Seminar

Irreversible, dissipative processes can be often naturally modeled as gradient flows; under certain assumptions, flow in deformable porous media is such a process. In this talk, we formulate the problem of linear poro-(visco-)elasticity as generalized gradient flow. By exploiting this structure, the analysis of well-posedness and construction of numerical solvers can be performed in a rather straight-forward and abstract manner. For this, we apply results from gradient flow theory and convex optimization.

Hopefully we know that $\displaystyle \sum_{i \geq 0} x^i $ $= \frac{1}{1-x}$. Similarly one computes that $ \displaystyle \sum_{i \leq 1} x^i $ $= \frac{x^2}{x-1}$. Interestingly, $ \frac{1}{1-x} + \frac{x^2}{x-1} = 1 + x$ which is the sum corresponding to the integers in the interval $[0, 1] = [0, \infty] \cap [-\infty, 1]$. We will explain generalization of this (called Brion's theorem) to integer points in convex polytopes of arbitrary dimension.

When a young mathematician told his advisor that he had found a way to turn a sphere inside out (without making any creases), his advisor told him that this was impossible to do so, and gave him a proof. His advisor was wrong. That young mathematician (Smale) went on to win the highest mathematical prize (the Fields Medal). His solution is very non-intuitive. Today, there are several excellent videos on YouTube that show how to turn a sphere inside out, and this talk will explain some of them. 

We will discuss telescoping, Tannery's Theorem and other techniques for calculating infinite sums.

Collisions between two masses $m_{1}$ and $m_{2}$ have two important quantities: momentum and energy, respectively given by

$$ p = m_{1}v_{1} + m_{2}v_{2}$$

and

$$E = \frac{1}{2}m_{1}v_{1}^2 + \frac{1}{2}m_{2}v_{2}^2.$$

In any normal linear algebra class you learn how to solve a system of linear equations using Gaussian elimination methods. But how would solve a system like the following:
$$x^2+y+z=1$$

$$x+y^2+z=1$$

$$x+y+z^2=1$$

In this talk I will go over how one can use Gr\"obner theory to solve a system of nonlinear equations and what assumptions might need to be made to make this possible. 

In this work we present and analyse a fully-mixed formulation for the nonlinear model given by the coupling of the Navier-Stokes and Darcy-Forchheimer equations with the Beavers-Joseph-Saffman condition on the interface. Our approach yields non-Hilbertian normed spaces and a twofold saddle point structure for the corresponding operator equation. Furthermore, since the convective term in the Navier-Stokes equation forces the velocity to live in a smaller space than usual, we augment the variational formulation with suitable Galerkin type terms.

Fluid models were developed as an alternative to the Navier-Stokes equations to avoid computational complexity especially in case of turbulent flows. Model errors due to the variation of a model parameter become an immediate concern in different aspects. In this presentation, we discuss two such aspects as the conservation of energy and enstrophy, and the reliability of the model given a parameter value for the so called Time Relaxation Model.

Using an oscillating ratchet geometry, an investigation of the 
net fluid transport through the ratchet is undertaken in both Newtonian 
and viscoelastic fluids. A detailed description of the numerical model 
that couples the moving immersed ratchet structure to the bulk fluid 
will be discussed. Numerical results presented will detail how 
viscoelasticity enhances the net fluid transport in the ratchet.