifang Wang. "Inhomogeneous Incompressible Navier–Stokes Equations on Thin Domains." Communications in Mathematics and Statistics 8, no. 2 (February 18, 2020): 239–53. http://dx.doi.org/10.1007/s40304-019-00202-6.

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28

Bae, Hantaek. "Analyticity of the inhomogeneous incompressible Navier–Stokes equations." Applied Mathematics Letters 83 (September 2018): 200–206. http://dx.doi.org/10.1016/j.aml.2018.04.001.

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29

Gupta, Murli M. "High accuracy solutions of incompressible Navier-Stokes equations." Journal of Computational Physics 89, no. 2 (August 1990): 488–89. http://dx.doi.org/10.1016/0021-9991(90)90157-v.

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30

Gupta, Murli M. "High accuracy solutions of incompressible Navier-Stokes equations." Journal of Computational Physics 93, no. 2 (April 1991): 343–59. http://dx.doi.org/10.1016/0021-9991(91)90188-q.

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31

Bardos, Claude W., Trinh T. Nguyen, Toan T. Nguyen, and Edriss S. Titi. "The inviscid limit for the 2D Navier-Stokes equations in bounded domains." Kinetic and Related Models 15, no. 3 (2022): 317. http://dx.doi.org/10.3934/krm.2022004.

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<p style='text-indent:20px;'>We prove the inviscid limit for the incompressible Navier-Stokes equations for data that are analytic only near the boundary in a general two-dimensional bounded domain. Our proof is direct, using the vorticity formulation with a nonlocal boundary condition, the explicit semigroup of the linear Stokes problem near the flatten boundary, and the standard wellposedness theory of Navier-Stokes equations in Sobolev spaces away from the boundary.</p>

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32

Germain, Pierre, Slim Ibrahim, and Nader Masmoudi. "Well-posedness of the Navier—Stokes—Maxwell equations." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 144, no. 1 (January 30, 2014): 71–86. http://dx.doi.org/10.1017/s0308210512001242.

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We study the local and global well-posedness of a full system of magnetohydrodynamic equations. The system is a coupling of the incompressible Navier—Stokes equations with the Maxwell equations through the Lorentz force and Ohm's law for the current. We show the local existence of mild solutions for arbitrarily large data in a space similar to the scale-invariant spaces classically used for Navier—Stokes. These solutions are global if the initial data are small enough. Our results not only simplify and unify the proofs for the space dimensions 2 and 3, but also refine those in [8]. The main simplification comes from an a prioriLt2 (Lx∞) estimate for solutions of the forced Navier—Stokes equations.

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33

Zhang, Xueying, Xin An, and C. S. Chen. "Local RBFs Based Collocation Methods for Unsteady Navier-Stokes Equations." Advances in Applied Mathematics and Mechanics 7, no. 4 (May 29, 2015): 430–40. http://dx.doi.org/10.4208/aamm.2013.m337.

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AbstractThe local RBFs based collocation methods (LRBFCM) is presented to solve two-dimensional incompressible Navier-Stokes equations. In avoiding the ill-conditioned problem, the weight coefficients of linear combination with respect to the function values and its derivatives can be obtained by solving low-order linear systems within local supporting domain. Then, we reformulate local matrix in the global and sparse matrix. The obtained large sparse linear systems can be directly solved instead of using more complicated iterative method. The numerical experiments have shown that the developed LRBFCM is suitable for solving the incompressible Navier-Stokes equations with high accuracy and efficiency.

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34

Bisi, Marzia. "Incompressible Navier–Stokes equations from Boltzmann equations for reacting mixtures." Journal of Physics A: Mathematical and Theoretical 47, no. 45 (October 29, 2014): 455203. http://dx.doi.org/10.1088/1751-8113/47/45/455203.

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35

Xiong, Linjie. "Incompressible Limit of isentropic Navier-Stokes equations with Navier-slip boundary." Kinetic & Related Models 11, no. 3 (2018): 469–90. http://dx.doi.org/10.3934/krm.2018021.

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36

Lu, Junxiang, Hong Xue, and Xianbao Duan. "An Adaptive Moving Mesh Method for Solving Optimal Control Problems in Viscous Incompressible Fluid." Symmetry 14, no. 4 (March 31, 2022): 707. http://dx.doi.org/10.3390/sym14040707.

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An adaptive moving mesh method for optimal control problems in viscous incompressible fluid is proposed with the incompressible Navier–Stokes system used to describe the motion of the fluid. The moving distance of nodes in the adopted mesh moving strategy is found by solving a diffusion equation with source terms, and an algorithm that fully considers the characteristics of the control problem is given with symmetry reduction to the incompressible Navier–Stokes equations. Numerical examples are provided to show that the proposed algorithm can solve the optimal control problem stably and efficiently on the premise of ensuring high precision.

