Introducción a la Mecánica de Fluidos

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3/3/2026 Prepared by: Dr. Vasco O. Duke-Walker 1 Universidad Tecnológica de Panamá Fluid Mechanics 1 Prepared by: Dr.Vasco O. Duke Walker Ph.D Prepared by: Dr. Vasco O. Duke-Walker Ph.D 5 Universidad Tecnológica de Panamá Course Policies – Engineering Course Attendance and Class Participation Communication Respect and Conduct Homework Policy Attendance is expected for all scheduled class sessions, as regular presence significantly supports comprehension and academic success. Attendance will be recorded during each session to ensure accountability, and students who are present will receive a corresponding attendance grade.

Students are required to come prepared to each session with the following materials: Course textbook Calculator Writing materials (notebook, pencil/pen) Active participation is strongly encouraged and is considered an essential component of a productive and engaging learning environment. All course-related communication will be conducted exclusively through the official UTP email account. Students are required to check their email daily for announcements, updates, and instructions. Phone calls and messaging applications (e.g., WhatsApp) are not authorized for course communication.

One student per class will be designated as a communication representative to support the dissemination of general announcements and to facilitate communication with the group in case of emergencies or special situations. Any work will be assigned and submitted through Microsoft Teams. Students are responsible for completing and submitting all assignments by the specified deadlines. A respectful and inclusive classroom environment is essential. Respect must be shown to the instructor, classmates, and yourself at all times. Disruptive behavior, including unauthorized cell phone use during class, will not be tolerated.

Phones should be silenced and stored away during instructional time unless explicitly allowed for a class activity. Language of Instruction The class will be conducted in Spanish to ensure clear understanding and participation. However, all instructional materials, including slides, handouts, assignments, and textbooks, will be shown in English. Two types of homework will be assigned in this course: non-graded homework and graded homework. All assignments will be managed and submitted through Microsoft Teams. Non-graded homework is mandatory and serves as evidence of continuous academic engagement.

These assignments are evaluated on a compliance basis: students who submit complete work by the specified deadline will receive full credit (100%), while failure to submit or submission of incomplete work will result in a non-compliance record (0%). Partial or insufficient submissions will not be accepted. Graded homework will be formally evaluated and must be submitted on the same day it is assigned, unless otherwise specified by the instructor. These assignments are expected to reflect individual effort, completeness, and adherence to academic standards. All submitted work must be original.

Any form of academic dishonesty, including copying or unauthorized collaboration, will result in a grade of zero (0%) and may lead to further disciplinary action in accordance with institutional regulations. Late submissions will not be accepted under any circumstances, except in cases of duly documented and justified causes approved in advance. Prepared by: Dr. Vasco O. Duke-Walker Ph.D 6 Universidad Tecnológica de Panamá Note-Taking Policy Class Recording Policy Recording of any kind is strictly prohibited during class sessions, including audio, video, or screen recording, without explicit authorization.

This policy applies to all recording devices, including smartphones, laptops, tablets, and wearable technology. Unauthorized recording is considered a violation of classroom conduct and may result in disciplinary action in accordance with university policy. Student Portfolio Requirement (Portafolio Estudiantil) Grading System Homework (graded and non- graded/compliance- based), laboratory work, and quizzes. 20 % Exams ( 1 – 3) 40 % Final Exam 40 % Grading Scale GRADE (GD): A - GD>91, B - 91>GD>81, C - 81> GD>71 D - 71> GD>61, F - 61> GD Students are responsible for taking accurate and complete notes during class.

Active note-taking is an essential component of the learning process and is expected of all students. Presentation slides may be provided at the instructor's discretion; however, students are expected to listen attentively and independently document key concepts, explanations, and examples discussed during lectures. The use of photographs (e.g., of slides or board work) is permitted as a supplementary resource. However, students must ensure that taking photographs does not interrupt the class, distract others, or replace active engagement and note-taking during the session.

Each student is required to compile and maintain a complete Student Portfolio (Portafolio Estudiantil) throughout the semester. The portfolio must include, at a minimum: Class notes Homework assignments Quizzes and exams Project documentation and reports It serves as an official record of academic progress delivered at the end of the semester. Compliance with this requirement may contribute up to five (5) additional points to the final course grade. It is used to support and clearly demonstrate the student's own work and problem-solving process.

