Elementary Fluid Mechanics CE 156 Undergraduate Course Notes

Elementary Fluid Mechanics (CE 156): Course Overview and Foundations

• Institution: KAAF University

• Department: Civil Engineering Department, Faculty of Civil and Computer Science

• Lecturer: Ing. Sampson Osei, PhD

Course Objectives and Learning Outcomes

• Aim of the Course:

  • To offer basic knowledge in fluid mechanics.

  • To obtain an understanding of the behavior of fluids.

  • To solve some simple problems of the type encountered in Engineering practice.

• Course Objectives: By the end of the course, students are expected to:

  • Define and use basic fluid properties.

  • Define and use basic concepts in fluid mechanics.

  • Perform simple calculations in hydrostatics and kinematics.

  • Make simple designs in hydraulics.

Mode of Delivery and Assessment

• Mode of Delivery:

  • Lectures

  • Tutorials (2 hours per week outside the usual schedule)

  • Laboratory sessions

• Course Assessment:

  • Examination: $70\%$

  • Combined Mid-semester Exams, Laboratory Reports, and Class/Home assignments: $30\%$

• Rules and Regulations:

  • Class attendance is compulsory.

  • Class assignments and quizzes will be unannounced.

• Laboratory Experiments:

  1. Pressure gauges

  2. Plane surfaces immersed in fluids

  3. Floating bodies

  • Note: Lab reports must be written by the group and defended on an agreed schedule.

Recommended Reading Materials

1. Fluid Mechanics (including Hydraulic Machines) – Dr. A. K. Jain, Khanna Publishers, Delhi, 2003.

2. Fluid Mechanics (6th edition) – Frank M. White; McGraw-Hill, 2008.

3. Introduction to Engineering Fluid Mechanics – J. A. Fox, 1985.

4. Fluid Mechanics (5th Edition) – J. F. Douglas; J. M. Gasiorek; J. A. Swaffield; Lynne B. Jack.

5. Hydraulics, Fluid Mechanics and Fluid Machines – S. Ramamrutham.

Definition and Nature of Fluids

• Molecular Structure:

  • Solids: Molecules are so closely packed that the attractive forces between them are large; solids tend to retain their shape unless compelled by external forces.

  • Fluids: Composed of molecules with relatively larger distances between them; attractive forces are smaller than in solids.

• Definition Perspectives:

  • Natural Form: A substance capable of flowing with no definite shape, assuming the shape of its container.

  • Deformation Characteristics: A fluid is a material which constantly deforms under the action of a shearing stress, no matter how small the stress.

  • Examples: Gases (air, LPG), liquids (water, kerosene).

Distinction Between Solids and Fluids

• Elastic Solids: These are deformable, but deformation stops in balance with the acting force. Upon release, the body recovers its original state.

• Plastic Solids: Deformed continuously during the application of force; once released, deformation stops (nominally).

• Fluids: Contrastingly, a fluid keeps deforming even when it is free from force.

Comparison of Liquids and Gases

• Liquids:

  • Composed of relatively close-packed molecules with strong cohesive forces.

  • Relatively incompressible.

  • A given mass occupies a definite volume if not subjected to extreme external pressures.

• Gases:

  • Widely spaced molecules with relatively small cohesive forces.

  • Expand to fill the entire volume of a container if external pressure is removed.

  • Readily compressible.

  • Equilibrium is achieved only when completely enclosed.

  • Volume and density are greatly affected by changes in pressure ($P$) and temperature ($T$).

Introduction to Fluid Mechanics

• Definition: The science of the mechanics of liquids and gases based on the same fundamental principles employed in solid mechanics. It concerns the behavior of fluids at rest and in motion.

• Scope: Accounts for fluid properties, flow patterns, internal forces, and interactions with boundaries.

• Fundamental Laws Applied:

  • Conservation of mass and energy.

  • Newton’s Law of Motion (force-momentum equation).

  • Laws of Thermodynamics.

• Subdivisions:

  • Fluid Statics: Study of fluids at rest. Since there are no shearing forces at rest, all considered forces are normal to the planes on which they act.

  • Fluid Kinematics: Deals with the geometry of motion (streamlines and velocities) without considering the forces causing the motion.

