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Microfluidics: Modeling, Mechanics and Mathematics by Bastian E. Rapp

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34.2.3.3 Instationary Flows

Momentum Equations. The second scenario we will consider involves instationary flows. For instationary flows we have to solve Eq. 34.1, Eq. 34.2, and Eq. 34.3 using a suitable numerical scheme for the timesteps vxt,vyt si121_e and vzt si122_e,. Usually, we would choose a forward Euler method (see section 27.2.2) for the time step. We could then rewrite Eq. 34.1, Eq. 34.2, and Eq. 34.3 as

vxt+vxvxx+vyvxy+vzvxz=1ρpx+ηρ2vxx2+2vxy2+2vxz2+kxρvyt+vxvyx+vyvyy+vzvyz=1ρpy+ηρ2vyx2+2vyy2+2vyz2+kyρvx

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