update to documentation. Update to sod notebook

Docs/sphinx/LBM.rst

@@ -51,3 +51,8 @@ where :math:`u` is the velocity vector, :math:`\rho` is the density, :math:`w_i`
    u(x, t) &= \frac{1}{\rho} \sum_{i} e_i f_i(x, t)
 
 MARBLES implements both the D2Q9 and D3Q27 LBM approaches for two dimensional and three dimensional flow, respectively.
+
+For the thermal version of MARBLES, please refer to additional theory related to the additional energy lattice and cross coupling, 
+in the PhD thesis  
+"Sawant, Nilesh. 'Kinetic Modeling of Reactive Flows.'. ETH Zurich, 2023. https://doi.org/10.3929/ETHZ-B-000607045."
+

Docs/sphinx/VandV.rst

@@ -24,3 +24,54 @@ This demonstrates MARBLES ability to simulate flow through complex, porous media
 
 .. image:: /figs/pine_box.gif
    :width: 600pt
+
+
+Viscosity verification
+----------------------
+
+This test case demonstrates MARBLES ability to maintain the correct viscosity over a wide range of temepratures. 
+Viscosity is computed from the simulation by tracking the maximum of velocity over time arising from an initial 
+condition consisting of a sinusoidal velocity perturbation.
+The temeprature is not restricted to 1/3, but it can be set to any value less than or equal to 1/3. 
+This feature has been enabled by incorporating product-form of the equilibrium distribition which 
+improves stability due to maximisation of the entropy and the incorporation of an extended lattice 
+Boltzmann approach which adds a non-local correction term to the pressure tensor to correct the insufficient 
+Galilean invariance of the standard lattice at the third order moment.
+
+Input and post processing files: `Tests/test_files/viscosityTest/`
+
+.. image:: /figs/kinematicViscosity.png
+   :width: 400pt   
+
+
+Thermal diffusivity verification
+---------------------------------
+
+This test case demonstrates MARBLES ability to maintain the correct thermal diffusivity over a wide range of temepratures, 
+as well as at different Prandtl numbers, adiabatic exponents and mean molecular masses.
+Themal diffusivity is computed from the simulation by tracking the maximum of temperature over time arising from an initial 
+condition consisting of a sinusoidal temperature perturbation.
+The capability to solve for temperature has been enabled by an additional lattice which conserves the total energy of the fluid.
+The new lattice representing energy has been two way coupled with the existing momentum lattice which had been augmented with 
+product-form equilibrium and extended pressure term to compensate the third order error. The new energy lattice adds an 
+energy equation, while the multi-relaxation on the energy lattice enables a variable Prandtl number and an adjustable adiabatic 
+exponent. The mean molecular mass is allowed to be a variable because the product form of the equilibrium on the 
+first lattice allows the variance of the equilibrium distribution to be RT instead of T.
+
+Input and post processing files: `Tests/test_files/thermalDiffusivityTest/`
+
+.. image:: /figs/thermalDiffusivity.png
+   :width: 400pt     
+
+Sod test tube
+----------------------
+
+This test case demonstrates MARBLES ability to solve compressible flows with realistic parameters of practical interest. 
+The correct position of the wave strutures confirm that the speed of sound is correct in the thermal version of the solver. The model 
+is the same as that built for the thermal diffusivity test. The sod shock test marks the completion of the MARBLES solver to the 
+extent of a contunuum limit, single-component, single-phase, transient computational fluid dynamics solver with mass, momentum and energy equations.    
+
+Input and post processing files: `Tests/test_files/sodTest/`
+
+.. image:: /figs/sodShock.png
+   :width: 600pt