Viscosity of a Fluid

It is defined as the internal resistance offered by one layer of fluid to the adjacent layer. In case of liquids, main reason of the viscosity is molecular bonding or cohesion. In case of gases main reason of viscosity is molecular collision. In case of liquids, due to increase in temperature the viscosity will decrease due to breaking of cohesive bonds. In case of gases, the viscosity will increase with temperature because of molecular collision increases. All fluids offer resistance to any force tending to cause one layer to move over another. Viscosity is the fluid property responsible for this resistance. Since relative motion between layers requires the application of shearing forces, that is, forces parallel to the surfaces over which they act, the resisting forces must be in exactly the opposite direction to the applied shear forces and so they too are parallel to the surfaces.

It is a matter of common experience that, under particular conditions, one fluid offers greater resistance to flow than another. Such liquids as tar, treacle and glycerine cannot be rapidly poured or easily stirred and are commonly spoken of as thick; on the other hand, thin liquids such as water, petrol and paraffin flow much more readily. (Lubricating oils with small viscosity are sometimes referred to as light, and those with large viscosity as heavy; but viscosity is not related to density). Gases as well as liquids have viscosity, although the viscosity of gases is less evident in everyday life.

Quantitative Definition of Viscosity

Consider two plates sufficiently large (so that edge conditions may be neglected) placed a small distance Y apart, the space between them being filled with fluid as shown in Fig.1. The lower plate is assumed to be at rest, while the upper one is moved parallel to it with a velocity ‘V’ by the application of a force ‘F’, corresponding to area ‘A’, of the moving plate in contact with the fluid. Particles of the fluid in contact with each plate will adhere to it and if the distance Y and velocity V are not too great, the velocity v at a distance y from the lower plate will vary uniformly from zero at the lower plate which is at rest, to V at the upper moving plate. Experiments show that for a large variety of fluids,

Fig.1 Fluid motion between two parallel plates

It may be seen from similar triangles in Fig.1 that the ratio V/Y can be replaced by the velocity gradient (dv/dy), which is the rate of angular deformation of the fluid.

If a constant of proportionality 'μ' (Greek ‘mu’) be introduced, the shear stress 'τ' (Greek ‘tau’) equal to (F/A) between any two thin sheets of fluid may be expressed as 

This equation is called Newton’s law of viscosity, it states that, for the straight and parallel motion of a given fluid, the tangential stress between two adjoining layers is proportional to the velocity gradient in a direction perpendicular to the layers.

In the transposed form, it serves to define the proportionality constant. which is called the coefficient of viscosity, or the dynamic viscosity (since it involves force), or simply viscosity of the fluid. Thus the dynamic viscosity μ, may be defined as the shear stress required to produce unit rate of angular deformation. In SI units μ is expressed in N.s/m², or kg/m.s. The dynamic viscosity μ is a property of the fluid and a scalar quantity.

In the metric gravitational system of units, μ is expressed in kg(f)-sec/m². In the metric absolute system of units μ is expressed in dyne-sec/m² or gm(mass)/cm-sec which is also called ‘poise’ after Poiseuille. The ‘centipoise’ is one hundredth of a poise. The numerical conversion from one system to another is as follows.

1 Ns/m² = 10 poise

In many problems involving viscosity, there frequently appears a term dynamic viscosity ‘μ’ divided by mass density ‘ρ’. The ratio of the dynamic viscosity μ and the mass density ρ is known as Kinematic viscosity and is denoted by the symbol ‘υ’ (Greek ‘nu’) so that

On analyzing the dimensions of the kinematic viscosity it will be observed that it involves only the magnitudes of length and time. The name kinematic viscosity has been given to the ratio (μ/ρ) because kinematics is defined as the study of motion without regard to the cause of the motion and hence it is concerned with length and time only.

In SI units υ is expressed in m²/s. In the metric system of units υ is expressed in cm²/sec or m²/sec. The unit cm²/sec is termed as ‘stoke’ after G.G. Stokes and its one-hundredth part is called ‘centistoke’. In the English system of units it is expressed in ft2/sec. The numerical conversion from one system to another is as follows.

1 m²/s = 10⁴ stokes

The dynamic viscosity μ of either a liquid or a gas is practically independent of the pressure for the range that is ordinarily encountered in practice. However, it varies widely with temperature. For gases, viscosity increases with increase in temperature while for liquids it decreases with increase in temperature. This is so because of their fundamentally different intermolecular characteristics. In liquids the viscosity is governed by the cohesive forces between the molecules of the liquid, whereas in gases the molecular activity plays a dominant role. The kinematic viscosity of liquids and of gases at a given pressure, is essentially a function of temperature.

Common fluids such as air, water, glycerine, kerosene etc., follow Newton’s law of viscosity. There are certain fluids which, however, do not follow Newton’s law of viscosity. Accordingly, fluids may be classified as Newtonian fluids and non-Newtonian fluids. In a Newtonian fluid there is a linear relation between the magnitude of shear stress and the resulting rate of deformation i.e., the constant of proportionality μ in the equation does not change with rate of deformation. In a non-Newtonian fluid there is a non-linear relation between the magnitude of the applied shear stress and the rate of angular deformation. In the case of a plastic substance which is a non-Newtonian fluid an initial yield stress is to be exceeded to cause a continuous deformation. An ideal plastic has a definite yield stress and a constant linear relation between shear stress and the rate of angular deformation. A thixotropic substance, which is a non-Newtonian fluid, has a non-linear relationship between the shear stress and the rate of angular deformation, beyond an initial yield stress. The printer’s ink is an example of a thixotropic liquid.

Fig. 2 Variation of shear stress with velocity gradient

An ideal fluid is defined as that having zero viscosity or in other words shear stress is always zero regardless of the motion of the fluid. Thus an ideal fluid is represented by the horizontal axis (τ = 0) in Fig. 2, which gives a diagrammatic representation of the Newtonian, non-Newtonian, plastic, thixotropic and ideal fluids. A true elastic solid may be represented by the vertical axis of the diagram. The fluids with which engineers most often have to deal are Newtonian, that is, their viscosity is not dependent on the rate of angular deformation, and the term ‘fluid-mechanics’ generally refers only to Newtonian fluids. The study of non-Newtonian fluids is termed as ‘rheology’.

Causes of Viscosity

For one possible cause of viscosity we may consider is the forces of attraction between molecules. Yet there is evidently also some other explanation, because gases have by no means negligible viscosity although their molecules are in general so far apart that no appreciable inter-molecular force exists. The individual molecules of a fluid are continuously in motion and this motion makes possible a process of exchange of momentum between different layers of the fluid.

In gases this interchange of molecules forms the principal cause of viscosity and the kinetic theory of gases (which deals with the random motions of the molecules) allows the predictions – borne out by experimental observations is that

1. The viscosity of a gas is independent of its pressure (except at very high or very low pressure) 
2. Because of the molecular motion increases with a rise of temperature, the viscosity also increases with a rise of temperature (unless the gas is so highly compressed that the kinetic theory is invalid).

The process of momentum exchange also occurs in liquids. There is, however, a second mechanism at play. The molecules of a liquid are sufficiently close together for there to be appreciable forces between them. Relative movement of layers in a liquid modifies these inter-molecular forces, thereby causing a net shear force which resists the relative movement. Consequently, the viscosity of a liquid is the resultant of two mechanisms, each of which depends on temperature, and so the variation of viscosity with temperature is much more complex than for a gas. The viscosity of nearly all liquids decreases with rise of temperature, but the rate of decrease also falls. Except at very high pressures, however, the viscosity of a liquid is independent of pressure.