Datasets:
Update README.md
Browse files
README.md
CHANGED
@@ -50,13 +50,15 @@ Rayleigh-Bénard convection involves fluid dynamics and thermodynamics, seen in
|
|
50 |
**Equation**:
|
51 |
|
52 |
While we solve equations in the frequency domain, the original time-domain problem is
|
53 |
-
|
|
|
54 |
\begin{align*}
|
55 |
\frac{\partial b}{\partial t} - \kappa\,\Delta b & = -u\nabla b\,,
|
56 |
\\
|
57 |
\frac{\partial u}{\partial t} - \nu\,\Delta u + \nabla p - b \vec{e}_z & = -u \nabla u\,,
|
58 |
\end{align*}
|
59 |
-
|
|
|
60 |
where \\(\Delta = \nabla \cdot \nabla\\) is the spatial Laplacian, \\(b\\) is the buoyancy, \\(u = (u_x,u_y)\\) the (horizontal and vertical) velocity, and \\(p\\) is the pressure, \\(\vec{e}_z\\) is the unit vector in the vertical direction, with the additional constraints \\(\int p = 0\\) (pressure gauge).
|
61 |
|
62 |
The boundary conditions vertically are as follows:
|
|
|
50 |
**Equation**:
|
51 |
|
52 |
While we solve equations in the frequency domain, the original time-domain problem is
|
53 |
+
|
54 |
+
$$
|
55 |
\begin{align*}
|
56 |
\frac{\partial b}{\partial t} - \kappa\,\Delta b & = -u\nabla b\,,
|
57 |
\\
|
58 |
\frac{\partial u}{\partial t} - \nu\,\Delta u + \nabla p - b \vec{e}_z & = -u \nabla u\,,
|
59 |
\end{align*}
|
60 |
+
$$
|
61 |
+
|
62 |
where \\(\Delta = \nabla \cdot \nabla\\) is the spatial Laplacian, \\(b\\) is the buoyancy, \\(u = (u_x,u_y)\\) the (horizontal and vertical) velocity, and \\(p\\) is the pressure, \\(\vec{e}_z\\) is the unit vector in the vertical direction, with the additional constraints \\(\int p = 0\\) (pressure gauge).
|
63 |
|
64 |
The boundary conditions vertically are as follows:
|