MMS05 manufactured solutions
----------------------------

The manufactured solutions for the Free-slip wall BC, single phase, 3D, curl-based functions are listed below.

Gas pressure:

.. math::
   :label: a1eq35

   p_{g} = p_{g0} &+ p_{\text{gx}}\cos\left( A_{p_{\text{gx}}}\text{Ï€x} \right) + p_{\text{gy}}\cos\left( A_{p_{\text{gy}}}\text{Ï€y} \right) + p_{\text{gxy}}\cos\left( A_{p_{\text{gxy}}}\text{Ï€xy} \right) \\
		  &+ p_{\text{gz}}\sin\left( A_{p_{\text{gz}}}\text{Ï€z} \right) + p_{\text{gyz}}\sin\left( A_{p_{\text{gyz}}}\text{Ï€yz} \right) + p_{\text{gzx}}\cos\left( A_{p_{\text{gzx}}}\text{Ï€zx} \right)

Gas velocity components:

.. math::
   :label: a1eq36

   u_{g} = x^{3}\big[& A_{w_{\text{gy}}}\pi w_{\text{gy}}\cos\left( A_{w_{\text{gy}}}\text{Ï€y} \right) + A_{w_{\text{gxy}}}\pi w_{\text{gxy}}x\cos\left( A_{w_{\text{gxy}}}\text{Ï€xy} \right) \\
	  &- A_{v_{\text{gyz}}}\pi v_{\text{gyz}}y\cos\left( A_{v_{\text{gyz}}}\text{Ï€yz} \right) + A_{w_{\text{gyz}}}\pi w_{\text{gyz}}z\cos\left( A_{w_{\text{gyz}}}\text{Ï€yz} \right) \\
	  &+ A_{v_{\text{gz}}}\pi v_{\text{gz}}\sin\left( A_{v_{\text{gz}}}\text{Ï€z} \right) + A_{v_{\text{gzx}}}\pi v_{\text{gzx}}x\sin\left( A_{v_{\text{gzx}}}\text{Ï€zx} \right) \big]


.. math::
   :label: a1eq37

   v_{g} =& v_{g0} + x^{3}\big[ - A_{w_{\text{gxy}}}\pi w_{\text{gxy}}y\cos\left( A_{w_{\text{gxy}}}\text{Ï€xy} \right) + A_{u_{\text{gyz}}}\pi u_{\text{gyz}}y\cos\left( A_{u_{\text{gyz}}}\text{Ï€yz} \right) \\ 
	  &+ A_{w_{\text{gx}}}\pi w_{\text{gx}}\sin\left( A_{w_{\text{gx}}}\text{Ï€x} \right) - A_{u_{\text{gz}}}\pi u_{\text{gz}}\sin\left( A_{u_{\text{gz}}}\text{Ï€z} \right) \\
	  &- A_{u_{\text{gzx}}}\pi u_{\text{gzx}}x\sin\left( A_{u_{\text{gzx}}}\text{Ï€zx} \right) + A_{w_{\text{gzx}}}\pi w_{\text{gzx}}z\sin\left( A_{w_{\text{gzx}}}\text{Ï€zx} \right) \big] \\
	  &+ 3x^{2} \big[ - w_{g0} - w_{\text{gx}}\cos\left( A_{w_{\text{gx}}}\text{Ï€x} \right) - w_{\text{gz}}\cos\left( A_{w_{\text{gz}}}\text{Ï€z} \right) \\
	  &- w_{\text{gzx}}\cos\left( A_{w_{\text{gzx}}}\text{Ï€zx} \right) - w_{\text{gy}}\sin\left( A_{w_{\text{gy}}}\text{Ï€y} \right) \\
	  &- w_{\text{gxy}}\sin\left( A_{w_{\text{gxy}}}\text{Ï€xy} \right) - w_{\text{gyz}}\sin\left( A_{w_{\text{gyz}}}\text{Ï€yz} \right) \big]

