The Stefan equation is the lubrication-theory relationship between squeezing force, viscosity, disk size, gap, and approach speed. For two parallel circular disks of radius (R) approaching one another, the equation given the half-gap convention (H(t)) is

[ F(t)=\frac{3\pi \mu U(t)R^4}{8H(t)^3}, ]

where (F) is the normal force, (\mu) is dynamic viscosity, (U=-dH/dt) is the approach speed of each disk, and (H) is the half-spacing between the disks. In an educational pilot plant, measuring (F), (H), and (U) allows students to test lubrication theory and estimate viscosity from force-displacement data.

The central result is the strong inverse-cube dependence on gap: reducing the half-gap increases the required squeezing force as (H^{-3}), while increasing disk radius increases the force as (R^4). The equation is reliable when the gap is narrow, the flow is slow, and viscous effects dominate inertia.

What the Stefan Equation Describes

The physical experiment

The experiment uses two parallel circular disks separated by a thin fluid layer. The disks move toward one another, forcing liquid to flow radially outward through the narrowing gap.

The setup represents a simplified squeeze-flow operation relevant to lubrication, polymer processing, coating, forming, and rheological measurement.

The key variables

The variables are:

(R): disk radius
(H(t)): half-spacing between the disks
(U(t)): approach speed of each disk
(F(t)): applied normal force
(\mu): dynamic viscosity
(\rho): fluid density
(r): radial distance from the disk center

The full fluid-layer thickness is (2H). This convention is important because alternative references often define the full gap as (h=2H), producing a different-looking numerical coefficient.

The governing assumptions

The derivation assumes:

Creeping flow, with negligible inertia
A narrow gap, (H_0/R \ll 1)
Newtonian, incompressible fluid behavior
No slip at the disk surfaces
Parallel disks and axisymmetric flow
Uniform approach speed across each disk
Negligible edge effects and fluid leakage

The relevant Reynolds number is

[ Re=\frac{H_0U_0}{\nu}\ll 1, ]

where (\nu=\mu/\rho) is the kinematic viscosity.

How the Equation Is Derived

Start with radial squeeze flow

As the disks approach, the fluid cannot move through the solid disks. It therefore moves outward in the radial direction.

Because the gap is small compared with the disk radius, velocity gradients across the gap dominate velocity gradients in the radial direction. This is the central approximation of lubrication theory.

Reduce the momentum equation

Under creeping-flow conditions, the inertial terms in the Navier-Stokes equations are negligible. The radial momentum balance reduces to

[ 0=-\frac{\partial P}{\partial r} +\mu\frac{\partial^2 u_r}{\partial z^2}, ]

where (u_r) is the radial velocity and (z) is the coordinate across the fluid layer.

The pressure varies mainly with radius, while the radial velocity varies strongly across the narrow gap.

Apply the disk boundary conditions

The no-slip condition requires the radial velocity to vanish at both disk surfaces:

[ u_r=0 \quad \text{at} \quad z=-H \text{ and } z=H. ]

Integrating the simplified momentum equation twice across the gap gives a parabolic radial velocity profile. The volume flow rate associated with this profile must match the fluid displaced by the approaching disks.

Use continuity

For two disks approaching symmetrically, the half-gap decreases at speed (U):

[ \frac{dH}{dt}=-U. ]

The changing gap displaces fluid at a rate proportional to the local disk area. Axisymmetric continuity then relates the radial flow to the rate of gap reduction.

Combining continuity with the parabolic velocity profile gives the radial pressure gradient

[ \frac{\partial P}{\partial r}

-\frac{3\mu U r}{2H^3}. ]

The negative sign indicates that pressure decreases from the center toward the outer edge.

Integrate the pressure field

Taking the pressure at the disk edge as the reference pressure, (P(R)=0), integration from (r) to (R) gives

[ P(r)=\frac{3\mu U}{4H^3}\left(R^2-r^2\right). ]

The pressure is therefore highest at the center and decreases quadratically toward the rim.

Integrate pressure to obtain force

The total normal force is the pressure integrated over the circular disk area:

[ F=2\pi\int_0^R P(r),r,dr. ]

Substituting the pressure distribution yields

[ F

2\pi\int_0^R \frac{3\mu U}{4H^3} \left(R^2-r^2\right)r,dr, ]

and therefore

[ \boxed{ F=\frac{3\pi\mu U R^4}{8H^3} }. ]

This is the Stefan equation under the stated half-gap and symmetric-approach convention.

Why the Scaling Matters

Force increases rapidly as the gap closes

The equation contains the factor

[ F\propto \frac{1}{H^3}. ]

Halving the half-gap increases the predicted force by a factor of eight if (U), (R), and (\mu) remain constant.

This strong dependence is why small errors in gap measurement can produce large errors in calculated viscosity or predicted force.

Disk radius has an even stronger effect

The force scales with the fourth power of radius:

[ F\propto R^4. ]

A larger disk produces substantially greater resistance because it displaces more fluid and increases the radial distance over which pressure acts.

Viscosity and approach speed have linear effects

The force is directly proportional to both viscosity and approach speed:

[ F\propto \mu U. ]

A more viscous liquid or faster compression requires a greater normal force, provided the fluid remains Newtonian and the creeping-flow assumptions remain valid.

Applying the Equation in a Pilot Plant

Prepare and inspect the apparatus

Install the circular disks, verify that they are parallel, and measure the disk radius. Fill the gap carefully to avoid trapped air, since air compliance can distort force measurements.

The temperature should be monitored because viscosity may vary significantly with temperature.

Control the motion

Set a known approach-speed profile (U(t)), or impose a controlled displacement history (H(t)). The controller should record the actual disk position rather than relying only on its commanded position.

