Applications of Ostwald Viscometer and Stoke’s Law of Terminal Velocity

An Ostwald Viscometer, which contains a vertical capillary tube T, is widely use for comparing the viscosities of two liquids. The Liquid is introduced at S, drawn by suction above P, and the time t_{1} taken for the liquid level to fall between the fixed marks, P, Q is observed. The experiment id then repeated with the same volume of a second liquid, and the time t_{2 }for the liquid level to fall from P to Q is noted.

Suppose the liquids have respective densities p_{1}, p_{2},. Then, since the average head h of liquid forcing it through T is the same in each case, the pressure excess between the ends of T = hp_{1}g respectively. If the volume between the marks P, Q is V, then, from Poiseuille’s formula, we have,

Thus knowing t_{1}, t_{2}, and the densities p_{1}, p_{2}, the coefficients of viscosity can be compared. Further, if a pure liquid of a known viscosity is used, the viscosity of a liquid. Since the viscosity varies with temperature, the viscometer should be used in a cylinder C and surrounded by water at a constant temperature. The arrangement can then also be used to investigate the variation of viscosity with temperature. In very accurate work a small correction is required in equation (iii). BARR, an authority on viscosity, estimates that nearly 90% of petroleum oil is tested by an Ostwald viscometer.

Experiment shows that the viscosity coefficient of a liquid diminishes as its temperature rises. Thus for water, ŋ at 15^{o}C is 1.1 x 10^{-3} N s m^{-2} at 30^{o}C it is 0.8 x 10^{-3 }N s m^{-2} and at 50^{o}C it is 0.6 x 10^{-3 }N s m^{-2}. Lubricating oils for motor engines which have the same coefficient of viscosity in summer and winter are known as viscostatic’ oils.

** Stoke’s Law On Terminal Velocity**

When a small object, such as a steel ball-bearing, is dropped into a viscous liquid like glycerin it accelerates at first, but its velocity soon reaches a steady value known as the terminal velocity. In this case the viscous force acting upwards, and the upthrust due to the liquid on the object, are together equal to its weight acting downwards, so that the resultant force on the object is zero. An object dropped from an aeroplane at first increases its speed v, but soon reaches its terminal speed. This illustration shows variation of V, with time as the terminal velocity v_{o} is reached.

Suppose a sphere of radius a is dropped into a viscous liquid of coefficient if viscosity ŋ, and its velocity at an instant is v. the frictional force, F, can be partly found by the method of dimensions. Thus suppose F = ka^{x}ŋ^{y}v^{z}, where k is a constant. The dimensions of F are MLT^{-2}; the dimension of a is L; the dimensions of ŋ are ML^{-1}T^{-1}; and the dimensions of v are LT^{-1}.

Hence, z = 1, x = 1, y = 1. Consequently F = kŋav. In 1850 STOKES showed mathematically that the constant k was 6π, and he arrived at the formula.

F = 6πaŋv – – – (1)

Stokes’ formula can be used to compare the coefficients of viscosity of very viscous liquids such as glycerine or treacle. A tall glass vessel G is filled with the liquid, and a small ball-bearing P is dropped gently into the liquid so that it falls along the axis of G. Towards the middle of the liquid P reaches its terminal velocity v_{o} which is measured by timing its fall through a distance AB or BC.

The upthrust , U, on P due to the liquid = 4πa^{3}όg/3, where a is the radius of P and ό is the density of the liquid. The weight, W, of P is 4πa^{3}pg/3, where p is density of the bearing’s material. The net download force is thus 4πa^{3}g(p-ό)/3. When the opposing frictional force grows to this magnitude, the resultant force on the bearing is zero. Thus for the terminal velocity v_{o}, we have,

Thus knowing v_{1}, v, p, ό_{1}, ό, the coefficients of viscosity can be compared. In very accurate work a correction to (iii) is required for the effect of the walls of the vessel containing the liquid.

** MOLECULAR THEORY OF VISCOSITY**

Viscosity forces are detected in gases as well as in liquids. Thus if a disc is spun round in a gas close to a suspended stationary disc, the latter rotates in the same direction. The gas hence transmits frictional forces. The flow of gas through pipes, particularly in long pipes as in transmission of natural gas from the North Sea area, is affected by the viscosity of the gas.

The viscosity of gases is explained by the transfer of momentum which takes place between neighboring layers of the gas as it flows in a particular direction. Fast-moving molecules in a layer X cross with their own velocity to a layer Y say where molecules are moving with a slower velocity. Molecules in Y likewise move to X. the net effect is an increase in momentum in Y and a corresponding decrease in X, although on the average the total number of molecules in the two layers is unchanged. Thus the layer Y speeds up and the layer X slows down, that is, a force acts on the layers of the gas while they move. This is the viscous force. We consider the movement of molecules in more detail shortly.

Although there is transfer of momentum as in the gas, the viscosity of a liquid is mainly due to the molecular attraction between molecules in neighboring layers. Energy is needed to drag one layer over the other against the force of attraction. Thus a shear is required to make the liquid move in laminar flow.

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