.. math::
   :label: a1eq38

   w_{g} =& w_{g0} + x^{3}\big[ A_{v_{\text{gx}}}\pi v_{\text{gx}}\cos\left( A_{v_{\text{gx}}}\text{Ï€x} \right) - A_{u_{\text{gyz}}}\pi u_{\text{gyz}}z\cos\left( A_{u_{\text{gyz}}}\text{Ï€yz} \right) \\
	  &+ A_{u_{\text{gxy}}}\pi u_{\text{gxy}}x\sin\left( A_{u_{\text{gxy}}}\text{Ï€xy} \right) + A_{u_{\text{gy}}}\pi u_{\text{gy}}\sin\left( A_{u_{\text{gy}}}\text{Ï€y} \right) \\
	  &- A_{v_{\text{gxy}}}\pi v_{\text{gxy}}y\sin\left( A_{v_{\text{gxy}}}\text{Ï€xy} \right) - A_{v_{\text{gzx}}}\pi v_{\text{gzx}}z\sin\left( A_{v_{\text{gzx}}}\text{Ï€zx} \right) \big] \\
	  &+ 3x^{2}\big[ v_{g0} + v_{\text{gy}}\cos\left( A_{v_{\text{gy}}}\text{Ï€y} \right) + v_{\text{gxy}}\cos\left( A_{v_{\text{gxy}}}\text{Ï€xy} \right) \\
	  &+ v_{\text{gz}}\cos\left( A_{v_{\text{gz}}}\text{Ï€z} \right) + v_{\text{gzx}}\cos\left( A_{v_{\text{gzx}}}\text{Ï€zx} \right) \\
	  &+ v_{\text{gx}}\sin\left( A_{v_{\text{gx}}}\text{Ï€x} \right) + v_{\text{gyz}}\sin\left( A_{v_{\text{gyz}}}\text{Ï€yz} \right) \big]


Gas volume fraction:

.. math::
   \varepsilon_{g} = 1.0
   :label: a1eq39

The parameters appearing in the manufactured solutions are as follows:

.. _tableA7:
.. csv-table:: Parameters in MMS05 manufactured solutions.
   :widths: auto

   ":math:`p_{g0}`", "100.0", ":math:`v_{g0}`", "9.0", ":math:`u_{g0}`", "7.0", ":math:`w_{g0}`", "8.0"
   ":math:`p_{\text{gx}}`", "20.0", ":math:`v_{\text{gx}}`", "-5.0", ":math:`u_{\text{gx}}`", "3.0", ":math:`w_{\text{gx}}`", "-4.0"
   ":math:`p_{\text{gy}}`", "-50.0", ":math:`v_{\text{gy}}`", "4.0", ":math:`u_{\text{gy}}`", "-4.0", ":math:`w_{\text{gy}}`", "3.5"
   ":math:`p_{\text{gz}}`", "20.0", ":math:`v_{\text{gz}}`", "5.0", ":math:`u_{\text{gz}}`", "-3.0", ":math:`w_{\text{gz}}`", "4.2"
   ":math:`p_{\text{gxy}}`", "-25.0", ":math:`v_{\text{gxy}}`", "-3.0", ":math:`u_{\text{gxy}}`", "2.0", ":math:`w_{\text{gxy}}`", "-2.2"
   ":math:`p_{\text{gyz}}`", "-10.0", ":math:`v_{\text{gyz}}`", "2.5", ":math:`u_{\text{gyz}}`", "1.5", ":math:`w_{\text{gyz}}`", "2.1"
   ":math:`p_{\text{gzx}}`", "10.0", ":math:`v_{\text{gzx}}`", "3.5", ":math:`u_{\text{gzx}}`", "-2.0", ":math:`w_{\text{gzx}}`", "2.5"
   ":math:`A_{p_{\text{gx}}}`", "0.4", ":math:`A_{v_{\text{gx}}}`", "0.8", ":math:`A_{u_{\text{gx}}}`", "0.5", ":math:`A_{w_{\text{gx}}}`", "0.85"
   ":math:`A_{p_{\text{gy}}}`", "0.45", ":math:`A_{v_{\text{gy}}}`", "0.8", ":math:`A_{u_{\text{gy}}}`", "0.85", ":math:`A_{w_{\text{gy}}}`", "0.9"
   ":math:`A_{p_{\text{gz}}}`", "0.85", ":math:`A_{v_{\text{gz}}}`", "0.5", ":math:`A_{u_{\text{gz}}}`", "0.4", ":math:`A_{w_{\text{gz}}}`", "0.5"
   ":math:`A_{p_{\text{gxy}}}`", "0.75", ":math:`A_{v_{\text{gxy}}}`", "0.9", ":math:`A_{u_{\text{gxy}}}`", "0.6", ":math:`A_{w_{\text{gxy}}}`", "0.4"
   ":math:`A_{p_{\text{gyz}}}`", "0.7", ":math:`A_{v_{\text{gyz}}}`", "0.4", ":math:`A_{u_{\text{gyz}}}`", "0.8", ":math:`A_{w_{\text{gyz}}}`", "0.8"
   ":math:`A_{p_{\text{gzx}}}`", "0.8", ":math:`A_{v_{\text{gzx}}}`", "0.6", ":math:`A_{u_{\text{gzx}}}`", "0.9", ":math:`A_{w_{\text{gzx}}}`", "0.75"