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# Viscosity of a Fluid

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A fluid at rest cannot resist shearing forces and if such forces act on a fluid in contact with a solid boundary as shown in below figure, the fluid will flow over the boundary in such a way that the particles immediately in contact with the boundary have the same velocity as the boundary, while successive layers of fluid parallel to the boundary move with increasing velocities. Shear stress opposing the relative motion of these layers are set up, their magnitude depending on the velocity gradient from layer to layer. For fluids obeying Newton’s law of viscosity, taking the direction of motion as the ‘x’ direction and vx as the velocity of the fluid in the ‘x’ direction at a distance ‘y’ from the boundary, the shear stress (τx) in the ‘x’ direction is given by the formula ### Coefficient of Dynamic Viscosity:

The coefficient of dynamic viscosity (μ) can be defined as the shear force per unit area (or shear stress τ) required to drag one layer of fluid with unit velocity passed another layer a unit distance away from it in the fluid. Rearranging equation, Units: newton seconds per square meter (N s m-2) or kilograms per meter per second (kg m-1 s-1). (But note that the coefficient of viscosity is often measured in poise (P); 10 P = 1 kg m-1 s-1.)
Dimensions: ML-1T-1.
Typical values: for water is 1.14 × 10-3 kg m-1 s-1, and for air is 1.78 × 10-5 kg m-1 s-1.

### Kinematic Viscosity:

The kinematic viscosity (v) is defined as the ratio of dynamic viscosity to mass density. Units: square meters per second (m2 s-1). (But note that kinematic viscosity is often measured in stokes (St); 104 St = 1 m2s-1.)
Dimensions: L2t-1.
Typical values: for water is 1.14 × 10-6 m2 s-1, and for air is 1.46 × 10-5 m2 s-1.

### Mechanism of Viscosity:

The viscosity of a fluid is caused mainly by two factors:

1. Intermolecular Force of Cohesion:

Due to strong cohesive forces between the molecules, any layer in a moving fluid tries to drag the adjacent layer to move with equal speed, and thus produces the effect of viscosity. Since cohesion decreases with temperature, the liquid viscosity does likewise.

2. Molecular Momentum Exchange:

As the random molecular motion of fluid particles increases with increases in temperature, the viscosity increases accordingly. Therefore, except for very special cases (e.g., at high-level pressure), the viscosity of both liquids and gases cease to be a function of temperature.

### Variation of Viscosity:

Viscosity of fluid varies with temperature and pressure both; the effect depends on the state of fluid (gas or liquid)

1. Changes in Viscosity of Liquids:

A higher temperature implies a more vigorous random motion, thereby weakening the effective intermolecular attraction. The viscosity of liquids is governed by intermolecular forces of attraction (i.e cohesive forces). Therefore, viscosity decreases with increasing temperature. For many liquids, the temperature dependence of viscosity can be represented reasonably well by the Arrhenius equation,

μ = AeB/T

where ‘T’ is the absolute temperature, and A and B are constants.

2. Changes in Viscosity of Gases:

Primarily because of lesser molecular density, the intermolecular attraction is not dominant in gases, and the viscosity originates mainly because of the transfer and exchange of molecular momentum. Thus, a viscosity of gases commonly increases with increase in temperature.