Fluid Mechanics Notes

Lecture 1: Fluid and Properties

This lecture introduces a new definition of matter states, focusing on the distinction between solids and fluids, and reviews fundamental concepts essential for studying fluid mechanics. Fluid mechanics is a branch of physics concerned with the mechanics of fluids (liquids, gases, and plasmas) and the forces on them. It has broad applications in various engineering disciplines, including civil, mechanical, chemical, and aerospace engineering.

Defining Solids and Fluids

Traditionally, matter exists in three states: solid, liquid, and gas. However, a more nuanced approach distinguishes between solids and fluids, acknowledging the ambiguity in classifying substances like peanut butter or toothpaste. This distinction relies on how materials respond to shear stress.

Shear Stress

Shear stress, defined as force per unit area, is the benchmark for differentiating solids and fluids. Shear stress occurs when a force is applied parallel to a surface. It is mathematically expressed as:

$Shear Stress = \frac{Force}{Area} \space [\frac{N}{m^2}]$

There are three types of stress:

Tensile: Pulling an object, resulting in elongation.
Compressive: Pressing an object, leading to shortening.
Shear: Sliding two objects against each other, causing deformation.

Solid vs. Fluid

Solid: Resists permanent deformation under shear stress, deforming only slightly before reaching equilibrium. When shear stress is applied, a solid deforms, but the deformation is limited and reversible once the stress is removed.
Fluid: Deforms continuously under even the smallest shear stress. A fluid will flow and not return to its original shape when shear stress is applied. This continuous deformation is a defining characteristic of fluids.

Friction force exemplifies shear stress. Applying force to a surface creates resistance, akin to shear stress. A fluid is defined as a substance that continuously deforms under any magnitude of shear stress. This definition leads to the classification of fluids as Newtonian or non-Newtonian.

Fluid Properties and Units

Studying fluid mechanics requires a review of fluid properties and the introduction of new concepts using proper units. Accurate and consistent use of units is essential for calculations and analysis.

SI Units

The preferred unit system in engineering is the SI (Système International) unit, also known as the MKS (meter-kilogram-second) system:

Mass: kilogram (kg)
Length: meter (m)
Time: second (s)
Temperature: Kelvin (K)

British Units

Another common system is the British unit, also referred to as the Imperial unit or AES (American Engineering System) unit:

Mass: pound (lb)
Length: foot (ft)
Time: second (s)
Temperature: degree Rankine (°R)

Unit Conversion

Using the correct units is crucial in engineering. Conversion factors are essential tools. For example, converting 65 miles per hour to kilometers per hour:

$65 \frac{miles}{hour} \times \frac{5280 \space feet}{3281 \space feet} = 65 \times 1.609 \frac{km}{hour}$

To convert to meters per second, additional conversion factors are required:

$65 \frac{km}{hour} \times \frac{1 \space hour}{3600 \space seconds} \times \frac{1000 \space meters}{1 \space km} = \frac{65 \times 1000}{3600} \frac{m}{s}$

Density

Density is defined as the mass per unit volume in the limit as the volume approaches zero. It is an intensive property that characterizes how much mass is contained in a given volume.

$\rho = \lim_{\Delta V \to 0} \frac{\Delta m}{\Delta V} \space [\frac{kg}{m^3} \space or \space \frac{g}{cm^3}]$

The conversion factor between $kg/m^3$ and $g/cm^3$ is 1000. For gases, density depends on pressure and temperature, as described by the ideal gas equation.

Ideal Gas Law

The ideal gas equation relates pressure, volume, and temperature for ideal gases. It is a fundamental equation in thermodynamics and fluid mechanics.

$PV = nRT$

Where:

$P$ is the pressure.
$V$ is the volume.
$n$ is the number of moles.
$R$ is the gas constant (8.31 J/mol·K).
$T$ is the absolute temperature in Kelvin.

