 Okay. The title of my talk is analysis of flow structural elements around obstacles in thermodynamically non-equilibrium media. As thermodynamically non-equilibrium media, we consider a stratified fluid, which is known to have diffusion molecular flux, which always exist in the direction opposite to external force action. And it's obvious that there is no homogeneous fluid in the natural systems, and each fluid is less or more non-homogeneous to presence of different dissolved substances, suspended particles, and uniformity of temperature, distribution, pressure, and so on. And we always live in the underaction of different external forces such as rotational, centrifugal, or electromagnetic forces. And underaction of these external forces, a certification is formed. And in such a fluid, in certified fluid, there can exist even fluid motions even in absence of pure mechanical reason. So we have a motionless obstacle and there can exist some fluid motions around these obstacles even without any mechanical reasons. So the main goal of our presentation is to classify different flow regimes, beginning starting from the low-speed diffusion induced flows, which are formed by molecular flux breaking on the obstacles, and to more complex transient multi-scale vortex flow regime, which are accompanied by a complex interaction of large and fine scale components between each other and with the free stream. So method of analysis is, main method of analysis is numerical simulation, and also we compare our results with analytical and laboratory data. So some illustration. So we have a non-homogeneous fluid and underaction of, as an external force where we consider gravity, underaction of gravity we have a redistribution of dissolved substances in the vertical direction and formation of a stable stratification. And in this fluid, molecular diffusion flux exists, which directed in this particular case from down to up in the direction opposite to gravity force action. And if we set an impermeable obstacle in such a fluid, it breaks the diffusion, molecular diffusion flux that leads to formation of zones of concentration, excess and deficiency of a stratifying agent and the formation of a set of fluid motion, which are called diffusion induced flows. This is an illustration of diffusion induced flows. So we have infinite slope and the fluid is going not down but up the hill. There is some analytical evaluations, but these evaluations are virtually one-dimensional. So we have infinite slope and these formulas are calculated only, dependent on only one coordinate normal to the wall. But in reality we understand that there is no infinite and infinite bodies. We must consider finite dimensions of bodies. So to take into account finite dimensions of the obstacle, we must apply numerical simulation. And there is some illustration what the diffusion induced flows. Here the smoke, you see the smoke from the factory chimney which is going in the opposite direction up the hills. There is a temperature stratified atmosphere in the evening when the ground is hotter than the surrounding air. And this is facility of Schlieren instrument in the laboratory of fluid mechanics institute for problems in mechanics, using which we can visualize diffusion induced flows and internal waves around obstacles. So let's go to problem formulation. We have an inclined plate with a certain length and thickness and it moves from right to left at a certain speed or as analogous formulation we have a free stream which is flow the plate. So the governing equation, we consider the equation of state incompressibility equation, Navier-Stock's equation with accounting for the gravity in busines approximation and diffusion equation. As a boundary condition, we consider no slip and no flux boundary conditions for velocity and salinity. And as initial condition, we consider the fields of diffusion induced flows. So when we consider the movement of the plate, the initial state is a diffusion induced flows which is formed on the immovable motionless obstacle. We use for our analysis direct numerical simulation. Initially, we developed our own programs in using programming language for Trun, but later we find very good package which allows to access to source code and we can implement our own solvers, our own programs inside this package for our particular needs. Our particular needs in this case is to account certification diffusion effects. Typical scales of the problem, it's time scale, buoyancy period, length scales, its length and thickness of the plates and attach internal waves and whiskers and diffusion micro scales. And our main goal is classification into fluid types and flow regimes. So we consider four basic fluid types. Strongest certified fluid which is typical for laboratory experiments. So such strong certification usually used in stratified tanks, brine solutions. Weekly certified, it's typical for ocean and sea media and potentially homogeneous fluid. So as I said before, there is no homogeneous fluid. Each fluid is homogeneous. So it's a real situation for pure water because there always exists some non-homogeneity. And actual homogeneous fluid, it's mathematical obstruction. So it's without taking account certification and diffusion efforts because in reality non-homogeneity is always present. And we also classified flow regimes. Diffusion induced flows which are the slowest one on the motionless obstacles. Extraslow regime with very slow speed of movement of the obstacle. Wave regime when internal waves appear, upstream perturbations and attached internal waves. Wattex regime when the intensive vortices appear in the wake and around the obstacles. And non-stationary regimes with relatively high Reynolds number which is the most complicated from this. So we begin our classification from diffusion induced flows. It's a schematic representation so we consider different forms of the obstacle. Climb