All content must reflect the student's individual effort and understanding; the direct use or reproduction of instructor-provided materials as a substitute for original work is strictly prohibited. Failure to comply with this policy will result in the portfolio or corresponding component being dismissed and assigned a grade of zero (0%). Basically, no additional points. The standard grading scale for this course is based on letter grades: A, B, C, D, and F, in accordance with institutional norms.

Grades of I (Incomplete) and N (No Grade) will not be assigned under normal circumstances and will only be granted in exceptional cases, subject to the instructor's discretion and supported by proper documentation. If a student remains officially enrolled in the course, they will be subject to evaluation under the established grading criteria. Students who are unable to continue with the course have the option to formally withdraw in accordance with institutional procedures. Course Policies – Engineering Course 3/3/2026 Prepared by: Dr. Vasco O. Duke-Walker 8 Universidad Tecnológica de Panamá Introduction Prepared by: Dr. Vasco O.

Duke-Walker Ph.D 9 Universidad Tecnológica de Panamá What is a fluid? Liquid and gas Unlike a solid, cannot withstand a "infinitesimal" shear stress unless it is in motion. – Couette flow example τ = μ y Newton's law of viscosity Homework problem 1 What is the SI unit of viscosity? Derive from the definition. Sometimes, the distinction between solids and fluids may not be well defined. Substances like paints, jelly, pitch, putty, polymer solutions, biological substances (egg whites may simultaneously display both solid and fluid properties - viscoelastic- Prepared by: Dr. Vasco O.

Duke-Walker Ph.D 10 Universidad Tecnológica de Panamá Difference between Solid and Fluid Solid: Applied tangential force/area (or shear stress) τ produces a proportional deformation angle (or strain) θ. τ = Gθ The constant of proportionality G is called the elastic modulus, and has the units of force/area. Fluid: Applied shear stress τ produces a proportional continuously- increasing deformation (or strain rate) ሶ θ τ = μ ሶ θ The constant of proportionality μ is called the viscosity, and has the units of force×time/area. Prepared by: Dr. Vasco O. Duke-Walker Ph.D 11 Universidad Tecnológica de Panamá We know: Fluids cannot resist shear stress Ans. Can a fluid resist normal stress?

(a) Yes, but only liquids (b) Yes, but only gases (c) Yes, both liquids and gases (d) No. Fluids cannot resist normal stress - Normal stress in a static fluid is called pressure Concept of a Fluid Prepared by: Dr. Vasco O. Duke-Walker Ph.D 12 Universidad Tecnológica de Panamá Dimensions and Units In fluid mechanics we use four Primary Dimensions: { } has dimensions Mass {M} Length {L} Time {T} Temperature {Θ} A unit is way of attaching a number to the dimension, using an arbitrary measure Prepared by: Dr. Vasco O.

Duke-Walker Ph.D 13 Universidad Tecnológica de Panamá Secondary Dimensions The most important secondary dimension is Force {F} Using Newton's Second Law: F = ma {F} = {M L T 2 } International System (SI): 1 N = 1 kg (1 m/s2 ) British Gravitational System: 1 lb = 1 slug (1 ft/s2 ) Important! "slug" is the unit of mass (not lb or lbm) 1 slug= 14.59 kg Energy has units of Joules J N m A Newton is defined by F=ma: N kg m s2 J kg m s2 m Dimensions: ML2 T2 Thus, energy/work/heat have dimensions {M L2 T 2 } Why does energy have dimensions {M L2 T 2 }? will be used later in lecture 4 in Dimensional Analysis Prepared by: Dr. Vasco O.

Duke-Walker Ph.D 14 Universidad Tecnológica de Panamá Example: Dimensional Consistency The dynamic viscosity (μ) of an oil is calculated by measuring the terminal velocity (V) of small spheres falling under the action of gravity ( ) For very slow flow ("Stokes Flow"): Confirm the dimensional consistency of this equation Homework problem 2 Prepared by: Dr. Vasco O.

