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In this work, a lattice Boltzmann (LB) model is proposed to study the capillary rise phenomena of two‐phase immiscible non‐Newtonian power‐law fluids. The main advantage of the LB model is that it not only allows us to solve the problems of two‐phase flow with a large density difference, but also can overcome the numerical instability caused by the change of relaxation time in the collision operator. In this model, two LB evolution equations are used to solve the conservative Allen–Cahn equation for capturing phase interface and incompressible Navier–Stokes equations for non‐Newtonian power‐law fluid dynamics. Some benchmark examples, including a droplet spreading on a smooth wall and two‐phase power‐law fluid flows between parallel plates, are used to test present LB model, and the results show that the present model is efficient and accurate in the study of two‐phase power‐law fluid flows. Furthermore, we pay attention to the phenomena of capillary rise of non‐Newtonian fluids, and carry out a parametric study on the effects of the gravity, viscosity, contact angle, power‐law index, and displaced fluids. The results show that the factors mentioned above have some significant influences on the rising process of the absorbed fluids.

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