### 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 |

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### Keywords

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

### ASJC Scopus subject areas

- Physics and Astronomy(all)
- Mathematical Physics

### Cite this

**Numerical investigations on self-similar solutions of the nonlinear diffusion equation.** / Li, Yibao; Kim, Junseok.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Numerical investigations on self-similar solutions of the nonlinear diffusion equation

AU - Li, Yibao

AU - Kim, Junseok

PY - 2013/11/1

Y1 - 2013/11/1

N2 - In this paper, we present the numerical investigations of self-similar solutions for the nonlinear diffusion equation ht=-(h 3hxxx)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)+ x0, t0)=f(α(t)(x-x0)) and α(t)=[1-4A(t-t0)]-1/4, where A and x0 are constants and t0 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 t0. 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 t0. Then, we take h(x,t0) as the self-similar solution f(x). Various numerical experiments are performed to show that f(x) is indeed a self-similar solution.

AB - In this paper, we present the numerical investigations of self-similar solutions for the nonlinear diffusion equation ht=-(h 3hxxx)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)+ x0, t0)=f(α(t)(x-x0)) and α(t)=[1-4A(t-t0)]-1/4, where A and x0 are constants and t0 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 t0. 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 t0. Then, we take h(x,t0) as the self-similar solution f(x). Various numerical experiments are performed to show that f(x) is indeed a self-similar solution.

KW - Nonlinear multigrid method

KW - Self-similar solution

KW - Thin film

UR - http://www.scopus.com/inward/record.url?scp=84882449695&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84882449695&partnerID=8YFLogxK

U2 - 10.1016/j.euromechflu.2013.05.003

DO - 10.1016/j.euromechflu.2013.05.003

M3 - Article

AN - SCOPUS:84882449695

VL - 42

SP - 30

EP - 36

JO - European Journal of Mechanics, B/Fluids

JF - European Journal of Mechanics, B/Fluids

SN - 0997-7546

ER -