Duke-Walker Ph.D 15 Universidad Tecnológica de Panamá – Statistical method molecular dynamics laws of mechanics and probability theory Theory is well developed for light gases, but it is incomplete for polyatomic gas molecules – Continuum approach Assuming fluid consists of continuous matter – mean free path of the molecules be very small compared with the smallest physical length scale of flow field At each point of the continuous fluid, there is supposed to be a unique value of the velocity, pressure, density, and other so called field variables. The continuous matter is then required to obey the conservation laws of mass, momentum and energy. Background: Prepared by: Dr. Vasco O.

Duke-Walker Ph.D 16 Universidad Tecnológica de Panamá Validity of Continuum Assumption One cubic millimeter of air under standard atmospheric conditions contains about 2.7 x1016 molecules; mean free path 68nm. Knudsen number : ratio of the molecular mean free path to a representative physical length scale. ~ ( , ) Breakdown – a foot-long satellite in the upper atmosphere or "in" a shock At normal P&T, 2 μ cube Gas: 2x108 molecules Liquid: 2x1011 molecules Where: 1. n is the number of molecules per unit volume 2. L is the smallest significant length scale in the flow field, which is usually called the macroscopic length scale. 3.

ε- small volume Then, the above condition states that the continuum concept will certainly be valid if some volume ε that is much larger than the volume occupied by a single molecule of the fluid but much smaller than the cube of the smallest macroscopic length scale (such as cylinder diameter) can be found. Statistical and Continuum Methods Continuum Assumption A gas is approximated by a continuous substance, so that the effects of the individual particles are averaged together. = = Knudsen Flow Regime Description Perfect Gas?

Kn<0.01 Continuum flow Molecules collide often; fluid mechanics apply Yes, perfect gas model works well 0.01<Kn<0.10 Slip flow Somemolecular effects appear Ideal gas OK with corrections 0.1<Kn< 10 Transition flow (Rarefied Flow) Continuum breaks down Ideal gas may not bevalid Typical assumptions? Steady State – Not changes in time Reversible Adiabatic Average or Uniformed properties. Prepared by: Dr. Vasco O. Duke-Walker Ph.D 17 Universidad Tecnológica de Panamá Local field Property Individual molecule in small volume ΔV having mass Δm and velocity v The field variables such as the density ρ and the velocity vector u will in general be functions of the spatial coordinates and time.

In symbolic form, this is written as ρ = ρ(x, t) and u = u(x, t), where x is the position vector whose Cartesian coordinates are x, y, and z. Pressure Pressure is defined as the force/area acting normal to a surface. A solid surface doesn't actually have to be present. The pressure can be defined at any point , , in the fluid, if we assume that an infinitesimally small surface Δ could be placed there at whim, giving a resulting normal force Δ . p = lim A 0 Consider a fluid element (or tiny "blob" of fluid) as it moves along. As it passes some point B, its instantaneous velocity is defined as the velocity at point B.

at a point = velocity of fluid element as it passes that point This velocity is a vector, with three separate components, and will in general vary between di erent points and di erent times. Velocity , , , = , , , Ƹ + , , , Ƹ + ( , , , )෠ = 2 + 2 + 2 Speed is the magnitude of the velocity vector ρ = lim V ε Σ m V u = lim V ε Σ V m m Prepared by: Dr. Vasco O. Duke-Walker Ph.D 18 Universidad Tecnológica de Panamá Steady and Unsteady Flows If the flow is steady, then , , don't change in time for any point, and hence can be given as ( , , ), ( , , ), ( , , ). If the flow is unsteady, then these quantities do change in time at some or all points.

For a steady flow, we can define a streamline, which is the path followed by some chosen fluid element. As shows three particular streamlines. Prepared by: Dr. Vasco O. Duke-Walker Ph.D 19 Universidad Tecnológica de Panamá One of the most important concepts in fluid mechanics! At a solid surface, fluid "sticks" to the surface Experimentally observed fact: Fluid has the same velocity as a solid surface If the surface is stationary: u=0 at y=0 Thus, at a solid stationary surface: No-slip Condition Prepared by: Dr. Vasco O.

Duke-Walker Ph.D 20 Universidad Tecnológica de Panamá No-slip Condition The no-slip condition provides the boundary condition for mathematical solutions of fluid flow Every day consequences of the no-slip condition: - Difficult to getting the last of the ketchup out of the bottle. - Your car is still dirty after driving fast. Dust particles are in a low velocity region due to no-slip https://www.youtube.com/ watch?v=cUTkqZeiMow Prepared by: Dr. Vasco O.

