Dynamics of fluids, viscosity

Stationary stream

Under a stationary state fluid if an obstacle is not present moves in parallel layers. Each fluid can flow stationary if certain conditions are fulfilled: the speed small enough and obstacles so that do not cause too big change in speed. If those conditions are not fulfilled, the flow of fluid is much more complex and is called turbulent flow.

The properties of stationary state will simply observe by the device Hele-Shaw. That device consists of two containers - for water and ink - in which small holes are pierced. Water or ink leak through holes into a very narrow space between the two glass plates. If water (or ink) stream slowly, the stream is stationary, and in the area between the two glass plates there are  parallel dark and light stripes coming as a result of the stream of water and ink: those layers do not mix, but slide parallel. The parallel dark and light stripes show so called streamers, in fact imagined lines which show the direction of the streaming of the fluid. Streamers can be defined more exactly, by explaining that those are the line whose tangent in every point is parallel to the direction of speed of motion of liquid.  What is important here is the stationary motion and it's temporal continuousness. The particles of fluid change its speed and direction, what can be seen in the example of narrowing the flowing area. But, streaming in one given point of the fluid is always done in the same direction, with the same speed. Each particle of the fluid that passes from the particular point of fluid passes with the same speed and in the same direction. That property is the basis of the fact that stationary stream can be presented mathematically through the function that does not depend upon time, but only on the position of that particular particle. That spatial function is called the potential of the streaming field of fluid and presents the solution of one differential equation analog to those we meet in electrostatics. Because of this property of stationary streaming we often call it potential streaming. It's connection to electrostatics will be explained deeper when studying it.

Bernoulli's equation

Each real fluid can in given conditions stream stationary. At higher speeds forces inside molecules in fluid are more expresses. Those forces are seen as friction among the layers of the fluid. When dealing with real fluids we ignore those forces: an ideal fluid is the fluid where the streaming of layers is done with no friction. The streaming of the ideal fluid is stationary as friction is an important condition for forming of whirls.

We will now derive the equation for streaming of fluid or Bernoulli's equation, which will relate pressure, speed and elevation in each point of a fluid.

The Bernoulli's equation says that when an ideal fluid streaming takes place the sum of hydrostatic and hydrodynamic pressure is constant. The static pressure in the liquid  that is streaming is less that in the liquid that is in the standstill state. As the speed of streaming is bigger the static pressure is smaller.

Viscosity

The inner friction or viscosity can be described as force used to make one layer friction on other layer of liquid. The molecules of each layer work on the molecules of the neighbouring layer with forces that are on such distances magnetic. Those forces will try to prevent and slow down the moving of layers and influence the inner friction. Viscosity is also meet with gases and liquids, but with liquids those forces are much stronger. The experimental inner friction we see as resistance to move through water or oil.

The coefficient of viscosity

The quantitative measuring of the inner friction or viscosity will be measured by the quantity called coefficient of viscosity. We will reach that quantity in the following way. Drown the waxed parallel glass plates into liquid. One plate is moving and the other is in standstill position. In that way only the layers closer to the plate move. The shift of layers is bigger as layers are closer to the glass plate. We must use certain work for that moving as layers resist to the moving by molecular forces. That is friction.

The whirls

For smaller speeds, the streaming of fluids is stationary. In that case the speed of fluid motion in each point is constant and does not change through time. When the speed grows and overpasses a certain value, called critical, the nature of streaming becomes more complex. There arouses the local irregular circulation, called whirls. The speed of fluid in the given point changes through time, and the profile of streaming is flat, the fluid does not move in laminas. This kind of streaming is called the turbulent streaming. The critical speed is expressed by the derivation : page 176.

The border layer. The forming of whirls under turbulent streaming can be explained by acting of border layer. The friction among fluids and membrane of the tube, in other words fluid and obstacle is the biggest in the narrow part of fluid closest to the obstacle. In that layer the motion of the fluid is the slowest. So there is a gradient of speed distribution; from the border layer to the inside of the fluid the speed is bigger.

Dynamic suppression

The bodies lighter then water float on it's surface. At the same time the bodies lighter then air rise up. The force that lifts them up is called buoyancy, and it is equal to the weight of the displaced fluid. That force is also called static thrust, and comes from the fact that hydrostatic pressure in fluid grows as depth grows. But, the plane flying is not kept in air because of static thrust: the plane that is not moving  falls down. So this kind of thrust will be called dynamic thrust. According to the Bernoulli's equation the body that relatively moves is under the influence of hydrodynamic pressure. If the motion of the fluid is symmetric to the position of the body, the pressure from both sides is annulled. But, if the streaming of fluid is such that the speed of streaming is higher above the body, than the one below the body, then according to Bernoulli there must be a resultant force upward present, responsible for the dynamic thrust.