2. Governing equations for fluid and structural domains

Every partitioned FSI solver is, at its core, a negotiation between two mathematical worlds. The fluid domain obeys the Navier–Stokes equations, which describe a continuum that deforms freely and carries momentum through convection. The structural domain obeys the equations of linear elasticity, which describe a solid that resists deformation and stores energy elastically. The coupling between them is enforced at a shared interface through kinematic and dynamic conditions. Understanding these three ingredients — and, critically, the assumptions baked into each — is what allows you to diagnose a diverging simulation instead of merely restarting it with a smaller time step.

The Fluid Domain: Navier–Stokes

For an incompressible Newtonian fluid with constant density ρ_f and dynamic viscosity μ, the governing equations are the momentum balance and the continuity constraint:

ρ_f (∂u/∂t + (u · ∇)u) = −∇p + μ ∇²u + f
∇ · u = 0

Here u is the fluid velocity vector, p is the pressure (defined up to an arbitrary constant in the incompressible formulation), and f represents body forces such as gravity. The term (u · ∇)u is the convective acceleration — the source of nonlinearity and the reason fluid solvers must iterate within each time step. The Reynolds number Re = ρ_f U L / μ, built from a characteristic velocity U and length L, tells you whether the flow is laminar, transitional, or turbulent, and therefore which turbulence treatment (if any) is required.

Two practical consequences matter for FSI. First, the incompressible constraint is enforced through a pressure Poisson equation, so pressure is a Lagrange multiplier rather than a thermodynamic state variable; this is why pressure fields can exhibit checkerboard oscillations on collocated grids and why Rhie–Chow interpolation exists. Second, when the fluid mesh moves to follow the structure, the convective term must be rewritten in arbitrary Lagrangian–Eulerian (ALE) form:

ρ_f (∂u/∂t |_χ + ((u − u_m) · ∇)u) = −∇p + μ ∇²u + f

where u_m is the mesh velocity and ∂u/∂t |_χ is the time derivative holding the reference (mesh) coordinate fixed. The mesh velocity must satisfy the geometric conservation law (GCL); violating it introduces spurious mass sources that masquerade as physical instability.

The Structural Domain: Linear Elasticity

For a solid with density ρ_s, the momentum balance in the absence of body forces is:

ρ_s ∂²d/∂t² = ∇ · σ

where d is the displacement vector and σ is the Cauchy stress tensor. The constitutive law for a linear isotropic elastic material is Hooke's law:

σ = λ tr(ε) I + 2μ_s ε
ε = ½ (∇d + ∇dᵀ)

with Lamé parameters λ and μ_s (the shear modulus, not to be confused with fluid viscosity), and ε the small-strain tensor. The linearization assumes displacements and rotations are small compared with the structural dimensions. This assumption is the single most common source of silent error in FSI: a flapping flag or a large-deflection membrane violates it, and the correct formulation is then geometrically nonlinear (total Lagrangian or updated Lagrangian), where σ is replaced by the second Piola–Kirchhoff stress and ε by the Green–Lagrange strain.

For a thin structure, the three-dimensional equations are often reduced. A beam follows the Euler–Bernoulli relation EI ∂⁴w/∂x⁴ + ρ_s A ∂²w/∂t² = q(x,t), while a plate follows the Kirchhoff or Mindlin–Reissner form depending on whether transverse shear is retained. These reduced models are attractive because they eliminate the need to mesh the structural thickness, but they must be matched to the fluid load correctly — a point load applied to a beam node is not the same as a distributed pressure.

Interface Conditions

The two domains are stitched together by two conditions on the shared boundary Γ_i. The kinematic condition states that the fluid and solid velocities match, and for a no-slip wall the fluid velocity equals the structural velocity:

u_f = ∂d_s/∂t   on Γ_i

The dynamic condition states that tractions balance. With the outward normal n defined consistently for both domains (a frequent source of sign errors), the fluid stress and structural stress must satisfy:

σ_f · n = −σ_s · n   on Γ_i

For an inviscid or high-Reynolds-number external flow, the fluid traction reduces to pressure alone, σ_f · n ≈ −p n, so the structural load becomes σ_s · n = p n. This simplification is valid for aerodynamics but not for internal flows where viscous shear dominates, such as blood vessels or lubricated contacts.

How the Equations Interact in a Partitioned Solver

In a partitioned scheme, the fluid and structural solvers are advanced separately and exchange interface data. The diagram below shows one coupling iteration: the structural displacement updates the fluid mesh, the fluid solve produces a pressure field, and that pressure is transferred back as a traction on the structural surface. The loop repeats until the interface residual falls below tolerance.

The order in which data is exchanged determines the scheme's stability. In a loosely coupled (explicit) scheme, the structural displacement is passed to the fluid once per time step, and the resulting pressure is applied without sub-iteration. This is cheap but conditionally stable: for a structure much denser than the fluid, the added-mass instability appears when the density ratio ρ_s/ρ_f drops below roughly 1–10, depending on geometry. In a strongly coupled (implicit) scheme, the interface residual is driven to zero within each time step using Aitken relaxation or a quasi-Newton method such as IQN-ILS. The trade-off is computational cost per step versus robustness.

Choosing the Right Formulation

Read the table as a set of assumptions, not prescriptions. The moment your problem leaves the small-strain, constant-density, no-slip regime, revisit the corresponding equation rather than tuning solver parameters. Most FSI failures are modeling failures wearing the costume of numerical ones.