The invariant sets and the solutions of the 1+2-dimensional generalized thin film equation are discussed. It is shown that there exists a class of solutions to the equations, which are invariant with respect to the se...The invariant sets and the solutions of the 1+2-dimensional generalized thin film equation are discussed. It is shown that there exists a class of solutions to the equations, which are invariant with respect to the set $$E_0 = \{ u:u_x = v_x F(u),u_y = v_y F(u)\} ,$$ where v is a smooth function of variables x, y and F is a smooth function of u. This extends the results of Galaktionov (2001) and for the 1+1-dimensional nonlinear evolution equations.展开更多
Let （M, g） be a complete non-compact Riemannian manifold without boundary. In this paper, we give the gradient estimates on positive solutions to the following elliptic equation with singular nonlinearity：△u（x）...Let （M, g） be a complete non-compact Riemannian manifold without boundary. In this paper, we give the gradient estimates on positive solutions to the following elliptic equation with singular nonlinearity：△u（x）＋cu^-a=0 in M,where a 〉 0, c are two real constants. When c 〈 0 and M is a bounded smooth domain in R^n, the above equation is known as the thin film equation, which describes a steady state of the thin film （see Guo-Wei [Manuscripta Math., 120, 193-209 （2006）]）. The results in this paper can be viewed as an supplement of that of J. Li [J. Funct. Anal., 100, 233-256 （1991）], where the nonlinearity is the positive power of u.展开更多
Classification and reduction of the generalized fourth-order nonlinear differential equations arising from theliquid films are considered.It is shown that these equations have solutions on subspaces of the polynomial,...Classification and reduction of the generalized fourth-order nonlinear differential equations arising from theliquid films are considered.It is shown that these equations have solutions on subspaces of the polynomial,exponential ortrigonometric form defined by linear nth-order ordinary differential equations with constant coefficients for n=4,...,9.Several examples of exact solutions are presented.展开更多
A second order accurate(in time)numerical scheme is analyzed for the slope-selection(SS)equation of the epitaxial thin film growth model,with Fourier pseudo-spectral discretization in space.To make the numerical schem...A second order accurate(in time)numerical scheme is analyzed for the slope-selection(SS)equation of the epitaxial thin film growth model,with Fourier pseudo-spectral discretization in space.To make the numerical scheme linear while preserving the nonlinear energy stability,we make use of the scalar auxiliary variable(SAV)approach,in which a modified Crank-Nicolson is applied for the surface diffusion part.The energy stability could be derived a modified form,in comparison with the standard Crank-Nicolson approximation to the surface diffusion term.Such an energy stability leads to an H2 bound for the numerical solution.In addition,this H2 bound is not sufficient for the optimal rate convergence analysis,and we establish a uniform-in-time H3 bound for the numerical solution,based on the higher order Sobolev norm estimate,combined with repeated applications of discrete H¨older inequality and nonlinear embeddings in the Fourier pseudo-spectral space.This discrete H3 bound for the numerical solution enables us to derive the optimal rate error estimate for this alternate SAV method.A few numerical experiments are also presented,which confirm the efficiency and accuracy of the proposed scheme.展开更多
Heat transport at the microscale is of vital importance in microtechnology applications. The heat transport equation is different from the traditional heat transport equation since a second order derivative of tempera...Heat transport at the microscale is of vital importance in microtechnology applications. The heat transport equation is different from the traditional heat transport equation since a second order derivative of temperature with respect to time and a third-order mixed derivative of temperature with respect to space and time are introduced. In this study, we develop a hybrid finite element-finite difference (FE-FD) scheme with two levels in time for the three dimensional heat transport equation in a cylindrical thin film with sub-microscale thickness. It is shown that the scheme is unconditionally stable. The scheme is then employed to obtain the temperature rise in a sub-microscale cylindrical gold film. The method can be applied to obtain the temperature rise in any thin films with sub-microscale thickness, where the geometry in the planar direction is arbitrary.展开更多