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37

Fazuruddin, Syed, Seelam Sreekanth, and G. Sankara Sekhar Raju. "Numerical Simulation of Slip effect on Lid-Driven Cavity Flow Problem for High Reynolds Number: Vorticity – Stream Function Approach." Mathematical Modelling of Engineering Problems 8, no. 3 (June 24, 2021): 418–24. http://dx.doi.org/10.18280/mmep.080311.

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Incompressible 2-D Navier-stokes equations for various values of Reynolds number with and without partial slip conditions are studied numerically. The Lid-Driven cavity (LDC) with uniform driven lid problem is employed with vorticity - Stream function (VSF) approach. The uniform mesh grid is used in finite difference approximation for solving the governing Navier-stokes equations and developed MATLAB code. The numerical method is validated with benchmark results. The present work is focused on the analysis of lid driven cavity flow of incompressible fluid with partial slip conditions (imposed on side walls of the cavity). The fluid flow patterns are studied with wide range of Reynolds number and slip parameters.

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38

STRODTBECK, J. P., J. M. McDONOUGH, and P. D. HISLOP. "CHARACTERIZATION OF THE DYNAMICAL BEHAVIOR OF THE COMPRESSIBLE "POOR MAN'S NAVIER–STOKES EQUATIONS"." International Journal of Bifurcation and Chaos 22, no. 01 (January 2012): 1230004. http://dx.doi.org/10.1142/s0218127412300042.

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The compressible "poor man's Navier–Stokes equations" (PMNS equations) are a discrete dynamical system derived from a Galerkin expansion of the compressible Navier–Stokes equations. Complete details of the derivation are presented, with attention given to the differences from the original, incompressible case. A thorough numerical investigation of the bifurcation behavior is given in the form of regime maps characterizing the different kinds of dynamical behavior, bifurcation sequences, power spectral density analysis, time series and phase portraits. As in the case of previously studied incompressible PMNS equations, the full range of dynamical behavior associated with physical turbulence is exhibited by the system of coupled maps. The conclusion is drawn that this system can be viable as a source of temporal fluctuations in synthetic-velocity subgrid-scale models for large-eddy simulation.

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39

Shin, Dongho, and John C. Strikwerda. "Fast solvers for finite difference approximations for the stokes and navier-stokes equations." Journal of the Australian Mathematical Society. Series B. Applied Mathematics 38, no. 2 (October 1996): 274–90. http://dx.doi.org/10.1017/s0334270000000655.

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AbstractWe consider several methods for solving the linear equations arising from finite difference discretizations of the Stokes equations. The two best methods, one presented here for the first time, apparently, and a second, presented by Bramble and Pasciak, are shown to have computational effort that grows slowly with the number of grid points. The methods work with second-order accurate discretizations. Computational results are shown for both the Stokes equations and incompressible Navier-Stokes equations at low Reynolds number.

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40

Aki, Gonca L., Wolfgang Dreyer, Jan Giesselmann, and Christiane Kraus. "A quasi-incompressible diffuse interface model with phase transition." Mathematical Models and Methods in Applied Sciences 24, no. 05 (March 3, 2014): 827–61. http://dx.doi.org/10.1142/s0218202513500693.

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This work introduces a new thermodynamically consistent diffuse model for two-component flows of incompressible fluids. For the introduced diffuse interface model, we investigate physically admissible sharp interface limits by matched asymptotic techniques. To this end, we consider two scaling regimes where in one case we recover the Euler equations and in the other case the Navier–Stokes equations in the bulk phases equipped with admissible interfacial conditions. For the Navier–Stokes regime, we further assume the densities of the fluids are close to each other in the sense of a small parameter which is related to the interfacial thickness of the diffuse model.

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41

Tan, Ning-Bo, Ting-Zhu Huang, and Ze-Jun Hu. "A Relaxed Splitting Preconditioner for the Incompressible Navier-Stokes Equations." Journal of Applied Mathematics 2012 (2012): 1–12. http://dx.doi.org/10.1155/2012/402490.

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A relaxed splitting preconditioner based on matrix splitting is introduced in this paper for linear systems of saddle point problem arising from numerical solution of the incompressible Navier-Stokes equations. Spectral analysis of the preconditioned matrix is presented, and numerical experiments are carried out to illustrate the convergence behavior of the preconditioner for solving both steady and unsteady incompressible flow problems.

