AGE 204: Basic Fluid Mechanics Study Notes

Course Overview and Outline

• Institution: Federal University of Technology, Akure.

• Department: Department of Agricultural Engineering.

• Course Code: AGE 204.

• Course Title: Basic Fluid Mechanics.

• Contact Information: Far Prints (08139628765, 07062173359).

• Detailed Course Outline:     1. Definition and properties of fluid.     2. Elements of fluid statics: density, pressure, surface tension, viscosity, and compressibility.     3. Hydrostatic forces on submerged surfaces due to incompressible fluid.     4. Introduction to fluid dynamics and conversion laws.     5. Introduction to viscous flows.     6. Illustrations and examples.

Fluid Properties and Basic Definitions

• Fluid Mechanics: This is the scientific study of the behavior of fluids both at rest (statics) and in motion (dynamics).

• Fluid Statics: The study of the forces keeping a fluid in static equilibrium.

• Fluid Dynamics: Deals with the motion of fluids.     * Hydrodynamics: The study of fluid flows where no density changes occur (incompressible flow).     * Hydraulics: A subdivision of hydrodynamics specifically studying the flow of liquids in pipes and open channels.     * Gas Dynamics: Deals with fluid flows where density changes are significant.

• Definition of a Fluid: A fluid is a substance that undergoes continuous deformation when subjected to a shear stress, regardless of how small that shear stress may be.

• Liquids vs. Gases:     * Liquid: Only slightly compressible; it possesses a free surface when placed in an open vessel and takes the shape of its container.     * Gas: Expands to fill the entire volume of its container.     * Vapour: A gas that is in a state near its liquid phase.

Shear Stress and Newton's Law of Viscosity

• Shear Force and Stress:     * A shear force is the component tangent to a surface.     * Average Shear Stress ($\tau$): Defined as the shear force divided by the area of the surface ($\tau = \frac{F}{A}$). It is measured in units such as $N/mm^2$ or $N/m^2$.

• Experimental Observation of Deformation:     * Consider a fluid between two large parallel plates. The lower plate is fixed; the upper plate (area $A$) is moved with a steady velocity ($U$) by a force ($F$).     * No-Slip Condition: Fluid in immediate contact with a solid boundary has the same velocity as the boundary.     * Velocity Variation: The velocity varies uniformly from zero at the stationary plate to $U$ at the moving plate.

• Proportionality in Fluid Flow: Experiments show that for a constant area ($A$), velocity ($U$), and distance between plates ($t$):     * $F \times \frac{t}{AU} = \text{constant}$.     * $F = \frac{\mu AU}{t}$, where $\mu$ is the proportionality factor known as viscosity.

• Newton's Law of Viscosity (Differential Form):     * $\tau = \frac{F}{A} = \frac{\mu U}{t}$.     * The term $\frac{du}{dy}$ (velocity gradient) represents the rate of angular deformation or the rate at which one layer moves relative to an adjacent layer.     * General formula: $\tau = μ \frac{du}{dy}$.

• Non-Fluid Substances:     * Plastic substance: Requires an initial yield shear stress to be exceeded before continuous deformation occurs.     * Elastic substance: Deforms a certain amount proportional to the force but does not deform continuously.     * Vacuum: Would result in an ever-increasing rate of motion, not a constant final rate.     * Sand: Requires a finite force to overcome dry friction for motion, thus it is not a fluid.

Fundamental Properties of Fluids

• Mass Density ($\rho$): Mass per unit volume. $\rho = \frac{m}{v}$. The S.I. unit is $kg/m^3$. For water at $4^{\circ}C$, $\rho = 1000\,kg/m^3$.

• Specific Weight ($w$): Weight per unit volume. $w = \frac{mg}{v}$.

• Relationship between $\rho$ and $w$: $w = \rho g$.

• Specific Gravity (Relative Density): The ratio of a fluid's density to the density of pure water at the same temperature.

• Bulk Modulus of Elasticity ($K$): A measure of compressibility.     * $K = \frac{\Delta P}{\Delta v/v}$, where $\Delta P$ is the increase in pressure and $\Delta v/v$ is the volumetric strain.     * Alternatively: $K = \frac{\Delta P}{\Delta \rho / \rho}$.

• Viscosity: The property determining resistance to shearing force, primarily due to molecular interaction. Static fluids cannot have shear stress.

• Surface Tension ($\sigma$): Due to cohesion between molecules at the surface. It allows water droplets to hang from taps, vessels to be filled slightly above the brim, and needles to float. It is the tensile force per unit length ($N/m$).

• Capillarity: The rise or fall of liquid in a narrow tube caused by surface tension. It depends on the balance of cohesion (liquid-liquid) and adhesion (liquid-solid).     * Adhesion > Cohesion: Liquid rises, forming a concave upward meniscus (e.g., water in glass).     * Cohesion > Adhesion: Liquid is depressed, forming a convex meniscus (e.g., mercury in glass).

