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Example · Two-Layer Flow Down an Inclined Plane

Problem statement

Two immiscible fluid films flow steadily down a plane inclined at angle \(\theta\): layer 1 of viscosity \(\mu_1\) and thickness \(h_1\) lies on the plane, layer 2 of viscosity \(\mu_2\) and thickness \(h_2\) lies on top, with a free surface above. Gravity drives the flow. Find the velocity profile in each layer.

Given information

  • Depths \(h_1\) (bottom layer), \(h_2\) (top layer); viscosities \(\mu_1\), \(\mu_2\)
  • Steady, fully developed, incompressible; no imposed pressure gradient along the plane

Solution strategy

In each layer the Navier–Stokes equation reduces to a gravity–viscosity balance:

\[0 = \mu_i \frac{d^2 u_i}{dy^2} + \rho g\sin\theta\]

Integrate twice per layer (4 constants) and spend the 4 boundary/matching conditions:

  1. No-slip at the plane: \(u_1(0) = 0\)
  2. Velocity continuity at the interface: \(u_1(h_1) = u_2(h_1)\)
  3. Shear-stress continuity at the interface: \(\mu_1\,u_1'(h_1) = \mu_2\,u_2'(h_1)\)
  4. Free surface: \(u_2'(h_1 + h_2) = 0\) (no shear from the air)

Step-by-step solution

Each layer integrates to \(u_i(y) = -\frac{\rho g\sin\theta}{2\mu_i}y^2 + C_i y + D_i\).

Working through the conditions (start from the free surface: condition 4 fixes the total stress profile — the shear at height \(y\) must carry the weight of all fluid above it, \(\tau(y) = \rho g\sin\theta\,(h_1 + h_2 - y)\), which automatically satisfies condition 3 for equal densities):

\[u_1(y) = \frac{\rho g\sin\theta}{\mu_1}\left[(h_1 + h_2)\,y - \frac{y^2}{2}\right]\]
\[u_2(y) = \frac{\rho g\sin\theta}{\mu_2}\left[(h_1+h_2)\,y - \frac{y^2}{2}\right] + \rho g \sin\theta\left[\frac{h_1^2}{2} - (h_1+h_2)h_1\right]\left(\frac{1}{\mu_2} - \frac{1}{\mu_1}\right)\]

Final answer

Two parabolic arcs, glued with equal velocity and equal shear stress at \(y = h_1\); the kink in slope at the interface is the viscosity ratio made visible: \(u_2'/u_1'|_{h_1} = \mu_1/\mu_2\).

Key takeaways

  • The interface conditions are the entire problem. Velocity continuous (no slip between liquids), stress continuous (Newton's third law) — and stress continuity means the velocity gradient jumps when viscosities differ.
  • The shear-stress profile is set by statics alone (weight of fluid above); viscosity only decides how much shearing that stress produces.
  • Limits check: \(\mu_1 = \mu_2\) recovers the single-film Nusselt profile; a very viscous bottom layer (\(\mu_1 \to \infty\)) becomes an effective solid floor for the top layer.

Couette flow · Viscosity · Navier–Stokes equation