Abstract

Keywords

۱٫ Introduction

۲٫ Derivation of the model

۲٫۱٫ Original domain

۲٫۲٫ Construction of the reference domain

۲٫۳٫ Navier-Stokes equations

۲٫۴٫ Asymptotic analysis

۳٫ A new generalized lubrication model

۴٫ A new thin fluid layer model

۵٫ Conclusions

Appendix A. Change of variable

Appendix B. Coefficients definition

Appendix C. Derivation of equations to calculate

References

Abstract

In this paper we study the behavior of an incompressible viscous fluid moving between two very close surfaces also in motion. Using the asymptotic expansion method we formally justify two models, a lubrication model and a shallow water model, depending on the boundary conditions imposed. Finally, we discuss under what conditions each of the models would be applicable.

۱٫ Introduction

The asymptotic analysis method is a mathematical tool that has been widely used to obtain and justify reduced models, both in solid and fluid mechanics, when one or two of the dimensions of the domain in which the model is formulated are much smaller than the others.

After the pioneering works of Friedrichs, Dressler and Goldenveizer (see [28] and [30]) the asymptotic development technique has been used successfully to justify beam, plate and shell theories (see, for example, [43], [16], [17], [15], [5], [54], and many others).

This same technique has also been used in fluid mechanics to justify various types of models, such as lubrication models, shallow water models, tube flow models, etc. (see, for example, [25], [24], [18], [3], [37], [55], [36], [31], [6], [2], [29], [26], [32], [33], [9], [23], [45], [46], [47], [48], [49], [50], [21], [22], [34], [35], [40], [41], [42], [10], [11], and many others).

In this work, we are interested in justifying, again using the asymptotic development technique, a lubrication model in a thin domain with curved mean surface. Following the steps of [3], but with a different starting point, we devote sections 2 and 3 to this justification. During the above process we have observed that, depending on the boundary conditions, other models can be obtained, which we show in section 4. In this section we derive a shallow water model changing the boundary conditions that we had imposed in section 3: instead of assuming that we know the velocities on the upper and lower boundaries of the domain, we assume that we know the tractions on these upper and lower boundaries.

Thus, two new models are presented in sections 3 and 4 of this article. These models can not be found in the literature, as far as we know. In addition, the method used to justify them allows us to answer the question of when each of them is applicable. In section 5 we discuss the models yielded, as well as the difference between one model and another depending on the boundary conditions, reaching the conclusion that the magnitude of the pressure differences at the lateral boundary of the domain is key when deciding which of the two models best describes the fluid behavior.