Lab 1 - Viscosity Lab Report PDF

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Viscosity Lab Report...

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Abstract The objective of this lab is to determine the dynamic viscosity of a commercial gear oil. The viscosity is obtained by measuring the terminal velocity of small spheres falling through the oil. [2] Introduction and Theory Viscosity is a fluid property that can be interpreted as the ‘thickness of a fluid. This property describes the level of resistance to the sliding motion of one layer of the fluid to another one. Viscosity is very important in many aspects. More specifically, lubricants have different viscosities to satisfy the needs of different machinery for which they are used, and without the correct viscosity, the machines will not be able to operate properly. A fluid’s dynamic viscosity is the resistance to an externally applied shear stress. Viscosity resists the fluid to change form. Because of this, if a solid object is dropped into a fluid of lesser density, the object will fall through the fluid, exerting a shear force on the fluid surrounding it. The shear stress (τ) is directly proportional to the velocity gradient within the fluid, for most common fluids. This relationship is described using the equation: [2] τ =μ

∂u ∂x

(1)

● µ is the proportionality constant (dynamic viscosity) The methods in this experiment determine the dynamic viscosity through dropping spheres of a known mass and density into an oil. As the sphere to falls through the oil, it will stop accelerating and reach a steady state velocity called terminal velocity. Different forces will be acting on the sphere but the sum of all the forces will be zero. ƩF=ma=0 FD+FB-Ws=0

(2)

Figure 1: Force balance on the sphere falling through a viscous liquid.

According to equation 2, the weight of a sphere (Ws) is equal to and opposite to the sum of the

i.

upward the drag force (FD) and the upward buoyancy force (FB). The weight force acts in the downward direction and is dependent on the acceleration due to gravity, the density of the sphere and the size of the sphere. The pressure exerted by the fluid on the sphere causes the buoyant force in the equation above. When a sphere is moving in a liquid, the viscous drag force will be exerted in the sphere.

In flows that are very slow, meaning the fluid has a very high viscosity, the viscous forces are very large in comparison to the inertial forces. This type of flow is called Creeping Flow or Stokes Flow. For a very slow flow, the mathematician, G.G. Stokes, obtained an analytical solution for the drag force on the sphere. The solution to the equations of the fluid motion gives that the drag force (FD) on a sphere moving at a constant velocity (U) through a still fluid is: F D =3 π μ UD

(3)

● D is the diameter of the sphere μ is the dynamic viscosity of the fluid

●

To understand Archimedes principle, we must have the basic knowledge of what a buoyant force is. This force “is defined as an upward force (with respect to gravity) on a body that is totally of partially submerged in a fluid.[1].” Archimedes principle is the general principle for buoyancy and can be applied to this experiment because it is valid for a single fluid of a uniform density. [1] Using this principle, the equation can be applied to find the amount of fluid displaced by finding the upward buoyant force through the following equation: F=ρf ∀g=ρf ρf

● ∀=

●

π D3 g 6

(4)

is the density of the fluid

π D3 is the volume of the sphere 6

Another force, which must be calculated and accounted for, is the force due to weight. The weight ( w s ) can be found through the formula: w s=m s g= ρs ∀g=ρ s ● ●

ms

ρs

π D3 g 6

(5)

is the mass of the sphere is the density of the sphere

1

Lastly, by subsisting the equations for all the forces present on the sphere during its free fall through the oil, the following equation is formed and can be utilized to calculate the dynamic velocity. Once equated, the equation can be used to calculate the dynamic viscosity of a fluid by using the density, diameter, and the terminal velocity of the sphere: 2

D (ρ s−ρf ) g μ= 18 U

(6)

Due to this fluid being a Stokes Flow, the following condition is met: ρf UD...