Today we continue with Physics, and more specifically the branch of "Classical Mechanics", in order to get into Viscosity and Turbulence.
So, without further ado, let's get straight into it!
Real Fluid
In the idealized models we ignored two very important properties of fluid flow, which are viscosity and turbulence. Real fluids differentiate from ideal fluids in exactly these phenomena. Viscosity can be thought of as the internal friction of fluid and turbulence as the mixing and swirling of the various layers caused mainly by obstruction and high speed. Turbulence, more or less, occurs due to the fluid's viscosity, and so these topics are very closely related.
Viscosity
Viscosity is basically the fluid's resistance to flow. The parallel layers of the fluid resist the motion of the fluid layers next to them, as well as the motion of solids that come in contact with them. As such, viscous but non-turbulent flow occurs in layers of different speed. If there was no viscosity the speed would be the same across the whole fluid.
It's due to this resistance of the fluid, also called drag, that the speed at the bottom is much less than the speed at the top. In the context of tubes, due to viscosity, the speed at the walls of the tube is basically zero, and maximum at the center.
If an obstruction comes up or the speed reaches a critical point, the layers start mixing and the flow becomes turbulent.
Measuring Viscosity
In order to end up with a measure of viscosity, we start by placing fluid between two parallel plates. The bottom plate is fixed (v = 0), whilst the top plate moves to the right with a velocity v, dragging the fluid along with it. The layers of fluid that are in contact with the plates don't move relative to these plates. So, the top layer moves at the same speed v as the top plate, and the bottom layer remains at rest.
A visible skew will occur as each layer tries to drag along the next layer and the speed variates from v to 0. The flow is carefully kept laminar, so that the layers don't mix. A continuous shearing motion thus takes place. Because fluids have zero shear strength the rate at which they are sheared is related to the geometrical factors A and L.
The shear stress between the layers is equal to the force F by the area A. The rate of shearing, also called strain rate or velocity gradient, is the velocity v by the length L. As such, we can now define viscosity as the ratio of these two quantities. Viscosity is represented by the Greek later η (eta) and given by:
Fluids that satisfy this equation are called Newtonian Fluids, and are a subset of real fluids.
Solving the equation for force gives us:
So, the greater the viscosity, the greater the force required.
The S.I. unit of viscosity is the N ⋅ s / m2 or Pa ⋅ s, but its more common to use a quantity related to cgs, the poise, which equals 1 dyn ⋅ s / cm2.
There are actually two types of viscosity. The one measured in poise is also called dynamic viscosity. Dividing it by the density gives us kinematic viscosity which is represented by the Greek letter ν (nu). Kinematic viscosity is measured in stokes (St) which equals 1 cm2 / s. The S.I. unit of 1 m2 / s is rarely used.
Poiseuille’s Law
As we already know, what causes flow is pressure difference. By also taking into account the resistance of fluids to flow, we can derive the following relationship:
As flow rate we consider the volume flow rate Q = dV / dt. Flow resistance is anything expect pressure that affects flow rate, such as viscosity. It's represented by capital R. As such, if p1 and p2 are the pressures at two points in a tube, the following will be true:
French scientist Poiseuille proved that the resistance to laminar flow in an incompressible fluid with viscosity η through a horizontal tube of radius r and length L is:
Combining these equations yields Poiseuille’s Law:
Stokes' Law
Again skipping the proof, let's mention yet another useful equation, which is Stokes' law. It gives us the drag force on a falling sphere:
When the sphere is fully submerged and falling with constant velocity, we can derive its viscosity. The buoyant force and drag force are equal to the weight, which gives:
Depending on what's known it's possible to calculate the viscosity (maybe even kinematic) or density of a sphere that is submerged into a liquid.
Turbulence and Reynolds number
Lastly, let's also get into an indicator of turbulence. For a uniform tube, Reynolds number NR is given by:
This quantity is dimension-less but experiments revealed that the flow is laminar for values below 2000, and turbulent for values above 3000. In-between values show unstable and chaotic behavior.