Duke-Walker Ph.D 21 Universidad Tecnológica de Panamá Fluid Properties Density, ρ Mass per unit volume: units kg/m3, slugs/ft3 Liquids can be considered incompressible for most engineering applications Liquid density is independent of pressure: ρ f(p) Liquid density is a weak function of temperature (due to thermal expansion) Table A.1 Density of liquid water as a function of temperature Density of gases at high temperatures and low pressures (T>> Tcritical and P<<Pcritical ) can be calculated using the ideal gas equation: For example, the density of air at room conditions (p = 100 kPa and T = 20 C) is: where R is the gas constant For air: R=287 J/kgK ρ = p RT Prepared by: Dr. Vasco O.

Duke-Walker Ph.D 22 Universidad Tecnológica de Panamá Fluid Properties Specific Gravity, SG Ratio of the density to the density of water (at 4 ° ) SG is dimensionless, no units SG indicates if substance will sink or float on water. If SG is less than 1.0, it will float, e.g. most oils Olive oil on water Specific Weight, γ Weight per unit volume; units N/m3 , lb/ft3 Example: Liquid water at 20 ° : ρ = 998 kg m3 = 1.937 slug/ft3 γ = ρg γwater = 998 kg m3 9.81 m s2 = 9790 N/m3 γwater = 1.937 slug ft3 32.14 ft s2 = 62.3 lb/ft3 We will use these values to calculate hydrostatic forces. Prepared by: Dr. Vasco O.

Duke-Walker Ph.D 23 Universidad Tecnológica de Panamá Fluid Properties Dynamic Viscosity*, μ (Greek lower case mu) Units, kg/(m s) or slug/(ft s) Viscosity is the fluid's resistance to flow, i.e. the resistance to applied shear stress High viscosity fluids, e.g., honey, engine oil Low viscosity fluids, e.g. water, air Kinematic Viscosity, ν (Greek lower case nu) Units, m2/s or ft2/s ν = μ ρ Viscosity For most fluids, the shear stress (τ) is linearly proportional to the rate of shear strain of the fluid element Shear strain rate is equal to the local velocity gradient: Using arc length formula: δθδy = δuδt For most fluids the relationship is linear: Prepared by: Dr. Vasco O.

Duke-Walker Ph.D 24 Universidad Tecnológica de Panamá Fluid Properties Dynamic viscosity (μ) is the constant of proportionality Sometimes called "Newton's Law of Viscosity" Dynamic Viscosity, μ For common fluids (air water, oils) viscosity is a constant; independent of shear rate Viscosity (μ) is the slope Such fluids are called Newtonian fluids In this course, we will assume that fluids are Newtonian Prepared by: Dr. Vasco O.

Duke-Walker Ph.D 25 Universidad Tecnológica de Panamá Dynamic Viscosity, μ Very weak function of pressure Viscosity is a function of temperature: - viscosity of liquids decrease as temperature increases - viscosity of gases increase as temperature increases Example: The Viscous Shear Force A plate in a machine is lubricated by a film of SAE50W oil at 20 ° with thickness t=1.5 mm. The plate has length L=30 mm and depth w=130 mm (into the page). Calculate the shear force (F) required to slide the plate at a velocity of V=2.1 m/s. FIGURE 2–29 Prepared by: Dr. Vasco O. Duke-Walker Ph.D 26 Universidad Tecnológica de Panamá SOLUTION How to estimate the velocity gradient?

Dynamic viscosity of SAE 50W at 20°C Linear approximation Direction of the fluid shear stress on the plate? Acts like opposes motion Prepared by: Dr. Vasco O. Duke-Walker Ph.D 27 Universidad Tecnológica de Panamá Some fluids, viscosity depends on shear rate and/or time Viscosity is not constant Such fluids are called non-Newtonian fluids Examples: - Blood is shear thinning Corn Starch and Water (Dilantant) A Non-Newtonian fluid - Recommend trying this a home Non-Newtonian Fluids Prepared by: Dr. Vasco O.