plate, slender, wave-shaped obstacle and there is a typical velocity profile with main jet which is going along the surface of an obstacle and compensatory backflow. And this results in numerical results for diffusion induced flows around slope and plate. So in case of a horizontal plate, we see here you see the streamlines. Streamlines and green color correspond to cyclonic rotation in circulating cells and blue color to correspond to anti-cyclonic rotation. And we see the structure is symmetric around the obstacles with a multiple level of circulation cells. But even a small deviation from the horizontal position we see that the symmetry is broken and the single circulation cell is formed around the obstacle and then at angles of inclination about 30 and more degrees the jet flows around the plate get thinner and the big anti-cyclonic rotation circulation cells are formed which are adjacent to the main jet. There is no main force diffusion induced flows, there is no flow caused by diffusion processes, only by diffusion processes. And this is comparison of our results on the slope and plate with the analytics. But as I said before analytics doesn't work when analytics works for infinite slope and it doesn't work when the angle of inclination tends to zero. So we have some differences in this area. We see that analytics tends to infinity when the angle tends to zero. So it's a thickness of the jet flow. This is the velocity of this jet flow. And it's a very interesting situation when we match a wedge-shaped obstacle in a certified fluid. In this situation also we have salinity excess under the wedge and salinity deficiency over it. But you see in the pressure field you see the intensive and extensive zone of negative pressure. This negative pressure, so we have a situation when a wedge is sucked by this negative zone. So it leads to self-motion of a wedge. There are a lot of experimental results which confirm this situation that a wedge begins to move horizontally in a certified medium. Also such experiments were performed in the Institute for Problems and Mechanics. This is a numerical simulation using the dynamic mesh. And here you see some comparison of numerical results. Here are the numerical results for horizontal plate and for sloping plate. And this is laboratory modeling. So we see that the numerical and experimental results are in the qualitative agreement. We see the prolongated interfaces which are attached to the edges of the plate. The same for the disc and for the cylinder. So also we see these prolongated layer structures which attach to the poles of the cylinder. And what happens when the obstacle begins to move? The structure of the floor changes dramatically. And we see formation of attached internal waves and upstream perturbations. So here are some numerical simulations using dynamic mesh techniques. So it's upstream perturbations which tends to horizontal. And here some attached internal waves are formed in the wake. This is relatively small velocity or movement 1 cm per second. Where do you see a non-symmetric data time? It's symmetric but it's a vertical component of velocity where you see a non-symmetric. Everything is symmetric. And you see a different internal wave structure for different velocity of movement. We see that with increase of velocity of movement the top area increases along with increase of length of internal waves. And here you see the synchronized images. Here the numerical simulation and here the Schlieren visualization. So it shows how the structure of internal waves change with change of fluid types with decrease of stratification. Let's go further. And now I would like to show you what happens to show you different floor regimes on the inclined plate. So here you see the diffusion induced flow. There is no flow, no movement. And here you see that when the plates begin to move with extremely slow velocity. Here you see the complex interaction of diffusion induced flow with the internal waves. Here you see the clearly manifested internal waves which are generated by the leading and trailing edge of the plate. And here you see the both internal waves and wake water system in the wake which coexist in this situation. And when the velocity of movement of plate movement is quite big we see only the intensive vortices which develop downstream. So and very well practically valuable situation when we have significantly great velocity of movement of the plate. And we consider two different fluid types, strongly certified fluid and potentially homogeneous fluid. It's velocity and velocity number up to 10 in 5 degree. So it's no time yes. So in this situation we have in intensive vortices which are produced by the leading edge of the plate. And I can show you what happens there. Vortices develop downstream from the upper and lower sides of the plate than they meet around the trailing edge of the plate. But this situation is very different when the thickness of the plate is another less than in this case there are no vortices which appear around the leading edge. And also only in the wake. And when angle of inclination is 5 degree we see that there is almost no vortices appear along the lower surface of the plate. And the structure of the vortices, the scale of the vortices get greater. With further increase of angle inclination the structure of water structure became multi scale. And there are more differences between flow structures in strongly certified or potentially homogeneous fluid. So here you see some, so it's different location. Okay. Comparison with Blasio solution we see that when the plate is quite thick there is no agreement with Blasio solution. But when we take a senior plate we have a good agreement but away from the edges. So there is my conclusion, there is no time to think. So thank you for attention. Yes diffusion, velocities of diffusion in this force is very slow. Yes, it's about 5 minus 5 degree.