  • Fluid Dynamics: Concerned with the relationship between velocities, accelerations, and the forces causing motion.

Engineering Applications

• Civil Engineering: Hydropower dams, river currents, erosion, hydraulic structures, pipes.

• Biomechanics: Flow of blood.

• Mechanical Engineering: Design of pumps, water turbines, gas turbines.

• Aeronautical Engineering: Airflow over aircraft to reduce drag and increase lift.

Units and Dimensions

• Base Units: Fundamental quantities such as Length ($L$), Mass ($M$), and Time ($T$) in an absolute system.

• Derived Units: Combinations of base units.

• SI Units (Systeme International d’Unites):

  • Area: $L^2$ (m$^2$)

  • Volume: $L^3$ (m$^3$)

  • Velocity: $L/T$ (m/s)

  • Density: $M/L^3$ (kg/m$^3$)

  • Pressure: $M/LT^2$ (N/m$^2$ or Pascal)

  • Work: $ML^2/T^2$ (N.m)

  • Power: $ML^2/T^3$ (J/s or Watt)

  • Viscosity: $M/LT$ (N.s/m$^2$)

  • Flow Rate: $L^3/T$ (m$^3$/s)

Forces Acting on Fluids

• Body Forces: Distributed forces acting on matter without direct contact (e.g., gravity, magnetic, inertia). Expressed as force per unit mass.

• Surface Forces: Forces arising from direct contact with surrounding media (e.g., pressure force, frictional force, surface tension).

Fluid Properties

• Property: A characteristic of a substance that is invariant when in a particular state. Properties uniquely determine the state of a system.

• Extensive Properties: Depend on the amount of substance present (weight, momentum, volume, energy).

• Intensive Properties: Independent of the amount of substance (temperature, pressure, viscosity, surface tension, mass density).

Density and Specific Gravity

• Density ($\rho$): Mass per unit volume.

  • $\rho = \frac{m}{V}$

  • At standard atmospheric pressure ($760\,mmHg$ or $101.3\,kPa$) and $4^{\circ} C$, $\rho = 1000\,kg/m^3$.

• Specific Volume ($v_s$): Reciprocal of density; volume per unit mass.

  • $v_s = \frac{1}{\rho}$

• Specific (Unit) Weight ($\gamma$): Weight per unit volume.

  • $\gamma = \frac{w}{V} = \frac{mg}{V} = \rho g$

• Specific Gravity (SG): Ratio of the weight of a substance to the weight of an equal volume of water at standard conditions.

Physical Properties Data (Select SI values at 15.6-20°C)

• Water ($15.6^{\circ} C$): $\rho = 999\,kg/m^3$, $\gamma = 9.80\,kN/m^3$, $\mu = 1.12 \times 10^{-3}\,N.s/m^2$.

• Mercury ($20^{\circ} C$): $\rho = 13,600\,kg/m^3$, $\gamma = 133\,kN/m^3$.

• Gasoline ($15.6^{\circ} C$): $\rho = 680\,kg/m^3$, $\gamma = 6.67\,kN/m^3$.

• Air ($15^{\circ} C$): $\rho = 1.23\,kg/m^3$, $\gamma = 1.20 \times 10^1\,N/m^3$, Gas constant $R = 2.869 \times 10^2\,J/kg\cdot K$.

Viscosity

• Definition: The property of a fluid to offer resistance to shear stress.

• Temperature Effects:

  • Liquids: Viscosity varies inversely with temperature.

  • Gases: Viscosity varies directly with temperature.

• Newton’s Law of Viscosity: Shear stress ($\tau$) is proportional to the rate of shear strain.

  • $\tau = \mu \frac{du}{dy}$

  • Dynamic Viscosity ($\mu$): Proportionality factor with units $N.s/m^2$.

  • Kinematic Viscosity ($\nu$): $\nu = \frac{\mu}{\rho}$, with units $m^2/s$.

• Fluid Classification:

  • Newtonian Fluids: Shear stress is directly proportional to the rate of angular deformation; $\mu$ is constant for fixed $T$ and $P$ (e.g., air, water, kerosene).

  • Non-Newtonian Fluids: Variable proportionality; may depend on time or magnitude of stress (e.g., plastics, paint, blood).

