TWO LIQUID PLANE COUETTE FLOW PASTA FLEXIBLE SURFACE

Abstract

The three layer configuration system consisting of a two liquid layers of plane couette flow of Newtonian fluids of different viscosities past through a flexible neo-Hookean solid surface. There are two different interfacial modes in this type of system. They are two fluids interfacial mode is due to viscosity stratification and fluid-solid interfacial mode. The stability of the system is determined by using the eigenvalues of the system. Here we found an eigenvalues for the two fluids of couette flow using the pseudospectral method and also we obtained eigenvalues for Orr-Sommerfeld equations of single Poiseuille flow and also for Orr-Sommerfeld equation of two fluids Poiseuille flow. In every system we used three methods. Those are D4, D2 and D methods. The D4 method does not gave correct results and this method is not applicable for complicated system contains soft layers and viscoelastic layers etc. D2 method gave exact values for every system. D method does not give exact values for two liquids couette flow and other systems. We compared the results of these three methods to each other and also compared with numerical solutions of others. The eigenvalues of the two fluids plane couette flow are compared with results of low wave number analysis of Shankar and Lalit Kumar [5] and. those are matching exactly with their results. The results of Orr-Sommerfeld equation for Poiseuille flow of single fluid and two fluids are also better agreement with numerical results of other reported. Finally we concluded that the D2 method gave more occurrences eigenvalues for two liquid couette flow past a flexible surface, after that the D4 method gave near results better than the D method. Here we observed that the eigenvalues are converging when we increasing the number of polynomials.