Viscosity and terminal velocity.

In hydrostatics, we’ve discussed surface tension and capillarity. In this unit, we’ll discuss viscosity in fluids and how terminal velocity is achieved under viscosity. We’ll perform a simple experiment to demonstrates viscosity as well as forces that contribute to the effect. So, let’s get started!

Defining Viscosity.

Viscosity in a fluid is the frictional force that exists between different layers of fluids. In simple terms, it is the property of a fluid which tends to prevent motion of one part over another.

The general concept of viscosity is the thickness corresponds to fluids. It relies on the molecular resistance associated with fluids. A common example of viscosity is noticed when a liquid is being poured from one container to another. You will notice higher viscosity in honey than water, because there is greater molecular force of attraction between the molecules of honey than water. Hence, Honey becomes thicker than water.

Moreover, viscosity also present in gasses. To give a few examples, Viscosity of the air causes a swinging pendulum to gradual die, it makes a piece of paper to fall to the ground less rapidly than a stone and causes the skydiver to jump out of airplane safely with the aid of parachute. When an object falls freely in air from great height, it first accelerates, and then gain speed. The force due to air resistance increases, leading to a decrease in the acceleration. Eventually, a velocity is attained at which the air resistance is just equal to the weight of the object such that acceleration becomes zero and the body thereafter falls with constant velocity. The point at which the constant velocity is attained is called the terminal velocity.

To determine the amount of viscosity will have liquids, we may compare how fast an object will reach the base of a container for different type of liquids. This is done by dropping a small ball bearing into a measuring cylinder containing a liquid with known length (d) and the time (t) to reach the base of measuring cylinder. The velocity of the body in the liquid is then determined as: Velocity = length of measuring cylinder d / time t. The greater the velocity of the ball bearing in the liquid, the lesser the viscosity of the liquid and vice versa.

Coefficient of viscosity 

The coefficient of viscosity of a fluid is a measure of viscosity of fluids and it’s defined as the frictional force per unit area of the liquid when it is in a region of velocity gradient.

Force F is inversely proportional to the thickness d of liquid. It’s is mathematically given as:

F α Av/d

F = μ Av/d

Where μ is the constant of proportionality and is the coefficient of viscosity of the liquid.

μ =Fd/Av = (F/A)/(V/d).

where (F/A) is the pressure and (V/d) is the velocity gradient. 

The SI unit of coefficient of viscosity is pascal-second (Pa.s).

Viscosity of liquid decrease with increase in temperature while that of gas increase with temperature. This knowledge is used in selection of appropriate liquids for lubrication. Heavy machines use thick oil for operation while light machine uses light oil.
Liquids whose viscosity is not affected seriously by change in temperature are called viscoelastic liquids.

The coefficient of viscosity is not affected by change in pressure. 

The comparison of solid friction and liquid friction (Viscosity).

Solid Friction	Fluid Friction (Viscosity)
Opposes motion	Opposes motion.
Depends on nature of solid surface	Depends on nature of liquid surface.
Independent of area of contact	Depends on area of contact
Independent on relative velocity	Depends on relative velocity.
Depends on normal reaction	Independent of normal reaction.

Terminal velocity.

Terminal velocity is the final velocity achieved by a body as it passes through a fluid. This can be done experimentally when an object is dropped into a vessel containing viscous liquid. The body will be under the influence of three forces:

i.  It own weight acting downwards,

ii. The viscous drag from the liquid acting upwards,

iii.  Upthrust acting upwards.

The diagram below shows what happens to a small bearing when drops into a liquid. The forces acting on the ball are shown in the diagram below:

Where: U is upthrust,

            V is the viscous force,

            Mg weight of the body.

The equation is given as:

Ma = mg – v – u

There are three stages that take place when the body drops: It first accelerates due to force of gravity.  As the speed of the ball increases, the viscous force of the liquid increases which in turn reduces the speed of the body. And finally attained a constant velocity as given in the equation below:

When the velocity is constant, the acceleration becomes zero a = 0. So, we have:

0 = mg – v – u. so, the terminal velocity is given as:

V = mg – u.

When the terminal velocity is attained, the two forces acting upwards are equal to the weight of the body acting downwards as given in the equation below:
mg = v + u.

Terminal velocity depends on shape and density of an object.

Velocity-Time graph of a body under the influence of viscosity.

The diagram below shows the velocity-time graph of an object falling through a fluid.