Coupled Inextensible Shell structure mechanics and TFMP model [equations, variables, boundary conditions, post-processing]

[wortiz]

Developers: Andrew Cochrane, August 2018
Description: Inextensible shells equations based on [2003 Carvalho], GT-027, and Pranck
dissertation are coupled to TFMP equations. Motion of the upper and lower bounding surfaces
is implemented in the TFMP equations. The target problem is a lubrication layer supported
web wrapped around a rotating roller. The results are comparable to the predictions of foil
bearing theory. See forthcoming article [2019 Cochrane].
\end{alltt}
\begin{alltt}
This inextensible shell equations of structural mechanics:
\end{alltt}
\begin{align*}
\frac{\partial T}{\partial s} + \kappa \frac{\partial }{ \partial s }( \kappa D ) + P_t &= 0 \

• \frac{\partial^2}{\partial s^2}(\kappa D) + \kappa T + P_n &= 0
  \end{align*}
  \begin{alltt}
  The equations of mesh position/deformation:
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  \begin{align*}
  %\frac{\partial^2 x}{\partial s ^2} + \kappa \frac{\partial y}{\partial s} &= 0 \label{eq:circle_d2x_ds2}\text{ or}\
  \frac{\partial^2 y}{\partial s ^2} - \kappa \frac{\partial x}{\partial s} &= 0
  \end{align*}
  \begin{align*}
  \frac{\partial^2 s}{\partial \xi^2} = 0
  \end{align*}
  \begin{alltt}
  The thin film multiphase flow equations, extended to include tangential motion of the
  confining boundaries:
  \end{alltt}
  \begin{align*}
  \frac{\partial Sh}{\partial t} + \frac{\partial}{\partial s} \left(h v_l + Sh\frac{u_a + u_b}{2}\right) &= 0 \
  \frac{\partial \rho_g(1-S)h}{\partial t} + \frac{\partial}{\partial s} \left(\rho_g h v_g + \rho_g (1-S)h\frac{u_a + u_b}{2}\right) &= 0
  \end{align*}