Couette Flow Simulator — Viscous Flow Between Parallel Plates
For steady laminar flow between parallel plates (top plate at speed U, bottom plate static), vary speed, gap, viscosity, and pressure gradient to see wall shear stress, unit-width flow rate, and the combined Couette + Poiseuille profile in real time.
Parameters
Top-plate speed U
m/s
Gap h
mm
Dynamic viscosity μ
Pa·s
Pressure gradient dp/dx
Pa/m
The fluid is assumed to be water (rho = 1000 kg/m^3). With dp/dx = 0 you get pure Couette flow; with U = 0 you get pure Poiseuille flow.
Results
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Mean wall shear stress
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Unit-width flow rate
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Mean velocity
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Reynolds number
Velocity profile between plates
Top: moving plate (speed U) / Bottom: static plate / Arrows = velocity vectors (Couette line + Poiseuille parabola superposed) / Reverse flow shown in red
Normalized profile u/U vs y/h
X = y/h [0, 1] / Y = u/U / Blue solid = combined profile / Gray dashed = pure Couette line (reference)
Theory & Key Formulas
For steady laminar flow between parallel plates (top plate at speed U, bottom plate static, gap h), the Navier-Stokes equations give an analytical solution that is the superposition of a Couette line and a Poiseuille parabola. The general solution including pressure gradient dp/dx is:
Reynolds number (using gap $h$ as length scale and $U$ as velocity scale):
$$Re = \frac{\rho\,U\,h}{\mu}$$
Here $U$ is the top-plate speed [m/s], $h$ the gap [m], $\mu$ the dynamic viscosity [Pa·s], $dp/dx$ the streamwise pressure gradient [Pa/m], $\rho$ the density [kg/m^3], and $y$ is the height above the bottom plate.
What is the Couette Flow Simulator?
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I am hearing the term "Couette flow" for the first time. How is it different from Hagen-Poiseuille?
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Poiseuille is the flow that gets pushed by a pressure difference; Couette is the flow that gets dragged by a moving boundary. Hold the bottom plate still and slide the top plate at speed U, and the fluid in between is dragged along by viscosity in clean horizontal layers. With zero pressure gradient the velocity goes from u = 0 at the bottom to u = U at the top in a straight line — that is pure Couette flow, $u(y) = U y/h$ in one short formula.
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Wait, a straight line and not a parabola? In a circular pipe it was a parabola, right?
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Right — the driving mechanism is different, so the shape is different. Couette flow is driven by the moving wall as a boundary condition, with no body force and no pressure gradient inside, so the momentum equation is just $\mu\,d^2u/dy^2 = 0$ — and that has a linear solution. Add a pressure gradient and a constant appears on the right-hand side, $\mu\,d^2u/dy^2 = dp/dx$, which gives a parabolic solution. This simulator shows the superposition $u(y) = U y/h - (1/(2\mu))(dp/dx)\,y(h-y)$. Set dp/dx = 0 for a line, set U = 0 for pure Poiseuille.
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When I move the pressure-gradient slider I see a "reverse flow" appear. What is that?
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When dp/dx is positive — that is, an adverse pressure gradient that pushes opposite to the top plate's motion — the Poiseuille component fights against the Couette drag. Near the bottom plate the adverse push wins, and the velocity goes negative. The simulator highlights this with red arrows and a red shaded band. This actually happens inside bearings and gears: parts of the lubricant film flow backward, in what is called a reverse-flow pocket. Hit Play to sweep dp/dx from -2000 to +2000 and you can watch the transition: pure Couette line, then Poiseuille bulge, then reverse flow.
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The wall shear stress shows 0.5 Pa. Is that big or small?
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With the defaults (U = 1 m/s, h = 2 mm, water) you get tau = mu U / h = 0.001 x 1 / 0.002 = 0.5 Pa. That is small, but shrink the gap to h = 0.02 mm — a typical lubricating film — and tau jumps to 50 Pa. Friction loss W = tau x U x area in bearing design comes straight from this formula. The same Couette-flow equation governs everything from the precision bearings inside a smartphone to the oil films in an automobile crankshaft.
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It says Re = 2000 — is the laminar analytical solution still appropriate?
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Good catch. Plane shear-driven flow can transition to turbulence above Re of about 1500 — quite different from the 2300 you remember from circular pipes. The default Re = 2000 is right at that boundary. Increase mu by a factor of 10 (light oil at 0.01 Pa s) and Re drops to 200 (definitely laminar); increase h by a factor of 10 (gap = 20 mm) and Re jumps to 20000 (firmly turbulent), where the laminar analytical solution becomes a reference value rather than a prediction.
Frequently Asked Questions
By the physical driving mechanism. Couette flow is driven by the motion of a boundary (a wall speed) and applies to oil films in bearings, gear gaps, rotational viscometers, and other "moving wall vs static wall" geometries. Poiseuille flow is driven by a pressure difference and applies to pipe flows, IV drips, and microfluidic chips where the fluid is pushed from one end. In practice the two often coexist — for example, an engine bearing has a Couette component from the rotating shaft plus a Poiseuille component from the externally pumped lubricant supply, and is analyzed as a superposition. This tool displays exactly that combined solution.
