Slip Wall
The Slip Wall boundary condition is used to model the effective nonideal wall conditions that exist in the slip-flow regime, when the Knudsen number is in the range from 0.001 to 0.1. The Slip Wall conditions is used for systems with very small geometrical dimensions or systems running at very low ambient pressures. This is, for example, relevant when modeling MEMS transducers and other microdevices. To model a slip wall on an interior condition, use the Interior Slip Wall condition.
In the slip flow regime, the linearized Navier-Stokes equations (the equations solved for thermoviscous acoustics) can still be used to model the acoustics, except within a thin layer of fluid adjacent to the walls. Within this layer, which is known as the Knudsen layer, the fluid is significantly rarefied. The effect of the Knudsen layer on the continuum part of the acoustic perturbation equations can be modeled by means of a slip model condition at the wall. In the slip wall formulation used here, the slip velocity depends linearly on the tangential wall stress (the traction) and also an a component related to thermal creep. The total acoustic velocity ut and total acoustic temperature variations Tt at the wall are in the general case given by:
where uw is the wall velocity (fixed or moving), σs is the viscous slip coefficient, λ0 is the mean free path, σT is the thermal slip coefficient, utherm is the thermal creep, and ζT is the temperature jump coefficient. The solved set of equations represents a linearized version of the equations solved in The Slip Flow Interface available with the Microfluidics Module.
Mechanical
Select the Mechanical condition as Fixed wall (the default) or Moving wall. For the Moving wall option select User defined (the default) enter the Velocity of the moving wall uw (SI unit: m/s). Use this option to define a source, for example, a vibrating structure. The fixed option represents uw = 0. Announced variables for a wall velocity can also be selected when applicable.
For setting up a coupling to a vibrating structure including the slip wall effects, select the announced solid velocity from the list, for example, Velocity (solid/lemm1). The coupling back to the solid is then achieved by applying a Boundary Load to the structure and selecting the announced Force per unit area variable, for example, Acoustic slip wall traction per unit area (ta/slw1).
Thermal
Select the Thermal condition as Isothermal (the default) or Temperature variation. For the Temperature variation option enter the Temperature variation of the wall Tw (SI unit: K). The isothermal option represents Tw = 0.
When Isothermal is selected choose to select (the default) or de-select the Neglect thermal creep option. When thermal creep is included, an additional contribution is added to the slip velocity. The effect is typically negligible for the default isothermal wall condition and when there are no gradients in the equilibrium temperature T0. Thermal creep is always included when a temperature variation is specified at the wall.
Slip Model
Select how to define the Slip coefficients used in the slip wall formulation. The options are to use Maxwell’s model (the default), User defined, or Slip length.
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For Maxwell’s model enter the Tangential momentum accommodation coefficient αv (default value: 0.9). This model is applicable for gases and is based on mean-free-path considerations.
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For User defined enter the three free parameters of the model: the Viscous slip coefficient σs (default value: 1.1), the Thermal slip coefficient σT (default value: 1.2), and the Temperature jump coefficient ζT (default value: 2.2). The default values correspond to the default values of Maxwell’s model for air at typical ambient conditions. For more details on the parameters see Ref. 10.
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For Slip length the slip velocity is defined through the classical slip length parameter. Enter the Slip length Ls (default value: 1 mm) and the two thermal coefficients if thermal creep is included.
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Theory Background for the Thermoviscous Acoustics Branch.
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Theory for the Slip Wall Boundary Condition
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Viscous Damping of a Microperforated Plate in the Slip Flow Regime. Application Library path Acoustics_Module/Tutorials,_Thermoviscous_Acoustics/viscous_damping_mpp
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MEMS Microphone with Slip Wall. Application Library path Acoustics_Module/Electroacoustic_Transducers/mems_microphone_slip_wall
Constraint Settings
The Constraint type for the Slip Wall is always to use Weak constraints, and the default for the Apply reaction terms on option is Individual dependent variables. With these default settings coupling a thermoviscous domain to a vibrating structure using the above mentioned steps will work consistently. It is also possible to choose All physics (symmetric) for the latter. This option does, however, not guarantee a consistent two-way coupling to other physics.