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Flow of mechanically incompressible, but thermally expnsible viscous fluids

Flow of mechanically incompressible, but thermally expnsible viscous fluids. A. Mikelic, A. Fasano, A. Farina Montecatini, Sept. 9 - 17. LECTURE 1. Basic mathemathical modelling LECTURE 2. Mathematical problem LECTURE 3. Stability.

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Flow of mechanically incompressible, but thermally expnsible viscous fluids

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  1. Flow of mechanically incompressible, but thermally expnsible viscous fluids • A. Mikelic, A. Fasano, A. Farina • Montecatini, Sept. 9 - 17

  2. LECTURE 1. • Basic mathemathical modelling • LECTURE 2. • Mathematical problem • LECTURE 3. • Stability

  3. Following standard mechanics arguments we have obteined:

  4. We now write explicitly the equations governing the flow. 1. Energy equation where and recall the constraint

  5. From definition y = e –T s, we have and

  6. This term gives the classical

  7. mechanical energy converted into heat by the internal friction

  8. Remark 1. The coefficient in front of represents, from the physical point of view, the isobaric specific heat. The fluid we are modelling admits only the isobaric specific heat. Indeed any change of body's temperature implies a change in volume. Hence it is not possible to work with the isochoric specific heat cv

  9. Remark 2. Experiments show that the variations of cpwith respect to pressure are generally quite small.Hence we imposethat cp (p,T) is constant with respect to the pressure field p. Thus we require b is of this form with TR reference temperature and bR=b(TR )

  10. As a consequence, from we have the following law for the density We will consider the linearized version, namely We however remark that, from the mathematical point of view such a Simplifcationis notcrucial and it is consistent with the data reported in theexperimental literature.

  11. Remark 3. We remark that in the framework of the mechanical incompressibility assumption, the term is necessarilycompensated by the mechanical work associated with dilation. Thus it does notappear in the energy balance. Indeed we have developed the theory assuming that the constraint response does not dissipate energy.

  12. Remark 4. Measuring cp we can reconstruct the Helmoltz free energy y. Indeed We have a method for “quantifying”y( T)

  13. 2. Momentum equation Next, we introduce the hydraulic head so that thus getting

  14. Concerning the viscosity mwe assume theVogel-Fulcher-Tamman's (VFT) formula In particular, m is monotonically decreasing with T. For more details we refer to [4], chapter 6. [4]. J.E. Shelby, Introduction to Glass Science and Technology, 2005.

  15. 3. Complete system

  16. x3 Gin H R (x3, f) Glat 0 Non-Dimensionalization The scaling of model (1) has to be operated paying particular attention to the specific problem we are interested in. We are considering a gravity driven flow of melted glassthrough a nozzle in the earlystage of a fiber manufacturing process. The inlet and outlet temperature of the fluid are prescribed. In particular, the fluidtemperature on Gin is higher than the one on Gout. Gout

  17. Concerning the temperature, we introduce so that Moreover we introduce also In the phenomena we are considering is small but not negligible. Typically is oforder 10-1.

  18. The characteristic of the problem we are analyzing is that there exists a reference velocityVR. This makes our approach different from the ones presented in [5] and in [6] where there is no velocity scale defined by exterior conditions. The flow takes place in a nozzle of radius R and length H, with R/H=O (1). Hence we takeHas length scale. Concerning the time scale we take tR=H/VR [5]. Rajagopal, Ruzicka, Srinivasa, M3AS 1996. [6]. Gallavotti, Foundations of Fluid Dynamics, 2002

  19. As the reference pressure PRwe take the point of view that flows of glass or polymer melts are essentially dominated by viscous effects. Accordingly we set Notice that PR0as VR tends to 0 and, as a consequence p tends to the hydrostatic pressure. This is consistent with the fact that P “measures” the deviation of the pressure from the hydrostatic-one due to the fluid motion.

  20. Summarizing, we have the following dimensionless quantities

  21. Suppressing tildas to keep notation simple, model (1) rewrites

  22. We may write We now list the non-dimensional characteristic numbers appearing in the previous model

  23. As mentioned, we are interested in studying vertical slow flows of very viscous heated fluids (molten glasses, polymer, etc.) which are thermally dilatable. So, introducingthe so-called expansivity coefficient (or thermal expansion coefficient) We will consider the mathematical system in the realistic situation in which theparameter ais small. Typically (e.g. for molten glass) In particular, can be rewritten as

  24. Next, we define the Archimedes' number So that the mathematical system rewrites

  25. We consider a flow regime such that and The terms in energy equation containing theEckert are dropped. So such an equation reads asfollow

  26. We consider the stationary version of system with the following BC

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