Viscosity (a) Consult a reference book to give and explain...

Viscosity (a) Consult a reference book to give and explain the definition of viscosity. (b) What are the dimensions of the coefficient of viscosity $\eta ?$ (c) What is the value of the viscosity of water at $20^{\circ} \mathrm{C} ?$ (d) Evaluate $(b)$ [see Eq. $(7.28)]$ for a sphere of radius $5 \mathrm{~cm}$ in a medium of viscosity $2.0$ centipoises. (See Prob. 13.) (e) If in $(d)$ the density of the sphere is $2.7 \mathrm{~g} / \mathrm{cc}$ and the density of the liquid is $1.1 \mathrm{~g} / \mathrm{cc}$, find the terminal velocity. Use the net vertical force, as explained in Prob. 5, and the relaxation time.

Viscosity
(a) Consult a reference book to give and explain the definition of viscosity.
(b) What are the dimensions of the coefficient of viscosity $\eta ?$
(c) What is the value of the viscosity of water at $20^{\circ} \mathrm{C} ?$
(d) Evaluate $(b)$ [see Eq. $(7.28)]$ for a sphere of radius $5 \mathrm{~cm}$ in a medium of viscosity $2.0$ centipoises. (See Prob. 13.)
(e) If in $(d)$ the density of the sphere is $2.7 \mathrm{~g} / \mathrm{cc}$ and the density of the liquid is $1.1 \mathrm{~g} / \mathrm{cc}$, find the terminal velocity. Use the net vertical force, as explained in Prob. 5, and the relaxation time.

Key Concepts

Relaxation Time

Relaxation time in fluid mechanics describes the time required for a particle or a system to reach its steady state, such as terminal velocity, following a disturbance. It is determined by factors like the mass of the particle and the drag coefficient arising from the fluid’s viscosity. This concept is key in dynamic systems where the transient behavior prior to equilibrium needs to be considered.

Stokes' Drag Force

Stokes' drag force refers to the resistive force experienced by a small spherical object moving slowly through a viscous fluid. It is commonly expressed as proportional to the sphere's radius, the fluid's viscosity, and the velocity of the sphere. This concept is central in determining the behavior of particles in a fluid, particularly under low Reynolds number conditions, and it lays the groundwork for calculating quantities like terminal velocity.

Terminal Velocity

Terminal velocity is the constant speed that a freely falling object eventually reaches when the net force acting on it becomes zero. In the context of a sphere moving through a viscous fluid, terminal velocity is achieved when the gravitational force (adjusted for buoyancy) is exactly balanced by the drag force. Understanding terminal velocity is essential in applications ranging from sedimentation processes to the design of particulate systems in fluids.

Viscosity

Viscosity is a measure of a fluid's resistance to deformation under shear or tensile stress. It quantifies the internal friction between adjacent layers of a fluid as they move relative to each other, and it plays a crucial role in determining how easily a fluid flows. High viscosity corresponds to a 'thicker' fluid that flows less readily, while low viscosity indicates a 'thinner,' more easily flowing fluid.

Dimensional Analysis of Viscosity

The coefficient of viscosity, or dynamic viscosity, has specific dimensions that can be derived from Newton's law of viscosity. In SI units, dynamic viscosity is measured in Pascal seconds (Pa·s) or equivalently in terms of mass, length, and time as kg·m?¹·s?¹. Dimensional analysis ensures that equations in fluid mechanics are dimensionally consistent and helps in deriving relationships between different physical quantities.

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