Provides the term to evaluate and multiply by the test function/test gradient function at each quadrature location. The Residual and Jacobians are in terms of

Q Test Function

\[F_q = \int_\Omega q \left( S \frac{\partial \rho}{\partial t} + \nabla \cdot \left(\rho \boldsymbol{u} \right)\right) d\Omega = 0\]

We know that \(\rho = \frac{p^{th}}{T}\).

\[S\frac{\partial \rho}{\partial t} \Rightarrow \frac{S}{T} \frac{\partial p^{th}}{\partial t}-\frac{Sp^{th}}{T^2}\frac{\partial T}{\partial t}\] \[\nabla \cdot \left(\rho \boldsymbol{u} \right) \Rightarrow \nabla \cdot \left(\frac{p^{th}}{T} \boldsymbol{u} \right) \Rightarrow p^{th} \left( \frac{1}{T} \nabla \cdot \boldsymbol{u} + \boldsymbol{u} \cdot \nabla \frac{1}{T} \right) \Rightarrow p^{th} \left( \frac{1}{T} \nabla \cdot \boldsymbol{u} - \frac{1}{T^2}\boldsymbol{u} \cdot \nabla T \right)\]

Combining the terms and assuming that \(\frac{\partial p^{th}}{\partial t} = 0\) results in

\[F_q = \int_\Omega q \left(-\frac{Sp^{th}}{T^2}\frac{\partial T}{\partial t} + \frac{p^{th}}{T} \nabla \cdot \boldsymbol{u} - \frac{p^{th}}{T^2}\boldsymbol{u} \cdot \nabla T \right) d\Omega\]