The number of moles, $n$ , can be expressed as:

$n = \frac{m}{MW}$

Where $m$ is mass and $MW$ is molecular weight. Density can be derived from the ideal gas equation:

$\rho = \frac{PMW}{RT}$

Liquids are generally considered incompressible, while gases are compressible. For liquids, changes in density with pressure are typically small and often negligible in many applications. However, for gases, density is highly sensitive to changes in both pressure and temperature.

Specific Weight

Specific weight ( $\gamma$ ) is the weight per unit volume:

$\gamma = \rho g \space [N/m^3]$

Where $g$ is the gravitational acceleration (9.8 m/s²).

Specific Gravity

Specific gravity (SG) is the ratio of a fluid's density to the density of water at standard conditions (4°C, where water density is 1000 kg/m³):

$SG = \frac{\rho{fluid}}{\rho{water}}$

For mercury, with $\rho_{mercury} = 13,600 \space kg/m^3$ , the specific gravity is:

$SG_{mercury} = \frac{13,600}{1,000} = 13.6$

Lecture 2: Viscosity of Fluid

This lecture introduces viscosity, a measure of a fluid's resistance to shear stress, and explores Newtonian and non-Newtonian fluids. Viscosity is a critical property in fluid mechanics, affecting how fluids flow and behave in various applications.

Viscosity

Viscosity quantifies a fluid's resistance to flow when subjected to shear stress. It is related to the internal friction within the fluid. Consider a fluid between two parallel plates, where the bottom plate is stationary and the top plate moves at a constant velocity, $v$ . This motion induces a shear stress, $\tau$ , within the fluid.

Newton's Law of Viscosity states that shear stress is proportional to the velocity gradient:

$\tau = \mu \frac{dv}{dy}$

Where:

$\tau$ is the shear stress.
$\mu$ is the dynamic viscosity.
$\frac{dv}{dy}$ is the velocity gradient or shear rate.

In this setup, the fluid's velocity at the bottom plate is zero, while at the top plate, it matches the plate's velocity. The velocity profile within the fluid is linear, and the viscosity, $\mu$ , remains constant for Newtonian fluids.

Newtonian Fluid

For a Newtonian fluid, the relationship between shear stress and shear rate is linear. The constant of proportionality is the viscosity, $\mu$ . The shear stress ( $\tau$ ) is the force applied to the board. The unit is same as pressure, Newton per meter square. $\mu$ is in Pascal second, and $\frac{dv}{dy}$ is called shear rate, and its unit is one on second. The viscosity, $\mu$ , remains constant regardless of changes in velocity or shear stress. Examples of Newtonian fluids include water, air, and thin oils.

Newton's Experiment

Newton's experiment in a swimming pool demonstrated this principle. A large surfing board moving at a constant velocity ( $\nu_x$ ) creates a velocity profile in the water. The shear stress applied to the board is related to the velocity gradient by the viscosity of the water.

Non-Newtonian Fluid

In Non-Newtonian fluids, viscosity changes with shear rate. Examples include toothpaste, peanut butter, and tomato sauce. These fluids do not exhibit a constant viscosity; their viscosity varies as the shear rate or deformation rate changes. Some exhibit shear thinning (viscosity decreases with increasing shear rate), while others show shear thickening (viscosity increases with increasing shear rate). This behavior is due to the complex molecular structures and interactions within these fluids.

Types of Non-Newtonian Fluids

Bingham Plastic

Bingham plastics behave like solids until a certain yield stress ( $\tau_0$ ) is exceeded, after which they flow like Newtonian fluids:

$\tau = \tau_0 + \mu \frac{dv}{dy}$

Toothpaste is a common example, resisting small shear stresses but flowing under larger ones. Other examples include drilling mud and sewage sludge.

Power Law Fluids

Power-law fluids relate shear stress and shear rate through a power-law relationship:

$\tau = k(\frac{dv}{dy})^n$

For shear-thinning fluids, n < 1.
For shear-thickening fluids, n > 1.
When $n = 1$ , the fluid is Newtonian.

