### Abstract

In this paper, we present the numerical investigations of self-similar solutions for the nonlinear diffusion equation h_{t}=-(h ^{3}h_{xxx})_{x}, which arises in the context of surface-tension-driven flow of a thin viscous liquid film. Here, h=h(x,t) is the liquid film height. A self-similar solution is h(x,t)=h(α(t)(x-x _{0})+ x_{0}, t_{0})=f(α(t)(x-x_{0})) and α(t)=[1-4A(t-t_{0})]^{-1/4}, where A and x_{0} are constants and t_{0} is a reference time. To discretize the governing equation, we use the Crank-Nicolson finite difference method, which is second-order accurate in time and space. The resulting discrete system of equations is solved by a nonlinear multigrid method. We also present efficient and accurate numerical algorithms for calculating the constants, A, x _{0}, and t_{0}. To find a self-similar solution for the equation, we numerically solve the partial differential equation with a simple step-function-like initial condition until the solution reaches the reference time t_{0}. Then, we take h(x,t_{0}) as the self-similar solution f(x). Various numerical experiments are performed to show that f(x) is indeed a self-similar solution.

Original language | English |
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Pages (from-to) | 30-36 |

Number of pages | 7 |

Journal | European Journal of Mechanics, B/Fluids |

Volume | 42 |

DOIs | |

Publication status | Published - 2013 Nov 1 |

### Keywords

- Nonlinear multigrid method
- Self-similar solution
- Thin film

### ASJC Scopus subject areas

- Physics and Astronomy(all)
- Mathematical Physics