The main objective of this study is to simulate the dynamics of a droplet of incompressible non-Newtonian fluids from a vertical capillary tube or an orifice into an ambient gas. The study simulates an axisymmetric drop with a free surface in suddenly applied time-dependent pressure gradients. The droplet is initially at rest, then a time-dependent pressure gradient is suddenly imposed on the fluid. The momentum equations are solved numerically by using Semi-Implicit Finite-Difference Method, Fractional Volume of Fluid Method and Fractional Area/Volume Obstacle Representation Method. To speed up the convergence of numerical iterations, we use Successive Over-Relaxation Method. Numerical solutions show that the evolution of free surface gives a comprehensive image to the capillary tube dynamics. It also shows the developing velocity and pressure profiles under different kinds of pressure gradients. The study also considers further the relation between volume of fluid and time to provide references to the flow control system of the micropipette. Keywords: Free Surface, Droplet, Finite-Difference Method, Fractional Volume Fluid Method.