Jacobians

\[F_{q,i} = \int -\phi_i \frac{Sp^{th}}{T^2}\frac{\partial T}{\partial t} + \phi_i \nabla \cdot \left(\frac{p^{th}}{T}\boldsymbol{u} \right) \\\] \[\begin{eqnarray} \frac{F_{q,i}}{\partial c_{T,j}} &=& \int -\phi_i \frac{\partial}{\partial c_{T,j}} \left(\frac{Sp^{th}}{T^2}\frac{\partial T}{\partial t}\right) + \phi_i \frac{\partial}{\partial c_{T,j}}\nabla \cdot \left(\frac{p^{th}}{T}\boldsymbol{u} \right)\\ &=& \int -\phi_i Sp^{th}\frac{\partial \frac{1}{T^2}}{\partial c_{T,j}}\frac{\partial T}{\partial t} + \phi_i p^{th} \frac{\partial}{\partial c_{T,j}} \left( \frac{1}{T}\nabla \cdot \boldsymbol{u} + \boldsymbol{u}\cdot \nabla \frac{1}{T} \right) \\ &=& \int \phi_i Sp^{th}\frac{\partial T}{\partial c_{T,j}} \frac{2}{T^3} \frac{\partial T}{\partial t} + \phi_i p^{th} \left( \frac{\partial \frac{1}{T}}{\partial c_{T,j}} \nabla \cdot \boldsymbol{u} + \boldsymbol{u}\cdot \frac{\partial}{\partial c_{T,j}} \nabla \frac{1}{T} \right) \\ &=& \int \phi_i \frac{2Sp^{th}}{T^3}\frac{\partial T}{\partial c_{T,j}} \frac{\partial T}{\partial t} + \phi_i p^{th} \left( - \frac{1}{T^2} \frac{\partial T}{\partial c_{T,j}} \nabla \cdot \boldsymbol{u} + \boldsymbol{u}\cdot \frac{\partial}{\partial c_{T,j}} \nabla \frac{1}{T} \right) \\ &=& \int \phi_i \frac{2Sp^{th}}{T^3}\frac{\partial T}{\partial c_{T,j}} \frac{\partial T}{\partial t} + \phi_i p^{th} \left( - \frac{1}{T^2} \frac{\partial T}{\partial c_{T,j}} \nabla \cdot \boldsymbol{u} - \boldsymbol{u}\cdot \frac{\partial}{\partial c_{T,j}} \frac{\nabla T}{T^2} \right) \\ &=& \int \phi_i \frac{2Sp^{th}}{T^3}\frac{\partial T}{\partial c_{T,j}} \frac{\partial T}{\partial t} + \phi_i p^{th} \left( - \frac{1}{T^2} \frac{\partial T}{\partial c_{T,j}} \nabla \cdot \boldsymbol{u} - \boldsymbol{u}\cdot \left(\frac{1}{T^2} \frac{\partial \nabla T}{\partial c_{T,j}} + \frac{\partial \frac{1}{T^2}}{\partial c_{T,j}} \nabla T \right)\right) \\ &=& \int \phi_i \frac{2Sp^{th}}{T^3}\frac{\partial T}{\partial c_{T,j}} \frac{\partial T}{\partial t} + \phi_i p^{th} \left( - \frac{1}{T^2} \frac{\partial T}{\partial c_{T,j}} \nabla \cdot \boldsymbol{u} - \boldsymbol{u}\cdot \left(\frac{1}{T^2} \frac{\partial \nabla T}{\partial c_{T,j}} - \frac{2}{T^3} \frac{\partial T}{\partial c_{T,j}} \nabla T \right)\right) \\ &=& \int \phi_i \frac{2Sp^{th}}{T^3}\psi_{T,j} \frac{\partial T}{\partial t} + \frac{\phi_i p^{th}}{T^2} \left( - \psi_{T,j} \nabla \cdot \boldsymbol{u} + \boldsymbol{u}\cdot \left(\frac{2}{T} \psi_{T,j} \nabla T - \nabla \psi_{T,j}\right) \right) \\ \end{eqnarray}\] \[\begin{eqnarray} \frac{F_{q,i}}{\partial c_{\frac{\partial T}{\partial t},j}} &=& \int \frac{-\phi_i S p^{th}}{T^2} \frac{\partial}{\partial c_{\frac{\partial T}{\partial t},j}}\frac{\partial T}{\partial t} \\ &=& \int \frac{-\phi_i S p^{th}}{T^2} \psi_j \\ \end{eqnarray}\] \[\begin{eqnarray} \frac{F_{q,i}}{\partial c_{u_c,j}} &=& \int \phi_i \frac{\partial}{\partial c_{u_c,j}}\nabla \cdot \left(\frac{p^{th}}{T}\boldsymbol{u} \right) \\ &=& \int \phi_i \frac{\partial}{\partial c_{u_c,j}} \left(\rho \nabla \cdot \boldsymbol{u} + \boldsymbol{u} \cdot \nabla \rho \right) \\ &=& \int \phi_i \left(\rho \frac{\partial \nabla \cdot \boldsymbol{u}}{\partial c_{u_c,j}} + \frac{\partial \boldsymbol{u}}{\partial c_{u_c,j}} \cdot \nabla \rho \right) \\ &=& \int \phi_i \rho \frac{\partial \psi_{u_c,j}}{\partial x_c} + \phi_i \psi_{u_c,j} \hat{e}_j \cdot \nabla \rho \\ &=& \int \phi_i \rho \frac{\partial \psi_{u_c,j}}{\partial x_c} + \phi_i \psi_{u_c,j} \frac{\partial \rho}{\partial x_c} \\ &=& \int \phi_i \frac{p^{th}}{T} \frac{\partial \psi_{u_c,j}}{\partial x_c} + \phi_i \psi_{u_c,j} p^{th} \frac{\partial \frac{1}{T}}{\partial x_c} \\ &=& \int \phi_i \frac{p^{th}}{T} \frac{\partial \psi_{u_c,j}}{\partial x_c} - \phi_i \psi_{u_c,j} \frac{p^{th}}{T^2} \frac{\partial T}{\partial x_c} \end{eqnarray}\]

