Fig 1: Microfludic biochip with microchannels and two hexagonal reservoirs, typical characteristic
length is |

Fig 2: A sharp jet is created by an interdigital transducer (IDT), placed on a piezoelectric substrate (lithium niobate in our case) and steers out surface acoustic waves (SAWs), which in turn makes the fluid to move (experimental setup). |
Fig 3: A cartoon with further elaboration of the phenomenon explained in left figure (courtesy: D. Köster [2006]). |

- The state equations represent a multiscale multiphysics problem consisting of the linearized
equations of
*piezoelectricity and compressible Navier-Stokes equations*. - A standard homogenization approach is applied, which results in a first order time periodic
linearized compressible Navier-Stokes equations with time scale [0,
*T*], where_{1}*T*is_{1}*O(ms)*(cf. Fig 5). - And a second order compressible Stokes system with time scale [0,
*T*] which results in an almost stationary pattern called acoustic streaming. Here_{2}*T*is_{2}*O(µs)*(cf. Fig 6).

Fig 4: Experimentally obtained acoustic streaming pattern. |

Surface Acoustic Wave
Fig 5: A SAW entering entering in the domain using an IDT placed at the bottom left corner. |
Fig 6: Stationary acoustic streaming pattern. |

Fig 7: Experimental filling of hexagonal shaped reservoirs which are part of microfluidic biochip. |

Fig 8: In order to ensure that the reservoirs are filled with precise amount of fluid we want to do shape optimization of the capillary barriers (marked in red) and since remaining part of the domain is large and fixed there we apply model reduction. As a result we have to solve a system of very small size as compared to a huge problem during optimization. |

- Matthias Heinkenschloss (Rice University)
- Ronald H. W. Hoppe (University of Houston)
- Achim Wixforth (University of Augsburg)

- Since the nonlinearities due to shape variation is localized in small part of the domain, we use domain decomposition and apply model reduction to large part of the domain which is fixed and use full order model on the variable domain.
- We have proven error estimated for the error between the solution of the original and the reduced order problem. The error estimate depend on the model reduction error estimate.
- The computational results are validated by the actual experiments, and may further lead to better manufacturing of the microfludic devices.

Fig 9: A finite element mesh generated for the microfluidic biochip using gmsh. |

Mathematics and Computers in Simulation (2010). link

Journal of Computational Mathematics (2010), 28(2):149--169. link