Interdisciplinary Applied Mathematics

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no-slip and no-penetration conditions are valid. Otherwise, the pressure boundary condition also needs to be modified, e.g., in electroosmotic flow with slip. Assuming now Stokes flow for simplicity, the rotational form of the boundary condition for the pressure in equation (14.4) is equivalent to the Laplacian form of the boundary condition


— = vn • V2vn+1 on dSl.    (14.5)


dn


The rotational form, unlike the Laplacian form, satisfies the compatibility condition (Karniadakis et al., 1991), and it also reinforces the incompressibility condition, since


V2v = V(V ■ v) — V x ш.


In addition, it leads to a stable approximation, since the magnitude of the boundary-divergence is directly controlled by the time step.


To illustrate the differences between the rotational (equation (14.4)) and Laplacian (equation (14.5)) forms of the pressure boundary condition, we consider the exact boundary condition at time step (n + 1)At, i.e.,


dpn+1


dn



v



dQn+l


dn



+1



s


where we have introduced u>s = n ■ Vx ш and Q = V ■ v. We can now expand шs in a Taylor series to obtain


dQn+1


dn



1 dpn+1


v dn



+ш:+At



du^


dt



+ ■ ■ ■ .



Inserting the Laplacian form (equation (14.5)) in the above equation, we obtain



dQn+1


dn



oc



dQn


dn



+ At



dt


which shows an accumulation of divergence flux at the boundary at every time step and therefore the possibility for instability. In contrast, if the rotational form in equation (14.4) is used, we obtain

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