W Test Function

\[\require{cancel} \begin{eqnarray} F_w &=& \int_\Omega w \rho C_p S \frac{\partial T}{\partial t} + w \rho C_p \boldsymbol{u} \cdot \nabla T + \nabla w \cdot \frac{k}{P} \nabla T - \cancelto{0}{w \Gamma \beta S T \frac{\partial p^{th}}{\partial t}} - wHSQ - \int_\Gamma w q_n = 0 \\ &=& \int_\Omega w \frac{C_p S p^{th}}{T} \frac{\partial T}{\partial t} + \ w \frac{C_p p^{th}}{T} \boldsymbol{u} \cdot \nabla T + \nabla w \cdot \frac{k}{P} \nabla T - wHSQ - \int_\Gamma w q_n \end{eqnarray}\]

Jacobians

\[\begin{eqnarray} \require{enclose} F_{w,i} &=& \int_\Omega \phi_i \frac{C_p S p^{th}}{T} \frac{\partial T}{\partial t} + \phi_i \frac{C_p p^{th}}{T} \boldsymbol{u} \cdot \nabla T + \nabla \phi_i \cdot \frac{k}{P} \nabla T - wHSQ - \int_\Gamma \phi_i q_n \\ &=& \int_\Omega \enclose{circle}{1} + \enclose{circle}{2} + \enclose{circle}{3} - wHSQ - \int_\Gamma \phi_i q_n \end{eqnarray}\] \[\begin{eqnarray} \frac{\partial F_{w,i}}{\partial c_{u_c,j}} &=& \frac{\partial}{\partial c_{u_c,j}} \phi_i \frac{C_p p^{th}}{T} \boldsymbol{u} \cdot \nabla T \\ &=& \phi_i \frac{C_p p^{th}}{T} \frac{\partial \boldsymbol{u}}{\partial c_{u_c,j}} \cdot \nabla T \\ &=& \phi_i \frac{C_p p^{th}}{T} \psi_{u_c,j} \hat{e}_c \cdot \nabla T \\ &=& \phi_i \frac{C_p p^{th}}{T} \psi_{u_c,j} \frac{\partial T}{\partial x_c} \end{eqnarray}\] \[\begin{eqnarray} \frac{\partial F_{w,i}}{\partial c_{T,j}} &=& \frac{\partial \enclose{circle}{1}}{\partial c_{T,j}} + \frac{\partial \enclose{circle}{2}}{\partial c_{T,j}} + \frac{\partial \enclose{circle}{3}}{\partial c_{T,j}}\\ \end{eqnarray}\] \[\begin{eqnarray} \frac{\partial \enclose{circle}{1}}{\partial c_{T,j}} &=& \frac{\partial }{\partial c_{T,j}} \phi_i \frac{C_p S p^{th}}{T} \frac{\partial T}{\partial t} \\ &=& \psi_i C_p S p^{th} \frac{\partial }{\partial c_{T,j}} \left(\frac{1}{T} \frac{\partial T}{\partial t} \right) \\ &=& \psi_i C_p S p^{th} \left(\frac{\partial T}{\partial t} \frac{\partial }{\partial c_{T,j}} \frac{1}{T} \right) \\ &=& -\psi_i C_p S p^{th} \left(\frac{\partial T}{\partial t} \frac{1}{T^2} \frac{\partial T}{\partial c_{T,j}} \right) \\ &=& -\psi_i