Compressibility and Surface Tension

• Compressibility: Measure of volume change when subjected to external force. Defined by Bulk Modulus ($K$):

  • $K = \frac{\Delta p}{\Delta V/V}$

• Incompressible Fluids: Density does not change due to external forces. Incompressible study is "Hydrodynamics"; compressible is "Gas Dynamics".

• Surface Tension ($\sigma$): Magnitude of force per unit length caused by unbalanced molecular attractive forces at the surface.

  • Force due to internal pressure: $P \pi r^2$

  • Force due to surface tension: $2 \pi r \sigma$

  • Equilibrium for a spherical drop: $p = \frac{2\sigma}{r}$

• Capillarity: Rise or fall of a fluid in a narrow tube.

  • Cohesive forces > Adhesive forces: Convex meniscus (capillary depression).

  • Adhesive forces > Cohesive forces: Concave meniscus (capillary rise).

  • Capillary height formula: $h = \frac{4\sigma \cos(\theta)}{\gamma d}$

Hydrostatics: Fluid at Rest

• Basic Principle: In a fluid at rest, shear stress is zero; only normal stresses (pressure) exist.

• Hydrostatic Pressure ($p$): Compressive stress acting along the inside normal to the area element. Magnitude is independent of surface orientation.

• Differential Equation of Hydrostatics:

  • $\frac{1}{\rho} dp = Xdx + Ydy + Zdz$

  • X, Y, Z are body force projections per unit mass.

  • A fluid is in equilibrium only when acted upon by potential forces ($U$).

• Integration of Basic Equation:

  • $p = p_0 + \rho(U - U_0)$

• Gravity as the Only Body Force:

  • $X=0, Y=0, Z=-g$

  • $dp = -\rho g dz$

  • Fundamental Equation: $p = p_0 + \gamma h$

  • Absolute Pressure ($p_{abs}$): Sum of external surface pressure and the pressure from the fluid column.

  • Gauge (Manometric) Pressure: Pressure excess above atmospheric ($p_{gauge} = \gamma h$).

Pascal’s Law and Hydraulic Press

• Pascal’s Law: Pressure applied to the surface of a liquid at rest is transmitted throughout the liquid in all directions without change.

• Hydraulic Press: Ability to produce large output forces ($F_2$) from small input forces ($F_1$).

  • $F_2 = p_1 A_2 = \left( \frac{F_1}{A_1} \right) A_2$

  • $F_2 = F_1 \frac{D^2}{d^2}$

Piezometric Height and Potential Energy

• Piezometric Head: Pressure at a point measured as a column of fluid.

  • Absolute head: $h_A = \frac{p_A}{\gamma}$

  • Gauge head: $h_{ex}$

• Potential Energy of Fluid at Rest:

  1. Energy by position: $(P.E.)_z = z \times G$

  2. Energy by pressure: $(P.E.)_p = h_{ex} \times G$

• Total Potential Head ($H$): Specific potential energy (energy per unit weight).

  • $H = z + h_{ex} = z + \frac{p}{\gamma}$

  • $z$ is Geometric Head; $p/\gamma$ is Pressure Head.

Pressure Variation in the Earth’s Atmosphere

• Perfect Gas Equation of State:

  • $pv = RT$

  • $\frac{p}{\rho} = RT$

  • $\gamma = \rho g = \frac{pg}{RT}$

• Process Equation:

  • $p v^n = \text{constant}$

  • $\frac{T_2}{T_1} = \left(\frac{v_1}{v_2}\right)^{n-1} = \left(\frac{p_2}{p_1}\right)^{\frac{n-1}{n}}$

Practice Examples and Quizzes

• Example (Oil/Water Tank): Oil ($SG=0.80, 0.90\,m$ deep) over water ($2.1\,m$ deep).

  • Pressure at bottom: $27.7\,kPa$.

• Example (Bubble Nozzle): Bubble diameter $2\,mm$, $\sigma = 72.7 \times 10^{-3}\,N/m$. Calculate required nozzle pressure excess.

• Example (Capillary Glass): Find diameter for water rise h < 1.0\,mm at $20^{\circ} C$.

  • Answer: $29.8\,mm$.

• Hydraulic Jack Problem: Hand force $F=100\,N$.

  • Support load $F_2 = 12.2\,kN$.