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42

Kim, Jae-Myoung. "Regularity for 3D Inhomogeneous Naiver–Stokes Equations in Vishik Spaces." Journal of Function Spaces 2022 (March 12, 2022): 1–4. http://dx.doi.org/10.1155/2022/7061004.

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43

DONATELLI, DONATELLA, and PIERANGELO MARCATI. "A DISPERSIVE APPROACH TO THE ARTIFICIAL COMPRESSIBILITY APPROXIMATIONS OF THE NAVIER–STOKES EQUATIONS IN 3D." Journal of Hyperbolic Differential Equations 03, no. 03 (September 2006): 575–88. http://dx.doi.org/10.1142/s0219891606000914.

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In this paper we study how to approximate the Leray weak solutions of the incompressible Navier–Stokes equations. In particular we describe an hyperbolic version of the so-called artificial compressibility method investigated by J. L. Lions and Temam. By exploiting the wave equation structure of the pressure of the approximating system we achieve the convergence of the approximating sequences by means of dispersive estimates of Strichartz type. We prove that the projection of the approximating velocity fields on the divergence free vectors is relatively compact and converges to a Leray weak solution of the incompressible Navier–Stokes equation.

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44

Hsu, C. H., Y. M. Chen, and C. H. Liu. "Preconditioned upwind methods to solve incompressible Navier-Stokes equations." AIAA Journal 30, no. 2 (February 1992): 550–52. http://dx.doi.org/10.2514/3.10951.

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45

De Michele, Carlo, Francesco Capuano, and Gennaro Coppola. "Fast-Projection Methods for the Incompressible Navier–Stokes Equations." Fluids 5, no. 4 (November 27, 2020): 222. http://dx.doi.org/10.3390/fluids5040222.

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An analysis of existing and newly derived fast-projection methods for the numerical integration of incompressible Navier–Stokes equations is proposed. Fast-projection methods are based on the explicit time integration of the semi-discretized Navier–Stokes equations with a Runge–Kutta (RK) method, in which only one Pressure Poisson Equation is solved at each time step. The methods are based on a class of interpolation formulas for the pseudo-pressure computed inside the stages of the RK procedure to enforce the divergence-free constraint on the velocity field. The procedure is independent of the particular multi-stage method, and numerical tests are performed on some of the most commonly employed RK schemes. The proposed methodology includes, as special cases, some fast-projection schemes already presented in the literature. An order-of-accuracy analysis of the family of interpolations here presented reveals that the method generally has second-order accuracy, though it is able to attain third-order accuracy only for specific interpolation schemes. Applications to wall-bounded 2D (driven cavity) and 3D (turbulent channel flow) cases are presented to assess the performances of the schemes in more realistic configurations.

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46

Capuano, Francesco, Gennaro Coppola, Matteo Chiatto, and Luigi de Luca. "Approximate Projection Method for the Incompressible Navier–Stokes Equations." AIAA Journal 54, no. 7 (July 2016): 2179–82. http://dx.doi.org/10.2514/1.j054569.

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47

Angermann, L. "Transport-stabilized Semidiscretizations of the Incompressible Navier—Stokes Equations." Computational Methods in Applied Mathematics 6, no. 3 (2006): 239–63. http://dx.doi.org/10.2478/cmam-2006-0013.

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AbstractWithin the framework of finite element methods, the paper investigates a general approximation technique for the nonlinear convective term of Navier — Stokes equations. The approach is based on an upwind method of the finite volume type. It has been proved that the discrete convective term satisfies the well-known collection of sufficient conditions for convergence of the finite element solution. For a particular nonconforming scheme, the assumptions have been verified in detail and the estimate of the semidiscrete velocity error has been proved.

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48

Neytcheva, Xin He and Maya. "Preconditioning the Incompressible Navier-Stokes Equations with Variable Viscosity." Journal of Computational Mathematics 30, no. 5 (June 2012): 461–82. http://dx.doi.org/10.4208/jcm.1201-m3848.

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49

Jun Choe, Hi, Hyea Hyun Kim, Do Wan Kim, and Yongsik Kim. "Meshless method for the stationary incompressible Navier-Stokes equations." Discrete & Continuous Dynamical Systems - B 1, no. 4 (2001): 495–526. http://dx.doi.org/10.3934/dcdsb.2001.1.495.

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50

Garcia, Salvador. "Incremental unknowns for solving the incompressible Navier–Stokes equations." Mathematics and Computers in Simulation 52, no. 5-6 (July 2000): 445–89. http://dx.doi.org/10.1016/s0378-4754(00)00158-0.

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