Newtonian and Non-Newtonian Fluids

• Newtonian Fluid: There is a linear relationship between applied shear stress and the resulting rate of deformation. Examples: water, gases, kerosene, alcohol.

• Non-Newtonian Fluid: The relationship between shear stress and angular deformation is non-linear. Examples: thick long-chained hydrocarbons like Alkenes.

• Ideal Plastic: Has a definite yield stress and a constant linear relation between $\tau$ and $\frac{du}{dy}$ after that yield point.

• Thixotropic Substance: Viscosity depends on the prior angular deformation (e.g., printer's ink); it tends to "set" when at rest.

• Ideal Fluid: Resistance to shearing deformation is zero (plotting coincides with the x-axis).

• Ideal Solid/Elastic Solid: No deformation occurs regardless of loading (plotting coincides with the y-axis).

• Viscosity vs. Temperature:     * Gases: Viscosity increases as temperature increases due to molecular momentum transfer.     * Liquids: Viscosity decreases as temperature increases because cohesion (the predominant cause of viscosity in liquids) decreases.

Dimensions and Units

• Absolute (Dynamic) Viscosity ($\mu$):     * Dimensions: $FL^{-2}T$ or $ML^{-1}T^{-1}$.     * Units: Poise or Centipoise ($1\,\text{poise} = 1\,\text{dyne-sec/cm}^2$), $lb-sec/ft^2$, or $slug/ft \cdot sec$.

• Kinematic Viscosity ($\nu$):     * The ratio of dynamic viscosity to mass density: $\nu = \frac{\mu}{\rho}$.     * Dimensions: $L^2T^{-1}$.

• Dimensional Table:     * Area ($A$): $L^2$     * Volume ($V$): $L^3$     * Velocity ($V$): $LT^{-1}$     * Acceleration ($a, g$): $LT^{-2}$     * Angular Velocity ($\omega$): $T^{-1}$     * Force ($F$): $F$ or $MLT^{-2}$     * Mass ($M$): $FT^2L^{-1}$ or $M$     * Density ($\rho$): $FT^2L^{-4}$ or $ML^{-3}$     * Pressure ($P$): $FL^{-2}$ or $ML^{-1}T^{-2}$     * Kinematic Viscosity ($\nu$): $L^2T^{-1}$     * Power ($P$): $FLT^{-1}$ or $ML^2T^{-3}$

Hydrostatics

• Hydrostatics defined: The study of fluids at rest. The main characteristic is the normal force exerted on boundaries.

• Pressure ($P$): The normal force per unit area. $P = \frac{F}{A}$. Units: $N/m^2$ or bar ($1\,\text{bar} = 10^3\,N/m$ - Note: Common standard is $10^5$, but transcript states $10^3$).

• Pressure-Depth Relationship: Pressure increases with depth ($y$) below the free surface.     * Derivation: Downward weight of fluid column ($\rho gAy$) must equal the upward pressure force ($PA$).     * Basic hydrostatic equation: $P = \rho gy$.

• Pressure on Plane Surfaces:     * The total force $F$ on a submerged plane area $A$ is given by $F = \rho g \sin(\theta) A \bar{l}$, where $\bar{l}$ is the distance to the centroid.     * The distance to the point of action (Center of Pressure) from the origin is $l_p = \frac{I}{A\bar{l}}$, where $I$ is the second moment of area.

Buoyancy and Archimedes' Principle

• Buoyant Force ($F_B$): The resultant upward force exerted by a static fluid on a submerged or floating body. It equals the weight of the displaced fluid.     * $F_B = V\gamma = V\rho g$, where $V$ is the displaced volume.

• Buoyancy Laws:     1. A body immersed in a fluid is buoyed up by a force equal to the weight of the fluid displaced.     2. A floating body displaces its own weight of the fluid.     3. The buoyant force is vertical and acts through the centroid of the displaced volume (Center of Buoyancy).

• Hydrometer Principle: Used to determine specific gravities ($S$) by measuring the submerged volume.     * $\Delta h = \frac{V_0}{a} \frac{S-1}{S}$, where $a$ is the stem cross-section and $V_0$ is the original submerged volume in water.

Dimensional Analysis and Hydraulic Similitude

• Similitude: Dimensional analysis simplifies experiments by reducing variables and establishing design principles for models.

• Geometric Similitude: Constant ratio of all corresponding linear dimensions between model ($m$) and prototype ($p$).

• Dynamic Similitude: Exists if the ratios of homologous forces (viscous, pressure, gravity, etc.) are the same in model and prototype.

• Key Dimensionless Numbers:     * Euler Number (Inertial/Pressure): $\frac{\rho V^2}{P}$     * Reynolds Number (Inertial/Viscous): $\frac{\rho VL}{\mu}$     * Froude Number (Inertial/Gravity): $\frac{V}{\sqrt{Lg}}$     * Mach Number (Inertial/Elasticity): $\frac{V}{\sqrt{E/\rho}}$     * Weber Number (Inertial/Surface Tension): $\frac{\rho L V^2}{\sigma}$

Principles of Fluid Flow

• Flow Classifications:     1. Steady vs. Unsteady: Depends on whether parameters (velocity, pressure) vary with time.     2. Uniform vs. Non-uniform: Depends on whether parameters vary with distance along the path.

