Differentially Heated Cavity
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In the differentially heated cavity, the fluid is driven by two opposite vertical walls that have different temperatures $T_A$ and $T_B$.
Since we use a nondimensional formulation, $T_A$ and $T_B$ are only used when rescaling back to dimensional quantities.

Governing Equations
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In a domain of dimensions $L \\times B \\times D$, the governing equations are given by

.. math:: \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} &= -\frac{1}{\rho_0} \nabla p + \nu \nabla^2 \mathbf{u} - \frac{\rho g}{\rho_0} e_z\\
          \nabla \cdot \mathbf{u} &= 0\\
          \frac{\partial T}{\partial t} + (\mathbf{u} \cdot \nabla) T &= \kappa_T \nabla^2 T

with boundary conditions

.. math:: x &= 0 &:~& u = v = w = 0, T = T_A\\
          x &= L &:~& u = v = w = 0, T = T_B\\
          y &= 0, B &:~& u = v = w = \frac{\partial T}{\partial y} = 0\\
          z &= 0, D &:~& u = v = w = \frac{\partial T}{\partial z} = 0

where $T_A$ is the temperature of the left wall and $T_B$ is the temperature of the right wall. See :ref:`symbols` for the definition of all quantities used here.

Nondimensional formulation
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In TransiFlow, we implement the following nondimensionalized equations

.. math:: \frac{\partial \tilde{\mathbf{u}}}{\partial \tilde t} + (\tilde{\mathbf{u}} \cdot \tilde \nabla) \tilde{\mathbf{u}} &= -\tilde \nabla \tilde p +\frac{1}{\mathrm{Gr}^{1/2}}\tilde \nabla^2 \tilde{\mathbf{u}} + \tilde T e_z\\
          \tilde\nabla \cdot \tilde{\mathbf{u}} &= 0\\
          \frac{\partial \tilde T}{\partial \tilde t} + (\tilde{\mathbf{u}} \cdot \tilde\nabla) \tilde T &= \frac{1}{\mathrm{Pr} \mathrm{Gr}^{1/2}} \tilde \nabla^2 \tilde T

with boundary conditions

.. math:: \tilde x &= 0 &:~& \tilde u = \tilde v = \tilde w = 0, \tilde T = \frac{1}{2}\\
          \tilde x &= A_x &:~& \tilde u = \tilde v = \tilde w = 0, \tilde T = -\frac{1}{2}\\
          \tilde y &= 0, A_y &:~& \tilde u = \tilde v = \tilde w = \frac{\partial \tilde T}{\partial \tilde y} = 0\\
          \tilde z &= 0, 1 &:~& \tilde u = \tilde v = \tilde w = \frac{\partial \tilde T}{\partial \tilde z} = 0

Here $x$, $y$ and $z$ are scaled by $D$, and hence $A_x = L / D$ and $A_y = B / D$. The other quantities are scaled using $u = \\hat u \\nu / D$, $t = \\hat t D^2 / \\nu$, $p = \\hat p (\\mu \\nu / D^2) - \\rho_0 g z$, $T = (T_B - T_A) \\hat T + 1 / 2 (T_A + T_B)$ and additionally $\\hat u = \\tilde u \\mathrm{Gr}^{1/2}$, $\\hat t = \\tilde t \\mathrm{Gr}^{-1/2}$, $\\hat p = \\tilde p \\mathrm{Gr}$, $\\hat T = \\tilde T$. Moreover, the Prandtl number is given by $\\mathrm{Pr} = \\nu / \\kappa_T$, the Rayleigh number by $\\mathrm{Ra} = (\\alpha_T g \\Delta T D^3) / (\\nu \\kappa_T)$ and the Grashof number by $\\mathrm{Gr} = \\mathrm{Ra} / \\mathrm{Pr}$. All other quantities are defined in :ref:`symbols`.

Parameters
----------
These are the relevant parameters in the ``parameters`` dictionary for this problem type.

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Parameter name        Default value Notes
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``'Problem Type'``                  Set to ``'Differentially Heated Cavity'``
``'Rayleigh Number'`` 1.0           Unused if Gr is defined
``'Prandtl Number'``  1.0
``'Grashof Number'``  Ra / Pr       Overrides Ra if defined
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