Analysis of Solutions, Asymptotic and Exact Profiles to an Eyring–Powell Fluid Modell
Abstract
:1. Introduction
2. Mathematical Model
3. Preliminaries
4. Existence and Uniqueness Analysis
5. Travelling Waves’ Existence and Regularity
5.1. Geometric Perturbation Theory
5.2. Travelling Waves’ Profiles
6. Numerical Validation Assessments
- The solver bvp4c in MATLAB was employed. This solver is based on a Runge–Kutta implicit approach with interpolant extensions [31]. The bvp4c collocation method requires specifying pseudo-boundary conditions. In this case, the left boundary is considered positive, , and the right boundary is given by the null critical state, . As the intention was to determine the exact coincidence along the profiles for which the exponential tail is given, the solutions were translated into the zero state by the standard vertical translation;
- The integration domain was assumed as , sufficiently large so as to hinder any potential effect of the pseudo-boundary conditions imposed by the collocation method involved in the bvp4c solver;
- The domain was split into 100,000 nodes with an absolute error of during the computation;
- An absolute error criterion was considered to stop the exploration criteria. The travelling wave speed for which both solutions, the numerically exact one and the analytical approach, were sufficiently close with an absolute error of , named as the critical . For this particular speed, The analytical solution in (39) can be regarded as a valid solution to the problem (34);
- The associated fluid constants in (34) were as one. The travelling wave speed a was the parameter used in the search for an analytical profile matching the error tolerance. In addition and with no loss of generality, . Note that this particular selection of constant values did not impact the ending conclusions, i.e., on the existence of an analytical exponential profile matching the exact solution for a certain value in the travelling wave speed.
7. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Díaz, J.L.; Rahman, S.U.; Sánchez Rodríguez, J.C.; Simón Rodríguez, M.A.; Filippone Capllonch, G.; Herrero Hernández, A. Analysis of Solutions, Asymptotic and Exact Profiles to an Eyring–Powell Fluid Modell. Mathematics 2022, 10, 660. https://doi.org/10.3390/math10040660
Díaz JL, Rahman SU, Sánchez Rodríguez JC, Simón Rodríguez MA, Filippone Capllonch G, Herrero Hernández A. Analysis of Solutions, Asymptotic and Exact Profiles to an Eyring–Powell Fluid Modell. Mathematics. 2022; 10(4):660. https://doi.org/10.3390/math10040660
Chicago/Turabian StyleDíaz, José Luis, Saeed Ur Rahman, Juan Carlos Sánchez Rodríguez, María Antonia Simón Rodríguez, Guillermo Filippone Capllonch, and Antonio Herrero Hernández. 2022. "Analysis of Solutions, Asymptotic and Exact Profiles to an Eyring–Powell Fluid Modell" Mathematics 10, no. 4: 660. https://doi.org/10.3390/math10040660
APA StyleDíaz, J. L., Rahman, S. U., Sánchez Rodríguez, J. C., Simón Rodríguez, M. A., Filippone Capllonch, G., & Herrero Hernández, A. (2022). Analysis of Solutions, Asymptotic and Exact Profiles to an Eyring–Powell Fluid Modell. Mathematics, 10(4), 660. https://doi.org/10.3390/math10040660