The half-gap is calculated from the measured disk spacing:

[ H(t)=\frac{\text{full disk spacing}}{2}. ]

Record force and displacement

During the squeeze step, record:

Normal force (F(t))
Disk position or gap (H(t))
Approach speed (U(t))
Fluid temperature
Time

A force-displacement curve is especially useful because it shows how the resistance rises as the gap narrows.

Estimate viscosity

Rearranging the Stefan equation gives

[ \boxed{ \mu

\frac{8F H^3}{3\pi U R^4} }. ]

Students can calculate an instantaneous apparent viscosity from each measured data point, or fit the model to the full force-time or force-gap dataset.

For a constant viscosity, the calculated values should remain approximately consistent over the valid operating range.

Validate the predicted scaling

The experiment can test the model by varying one parameter at a time:

Plot (F) against (U) to test linear speed dependence.
Plot (F) against (\mu) using fluids of different viscosities.
Plot (F) against (H^{-3}) to test the gap dependence.
Compare disks of different radii to test the (R^4) dependence.

A log-log plot can make the predicted power-law exponents easier to identify experimentally.

Understanding the Trade-offs

The narrow-gap approximation can fail

The model requires

[ \frac{H_0}{R}\ll 1. ]

If the gap is not sufficiently small relative to the disk radius, radial and edge effects become important and the one-dimensional lubrication approximation loses accuracy.

Inertia can contaminate high-speed tests

The creeping-flow condition requires

[ Re=\frac{H_0U_0}{\nu}\ll 1. ]

At high approach speeds or with low-viscosity fluids, inertial forces may become measurable. The applied squeezing force should satisfy the stated practical condition

[ F\ll \frac{\mu^2}{\rho} \left(\frac{R}{H_0}\right)^4, ]

which helps keep inertial interference small in the pilot-plant experiment.

Edge leakage reduces the measured force

The ideal derivation assumes a well-defined circular boundary and a pressure equal to the reference pressure at the disk edge. If fluid escapes unevenly, the pressure field and measured force will be lower than predicted.

Poor parallelism can produce the same type of error by creating a nonuniform gap.

Newtonian behavior may not apply

The Stefan equation uses a constant dynamic viscosity. Polymer melts, concentrated suspensions, gels, and many structured fluids may be shear-thinning, shear-thickening, viscoelastic, or yield-stress fluids.

For those materials, the measured force may represent an apparent or rate-dependent viscosity rather than a single material constant.

The gap measurement is especially sensitive

Because force varies with (H^{-3}), a small zeroing error or uncertainty in disk separation can produce a large viscosity error.

The apparatus should be calibrated for displacement, force, disk parallelism, and the initial gap before collecting data.

Transient effects can matter

The equation describes the viscous squeeze response under quasi-steady assumptions. At the start and end of a motion cycle, actuator acceleration, instrument response time, trapped air, and fluid inertia can create transient deviations.

Data should be interpreted over a region where the commanded motion and measured motion agree.

How to Apply This to Your Project

The most reliable educational experiment combines controlled motion, accurate gap measurement, temperature control, and dimensionless checks before interpreting the force data.

If your primary focus is validating lubrication theory: Use a low-viscosity or well-characterized Newtonian fluid, a small gap-to-radius ratio, and several approach speeds to test (F\propto U) and (F\propto H^{-3}).
If your primary focus is measuring viscosity: Rearrange the equation to calculate (\mu=8FH^3/(3\pi U R^4)), then compare the result with an independent viscometer measurement.
If your primary focus is studying scale effects: Change disk radius or initial gap systematically and compare the measured force with the predicted (R^4) and (H^{-3}) dependencies.
If your primary focus is polymer or non-Newtonian processing: Treat the result as an apparent viscosity measurement and examine how the inferred viscosity changes with approach speed and gap.
If your primary focus is obtaining defensible data: Confirm (Re\ll1), (H_0/R\ll1), disk parallelism, fluid confinement, and temperature stability before fitting the Stefan model.

With the geometry and operating conditions kept within its assumptions, the Stefan equation provides a direct bridge between measured squeezing force and the viscosity-driven physics of narrow-gap flow.

Summary Table:

Topic	Key Information
Equation	F = 3πμUR⁴/(8H³)
Main variables	F: force; μ: viscosity; U: approach speed; R: disk radius; H: half-gap
Core scaling	Force increases with μ, U, and R⁴, and rises sharply as H⁻³
Main assumptions	Creeping flow, narrow gap, Newtonian fluid, no slip, parallel disks, negligible leakage
Experimental measurements	Normal force, disk spacing, approach speed, temperature, and time
Viscosity calculation	μ = 8FH³/(3πUR⁴)
Validation tests	Vary speed, viscosity, gap, or disk radius and compare measured power-law trends
Common error sources	Trapped air, poor parallelism, gap-measurement errors, edge leakage, inertia, and temperature changes
Educational applications	Lubrication theory, rheology, polymer processing, coating, forming, and fluid mechanics training

Apply Squeeze-Flow Theory with LABPARK

Bring the Stefan equation from the textbook into a controlled, measurable experiment. LABPARK provides Educational and Vocational Unit Operations Pilot Plants for chemical engineering, bioprocess and biotechnology, and environmental and water treatment education.

Our pilot plants help universities, research institutes, and enterprises teach and evaluate fluid mechanics through controlled motion, force and displacement measurement, temperature monitoring, and repeatable operating conditions. Students and researchers can validate lubrication theory, estimate viscosity, investigate Newtonian and non-Newtonian behavior, and understand how gap, disk radius, and approach speed affect process performance.

Contact LABPARK to discuss a pilot-plant solution tailored to your teaching, research, or industrial training goals.