What makes fluid move?

Change pressure, viscosity or wall speed and watch the velocity profile take shape.

Flow between two plates

Choose a preset, then play.

Inside the channel

Side view · 1 m gap

Dots trace the fluid. Arrows show its direction and speed.

Velocity across the gap

CurrentSteady state

Time 1.05 sCentreline velocity 0.30 m/s

A pressure drop drives fluid to the right. Drag at both fixed walls leaves the centre moving fastest.

Pressure drive0.32 m/s²

Positive pushes right. Negative pushes left.

Kinematic viscosity0.06 m²/s

Controls how quickly momentum spreads across the gap.

Top wall speed0.00 m/s

The fluid at the top wall moves with it.

The diagrams show a saved snapshot. Enable JavaScript to run the simulation and change the settings.

From the equation to the motion

Imagine two very long plates with a fluid between them. We look away from the entrance and assume the flow is laminar and identical along the channel, with constant density and viscosity and no body force. The fluid moves along the plates, with no motion across the gap, so the momentum advection term is zero. That leaves pressure driving the flow and viscosity spreading momentum across the gap.

\underbrace{\frac{\partial u}{\partial t}}_{\text{change in velocity}} = \underbrace{-\frac{1}{\rho}\frac{\partial p}{\partial x}}_{\text{pressure drive}} + \underbrace{\nu\frac{\partial^2u}{\partial y^2}}_{\text{viscous diffusion}}

Pressure drives the flow: With both walls fixed, use the pressure preset. The centre speeds up while the fluid at each wall stays at rest.
Viscosity spreads momentum: Use the moving wall preset and turn viscosity up. Watch how quickly motion reaches the lower half of the channel.
Walls set the boundary values: The no-slip condition ties the fluid speed at each wall to the wall speed. Opposing drives can make the fluid move in both directions within one gap.

How the simulation calculates the curves

The solid curve comes from a one-dimensional finite-difference calculation on 48 equal intervals across the gap. Each step advances 0.01 simulated seconds using backward Euler in time and a centred second derivative in space. Restart sets the fluid interior to rest and applies the chosen wall speeds.

-r u_{j-1}^{n+1}+(1+2r)u_j^{n+1}-r u_{j+1}^{n+1}=u_j^n+a\Delta t,\qquad r=\frac{\nu\Delta t}{\Delta y^2}.

Each interior point j depends on its two neighbours at the new time level n + 1. The solver calculates those new velocities together using the two wall speeds as boundary values.

The dashed curve is the analytic steady solution. Here H is the gap, U_w is top wall speed, a is pressure drive and ν is kinematic viscosity.

u_{\infty}(y)=U_w\frac{y}{H}+\frac{a}{2\nu}y(H-y),\qquad a=-\frac{1}{\rho}\frac{\partial p}{\partial x}.

The model represents a cross-section of fully developed flow. The moving dots repeat at the edges of the illustration to keep the flow visible.