C_p S p^{th} \frac{\partial T}{\partial t} \frac{1}{T^2} \psi_{T,j} \end{eqnarray}\] \[\begin{eqnarray} \frac{\partial \enclose{circle}{2}}{\partial c_{T,j}} &=& \frac{\partial }{\partial c_{T,j}} \phi_i \frac{C_p p^{th}}{T} \boldsymbol{u} \cdot \nabla T \\ &=& \psi_i C_p p^{th} \boldsymbol{u} \cdot \frac{\partial}{\partial c_{T,j}}\left(\frac{\nabla T}{T} \right) \\ &=& \psi_i C_p p^{th} \boldsymbol{u} \cdot \left( \frac{1}{T} \frac{\partial \nabla T}{\partial c_{T,j}} + \nabla T \frac{\partial \frac{1}{T}}{\partial c_{T,j}} \right) \\ &=& \psi_i C_p p^{th} \boldsymbol{u} \cdot \left( \frac{1}{T} \frac{\partial \nabla T}{\partial c_{T,j}} - \frac{\nabla T}{T^2} \frac{\partial T}{\partial c_{T,j}} \right) \\ &=& \psi_i C_p p^{th} \boldsymbol{u} \cdot \left( \frac{1}{T} \nabla \psi_{T,j} - \frac{\nabla T}{T^2} \psi_{T,j} \right) \end{eqnarray}\] \[\begin{eqnarray} \frac{\partial \enclose{circle}{3}}{\partial c_{T,j}} &=& \frac{\partial \enclose{circle}{3}}{\partial c_{T,j}} \nabla \phi_i \cdot \frac{k}{P} \nabla T \\ &=& \frac{k}{P} \nabla \phi_i \cdot \frac{\partial \nabla T}{\partial c_{T,j}} \\ &=& \frac{k}{P} \nabla \phi_i \cdot \nabla \psi_{T,j} \end{eqnarray}\] \[\therefore \frac{\partial F_{w,i}}{\partial c_{T,j}} = - \phi_i C_p S p^{th} \frac{\partial T}{\partial t} \frac{1}{T^2} \psi_{T,j} + \phi_i C_p p^{th} \boldsymbol{u} \cdot \left( \frac{1}{T} \nabla \psi_{T,j} - \frac{\nabla T}{T^2} \psi_{T,j} \right) + \frac{k}{P} \nabla \phi_i \cdot \nabla \psi_{T,j}\] \[\begin{eqnarray} \frac{\partial F_{w,i}}{\partial c_{\frac{\partial T}{\partial t},j}} &=& \frac{\partial }{\partial c_{\frac{\partial T}{\partial t},j}} \phi_i \frac{C_p S p^{th}}{T} \frac{\partial T}{\partial t} \\ &=& \psi_i C_p S p^{th} \frac{\partial }{\partial c_{\frac{\partial T}{\partial t},j}} \left(\frac{1}{T} \frac{\partial T}{\partial t} \right) \\ &=& \psi_i C_p S p^{th} \left(\frac{1}{T} \frac{\partial }{\partial c_{\frac{\partial T}{\partial t},j}} \frac{\partial T}{\partial t} \right) \\ &=& \psi_i C_p S p^{th} \left(\frac{1}{T} \psi_{j} \right) \\ &=& \psi_i C_p S p^{th} \frac{1}{T} \psi_{j}\\ \end{eqnarray}\]

