Browsing by Author "Rahman, S."
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Item Analysis and profiles of travelling wave solutions to a Darcy-Forchheimer fluid formulated with a non-linear diffusion.(AIMS Mathematics, 2022) Rahman, S.; Díaz Palencia, José Luis; Roa González, J.The intention along the presented analysis is to explore existence, uniqueness, regularity of solutions and travelling waves profiles to a Darcy-Forchheimer fluid flow formulated with a non-linear di usion. Such formulation is the main novelty of the present study and requires the introduction of an appropriate mathematical treatment to deal with the introduced degenerate di usivity. Firstly, the analysis on existence, regularity and uniqueness is shown upon definition of an appropriate test function. Afterwards, the problem is formulated within the travelling wave domain and analyzed close the critical points with the Geometric Perturbation Theory. Based on this theory, exact and asymptotic travelling wave profiles are obtained. In addition, the Geometric Perturbation Theory is used to provide evidences of the normal hyperbolicity in the involved manifolds that are used to get the associated travelling wave solutions. The main finding, which is not trivial in the non-linear di usion case, is related with the existence of an exponential profile along the travelling frame. Eventually, a numerical exercise is introduced to validate the analytical solutions obtained.Item Geometric Perturbation Theory and Travelling Waves profiles analysis in a Darcy–Forchheimer fluid model.(Journal of Nonlinear Mathematical Physics, 2022) Díaz Palencia, José Luis; Rahman, S.The intention along the presented analysis is to develop existence, uniqueness and asymptotic analysis of solutions to a magnetohydrodynamic (MHD) flow saturating porous medium. The influence of a porous medium is provided by the Darcy–Forchheimer conditions. Firstly, the existence and uniqueness topics are developed making used of a weak formulation. Once solutions are shown to exist regularly, the problem is converted into the Travelling Waves (TW) domain to study the asymptotic behaviour supported by the Geometric Perturbation Theory (GPT). Based on this, analytical expressions are constructed to the velocity profile for the mentioned Darcy–Forchheimer flow. Afterwards, the approximated solutions based on the GPT approach are shown to be sufficiently accurate for a range of travelling waves speeds in the interval [2.5, 2.8].Item Regularity Criterion for a Two Dimensional Carreau Fluid Flow.(Journal of Nonlinear Mathematical Physics, 2022) Díaz Palencia, José Luis; Rahman, S.; Khan, M.; Yin, Guang-ZhongCarreau fuids are a source of research from both theoretical and applied approaches. They have been considered to model diferent non-newtonian phenomena such as blood fow, plasma and viscoeslastic materials. The purpose of this study is to develop the global regularity criteria for a Carreau fuid in two dimensions fowing in a strip. Firstly, a regularity criteria is shown for the initial set ( u10, u20) ∈ H1(Ω) where Ω = [0, L]× [0, ∞). Secondly, the analysis focuses on a regularity criteria when ( u10, u20) ∈ L4(Ω) and, lastly, similar results are obtained for ( u10, u20) ∈ H2(Ω) while the fuid velocity vertical component, u2(x, y), is such that 휕u2 휕x ∈ L4(Ω) and (휕∇u2 휕y , Δu2 ) ∈ L2(Ω).