Question: Help Solving PDE System (Eqs. 9–11) Using ICs and BCs for H- and Square-Shaped Cavities

Hello everyone,

I am working on reproducing Fig. 12 (Isotherms) and Fig. 13 (Streamlines) from the attached paper, cacity_paper_work.pdf which deals with natural convection of GO–MgO/silicone oil hybrid nanofluid inside square and H-shaped cavities under a periodic magnetic field.

Equations (9–11) in the paper describe the steady-state 2D flow and heat transfer in terms of u(x,y), v(x,y), and θ(x,y). I have already set up the full PDE system, parameters, and boundary conditions (BCs) in Maple.

Now, iusing pdsolve(..., numeric, method=fd), 

Solve these equations for both square and H-shaped cavities using the given ICs and BCs, and

Generate Isotherm and Streamline plots (similar to Figs. 12 and 13 in the paper).

I have attached:

The PDF (showing equations and figures), and

My Maple worksheet containing the PDE definitions and parameters.

I would really appreciate guidance on:

Whether dsolve can be used effectively for this PDE system,

How to define the BCs correctly for the H-shaped cavity (especially the partially heated wall, θ=1 for 0.4 < y < 0.6), and

How to extract and plot the temperature (θ) and streamfunction (ψ) fields similar to the paper’s plots.

cavity_work_error.mw