V Test Function

\[\require{cancel} \begin{eqnarray} F_\boldsymbol{v} &=& \int_\Omega \boldsymbol{v} \cdot \rho S \frac{\partial \boldsymbol{u}}{\partial t} + \boldsymbol{v} \cdot \rho \boldsymbol{u} \cdot \nabla \boldsymbol{u} + \nabla^S \boldsymbol{v} \cdot \frac{2 \mu}{R} \boldsymbol{\epsilon}'(\boldsymbol{u}) - p \nabla \cdot \boldsymbol{v} + \frac{\rho \hat{\boldsymbol{z}}}{F^2} \cdot \boldsymbol{v} - \int_\Gamma \boldsymbol{t_n} \cdot \boldsymbol{v} = 0 \end{eqnarray}\]

Jacobians

\[\begin{eqnarray} F_{\boldsymbol{v}_i} &=& \int_\Omega \boldsymbol{\phi_i} \cdot \rho S \frac{\partial \boldsymbol{u}}{\partial t} + \boldsymbol{\phi_i} \cdot \rho \boldsymbol{u} \cdot \nabla \boldsymbol{u} + \nabla^S \boldsymbol{\phi_i} \cdot \frac{2 \mu}{R} \boldsymbol{\epsilon}'(\boldsymbol{u}) - p \nabla \cdot \boldsymbol{\phi_i} + \frac{\rho \hat{\boldsymbol{z}}}{F^2} \cdot \boldsymbol{\phi_i} - \int_\Gamma \boldsymbol{t_n} \cdot \boldsymbol{\phi_i} = 0 \end{eqnarray}\] \[\begin{eqnarray} \frac{\partial F_{\boldsymbol{v}_i}}{\partial c_{p,j}} &=& \int - \frac{\partial}{\partial c_{p,j}} p \nabla \cdot \boldsymbol{\phi_i} \\ &=&\int - \frac{\partial p}{\partial c_{p,j}} \nabla \cdot \boldsymbol{\phi_i} \\ &=&\int - \psi_{p,j} \nabla \cdot \boldsymbol{\phi_i} \\ \end{eqnarray}\] \[\begin{eqnarray} \frac{\partial F_{\boldsymbol{v}_i}}{\partial c_{T,j}} &=& \int \boldsymbol{\phi_i} \cdot \frac{\partial \rho}{\partial c_{T,j}} S \frac{\partial \boldsymbol{u}}{\partial t} + \boldsymbol{\phi_i} \cdot \frac{\partial \rho}{\partial c_{T,j}} \boldsymbol{u} \cdot \nabla \boldsymbol{u} + \frac{\partial \rho}{\partial c_{T,j}} \frac{\hat{\boldsymbol{z}}}{F^2} \cdot \boldsymbol{\phi_i} \\ &=& - \int \boldsymbol{\phi_i} \cdot \frac{p^{th}}{T^2} \frac{\partial T}{\partial c_{T,j}} S \frac{\partial \boldsymbol{u}}{\partial t} - \boldsymbol{\phi_i} \cdot \frac{p^{th}}{T^2} \frac{\partial T}{\partial c_{T,j}} \boldsymbol{u} \cdot \nabla \boldsymbol{u} - \frac{p^{th}}{T^2} \frac{\partial T}{\partial c_{T,j}} \frac{\hat{\boldsymbol{z}}}{F^2} \cdot \boldsymbol{\phi_i} \\ &=& - \int \boldsymbol{\phi_i} \cdot \frac{p^{th}}{T^2} \psi_{T,j} S \frac{\partial \boldsymbol{u}}{\partial t} - \boldsymbol{\phi_i} \cdot \frac{p^{th}}{T^2} \psi_{T,j} \boldsymbol{u} \cdot \nabla \boldsymbol{u} - \frac{p^{th}}{T^2} \psi_{T,j} \frac{\hat{\boldsymbol{z}}}{F^2} \cdot \boldsymbol{\phi_i} \end{eqnarray}\] \[\begin{eqnarray} \frac{\partial F_{\boldsymbol{v}_i}}{\partial c_{u_c,j}} &=& \int_\Omega \frac{\partial}{\partial c_{u_c,j}}\boldsymbol{\phi_i} \cdot \rho \boldsymbol{u} \cdot \nabla \boldsymbol{u} + \frac{\partial }{\partial c_{u_c,j}}\nabla^S \boldsymbol{\phi_i} \cdot \frac{2 \mu}{R} \boldsymbol{\epsilon}'(\boldsymbol{u}) \end{eqnarray}\] \[\begin{eqnarray} \frac{\partial}{\partial c_{u_c,j}} \boldsymbol{\phi_i} \cdot \rho S \frac{\partial \boldsymbol{u}}{\partial t} &=&\boldsymbol{\phi_i} \cdot \rho S \frac{\partial}{\partial c_{u_c,j}} \frac{\partial \boldsymbol{u}}{\partial t} \end{eqnarray}\] \[\begin{eqnarray} \frac{\partial}{\partial c_{u_c,j}}\boldsymbol{\phi_i} \cdot \rho \boldsymbol{u} \cdot \nabla \boldsymbol{u} &=& \boldsymbol{\phi_i} \cdot \left(\rho \frac{\partial\boldsymbol{u}}{\partial c_{u_c,j}} \cdot \nabla \boldsymbol{u} + \rho \boldsymbol{u} \cdot \frac{\partial \nabla \boldsymbol{u}}{\partial c_{u_c,j}} \right) \\ &=& \boldsymbol{\phi_i} \cdot \left(\rho \psi_j \hat{e}_c \cdot \hat{e}_l \frac{\partial u_k}{\partial x_l}\hat{e}_k + \rho \boldsymbol{u} \cdot \hat{e}_l \frac{\partial \psi_j}{\partial x_l}\hat{e}_c \right) \\ &=& \boldsymbol{\phi_i} \cdot \left(\rho \psi_j \frac{\partial u_k}{\partial x_c}\hat{e}_k + \rho u_c \nabla \psi_j \right) \\ \end{eqnarray}\] \[\begin{eqnarray} \frac{\partial}{\partial c_{u_c,j}}\nabla^S \boldsymbol{\phi_i} \cdot \frac{2 \mu}{R} \boldsymbol{\epsilon}'(\boldsymbol{u}) &=& \nabla^S \boldsymbol{\phi_i} \cdot \frac{2 \mu}{R} \frac{\partial \boldsymbol{\epsilon}'(\boldsymbol{u})}{\partial c_{u_c,j}} \\ &=& \nabla^S \boldsymbol{\phi_i} \cdot \frac{2 \mu}{R} \frac{\partial }{\partial c_{u_c,j}} \left( \frac{1}{2}\left(\nabla \boldsymbol{u} + \nabla\boldsymbol{u}^T \right) - \frac{1}{3}(\nabla \cdot \boldsymbol{u} \boldsymbol{I}) \right) \\ &=& \nabla^S \boldsymbol{\phi_i} \cdot \frac{2 \mu}{R} \left( \frac{1}{2}\left(\frac{\partial \nabla \boldsymbol{u}}{\partial c_{u_c,j}} + \frac{\partial \nabla \boldsymbol{u}^T}{\partial c_{u_c,j}} \right) - \frac{1}{3} \frac{\partial \nabla \cdot \boldsymbol{u}}{\partial c_{u_c,j}} \boldsymbol{I} \right) \\ &=& \nabla^S \boldsymbol{\phi_i} \cdot \frac{2 \mu}{R} \left( \frac{1}{2}\left( \hat{e}_l \frac{\partial \psi_j}{\partial x_l}\hat{e}_c + \hat{e}_c \frac{\partial \psi_j}{\partial x_l}\hat{e}_l \right) - \frac{1}{3} \frac{\partial \psi_j}{\partial x_c} \boldsymbol{I} \right) \\ \end{eqnarray}\] \[\therefore \frac{\partial F_{\boldsymbol{v}_i}}{\partial c_{u_c,j}} = \int \boldsymbol{\phi_i} \cdot \left(\rho \psi_j \frac{\partial u_k}{\partial x_c}\hat{e}_k + \rho u_c \nabla \psi_j \right) + \nabla^S \boldsymbol{\phi_i} \cdot \frac{2 \mu}{R} \left( \frac{1}{2}\left( \hat{e}_l \frac{\partial \psi_j}{\partial x_l}\hat{e}_c + \hat{e}_c \frac{\partial \psi_j}{\partial x_l}\hat{e}_l \right) - \frac{1}{3} \frac{\partial \psi_j}{\partial x_c} \boldsymbol{I} \right)\] \[\begin{eqnarray} \frac{\partial F_{\boldsymbol{v}_i}}{\partial c_{\frac{\partial u_c}{\partial t},j}} &=& \int_\Omega \frac{\partial }{\partial c_{\frac{\partial u_c}{\partial t},j}} \boldsymbol{\phi_i} \cdot \rho S \frac{\partial \boldsymbol{u}}{\partial t} \\ &=& \int_\Omega \boldsymbol{\phi_i} \cdot \rho S \frac{\partial \frac{\partial \boldsymbol{u}}{\partial t}}{\partial c_{\frac{\partial u_c}{\partial t},j}} \\ &=& \int_\Omega \boldsymbol{\phi_i} \cdot \rho S \psi_j \\ \end{eqnarray}\]