If an average inlet velocity or inlet volume flow is specified instead of the pressure,

COMSOL Multiphysics adds an ODE that calculates a pressure, pentr, such that the desired inlet velocity or volume flow is obtained.

Theory for the Laminar Outflow Condition

In order to prescribe an outlet velocity profile, this boundary conditions adds a weak form contribution corresponding to one-dimensional Navier-Stokes equations projected on the boundary. The applied condition corresponds to the situation shown in Figure 4-3: assume that a fictitious domain of length Lexit is attached to the outlet of the computational domain. This boundary condition uses the assumption that the flow in this fictitious domain is laminar plug flow. If the option is selected that constrains outer edges or endpoints to zero, the assumption is instead that the flow in the fictitious domain is fully developed laminar channel flow (in 2D) or fully developed laminar internal flow (in 3D). This does not affect the boundary condition in the real domain, , where the boundary conditions are always fulfilled.

Lexit

Figure 4-3: Example of the physical situation simulated when using Laminar outflow boundary condition. is the actual computational domain while the dashed domain is a fictitious domain.

If the average outlet velocity or outlet volume flow is specified instead of the pressure,

the software adds an ODE that calculates pexit such that the desired outlet velocity or volume flow is obtained.

Theory for the Slip Velocity Wall Boundary Condition

In the microscale range, the flow at a boundary is seldom strictly no slip or slip. Instead, the boundary condition is something in between, and there is a Slip velocity at the boundary. Two phenomena account for this velocity: the noncontinuum effect in viscosity and flow induced by a thermal gradient along the boundary.

The following equation relates the viscosity-induced jump in tangential velocity to the tangential shear stress along the boundary:

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1

u = -- n t

For gaseous fluids, the coefficient equals

where is the fluid’s dynamic viscosity (SI unit: Pa·s), v represents the tangential momentum accommodation coefficient (TMAC) (unitless), and is the molecules’ mean free path (SI unit: m). The tangential accommodation coefficients are typically in the range of 0.85 to 1.0 and can be found in Ref. 1.

A simpler expression for is

= -----

Ls

where Ls, the slip length (SI unit: m), is a straight channel measure of the distance from the boundary to the point where the flow profile extrapolates to zero at distance. This equation holds for both liquids and gases.

Thermal creep results from a temperature gradient along the boundary. The following equation relates the thermally-induced jump in tangential velocity to the tangential gradient of the natural logarithm of the temperature along the boundary:

u = T-- tlogT

where T is the thermal slip coefficient (unitless) and is the fluid’s density. The thermal slip coefficients range between 0.3 and 1.0 and can be found in Ref. 1.

Combining the previous relationships results in the following equation:

Relate the tangential shear stress to viscous boundary force by

n t = K – n K n

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