• Examples:     * Steady Uniform Flow: Constant discharge, constant cross-section (e.g., flow in a long straight uniform pipe).     * Steady Non-Uniform Flow: Constant discharge, varying cross-section (e.g., tapering pipe, river flow).     * Unsteady Uniform Flow: Constant cross-section, discharge varies with time (e.g., pressure surge).     * Unsteady Non-Uniform Flow: Both cross-section and discharge vary with time and distance (e.g., flood wave in a natural channel).

• Streamlines: Lines tangential to velocity vectors; there is no flow across a streamline.

• Streamtube: A bundle of streamlines forming an imaginary tube.

Fundamental Conservation Laws

1. Conservation of Mass (Continuity Equation): Matter is neither created nor destroyed. For incompressible steady flow:     * $Q_{in} = Q_{out}$     * $U_1 A_1 = U_2 A_2$

2. Conservation of Energy (Bernoulli Equation): Energy is conserved, though it can change forms (Potential, Kinetic, Pressure).     * Euler Equation: $\frac{1}{\rho} \frac{dp}{dL} + U \frac{du}{dL} + g \frac{dz}{dL} = 0$     * Bernoulli Equation: $\frac{P}{\rho g} + \frac{U^2}{2g} + z = \text{constant}$     * Terms: $\frac{P}{\rho g}$ (Pressure Head), $\frac{U^2}{2g}$ (Velocity Head), $z$ (Elevation Head).

3. Conservation of Momentum (Momentum Equation): Based on Newton's Second Law where force equals the rate of change of momentum.     * $F = ρQ(V_2 - V_1)$     * Includes pressure forces ($F_P$) and reaction forces ($F_R$).

Flow Measurement and Viscous Flow Applications

• Velocity Measurement:     * Pitot Tube: Measures stagnation pressure. By comparing stagnation and static pressure, velocity is found: $U = \sqrt{2gh}$.

• Discharge Measurement in Pipelines:     * Venturimeter: Consists of a narrowing throat where velocity increases and pressure drops. $Q_{ideal} = A_1 \frac{1}{\sqrt{(\frac{A_1}{A_2})^2 - 1}} \sqrt{2gh^*}$.     * Orifice Plate: Similar to venturimeter but simpler/cheaper; creates larger energy losses via turbulent eddies. $C_d ≈ 0.85$.     * Small Orifice Discharge: $U = \sqrt{2gh}$ (Torricelli theorem). Actual discharge incorporates contraction ($C_c$) and velocity ($C_v$) coefficients: $Q_{actual} = C_d A_0 \sqrt{2gh}$.

• Hagen-Poiseuille Law: Governs steady laminar flow in circular pipes.     * Shear stress distribution: $\tau = \frac{r \cdot dp}{2dx}$.     * Velocity profile for laminar flow is parabolic: $u = \frac{1}{4\mu} (\frac{-dp}{dx}) (R^2 - r^2)$.     * Discharge formula: $Q = \frac{\pi R^4 \Delta P}{8\mu L} = \frac{\pi d^4 \Delta P}{128\mu L}$.

• Reynolds Number Significance: Determines if flow is laminar or turbulent. High $Re$ means inertial forces dominate (turbulent); low $Re$ means viscous forces dominate (laminar).

Questions & Discussion

• Q1: What is a fluid? Distinguish Newtonian vs. Non-Newtonian. Explain Viscosity and Vapour Pressure.

• Q2: Define Buoyant Force and state Buoyancy Laws. Calculate drift of a hydrometer (0.002157 N weight, 2.794 mm diameter stem) between oil (S=0.780) and alcohol (S=0.821).

• Q3: Define Steady, Unsteady, Uniform, Non-Uniform, and Ideal flow. State/prove Bernoulli's theorem. Calculate venturimeter discharge with a mercury gauge reading 0.1778 m, pipe diam 0.1524 m, throat 0.0762 m, and $C_d = 0.97$.

• Q4: Show discharge for a venturimeter. Find horsepower for a water jet at 4 m/s with 10 cm diameter.

• Q5: Calculate flow rate for a venturimeter (oil S=0.9, 75 mm pipe, pressure diff 34.5 kN/m², area ratio 4, $C_d=0.97$).

• Q6: Derive Kinetic energy correction factor. Determine coefficients for a 25 mm nozzle discharging 0.76 m³/min at 60 m head (jet diam 22.5 mm).

• Q7: State Hagen-Poiseuille formula. Solve for flow rate, velocity, kinematic viscosity, and Reynolds number for oil (μ=8 poise, S=0.9) in a 50 mm pipe with a 2000 kN/m² drop over 100 m.

• Q8: Show from first principles that $Q = \frac{\pi R^4 dp}{8\mu dx}$.