docs: add EL bubble translational-motion equations; fix IB patch-types table

- equations.md 6.2: document the moving-bubble equation of motion -- tracer
  (dx/dt = u_l) and Newton's 2nd law (m dv/dt = F_D + F_p + F_g) -- with the
  non-dimensional drag/pressure/gravity force laws and their citations. This is
  the canonical @ref equations home; case.md keeps the concise inline version.
- case.md: fix the Immersed Boundary Patch Types table -- the 3D Cylinder row
  carried a 4th cell with no matching header column, which broke the table
  render. Added a Notes column.

Author: sbryngelson

docs/documentation/case.md

@@ -1320,14 +1320,14 @@ Boundary is at polar angle \f$\theta = \mathrm{atan2}(y - y_{\mathrm{centroid}},
 
 ### Immersed Boundary Patch Types {#immersed-boundary-patch-types}
 
-| #    | Name               | Dim.   |
-| ---: | :----:             | :---   |
-| 2    | 2D Circle          | 2      |
-| 3    | 2D Rectangle       | 2      |
-| 4    | 2D Airfoil         | 2      |
-| 8    | 3D Sphere          | 3      |
+| #    | Name               | Dim.   | Notes                                             |
+| ---: | :----:             | :---:  | :---                                              |
+| 2    | 2D Circle          | 2      |                                                   |
+| 3    | 2D Rectangle       | 2      |                                                   |
+| 4    | 2D Airfoil         | 2      |                                                   |
+| 8    | 3D Sphere          | 3      |                                                   |
 | 10   | 3D Cylinder        | 3      | `length_x` sets the axial length of the cylinder. |
-| 11   | 3D Airfoil         | 3      |
+| 11   | 3D Airfoil         | 3      |                                                   |
 
 ### Acoustic Supports {#acoustic-supports}
 

docs/documentation/equations.md

@@ -522,6 +522,25 @@ with \f$\sigma = \varepsilon_b \max(\Delta x^{1/3}_\text{cell},\;R_\text{bubble}
 
 Each bubble is tracked individually with Keller-Miksis dynamics and 4th-order adaptive Runge-Kutta time integration.
 
+**Translational motion (`vel_model > 0`):**
+
+Bubbles may also translate through the carrier flow. Let \f$\mathbf{x}_b\f$ be the bubble position, \f$\mathbf{u}_b\f$ its velocity, \f$a\f$ its radius, and \f$\mathbf{u}_l(\mathbf{x}_b)\f$ the carrier velocity interpolated to the bubble location (a Lagrange polynomial of order `fd_order`, which must be set when `vel_model > 0`).
+
+- **Tracer bubbles (`vel_model = 1`)** follow the local carrier velocity:
+  \f[\frac{d\mathbf{x}_b}{dt} = \mathbf{u}_l(\mathbf{x}_b).\f]
+
+- **Newton's second law (`vel_model = 2`)** integrates the bubble momentum:
+  \f[m_b\,\frac{d\mathbf{u}_b}{dt} = \mathbf{F}_D + \mathbf{F}_p + \mathbf{F}_g, \qquad \frac{d\mathbf{x}_b}{dt} = \mathbf{u}_b,\f]
+  with bubble mass \f$m_b\f$ and forces acting on the slip velocity \f$\mathbf{u}_\text{rel} = \mathbf{u}_b - \mathbf{u}_l\f$:
+
+| Force | `case.py` control | Non-dimensional form |
+|---|---|---|
+| Drag \f$\mathbf{F}_D\f$ | `drag_model` | \f$-\,c_D\,\pi\,a\,\mathbf{u}_\text{rel} / \text{Re}\f$, with \f$c_D = 4\f$ free-slip (\cite Hadamard1911; \cite Rybczynski1911), \f$c_D = 6\f$ no-slip Stokes (\cite Stokes1851), \f$c_D = 12\f$ Levich (\cite Levich1962) |
+| Pressure \f$\mathbf{F}_p\f$ | `pressure_force` | \f$-V_b\,\nabla p\f$, with bubble volume \f$V_b = \frac{4}{3}\pi a^3\f$ |
+| Gravity \f$\mathbf{F}_g\f$ | `gravity_force` | \f$m_b\,\mathbf{g}\f$, with \f$\mathbf{g}\f$ the body-force acceleration |
+
+Here \f$\text{Re}\f$ is the mixture Reynolds number — the same `fluid_pp%%Re(1)` that scales the viscous stress tensor (@ref sec-two-viscosities) — so the drag scales with the liquid viscosity. The three drag models increase in magnitude free-slip \f$<\f$ no-slip \f$<\f$ Levich; see \cite Magnaudet2000 for a review of these bubble-drag regimes.
+
 ---
 
 ## 7. Fluid-Structure Interaction