The two-dimensional spreading under gravity of a thin fluid film with suction (fluid leak-off) or blowing (fluid injection) at the base is considered. The thin fluid film approximation is imposed. The height of the th...The two-dimensional spreading under gravity of a thin fluid film with suction (fluid leak-off) or blowing (fluid injection) at the base is considered. The thin fluid film approximation is imposed. The height of the thin film satisfies a nonlinear diffusion equation with a source/sink term. The Lie point symmetries of the nonlinear diffusion equation are derived and exist, which provided the fluid velocity at the base, <em>v<sub>n</sub></em> satisfies a first order linear partial differential equation. The general form has algebraic time dependence while a special case has exponential time dependence. The solution in which <em>v<sub>n</sub></em> is proportional to the height of the thin film is studied. The width of the base always increases with time even for suction while the height decreases with time for sufficiently weak blowing. The streamlines of the fluid flow inside the thin film are plotted by first solving a cubic equation. For sufficiently weak blowing there is a dividing streamline, emanating from the stagnation point on the centre line which separates the fluid flow into two regions, a lower region consisting of rising fluid and dominated by fluid injection at the base and an upper region consisting of descending fluid and dominated by spreading due to gravity. For sufficiently strong blowing the lower region expands to completely fill the whole thin film.展开更多
A fourth-order degenerate parabolic equation with a viscous term: ?is studied with the initial-boundary conditions ux=wx=0?on {-1,1}×(0,T), u(x,0)=u0(x)?in (-1,1). It can be taken as a thin film equation or a Cah...A fourth-order degenerate parabolic equation with a viscous term: ?is studied with the initial-boundary conditions ux=wx=0?on {-1,1}×(0,T), u(x,0)=u0(x)?in (-1,1). It can be taken as a thin film equation or a Cahn-Hilliard equation with a degenerate mobility. The entropy functional method is introduced to overcome the difficulties that arise from the degenerate mobility m(u)?and the viscosity term. The existence of nonnegative weak solution is obtained.展开更多
The authors study a generalized thin film equation. Under some assumptions on the initial value, the existence of weak solutions is established by the time-discrete method.The uniqueness and asymptotic behavior of sol...The authors study a generalized thin film equation. Under some assumptions on the initial value, the existence of weak solutions is established by the time-discrete method.The uniqueness and asymptotic behavior of solutions are also discussed.展开更多
Thin film and elastohydrodynamic lubrication regimes are rather young domains of tribology and they are still facing unresolved issues.As they rely upon a full separation of the moving surfaces by a thin (or very thin...Thin film and elastohydrodynamic lubrication regimes are rather young domains of tribology and they are still facing unresolved issues.As they rely upon a full separation of the moving surfaces by a thin (or very thin) fluid film,the knowledge of its thickness is of paramount importance,as for instance to developing lubricated mechanisms with long lasting and efficient designs.As a consequence,a large collection of formulae for point contacts have been proposed in the last 40 years.However,their accuracy and validity have rarely been investigated.The purpose of this paper is to offer an evaluation of the most widespread analytical formulae and to define whether they can be used as qualitative or quantitative predictions.The methodology is based on comparisons with a numerical model for two configurations,circular and elliptical,considering both central and minimum film thicknesses.展开更多
基金This work was supported by the National Natural Science Foundation of China (Grant No. 10671156)the Program for New Century Excellent Talents in Universities (Grant No. NCET-04-0968)
文摘The invariant sets and the solutions of the 1+2-dimensional generalized thin film equation are discussed. It is shown that there exists a class of solutions to the equations, which are invariant with respect to the set $$E_0 = \{ u:u_x = v_x F(u),u_y = v_y F(u)\} ,$$ where v is a smooth function of variables x, y and F is a smooth function of u. This extends the results of Galaktionov (2001) and for the 1+1-dimensional nonlinear evolution equations.
基金Partly supported by National Natural Science Foundation of China (Grant Nos. 1060106, 10811120558) the program for NCET
文摘Let （M, g） be a complete non-compact Riemannian manifold without boundary. In this paper, we give the gradient estimates on positive solutions to the following elliptic equation with singular nonlinearity：△u（x）＋cu^-a=0 in M,where a 〉 0, c are two real constants. When c 〈 0 and M is a bounded smooth domain in R^n, the above equation is known as the thin film equation, which describes a steady state of the thin film （see Guo-Wei [Manuscripta Math., 120, 193-209 （2006）]）. The results in this paper can be viewed as an supplement of that of J. Li [J. Funct. Anal., 100, 233-256 （1991）], where the nonlinearity is the positive power of u.
基金Supported by the National Natural Science Foundation of China under Grant No.10671156the Northwest University Graduate Innovation and Creativity Funds under Grant No.07YZZ15
文摘Classification and reduction of the generalized fourth-order nonlinear differential equations arising from theliquid films are considered.It is shown that these equations have solutions on subspaces of the polynomial,exponential ortrigonometric form defined by linear nth-order ordinary differential equations with constant coefficients for n=4,...,9.Several examples of exact solutions are presented.
