FLUID136 is used to model the thin-film fluid behavior between flat surfaces and is based on the generalized nonlinear Reynolds equation known from lubrication theory.

(7–1)

where:

d = local gap separation

ρ = density

t = time

= divergence operator

= gradient operator

η = dynamic viscosity

Pabs = absolute pressure

Assuming an ideal gas:

(7–2)

where:

Substituting Equation 7–2 into Equation 7–1 gives:

(7–3)

After substituting ambient pressure plus the pressure for the absolute pressure (P_abs = P₀ + P) this equation becomes:

(7–4)

Equation 7–4 is valid for large displacements and large pressure changes (KEYOPT(4) = 1). Pressure and velocity degrees of freedom must be activated (KEYOPT(3) = 1 or 2).

For small pressure changes (P/P₀ << 1), Equation 7–4 becomes:

(7–5)

where ν_z = wall velocity in the normal direction. That is:

(7–6)

Equation 7–5 is valid for large displacements and small pressure changes (KEYOPT(4) = 0). Pressure and velocity degrees of freedom must be activated (KEYOPT(3) = 1 or 2).

For small displacements (d/d₀ << 1) and small pressure changes (P/P₀ << 1), Equation 7–5 becomes:

(7–7)

where

This equation applies when pressure is the only degree of freedom (KEYOPT(3) = 0).

For incompressible flows (ρ is constant), the generalized nonlinear Reynolds equation (Equation 7–1) reduces to:

(7–8)

This equation applies for incompressible flow (KEYOPT(4) = 2). Pressure and velocity degrees of freedom must be activated (KEYOPT(3) = 1 or 2).

Reynolds squeeze film equations are restricted to structures with lateral dimensions much larger than the gap separation. Futhermore, viscous friction may not cause a significant temperature change. Continuum theory (KEYOPT(1) = 0) is valid for Knudsen numbers smaller than 0.01.

The Knudsen number Kn of the squeeze film problem can be estimated by:

(7–9)

where:

For small pressure changes, P_abs is approximately equal to P₀ and the Knudsen number can be estimated by:

(7–10)

For systems that operate at Knudsen numbers <0.01, the continuum theory is valid (KEYOPT(1) = 0). The effective viscosity η_eff is then equal to the dynamic viscosity η.

For systems which operate at higher Knudsen numbers (KEYOPT(1) = 1), an effective viscosity η_eff considers slip flow boundary conditions and models derived from Boltzmann equation. This assumption holds for Knudsen numbers up to 880 (Veijola([339])):

(7–11)

For micromachined surfaces, specular reflection decreases the effective viscosity at high Knudsen numbers compared to diffuse reflection. Surface accommodation factors, α, distinguish between diffuse reflection (α = 1), specular reflection (α = 0), and molecular reflection (0 < α < 1) of the molecules at the walls of the squeeze film. Typical accommodation factors for silicon are reported between 0.8 and 0.9, those of metal surfaces are almost 1. Different accommodation factors can be specified for each wall by using α ₁ and α ₂ (input as A1 and A2 on R command). α ₁ is the coefficient associated with the top moving surface and α ₂ is the coefficient associated with the bottom metallic surface. Results for high Knudsen numbers with accommodation factors (KEYOPT(1) =2) are not expected to be the same as those for high Knudsen numbers without accommodation factors (KEYOPT(1) =1).

The effective viscosity equations for high Knudsen numbers are based on empirical correlations. Fit functions for the effective viscosity of micromachined surfaces are found in Veijola([339]). The effective viscosity is given by the following equation if α ₁ = α ₂:

(7–12)

and by the following equation if α ₁ ≠α ₂:

(7–13)

where D is the inverse Knudsen number:

and Q₁, Q₂, and Q₃ are Poiseuille flow rate coefficients:

for p = 1, 2, or 3.

If both surfaces are the same (α ₁ = α ₂), the Poiseuille flow rate coefficient is given by:

If the bottom fixed plate is metallic (α ₂ = 1) and the top moving plate is not metallic (α ₁ ≠1), the Poiseuille flow rate coefficient is given by:

The general solution valid for arbitrary α ₁ and α ₂ is a simple linear combination of Q₁ and Q₂:

FLUID138 can be used to model the fluid flow though short circular and rectangular channels of micrometer size. The element assumes isothermal viscous flow at low Reynolds numbers, the channel length to be small compared to the acoustic wave length, and a small pressure drop with respect to ambient pressure.

In contrast to FLUID116, FLUID138 considers gas rarefaction, is more accurate for channels of rectangular cross sections, allows channel dimensions to be small compared to the mean free path, allows evacuated systems, and considers fringe effects at the inlet and outlet which considerably increase the damping force in case of short channel length. FLUID138 can be used to model the stiffening and damping effects of fluid flow in channels of micro-electromechanical systems (MEMS).

Using continuum theory (KEYOPT(1) = 0) the flow rate Q of channels with circular cross-section (KEYOPT(3) = 0) is given by the Hagen-Poiseuille equation:

(7–14)

This assumption holds for low Reynolds numbers (Re < 2300), for l >> r and r >> Lm where Lm is the mean free path at the current pressure.

(7–15)

In case of rectangular cross sections (KEYOPT(3) = 1) the channel resistance depends on the aspect ratio of channel. The flow rate is defined by:

(7–16)

where:

The hydraulic radius is defined by:

(7–17)

and the friction factor χ is approximated by:

(7–18)

where:

A special treatment is necessary to consider high Knudsen numbers and short channel length (KEYOPT(1) = 1) (Sharipov([340])).