Research

My major interests lie in modelling real phenomena, including the way how to formulate thermodynamically consistent models (non-equilibrium thermodynamics provides a means for such formulations; CIT, EIT, GENERIC) and how to analyse them (stability, symmetries of DEs, asymptotic methods), and the emergence of spatial organisation (diffusion-driven instability and beyond).

Research articles typically automatically appear with some slight delay on researchgate.

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Research interests can be summarised as (a version with links to actual research papers is available here)

1. Formulation of thermodynamically consistent models for various non-equilibrium processes. To this end VK develops the general theory of non-equilibrium thermodynamics (an overview of our viewpoint of multiscale thermodynamics is in our monograph). In detail they are

• Contributions to frameworks of non-equilibrium thermodynamics. :

• mixtures, multiscale thermodynamics  including  a discussion of role and relevance of entropy and entropy production at different scales 

• functional restrictions of phenomenological coefficients and their relation to Onsager-Casimir relations

• a systematic study of irreversibility and dissipation in evolution equations

• generalisation of the widely used exergy analysis pointing out its limits

• a method for the identification of dissipation compliant with purely reversible dynamics 

• discussion of a link between stability and dissipation together with symplectic numerical integrators

• a continuum Hamiltonian mechanics formulation containing Poisson brackets with deformation gradients including a link between invariants and conservation laws 

• an alternative view on critical phenomena across scales 

• a technique providing plausible evolution equations on a lower level of description (dynamic MaxEnt reduction)

• generalization of the dynamical lack-of-fit reduction 

• a general viewpoint on changing levels of description.
  
  

2. Self-organisation in nature. Systematic description of self-organisation in nature was initiated by Turing (from mathematical perspective) and then later by Prigogine (from non-equilibrium thermodynamics perspective). Nowadays, it is a widely recognised phenomena ranging across many disciplines. VK's long-term aim is to assess and understand the behaviour and robustness Turing's approach and to amend it if necessary by physically plausible extensions:

• the role of non-diffusibles in Turing's model, large wave number behaviour and the issue of reductionism

• history dependence due to growing domains and a breakdown of the continuum description, a general result concerning dilatationally growing domains

• effective diffusion (including interactions with a substrate) rather than actual physical diffusion play a role

• the effect of heterogeneity - the surprising formation of travelling waves, piece-wise constant kinetics, WKBJ study implying local nature of patterning conditions

• the role of advection in a RD system

• pattern formation in a layered system mimicking experimental set-ups

• isolating a rather robust pattern away form boundary by the choice of boundary conditions

• revisiting RD model from non-equilibrium thermodynamics perspective yielding Burger's type equation

• exploration of intrinsic non-normality of Turing models

• a recent review of Turing model analysis within a theme PTRSA issue

• application to understanding hair follicle patterning or to Belousov-Zhabotinsky reaction

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3. Biomechanics - bone and cartilage. 
V.K.'s interest in biomechanics dates back to his PhD and Master's thesis (under the supervision of prof F. Maršík) although the main aims have shifted as indicated in the first point above. The primary interest has been in bone adaptation.
Since 2014 V.K. has been working with EA Gaffney (MI platform grant and recently a MCSA-IF project) on cartilage modelling project with emphasis on pathology and experiments (closely discussed with CP Brown). This includes an overview of multiphasic and mixture models in cartilage applications, followed by a study pointing out the importance of the appropriate inclusion of heterogeneity of the problem into the model together with the initial and boundary conditions even in standard mechanical tests.

4. PEM fuel cells, membranes. In cooperation with a very skilled experimentalist JB Benziger (Princeton) we try to isolate particular phenomena in PEM fuel cells such as transport across thin membranes and to understand them from both theoretical and experimental perspective.

5. Chemical kinetics. The need of understanding mechano-chemical coupling for plausible modelling of bone adaptation led to a development of a rather general theory on this topic.

    
 6. Other research projects include a result based on MM study group describing placenta development

Some recent works with a sort of public outreach (thanks to Andrew Krause): constraining nonequilibrium physics, how reaction-diffusion models generate complex patterns or see the editorial for a recent theme issue.