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 RAUF*, QAMMAR RUBBAB, DUMITRU VIERU & ALI MAJEED ABSTRACT One-dimensional transient fows of two layers immiscible fractional Maxwell fuids in a rectangular channel is in- vestigated. The studied problem is based on a mathematical model focused on the fuids with memory described by a constitutive equation with time-fractional Caputo derivative. The fow domain is considered two regions namely one clear region and another flled with a homogeneous porous medium saturated by a generalized Maxwell fuid. Semi- analytical and analytical solutions to the problem with initial-boundary conditions and interface fuid-fuid conditions are determined by employing the integral transform method (the Laplace transform and the fnite sine-Fourier trans- form). Talbot’s algorithm for the numerical inversion of the Laplace transforms is employed. The memory efects and the infuence of the porosity coefcient on the fuid motion are studied. Numerical results and graphical illustrations obtained using the Mathcad software are utilised to analyze the fuid behavior. The infuence of the memory on the fuid motion is signifcant at the beginning of motion and it is attenuated as time passes by. Keywords: Analytical and semi-analytical solutions; fractional Maxwell fuids; memory efects; simultaneous clear and porous medium; two-layered immiscible fuids 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 grafk yang diperoleh menggunakan perisian Mathcad digunakan untuk menganalisis telatah bendalir. Pengaruh memori pada gerakan bendalir adalah signifkan 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 INTRODUCTION The study of simultaneous fow of two or more immiscible fuids in porous as well as in clear medium is signifcant due to its wide applications in science, medical, geophysics, industry, petroleum engineering, and hydrogeology (Bear 2013; Dullien 2012; Lake 1989; Satpathi et al. 2003). Various applications include oil recovery, blood flow through capillary vessels, equipment cleaning, bioflms, and mucus fow in living cells, removal of carbon dioxide from the atmosphere, groundwater management, crude oil fow through pipelines, bubble generation in microfuidics and bubble trains fow in various complex porous systems. Several researchers have studied the stability/instability of two-layer or multi-layer immiscible fuids fow (Gin & Daripa 2015; Papaefthymiou & Papageorgiou 2017; Ward et al. 2019). The linear stability of the viscoelastic two-