Couette flow is one of the most elegant validation problems in computational fluid dynamics (CFD). It strips the Navier–Stokes equations down to their viscous core and gives us a linear velocity profile that can be derived analytically and verified numerically.

In this post, we:

• Derive the governing equation from first principles

• Prove the finite difference discretization of the second-order derivative

• Implement the solver in Julia

• Verify the numerical solution against the analytical result

---

1. Physical Setup: Plane Couette Flow

Two infinite parallel plates separated by distance $H$:

• Bottom plate: stationary

• Top plate: moving with velocity $U$

• Flow is steady

• Incompressible

• No pressure gradient

• One-dimensional velocity field

Velocity field:

$$\mathbf{u} = (u(y), 0, 0)$$

---

2. Reduction of Navier–Stokes

The incompressible Navier–Stokes equation is:

$$

\rho (\mathbf{u} \cdot \nabla)\mathbf{u}

$$

• \nabla p

\mu \nabla^2 \mathbf{u} $$

For steady, fully developed Couette flow:

• Convective term = 0

• Pressure gradient = 0

• Only viscous diffusion remains

So we obtain:

$$\mu \frac{d^2 u}{dy^2} = 0$$

Since \(\mu \neq 0\):

$$\frac{d^2 u}{dy^2} = 0$$

This is a second-order ordinary differential equation.

---

3. Analytical Solution

Integrate once:

$$\frac{du}{dy} = C_1$$

Integrate again:

$$u(y) = C_1 y + C_2$$

Apply boundary conditions:

$$u(0) = 0$$

$$u(H) = U$$

Solving gives:

$$u(y) = \frac{U}{H} y$$

The velocity profile is linear.

---

4. Mathematical Proof of Finite Difference Discretization

We now discretize:

$$\frac{d^2 u}{dy^2}$$

using Taylor series expansion.

---

4.1 Taylor Expansion

Expand \(u(y + \Delta y)\):

$$ u_{i+1}

u_i + \Delta y u'_i + \frac{\Delta y^2}{2} u''_i + \frac{\Delta y^3}{6} u'''_i + O(\Delta y^4) $$

Expand \(u(y - \Delta y)\):

$$ u_{i-1}

u_i

\Delta y u'_i + \frac{\Delta y^2}{2} u''_i

\frac{\Delta y^3}{6} u'''_i + O(\Delta y^4) $$

---

4.2 Add the Two Expansions

$$ u_{i+1} + u_{i-1}

2u_i + \Delta y^2 u''_i + O(\Delta y^4) $$

Rearranging:

$$ u''_i

\frac{u_{i+1} - 2u_i + u_{i-1}}{\Delta y^2} + O(\Delta y^2) $$

---

4.3 Conclusion

The central difference formula:

$$\frac{d^2 u}{dy^2} \approx \frac{u_{i+1} - 2u_i + u_{i-1}}{\Delta y^2}$$

is:

• Second-order accurate

• Symmetric

• Consistent with diffusion physics

---

5. Discrete System

Since:

$$\frac{d^2 u}{dy^2} = 0$$

We obtain:

$$u_{i+1} - 2u_i + u_{i-1} = 0$$

Or equivalently:

$$u_i = \frac{u_{i+1} + u_{i-1}}{2}$$

This forms a linear algebra system:

$$A u = b$$

with boundary conditions embedded.

---

6. Julia Implementation

using LinearAlgebra
using Plots

H = 1.0
U = 1.0
N = 50

dy = H/(N-1)

A = zeros(N, N)
b = zeros(N)

# Boundary conditions
A[1,1] = 1
b[1] = 0.0

A[N,N] = 1
b[N] = U

# Interior nodes
for i in 2:N-1
    A[i,i-1] = 1
    A[i,i]   = -2
    A[i,i+1] = 1
end

u = A\b

y = LinRange(0, H, N)
u_exact = U .* y ./ H

p = plot(u, y, label="Numerical", linewidth=3)
plot!(p, u_exact, y, linestyle=:dash, label="Analytical")

xlabel!("Velocity")
ylabel!("y")
title!("Couette Flow Verification")

display(p)

error = sqrt(sum((u .- u_exact).^2)/N)
println("L2 Error = ", error)