The coefficient of viscosity and its definition:

The coefficient of viscosity, denoted by η, is a measure of the internal friction within a fluid. It quantifies how resistant a fluid is to flow. It is defined as the tangential force per unit area required to maintain a unit velocity gradient across a fluid layer. Mathematically, it can be expressed as:

η = (F/A) / (∆v/∆x)

Where:
- η is the coefficient of viscosity
- F is the tangential force acting on the layer
- A is the area of the layer
- ∆v is the change in velocity across the layer
- ∆x is the distance across the layer

Motion of a sphere through a viscous fluid:

When a sphere of radius a moves through a viscous fluid under the influence of a viscous force F, it experiences two opposing forces: the viscous force and the gravitational force.

The viscous force acting on the sphere is given by Stokes' law:

F = 6πηav

Where:
- F is the viscous force
- η is the coefficient of viscosity
- a is the radius of the sphere
- v is the velocity of the sphere

Attaining constant velocity:

In the absence of any external forces, the sphere will continue to accelerate until the viscous force becomes equal in magnitude to the gravitational force. At this point, the sphere attains a constant velocity.

Equating the viscous force and gravitational force:

6πηav = 4/3πa³ρg

Where:
- ρ is the density of the fluid
- g is the acceleration due to gravity

Simplifying the equation:

v = (2/9)(a²g/η)

Hence, the constant velocity v attained by the sphere moving through the viscous fluid is given by (2/9)(a²g/η).

Explanation:

The coefficient of viscosity measures the internal friction within a fluid. In the case of a sphere moving through a viscous fluid, the viscous force acting on the sphere opposes its motion. As the sphere accelerates, the viscous force increases until it becomes equal to the gravitational force. At this point, the sphere attains a constant velocity.

The constant velocity v can be calculated by equating the viscous force to the gravitational force using Stokes' law. By rearranging the equation, the expression for v in terms of the radius of the sphere, density of the fluid, acceleration due to gravity, and the coefficient of viscosity is obtained.

This equation provides a quantitative relationship between the physical properties of the sphere, the fluid, and the coefficient of viscosity, allowing for the calculation of the constant velocity attained by the sphere.

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