Newtonian Fluids

Newtonian fluids are simple liquids that follow Newton’s law of direct proportionality between shear stress and shear rate. In these fluids, viscosity remains independent of both shear stress and shear rate. The viscosity of simple liquids (which include pure liquids with small molecules and solutions where both solute and solvent are small molecules) depends solely on composition, temperature, and pressure. Viscosity increases moderately with rising pressure and significantly with decreasing temperature. For solutions with solid solutes, viscosity generally increases with concentration.

Rheograms or flow curves of Newtonian liquids.
Fig. 1: Rheograms or flow curves of Newtonian liquids.

Newton’s Law of Flow

Imagine a liquid contained between two very large, parallel plates, divided into a stack of very thin, parallel layers, similar to a deck of cards. Shear is applied to the liquid by pulling or pushing the top plate with a constant force F’ per unit area A, denoted as F’/A or F, while the bottom plate remains stationary. The top liquid layer, in contact with the moving plate, sticks to it and moves at the same velocity as the plate.

The flow of a liquid between to parallel plates.( newtonIan FluIds)
Fig. 2: The flow of a liquid between to parallel plates.

The second layer, adjacent to the top one, is dragged along by friction, but its velocity decreases due to the resistance from the layers beneath it. Each layer is pulled forward by the layer above it but is held back by the layer below it, over which it moves and which it drags along. The farther the liquid layers are from the moving plate, the slower their velocities. The difference in velocity, dv, between two planes of liquid separated by an infinitesimal distance, dr, is the velocity gradient or rate of shear, dv/dr.

The higher the viscosity of a liquid, the greater the force per unit area (shearing stress) required to achieve a specific rate of shear. The rate of shear, denoted by the symbol G, should therefore be directly proportional to the shearing stress. This relationship can be expressed as:

$$ \frac{F’}A=\eta\frac{d\nu}{dr} $$

or.

$$ \eta=\frac FG $$

where η is the coefficient of viscosity, usually referred to simply as viscosity, F=F’/A and G=dv/dr. A representative flow curve, or rheogram, obtained by plotting F versus G for a Newtonian system is shown in figure 2. As implied by equation η=F/G, a straight line passing through the origin is obtained.

The unit of viscosity

The unit of viscosity is the poise, defined as the shearing force required to produce a velocity of 1 cm/sec between two parallel planes of liquid, each 1 cm² in area and separated by a distance of 1 cm. The cgs units for poise are dyne·sec·cm⁻² (i.e., dyne·sec/cm²) or g·cm⁻¹·sec⁻¹ (i.e., g/cm·sec). These units are readily obtained by a dimensional analysis of the viscosity coefficient.

$$ \eta=\frac{F’dr}{Adv}=\frac{dyne\times cm}{cm^2\times{\displaystyle\frac{cm}{sec}}}=\frac{dyne\;sec}{cm^2} $$

gives the result

$$ \frac{dyne\;sec}{cm^2}=\frac{g\times{\displaystyle\frac{cm}{sec^2}}\times sec}{cm^2}=\frac g{cm\;sec} $$

A more convenient unit for most work is the centipoise(cp, plural cps), 1cp being equal to 0.01poise.

The Effect of Temperature on Viscosity

While the viscosity of a gas increases with temperature, the viscosity of a liquid decreases as temperature rises. Consequently, the fluidity of a liquid (the reciprocal of viscosity) increases with temperature. For many substances, the dependence of a liquid’s viscosity on temperature can be approximated by an equation similar to the Arrhenius equation used in chemical kinetics:

$$ \eta=Ae^\frac{E_a}{RT} $$

or.

$$ \ln\left(\eta\right)=\ln\left(A\right)+\ln\left(\frac{E_a}{RT}\right) $$

where A is a constant that depends on the molecular weight and molar volume of the liquid, and Ea is the “activation energy” required to initiate flow between molecules.

Reference:

  • Sinko, P. (2011). Martin’s Physical Pharmacy and Pharmaceutical Sciences. Baltimore, : Lippincott Williams & Wilkins, a Wolters Kluwer business.
  • Felton. L. (2013). Remington Essentials of Pharmaceutics. London. UK: Pharmaceutical Press.