Fluid mechanics

The Navier–Stokes equations

The Navier–Stokes equations describe momentum balance in a viscous fluid, combining inertia, pressure, body forces, and viscous stress.

Navier–Stokes applies Newton’s second law to fluid elements. Its form depends on the constitutive model and assumptions; below is the differential form for a homogeneous, incompressible Newtonian fluid with constant viscosity.

\rho\left(\frac{\partial\mathbf u}{\partial t}+(\mathbf u\cdot\nabla)\mathbf u\right)=-\nabla p+\mu\nabla^2\mathbf u+\rho\mathbf f,\qquad\nabla\cdot\mathbf u=0

Definition: Momentum terms

$\rho$ is density, $\mathbf u$ velocity, $p$ pressure, $\mu$ dynamic viscosity, and $\mathbf f$ body force per unit mass. The left side is local plus advective acceleration; the right side contains pressure gradient, viscous diffusion, and body force.

Explore the Couette velocity profile between plates and viscosity effects; this uses a simple laminar solution, not a general solution of Navier–Stokes.

Momentum balance

The material acceleration combines local change $\partial_t\mathbf u$ with advection $(\mathbf u\cdot\nabla)\mathbf u$ . At low $Re$ , viscosity smooths velocity gradients; at high $Re$ , advection and inertia are more important, often making the flow harder to predict.

A benchmark solution: Couette flow

u_x(y)=U\frac{y}{H},\qquad 0\le y\le H

Between parallel plates separated by $H$ , with the lower plate fixed and upper plate moving at speed $U$ , a Newtonian fluid with no streamwise pressure gradient has a linear steady laminar profile. No-slip boundary conditions match fluid velocity to each plate.

Example: Couette speed across a gap

Two plates are separated by $H=2.0\,\mathrm{mm}$ ; the upper one moves at $U=0.10\,\mathrm{m/s}$ . Find the midpoint speed for Couette flow.

Solution

$u(H/2)=U(H/2)/H=U/2=0.050\,\mathrm{m/s}$ .

Quick check

Each term has a physical meaning and units of force per volume: $ho Dmathbf u/Dt$ is inertia, $- abla p$ is pressure force, $mu abla^2mathbf u$ is viscous momentum diffusion, and $homathbf f$ is body force per volume. For an incompressible fluid, the equation must be paired with $ablacdotmathbf u=0$ ; without boundary conditions such as no slip at a wall, the solution is not determined.

Which fluid model underlies the incompressible equation shown?

What is the Couette speed at mid-gap in the example?

References

Frank M. White (2016). Fluid Mechanics
G. K. Batchelor (1967). An Introduction to Fluid Dynamics