Frequency domain analyses of a thin liquid film surface by repetitively applied stress (dynamic response analyses by the long wave equation)

Abstract

Numerical results were calculated for the dynamic behavior of an ultrathin liquid (lubricant) surface resulting from repetitively applied pressure and shear stress using the frequency domain equation and compared with those obtained using the time domain equation. Frequency domain analyses of the dynamic behavior of the ultrathin liquid (lubricant) surface produced by sinusoidally applied pressure and shear stress clarified the dependence of the liquid surface deformation on the frequency of the stresses and the disk speed. The dynamic behavior resulting from sinusoidally applied pressure and shear stress calculated using the time domain equation were found to gradually coincide with those obtained using the frequency domain equation.

Appendix: Long wave equation for a liquid thin film

The equation for liquid film deformation used for the frequency domain analyses is the long wave equation (Oron et al. 1997), which is derived from the Navier–Stokes equation by application of the lubrication theory and consideration of the force balance at the liquid–gas interface:

$$ \frac{{\partial h_{L} }}{\partial t} = \frac{\partial }{\partial x}\left\{ {\frac{{h_{L}^{3} }}{{3\mu_{L} }}\frac{\partial }{\partial x}\left( {p_{G} - \gamma_{GL} \frac{{\partial h_{L} }}{{\partial x^{2} }} - p_{dis} } \right)} \right\} - \frac{\partial }{\partial x}\left( {\frac{{h_{L}^{2} }}{{2\mu_{L} }}\tau_{Gx} } \right) - u_{D} \frac{{\partial h_{L} }}{\partial x} , $$

(A1)

where the x-coordinate is fixed in space, and the surface tension γ _GL at the gas–liquid interface and the viscosity of the liquid μ _L are both assumed to be constant. The quantities p _G and τ _Gx are the external gas pressure and the shear stress acting at the gas–liquid interface, respectively, and u _D is the disk velocity.

The quantity p _dis is the disjoining pressure and is obtained as follows:

$$ p_{dis} = - A_{123} /6\pi h_{L}^{3} , $$

(A2)

where A ₁₂₃ is the Hamaker constant (Israelachvili 1992).

Deviations of liquid film thickness, external gas pressure, and external gas shear stress are assumed to be small compared to the characteristic values. Thus, liquid deformation can be expressed as the sum of the static deformation produced by the static stress and the dynamic deformation produced by dynamic or repetitive stresses. Therefore, the liquid film deformation h _L, gas pressure p _G, and gas shear stress τ _GX, are respectively obtained as follows:

$$ \begin{aligned} h_{L} &= h_{L00} \left\{ {1 + H_{L0} (X)\; + \eta (X,T)} \right\},\quad H_{L0} , \;\eta < < 1 \\ p_{G} &= p_{a} \left\{ {1 + P_{G0} (X) \; + \psi (X,T)} \right\},\quad P_{G0} ,\psi < < 1 \\ \tau_{GX}& = p_{a} \left\{ {T_{GX0} (X) + \delta (X,T)} \right\},\quad T_{GX0} , \delta < <1, \\ \end{aligned} $$

(A3)

where h _L00 is the average film thickness and p _a is the atmospheric pressure. The quantities H _L0, P _G0, and T _GX0 are the static components, and η, ψ, and δ are the dynamic components. Nondimensional variables X and T are the coordinate X = x/l and time T = ω ₀ t, respectively.

Substituting Eq. A3 into Eq. A1 and neglecting terms with orders higher than the second order for H _L0, P _G0, T _GX0, η, ψ, and δ, nondimensional H _L0 and η are given by Eqs. A4 and A5, respectively.

Equation to evaluate static liquid deformation H _L0

$$ - 2\left\{ {\frac{{\partial^{2} P_{G0} }}{{\partial X^{2} }} - \varepsilon_{L} \Upgamma_{GL} \frac{{\partial^{4} H_{L0} }}{{\partial X^{4} }} - \frac{{\tilde{A}}}{2}\frac{{\partial^{2} H_{L0} }}{{\partial X^{2} }}} \right\} + \frac{3}{{\varepsilon_{L} }}\frac{{\partial T_{GX0} }}{\partial X} + \Uplambda_{L} \frac{{\partial H_{L0} }}{\partial X} = 0 , $$

(A4)

Equation to evaluate dynamic liquid deformation η

$$ \sigma_{L} \frac{\partial \eta }{\partial T} - 4\left\{ {\frac{{\partial^{2} \psi }}{{\partial X^{2} }} - \varepsilon_{L} \Upgamma_{GL} \frac{{\partial^{4} \eta }}{{\partial X^{4} }} - \frac{{\tilde{A}}}{2}\frac{{\partial^{2} \eta }}{{\partial X^{2} }}} \right\} + \frac{6}{{\varepsilon_{L} }}\frac{\partial \delta }{\partial X} + 2\Uplambda_{L} \frac{\partial \eta }{\partial X} = 0 , $$

(A5)

where Λ_L, σ _L, and Γ_GL are the bearing number, squeeze number, and the nondimensional surface tension at the liquid surface, respectively. Parameters ε _L and $\tilde{A}$ are given as ε _L  = h _L00/l and $\tilde{A} = A_{123} /\pi p_{a} h_{L00}^{3}$.

These equations are nondimensional forms of the static and dynamic components of the linearized equation. In Eq. A5, the first term is the time term, the second term is the gas pressure term, the third term is the surface tension term, the fourth term is the disjoining pressure term, the fifth term is the shear stress term, and the sixth term is the velocity term. On the other hand, Eq. A4 for static deformation H _L0 is calculated for a given static pressure distribution P _G0, and static shear stress distribution T _GX0. The actual liquid film deformation is the superposition of H _L0 and η obtained from Eqs. A4 and A5, respectively.