Laminar Boundary Layer — Similarity Solution & Convective Heat Transfer

Stokes' First Problem — Penetration Depth

• Solved for the momentum penetration depth using the error function (erf) and its inverse (erfinv)
• Compared penetration depth across velocity ratios (u/U₀) from 0.001 to 0.1
• Solved the transient velocity profile for time steps of 10, 20, 30, 40 and 50 seconds

Fig. 1 — Penetration depth at t = 10–50 s

Blasius Similarity Solution — Flat‑Plate Boundary Layer

• Reduced the momentum similarity equation 2f‴ + ff″ = 0 and thermal equation θ″ + (Pr/2)fθ′ = 0 to first‑order systems
• Solved both with the shooting method in MATLAB (ode45 + fsolve to close the unknown boundary condition)
• Compared velocity profiles along the plate for free‑stream velocities of 1 m/s and 3 m/s
• Derived the stream function and plotted streamlines for uniform flow over the flat plate
• Derived the heat function (Morega & Bejan method) and plotted heatlines to visualize the heat‑transfer path

Fig. 2‑1 — Momentum similarity solution Fig. 2‑2 — Velocity profile, U∞ = 1 m/s Fig. 2‑3 — Velocity profile, U∞ = 3 m/s Fig. 2‑4 — Streamlines Fig. 2‑5 — Thermal similarity solution Fig. 2‑6 — Heatlines

MATLAB Script