Sains Malaysiana 49(11)(2020): 2871-2880

http://dx.doi.org/10.17576/jsm-2020-4911-25

Simultaneous Flow of Two Immiscible Fractional Maxwell Fluids with the Clear Region and Homogeneous Porous Medium

(Aliran Serentak bagi Dua Bendalir Maxwell Pecahan tak Tercampur dengan Rantau Jernih dan Medium Berliang Homogen)

ABDUL RAUF¹*, QAMMAR RUBBAB², DUMITRU VIERU³ & ALI MAJEED¹

¹Department of Computer Science and Engineering, Air University Multan Campus, Multan, 60000, Pakistan

²Department of Mathematics, The Woman University Multan, Pakistan

³Department of Theoretical Mechanics, Technical University of Iasi 700050, Romania

Received: 2 December 2019/Accepted: 31 May 2020

ABSTRACT

One-dimensional transient flows of two layers immiscible fractional Maxwell fluids in a rectangular channel is investigated. The studied problem is based on a mathematical model focused on the fluids with memory described by a constitutive equation with time-fractional Caputo derivative. The flow domain is considered two regions namely one clear region and another filled with a homogeneous porous medium saturated by a generalized Maxwell fluid. Semi-analytical and analytical solutions to the problem with initial-boundary conditions and interface fluid-fluid conditions are determined by employing the integral transform method (the Laplace transforms and the finite sine-Fourier transform). Talbot’s algorithm for the numerical inversion of the Laplace transforms is employed. The memory effects and the influence of the porosity coefficient on the fluid motion are studied. Numerical results and graphical illustrations obtained using the Mathcad software are utilised to analyze the fluid behavior. The influence of the memory on the fluid motion is significant at the beginning of motion and it is attenuated as time passes by.

Keywords: Analytical and semi-analytical solutions; fractional Maxwell fluids; memory effects; simultaneous clear and porous medium; two-layered immiscible fluids

ABSTRAK

Aliran sementara satu dimensi bagi dua lapisan bendalir Maxwell pecahan yang tidak tercampur dalam saluran segi empat dikaji. Masalah yang dikaji berdasarkan model matematik yang berfokus pada bendalir dengan memori yang diperihalkan oleh persamaan juzuk dengan terbitan Caputo pecahan masa. Domain aliran dianggap dua rantau iaitu satu rantau jernih dan satu lagi diisi dengan medium berliang homogen yang tepu oleh bendalir Maxwell teritlak. Penyelesaian semi-analitik dan analitis untuk masalah dengan keadaan sempadan awal dan keadaan antara muka bendalir ditentukan dengan menggunakan kaedah penjelmaan kamiran (jelmaan Laplace dan jelmaan sinus-Fourier terhingga). Algoritma Talbot untuk songsangan berangka bagi jelmaan ‘Laplace’ digunakan. Kesan memori dan pengaruh pekali keliangan pada pergerakan bendalir dikaji. Hasil berangka dan ilustrasi grafik yang diperoleh menggunakan perisian Mathcad digunakan untuk menganalisis telatah bendalir. Pengaruh memori pada gerakan bendalir adalah signifikan pada awal gerakan dan ia dilemahkan apabila masa berlalu.

Kata kunci: Bendalir Maxwell pecahan; dua lapisan bendalir tak tercampur; kesan memori; penyelesaian analitik dan semi-analitik; serentak jernih dan medium berliang

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*Corresponding author; email: attari_ab092@yahoo.com