Instability Phenomena

Projects

Three aspects constitute the focus of the network: The first aspect is on the derivation and validation of the governing model equations as asymptotic limit equations from the Navier–Stokes system, respectively the Euler equations. With methods from nonlinear analysis we then study different solution concepts for the respective model equations, such as existence and uniqueness of classical strong solutions and globally in time existing weak solutions. Particular attention will be turned to travelling waves as special solutions of the respective equations. The third aspect is on understanding stability and instability phenomena, depending on involved physical parameters such as gravity and surface tension. These questions will be addressed for the following fluid mechanical problems:

(A) Stability of thin-film flows on linclined planes

The overarching goal of the proposed project is to study the existence and stability of front solutions for the integral boundary layer (IBL) equation and the full Navier–Stokes system as models for the dynamics of liquid flows down an inclined plane. Moreover, transversal (in)stability in two-dimensional IBL equations will be investigated.

The main steps of the project are

Existence and stability of front solutions for the IBL equation
Existence and stability of front solutions for the Navier-Stokes system
Transversal stability and instability for two-dimensional IBL models

Publications

(B) Rimming flows

The overarching goal of the proposed project is to understand the dynamics of rimming-flow problems in the limit of ‘thin’ fluid layers, depending on the interplay of the involved physical parameters.

The main steps of the project are

Derivation of two-dimensional rimming-flow equations
Existence and stability of solutions with different solution concepts such as weak solutions, strong solutions, travelling waves, steady states, . . .
Stability and instability of travelling waves and steady states propagating in x-direction with respect to perturbations transverse to the cross section of the cylinder.

Publications

J. Joussen, J. Laudien, C. Lienstromberg, and J. Velázquez: Analysis of a three-dimensional fluid flow in rotating cylinders

(C) Transverse (in)stability in water-wave model equations

The overarching goal of the proposed project is to understand the existence and stability of traveling wave solutions for asymptotic models appearing in water waves. Furthermore, the valitidy of the model equations shall be investigated.

The main steps of the project are

Existence of fully-localized solutions for nonlocal water-wave model equations
Transverse stability of traveling waves in two-dimensional models
Existence of two-dimensional highest waves
Validity of asymptotic models in water waves

Publications

Funding acknowledgement

In all project related publications, we acknowledge the funding by the DFG as follows:

"Funded by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) — project number 545145736.“

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