In pure Couette flow (dp/dx = 0) there are no body forces and no pressure gradient on the fluid between the plates, so the momentum equation collapses to the trivial form $\mu\,d^2u/dy^2 = 0$ — second derivative equals zero. With boundary conditions u(0) = 0 and u(h) = U the unique solution is $u(y) = U y/h$. Physically, the moving top plate transmits a uniform shear stress through the fluid down to the static bottom plate, and each layer slides past the next at the same constant rate du/dy = U/h, like a deck of cards being sheared sideways. The shear stress is the same at every depth, and this is the foundation of standard design formulas.
Adding dp/dx not equal to 0 turns the momentum equation into $\mu\,d^2u/dy^2 = dp/dx$, and a parabolic component is superposed on the linear Couette part. The general solution is $u(y) = U y/h - (1/(2\mu))(dp/dx)\,y(h-y)$. The wall shear stresses on the top and bottom plates become asymmetric: tau_top = mu U/h - (h/2) dp/dx and tau_bot = mu U/h + (h/2) dp/dx. When dp/dx is large and positive (adverse), reverse flow appears near the bottom plate, which is connected to bearing cavitation and the instability of lubricant films. The Reynolds equation, the foundation of fluid lubrication theory, is derived from this combined Couette + Poiseuille solution.
A concentric-cylinder viscometer (Couette-type rheometer) is exactly an application of Couette flow. The inner cylinder (or sometimes the outer one) is rotated at a fixed angular speed and the test liquid is sheared in the thin annular gap between the two cylinders, with viscosity inferred from the rotation torque. When the gap h is small compared to the cylinder radius, the cylindrical surfaces are approximated as parallel plates and the formula tau = mu U / h gives mu directly. A cone-and-plate rheometer works by the same principle and keeps du/dy uniform throughout the sample, allowing precise measurement of the shear-rate-dependent viscosity of non-Newtonian fluids. These instruments are widely used in cosmetics, food, paint, and blood research.
Real-World Applications
Lubrication and tribology: Lubricant flow in journal bearings, ball bearings, and gear gaps is treated as a combined Couette + Poiseuille flow. The Reynolds equation, the central equation of fluid lubrication theory, is derived starting from the same general solution $u(y) = U y/h - (1/(2\mu))(dp/dx)\,y(h-y)$ that this tool displays. Friction loss W = tau x U x area, heat generation, minimum film thickness, and load capacity of a bearing are all computed from it. Engine bearings, the precision motors inside a smartphone, and the crankshaft of a high-speed train all rely on this lubricated-film analysis.
Rheology and viscometry: Couette-type cylindrical viscometers and cone-and-plate rheometers both operate on the Couette-flow principle. By keeping du/dy uniform across the sample, they measure Newtonian viscosity precisely and characterize the shear-rate-dependent viscosity mu(gamma_dot) of non-Newtonian fluids (thixotropic, Bingham, power-law, etc.). They are essential for the quality control of cosmetics, food, paint, printing inks, blood, and polymer solutions in industry.
Microfluidics and lab-on-a-chip: In microfluidic chips with channel gaps of about 10-100 micrometers, electroosmotic flow generates a thin Couette-like layer near the wall while pressure-driven Poiseuille flow acts on the bulk; both components coexist. The combined profile is used to design devices for cell sorting, protein separation, and PCR. The "reverse flow" condition shown by this tool is also used deliberately in some channel designs to enhance mixing.
Geodynamics and glacier flow: Glaciers, mantle convection, and tectonic-plate motion are all treated as viscous flows on long time scales. Glacier flow has different boundary conditions (fixed bedrock at the bottom, free surface at the top), but the velocity profile near the bed is qualitatively similar to a Couette profile. With viscosity mu of about 10^15 Pa s and very slow flow (meters per year), the same momentum equation that drives this tool is at work in geophysics.
Common Misconceptions and Pitfalls
The most common mistake is to assume Couette flow always has a linear profile. The profile is linear only for the special case dp/dx = 0. In real bearings, gears, and rheometers there is almost always a pressure gradient: in a bearing, for example, a pressure distribution must develop in the film to support the load, and that contributes a Poiseuille component. Move the dp/dx slider in this simulator and you can feel the profile bulge from the line into a parabola, and then exhibit reverse flow. Standard bearing design uses the Reynolds equation with this pressure distribution included.
The next pitfall is to assume the critical Re for parallel-plate flow is the same 2300 you know from circular pipes. The famous result for plane shear-driven (pure Couette) flow is that it is linearly stable at all Reynolds numbers, yet experimentally it transitions to turbulence around Re = 360 to 1500 (subcritical transition). This tool always displays the laminar analytical solution, so for high Re it diverges from the real flow. Also remember that critical Re differs by problem: about 2300 for pipe flow, about 5 x 10^5 for boundary-layer flow over a flat plate, and so on.
Finally, watch out for the confusion between unit-width flow rate Q' (m^2/s) and ordinary volume flow rate Q (m^3/s). Plane parallel-plate flow is treated as a 2D problem where the channel extends infinitely in the depth direction, so the flow rate is per unit depth and has units of m^2/s. To get the actual 3D volume flow rate just multiply by the channel width W: Q = Q' x W in m^3/s. This tool reports both Q' (unit-width flow rate) and V_avg (mean velocity), so use the right one for your purpose. Mixing 2D assumptions with 3D conversions is a classic CAE bug that gives a flow-rate error by a factor of W.