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In this paper, steady motion of non-Newtonian falling drop through a Newtonian fluid at low Reynolds number is investigated analytically. Here, the Upper Convected Maxwell model (UCM) is used for drop phase and Newtonian model is considered for external fluid. During the past few decades, studies relating to non-Newtonian instabilities especially those involving free surfaces are amongst the most striking. These types of studies can be used to optimize design processes in, for example, the petroleum and medicine related processes, metal extraction, and paint and power-plant related fields. Analytical solution is obtained using the perturbation method. Reynolds and Deborah numbers are used to linearize the equations governing the problem in analytical method. Deborah number indicates the elastic effect of drop. The drag force increases by the growth of the elastic effect of non-Newtonian Drop’s. The non-Newtonian drop loses its shape and exchanges to an oblate form. Increment in Deborah number enhances the dimple at the bottom of the drop and results in an increment in its drag force and as a consequence its terminal velocity decreases. A hole is created at the rear of the drop due to the presence of inertia force and focus of normal component of stress at the rear of the drop. The novelty of this study is to consider the convection (non-linear) term of the momentum equations which was neglected in the previous studies due to the creeping flow.

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In this paper, Saffman-Taylor instability of an immiscible displacement in a Hell-Shaw cell is studied numerically for the first time. The VOF method is used for two phases flow simulation. Viscoelastic fluid with less viscosity is considered as the displacing fluid and Newtonian fluid with high viscosity is used as the displaced fluid. The upper convected Maxwell constitutive equation is applied to simulate the viscoelastic fluid. In this research, the effects of dimensionless parameters consisting of the mobility ratio, elasticity number and capillary number are studied and the sweep efficiency diagram is depicted. The results show that, increasing the elasticity number and capillary number, and decreasing the mobility ratio can stabilize the flow. It is also found that, changing these parameters has a significant effect on the phase contours and mechanisms of viscous fingering patterns. The results of this numerical study could be helpful for enhanced oil recovery process, especially in polymer flooding technique. As a main consequence, it is concluded that, the elastic properties of displacing viscoelastic fluid in the presence of capillary forces has a stabilizing effect on the flow instability.