Chapter 15FLUIDS 15.1 Fluid and the World Around Us 1.A fluid is a substance that cannot support a shearing stress. 2.Both gases and liquids are fluids. 3.Fluids in some cases can be regarded as incompressible and no viscosity. These kinds of fluids are regarded as ideal fluids.

Hydrostatics: Fluid statics is concerned with fluids in which the center of mass of each fluid particle has zero velocity and zero acceleration. A fluid that is at rest is said to be in hydrostatic equilibrium Pressure  Depth Relation The internal stress is normal and compressive. It is called pressure: The unit of pressure is also pascal in SI. 1 atm = 760 mmHg =  10 4 N/m 2

Take an imagined right triangle column: We get:

The pressure in equal-heights It gives out: Then, the pressures at the same depth of a static fluid is equal in magnitude.

The relationship between pressure and depth:

A volume force is the force exerted on each volume element: The equal-pressure surface consists of equal height positions. It is found that gravity is perpendicular to the equal-pressure surface. D EDUCTION : Volume force is always perpendicular to the equal-pressure surfaces.

Archimedes’ Principle: When a body is fully or partially submerged in a fluid, a buoyant force from surrounding fluid acts on the body. The force is directed upward and has a magnitude equal to the weight of the fluid that has been displaced by the force. Pascal’s Principle: A change in the pressure applied to an incompressible fluid is transmitted undiminished to every portion of the fluid and to the walls of its container.

15.3 Fluid Dynamics Euler’s method: Therefore, the concepts of stream line and flow tube are used to describe the fluids in motion. Steady flows: Ideal fluids:1. Incompressible flow 2. Nonviscous flow Ideal fluids in motion

15.3.2The equation of continuity Mass into segment: Mass out segment: We get This is called the equation of continuity for steady and impressible flow.

15.3.3Bernoulli’s Equation

The work done by the pressure force p 1 A 1 is p 1 A 1  l 1 ; The work done by the pressure force p 2 A 2 is p 2 A 2  l 2 ; The work done by gravity is  mg(y 2  y 1 ). Therefore, the work done by the resultant force is The change in kinetic energy is We get

We rearrange it to read This is called Bernoulli’s equation for steady, nonviscous, incompressible flows only. There are many applications associated with this equation.

15.4 Viscosity In fluids, the analog of the shear modulus for solids is the ratio of the shear stress to the rate of strain. This ratio is termed the dynamic viscosity That is to say, internal friction (viscous force) is proportional to velocity gradient and the area of contact layer.

15.5 Turbulence Laminar flow is a stable streamline flow in a viscous fluid of which flow speed is small. A random and irregular motion of the fluid is referred to as turbulent flow. Reynolds number: Where L is some characteristic length associated with the flow. When a flow becomes unstable and turbulent, the value of the Reynolds number is called the critical Reynolds number.

Poiseuille equation for horizontal and pipe-like laminar flows: Volume flux: Velocity:

Problems: (on page 341), , , , ,