Fluid flow and viscosity

In a nutshell

Viscosity describes a liquid's resistance to flow, the more viscous an object is, the more resistance there is to its flow. 

Equations

Variables

Upthrust

A fluid exerts an upward force on a body submerged in water. This force is called upthrust. If the weight is equal to this upthrust, the body will float. If the weight is greater, the body will sink. The deeper the object goes into water, the more fluid is displaced to make room for the object. 

Note: Archimedes' principle states that 'The upthrust exerted on a body immersed in fluid, whether partially or fully submerged, is equal to the weight of the fluid that the body displaces.'

Viscosity

Viscosity is defined as the resistance to flow. An example of a viscous liquid would be something thick, like honey. The higher the viscosity, the more opposition to the flow there is. The unit for viscosity is $$Pas$$ or $$kgm^{-2}s$$. 

Viscosity for liquids usually decreases as temperature increases, however in gases viscosity usually increases as temperature decreases. This is because as temperature increases, kinetic energy also increases, which increases the mobility of molecules. The attractive binding energy decreases, and therefore the viscosity of the liquid decreases. In the animation below, 1. represents a high viscosity substance and 2. is a low viscosity substance.

Viscous drag

Viscous drag is the frictional force between an object and a fluid which opposes motion between the two. It is calculated using Stoke's law:

​$$F = 6 \pi \eta rv$$

Conditions for Stoke's equation

• The flow is laminar
  
• The object is small
• The object is spherical
• It is at a slow speed

The coefficient of viscosity is a property of a liquid which states how much it will resist flow. The rate of flow is inversely proportional to the coefficient of viscosity.

Laminar flow

As an object moves through a liquid, layers of the liquid are created. In laminar flow, these layers move in the same direction and do not mix.

In turbulent flow, as an object moves through the liquid, the layers move in different directions and mix.

Example

An object of radius $$5 \ cm$$ is placed into a liquid with viscosity $$2 \ Nsm^{-2}$$. The terminal velocity of the ball was $$0.3 \ ms^{-1}$$, calculate the force acting on it.

First state the variables in the question:

​$$r = 5 \ cm, \ \ \eta= 2 \ Nsm^{-2}, \ \ v = 0.3 \ ms^{-1}$$

Next state the equation needed:

​$$F = 6 \pi \eta rv$$

Substitute into the equation and solve:

$$F = 6 \pi \times 2 \times 5 \times 10^{-2} \times 0.3\newline \\[0.1in] F = 0.5654 \ N\newline \\[0.1in] F = 0.57 \ N \ (2sf)$$​​

The force on the ball is $$\underline{0.57 \ N}$$.