--- title: "Reduce a flowing layer to a height equation" description: "Derive film flux from a velocity profile and see why thickness enters the flow rate as a cube." canonical_url: "https://sandboxphysics.com/stories/dynamic-interfaces/lubrication-flow/" language: "en" page_type: "story" alternate_languages: zh-CN: "https://sandboxphysics.com/zh/stories/dynamic-interfaces/lubrication-flow/" last_modified: "2026-10-28" --- [Dynamic interfaces](https://sandboxphysics.com/stories/dynamic-interfaces/) L2 · Chapter 2 # Reduce a flowing layer to a height equation Sandbox Physics · 2026-10-28 A thin liquid layer flows when pressure or surface tension varies along it. If the layer is shallow and its surface changes gradually, we can calculate the velocity across each section, integrate it through the depth, and follow the height through the imbalance of inflow and outflow. ## Before you begin Curvature pressure from Chapter 1, derivatives and definite integrals. Partial derivatives and flux per unit width are introduced here. The liquid is incompressible and Newtonian, with negligible inertia. - Derive a velocity profile and film flux using explicit boundary conditions. - Use conservation to recognize a simulation that creates or loses liquid. ## State what thin and slow mean Take $x$ to the right along the substrate and $z$ upwards from it. The surface is $h(x,t)$. A typical thickness $H$ is much smaller than the horizontal variation scale $L$, and surface slopes are small. The derivative $h_t$ follows height at a fixed position; $h_x$ measures slope at a fixed time. The dominant viscous shear comes from velocity differences through the depth. With slow motion, this shear balances the horizontal pressure gradient. Pressure is nearly uniform through a section. This reduction is the lubrication approximation; the name describes the geometry, not a requirement to use lubricating oil. $$ \varepsilon=\frac HL\ll1,\qquad 0=-p_x+\eta u_{zz} $$ We also assume a horizontal substrate, positive viscosity and no body force. Sources: Oron · Bowen ## Integrate twice and use both boundaries The solid is stationary, so no slip gives $u(0)=0$. Uniform surface tension and negligible gas shear give $u_z(h)=0$. Integrating the viscous balance twice produces two constants, which these conditions determine. Integrate the resulting velocity from the solid to the surface. This gives volume flux per unit width, $q$, measured in $\mathrm{m^2\,s^{-1}}$. A pressure that rises to the right produces a negative flux: liquid travels towards lower pressure. $$ u(z)=\frac{p_x}{2\eta}(z^2-2hz) $$ The profile has zero velocity at the solid and zero shear rate at the free surface. $$ q=\int_0^h u\,\mathrm dz=-\frac{h^3}{3\eta}p_x $$ Velocity contributes two powers of thickness; integrating contributes the third. ## Height falls where more liquid leaves than enters Consider a narrow interval on the substrate. Its liquid volume per unit width is height times interval length. Subtract the rightward outflow from the leftward inflow, then shrink the interval. This gives the local continuity equation. With periodic boundaries, the end fluxes cancel and total volume stays fixed. A numerical grid should preserve the same cancellation: adjacent cells share one face flux. Adjusting cell heights independently can change the total volume. $$ h_t+q_x=0,\qquad \frac{\mathrm d}{\mathrm dt}\int_0^L h\,\mathrm dx=q(0)-q(L)=0 $$ Periodic boundaries or sealed ends conserve volume. Evaporation and injection require source terms. ## Curvature drives a bump sideways For small slopes the capillary pressure is $p=-\gamma h_{xx}$. A bump has a negative second derivative and elevated pressure. Liquid flows away from it. Insert this pressure into the flux and then continuity to obtain a fourth-order height equation. Worked example: start from $h=H+a\cos(kx)$ with $a\ll H$. Replace height in the mobility by $H$ at linear order. Each pair of spatial derivatives contributes a squared wavenumber. Doubling the wavenumber makes the decay rate sixteen times larger, so fine wrinkles disappear quickly. $$ h_t=-\partial_x\!\left(\frac{\gamma h^3}{3\eta}h_{xxx}\right) $$ Capillary levelling without gravity, disjoining pressure or a tension gradient. $$ a(t)=a(0)e^{-t/\tau},\qquad \tau^{-1}=\frac{\gamma H^3k^4}{3\eta} $$ This is overdamped levelling. Oscillating capillary waves on deep water require inertia and obey a different relation. ## Predict, change and explain ### [Watch curvature level a film](https://sandboxphysics.com/labs/thin-film-instability/) 1. Choose capillary levelling and locate shorter and longer features in the initial profile. 2. Compare their decay and inspect volume conservation while the height redistributes. ## Exercises and explanations 1. At fixed pressure gradient and viscosity, what does doubling film thickness do to flux? Read the explanation It multiplies flux by eight. Greater depth both raises the velocity and enlarges the cross-section available for transport. 2. What does doubling viscosity do to the capillary decay time? Read the explanation It doubles the time. The same driving pressure produces half the flux, so height evolves half as quickly. 3. The mean height of a periodic simulation keeps decreasing. What should you inspect first? Read the explanation Check shared face fluxes, time stepping and boundary handling. Without evaporation, spreading redistributes the existing liquid while its mean height stays fixed. [Previous: Why surface tension pulls water into beads](https://sandboxphysics.com/stories/dynamic-interfaces/surface-energy/) [Next: Which wavelength grows out of noise first?](https://sandboxphysics.com/stories/dynamic-interfaces/mode-selection/) ## Sources and calculation scope Examples and diagrams are newly authored. The paper comparison recalculates analytic relations; the research reading distinguishes those curves from experimental photographs and molecular simulations. 1. [Oron, Davis & Bankoff, Long-scale evolution of thin liquid films, Rev. Mod. Phys. 69, 931 (1997)](https://doi.org/10.1103/RevModPhys.69.931) Published review and framework for long-wave reductions. Specific implemented equations are cross-checked against the accessible author papers below. 2. [Bowen, Sur, Bertozzi & Behringer, Nonlinear dynamics of two-dimensional undercompressive shocks, Physica D 209, 36–48 (2005)](https://www.math.ucla.edu/~bertozzi/papers/BowenPhysicaD05.pdf) Section 2, equation (2.1), identifies the thermal-stress, gravity and capillary fluxes. The horizontal, prescribed-temperature browser model retains the thermal and capillary terms. The paper’s inclined fronts and experimental images are not reproduced.