文摘A second order accurate(in time)numerical scheme is analyzed for the slope-selection(SS)equation of the epitaxial thin film growth model,with Fourier pseudo-spectral discretization in space.To make the numerical scheme linear while preserving the nonlinear energy stability,we make use of the scalar auxiliary variable(SAV)approach,in which a modified Crank-Nicolson is applied for the surface diffusion part.The energy stability could be derived a modified form,in comparison with the standard Crank-Nicolson approximation to the surface diffusion term.Such an energy stability leads to an H2 bound for the numerical solution.In addition,this H2 bound is not sufficient for the optimal rate convergence analysis,and we establish a uniform-in-time H3 bound for the numerical solution,based on the higher order Sobolev norm estimate,combined with repeated applications of discrete H¨older inequality and nonlinear embeddings in the Fourier pseudo-spectral space.This discrete H3 bound for the numerical solution enables us to derive the optimal rate error estimate for this alternate SAV method.A few numerical experiments are also presented,which confirm the efficiency and accuracy of the proposed scheme.
文摘Heat transport at the microscale is of vital importance in microtechnology applications. The heat transport equation is different from the traditional heat transport equation since a second order derivative of temperature with respect to time and a third-order mixed derivative of temperature with respect to space and time are introduced. In this study, we develop a hybrid finite element-finite difference (FE-FD) scheme with two levels in time for the three dimensional heat transport equation in a cylindrical thin film with sub-microscale thickness. It is shown that the scheme is unconditionally stable. The scheme is then employed to obtain the temperature rise in a sub-microscale cylindrical gold film. The method can be applied to obtain the temperature rise in any thin films with sub-microscale thickness, where the geometry in the planar direction is arbitrary.
文摘The two-dimensional spreading under gravity of a thin fluid film with suction (fluid leak-off) or blowing (fluid injection) at the base is considered. The thin fluid film approximation is imposed. The height of the thin film satisfies a nonlinear diffusion equation with a source/sink term. The Lie point symmetries of the nonlinear diffusion equation are derived and exist, which provided the fluid velocity at the base, <em>v<sub>n</sub></em> satisfies a first order linear partial differential equation. The general form has algebraic time dependence while a special case has exponential time dependence. The solution in which <em>v<sub>n</sub></em> is proportional to the height of the thin film is studied. The width of the base always increases with time even for suction while the height decreases with time for sufficiently weak blowing. The streamlines of the fluid flow inside the thin film are plotted by first solving a cubic equation. For sufficiently weak blowing there is a dividing streamline, emanating from the stagnation point on the centre line which separates the fluid flow into two regions, a lower region consisting of rising fluid and dominated by fluid injection at the base and an upper region consisting of descending fluid and dominated by spreading due to gravity. For sufficiently strong blowing the lower region expands to completely fill the whole thin film.
文摘A fourth-order degenerate parabolic equation with a viscous term: ?is studied with the initial-boundary conditions ux=wx=0?on {-1,1}×(0,T), u(x,0)=u0(x)?in (-1,1). It can be taken as a thin film equation or a Cahn-Hilliard equation with a degenerate mobility. The entropy functional method is introduced to overcome the difficulties that arise from the degenerate mobility m(u)?and the viscosity term. The existence of nonnegative weak solution is obtained.
基金Project supported by the 973 Project of the Ministry of Science and Technology of China and the National Natural Science Foundation of China (No.10125107)
文摘The authors study a generalized thin film equation. Under some assumptions on the initial value, the existence of weak solutions is established by the time-discrete method.The uniqueness and asymptotic behavior of solutions are also discussed.
文摘Thin film and elastohydrodynamic lubrication regimes are rather young domains of tribology and they are still facing unresolved issues.As they rely upon a full separation of the moving surfaces by a thin (or very thin) fluid film,the knowledge of its thickness is of paramount importance,as for instance to developing lubricated mechanisms with long lasting and efficient designs.As a consequence,a large collection of formulae for point contacts have been proposed in the last 40 years.However,their accuracy and validity have rarely been investigated.The purpose of this paper is to offer an evaluation of the most widespread analytical formulae and to define whether they can be used as qualitative or quantitative predictions.The methodology is based on comparisons with a numerical model for two configurations,circular and elliptical,considering both central and minimum film thicknesses.