Examples

Shear-thinning: Tomato sauce and chili sauce exhibit decreased viscosity with increasing shear rate. These fluids are also known as pseudoplastic fluids.
Shear-thickening: Cornstarch and water mixtures become more viscous with increasing shear rate. These fluids are also known as dilatant fluids.

Lecture 3: Unit of Viscosity

This lecture delves into the units of viscosity and explores the behavior of viscosity in gases and liquids.

Units of Viscosity

The SI unit of viscosity is the Pascal-second (Pa·s) or N·s/m²:

$[\mu] = Pa \cdot s = \frac{N}{m^2} \cdot s$

However, this unit is often too large for practical use, leading to the adoption of smaller units like centipoise (cP) from the CGS (centimeter-gram-second) system. In the CGS system, the unit of viscosity is poise (P), with 1 P = 0.1 Pa·s, and 1 cP = 0.001 Pa·s. Water at 20°C has a viscosity of approximately 1 cP.

Conversion Table

$1 \space P = 1 \space dyne \cdot s/cm^2$

Measuring Viscosity

Viscosity can be measured using devices like inclined planes, where the rate at which a weight slides down the plane is related to the fluid's viscosity. The shear stress on the fluid is constant and can be calculated from the weight and the angle of the slope. Other viscometers include rotational viscometers, capillary viscometers, and falling ball viscometers.

Viscosity and Temperature/Pressure

Gases

The viscosity of gases generally increases with temperature because higher temperatures lead to more frequent and energetic molecular collisions, increasing internal friction. Viscosity of gas will also increase with pressure. There are empirical relations for the relation between viscosity and the absolute temperature Kelvin. These relations are often based on kinetic theory and experimental data.

Liquids

Conversely, the viscosity of liquids typically decreases with increasing temperature. Higher temperatures weaken the cohesive forces between molecules, reducing resistance to flow. The viscosity of liquid is virtually independent of pressure. However, at very high pressures, the viscosity of liquids can increase.

Empirical Relations

Various empirical relations describe the temperature dependence of viscosity for gases and liquids. These relations often involve absolute temperature (Kelvin) and empirical constants specific to each substance. Examples include the Andrade equation for liquids and the Sutherland equation for gases.

Lecture 4: Kinematic Viscosity

This lecture introduces kinematic viscosity and its significance in fluid dynamics.

Dynamic vs. Kinematic Viscosity

Dynamic viscosity ( $\mu$ ), also known as absolute viscosity, is contrasted with kinematic viscosity ( $\nu$ ), which is the ratio of dynamic viscosity to density:

$\nu = \frac{\mu}{\rho} \space [m^2/s]$

Kinematic viscosity has units of m²/s, similar to diffusivity in Fick's law:

$J = -D \frac{dC}{dy}$

Where $D$ is diffusivity. The kinematic viscosity is a better indication of momentum transfer and molecular interactions under the influence of gravity.

Units of Kinematic Viscosity

The unit for kinematic viscosity is often expressed in stokes (St) or centistokes (cSt), with 1 cSt = 10⁻⁶ m²/s:

$1 \space cSt = 1 \space cP / (1 \space g/cm^3)$

Water vs. Air

Air has smaller dynamic viscosity than the water. But air has ahigher kinematic viscosity than water, because water has higher density. Kinematic viscosity is a good indication of the fluid flowing behavior under gravity.

Lecture 5: Pressure in Fluid Statics

This lecture introduces the concept of pressure and its measurement in fluid statics.

Pressure

Pressure is defined as the normal force per unit area:

$P = \frac{F}{A} \space [N/m^2 \space or \space Pa]$

Pressure has a complicated unit because the force is alwyas normal to the force.

Units of Pressure

Common units of pressure include Pascal (Pa), bar, mmHg (torr), atmosphere (atm), and pounds per square inch (PSI). Conversion between these units can be complex.

Absolute vs. Gauge Pressure

Absolute pressure is the actual pressure at a given point.
Gauge pressure is the difference between the absolute pressure and atmospheric pressure.

$P{gauge} = P{absolute} - P_{atmospheric}$

At room pressure the gauge pressure is zero. On earth, atmosphere pressure is one ATM. The atmosphere pressure on the moon is zero.

Pressure in Liquids

The pressure at a certain depth in a liquid is equal in all directions. The relationship between pressure and depth is given by:

$\frac{dP}{dz} = -\rho g$

Where:

$\frac{dP}{dz}$ is the change in pressure with respect to depth.
$\rho$ is the density of the liquid.
$g$ is the gravitational acceleration.

Pressure Head

Pressure head (h) is defined as the height of a column of fluid:

$h = \frac{\Delta P}{\rho g} \space [m \space or \space feet]$

The pressure difference between two points in a fluid is related to the height of the fluid column between those points.

Free Surface

The interface between a liquid and air is a free surface, where the pressure equals the gas pressure. At a depth of H the pressure is:

$P = \rho g H + P_{atmospheric}$

Pressure Measurement

Traditional methods to measure pressure include U-tube manometers. They are popular to measure in the industry. The mercury barometer uses a glass tube filled with mercury to measure atmospheric pressure.

U-Tube Manometer

A U-tube manometer measures the pressure difference between a gas chamber and atmospheric pressure by observing the height difference in the liquid column.

Lecture 6: Flow velocity and Regimes

This lecture introduces the different flow regimes and the concept of fluid velocity fields.

Flow Regimes

Fluid flow can be characterized as laminar, streamline, or turbulent. Laminar flow is characterized by smooth, parallel layers of fluid, while turbulent flow is characterized by chaotic, irregular motion. The transition between these regimes is governed by the Reynolds number.

Reynolds theory and experiment provide the fundamental explanation for this flow regime.

Equation of Reynolds Number

Reynolds number (Re) is a dimensionless number that characterizes the flow regime:

$Re = \frac{\rho V D}{\mu}$

Where:

$\rho$ is the density of the fluid.
$V$ is the velocity of the fluid.
$D$ is the diameter of the pipe.
$\mu$ is the dynamic viscosity of the fluid.

Values of the Reynolds Number

Re < 2000: Laminar flow.
Re > 4000: Turbulent flow.
2000 < Re < 4000: Transition zone.

Significance of the renal number is non dimensional. Its a pure number, and its a property of the fluid kinematics.

Momentum of Fluids and Forces

Renolds number is the ratio of mass and velocity to the viscous of the fluid.

A flow force increases mass times velocity. The viscosity forces try to stop the fluids. This means it increases flow for force is trying to push the liquid. If viscosity increases, they are stopping the fluid.

We increase the Reynolds number.

The experiment results from Reynolds show the pressure can be ploted on the graph. The straightline the DPTX is the straight line to the valumetric flow. If we rise the value above 4000. We see a parabola. This means we are reaching the turbulent regime. We increased the velocity of the volume the friction increased. Exponential increase in this region.

Internal vs. External Flow

Internal: The water flow from a pipe.
External: Flight path of airplanes in the air.

Lecture 7: Volumetric Flow Rate

This lecture extends the discussion of fluid flow, focusing on velocity fields and flow patterns. Volumetric flow rate is a measure of the volume of fluid passing through a cross-sectional area per unit time.

Velocity Field

Velocity is a vector quantity with magnitude and direction. A velocity field describes the velocity of a fluid at every point in space.

In two-dimensional space, the velocity field is described by two components, $\nux$ and $\nuy$ , which are functions of x and y:

$V = f(Vx, Vy)$

Magnitude

The magnitude of the velocity is

$|V| = \sqrt{Vx^2 + Vy^2}$

Lecture 8: Overview

This review over fluid staticts and fluids kinematics.

In this part we learn that the statics pressure equals rho gh at the hieght, and for kinmatics study it at Reynold number.