Duke-Walker Ph.D 28 Universidad Tecnológica de Panamá In turbulent flow, fluid "particles" move in irregular paths The velocity at a fixed point varies randomly (Turbulent flow is a stochastic process; requiring statistical analysis) Enhances convective heat transfer, good mixing Turbulent flows are most common While there are some good engineering models, turbulence remains an unsolved problem in modern physics Laminar & Turbulent Flows In laminar flow the fluid "particles" move along smooth paths (from laminae, meaning thin layers) Velocity at a point is constant in steady flow Laminar flow occurs a low fluid velocities, over small objects or in highly viscous fluids Fluid viscosity dampens out the eddies (vortices) associated with turbulent flow https://www.youtube.com/watch?v=5YwnY0wPphA Prepared by: Dr.

Vasco O. Duke-Walker Ph.D 29 Universidad Tecnológica de Panamá The character of a flow, laminar or turbulent, depends mainly upon a dimensionless parameter called the Reynolds number (Re): (e.g. Re<2300 laminar pipe flow 4000>Re>2300 transition Re>4000 Turbulent ) Transition from Laminar to Turbulent Flow Large eddy simulation of a pitching airfoil undergoing deep dynamic stall Re = ρVD μ ρ fluid density μ fluid viscosity V fluid velocity D pipe diameter Prepared by: Dr. Vasco O. Duke-Walker Ph.D 30 Universidad Tecnológica de Panamá Compressibility?

Incompressible M < 0.3 Subsonic Regime M < 0.8 Subsonic Definition M <1.0 Transonic Regime 0.8<M <1.2 Where: = = = = Thermodynamically: Compressibility Factor Z If : = 1 the gas behaves ideal gas If < 1: the gas is more compressible than an ideal gas (dominant attractive forces) If > 1 : the gas is less compressible than an ideal gas (dominant repulsive forces) Supersonic Definition M>1.0 Supersonic Regime M>1.2 Hypersonic Definition M> 5 Define with Mach: = Which EOS (Equation of State) are used for real gases? EOS for ideal gases = = For ideal gases: = ; gamma = Mach number: https://www.youtube.com/watch?v=OwDiONfCt2s For more check, sonic boom Prepared by: Dr. Vasco O.

Duke-Walker Ph.D 31 Universidad Tecnológica de Panamá A liquid in an open container will evaporate Some molecules have enough momentum to overcome the intermolecular cohesion Evaporation rate increases as temperature increases more molecular kinetic energy With a lid, the molecules will build up in the vapor until the number of molecules entering and leaving the liquid surface are EQUAL An equilibrium is reached.

Mixture is saturated The pressure that builds up in the vapor is called the vapor pressure, Vapor pressure increases with temperature A liquid with a higher vapor pressure will evaporate at a higher rate Vapor Pressure (A measure of volatility) Another way to look at it: Vapor pressure is the pressure at which a liquid boils for a given temperature For example: At atmospheric pressure, = 101.3 Water boils at 100 ° As pressure decreases, water boils at a lower temperature. Water can boil at 0 ° near vacuum. Vapor Pressure of Water with Temperature Prepared by: Dr. Vasco O.

Duke-Walker Ph.D 32 Universidad Tecnológica de Panamá Surface Tension, ϒ (Greek letter Upsilon) Liquid molecules have cohesive forces. At an interface with a gas there appears to be an elastic membrane at the surface Units, N/m or lb/ft Modelled as a membrane, where ϒ is the force per unit length of the (fictitious) membrane Water-air interface, ϒ=0.073 N/m (at 20C): Enough tension to support an insect and small metal objects at a surface Surface tension is caused by the cohesive force between liquid molecules Molecules away from the interface are pulled uniformly in all direction At the interface the outward force is weak because The molecules in the gas are more dispersed Prepared by: Dr.

Vasco O. Duke-Walker Ph.D 33 Universidad Tecnológica de Panamá Surface Tension, ϒ Surface tension is a function of temperature Decreases with temperature Surface tension can be strongly affected by contaminants Detergents reduce the surface tension Surface tension is the cause of capillary action Wetting liquid/solid pair < 90 Liquid drawn up into a small tube Non-wetting liquid/solid pair > 90 Liquid drawn down into a small tube Capillary Effect Liquids are said to "find their own level" Not true for small diameter tubes For capillary effects to be negligible Tube diameter must be > 5mm (approx.) Capillary rise: h = 2σs ρgR cos ϕ (R = constant) Prepared by: Dr. Vasco O.

Duke-Walker Ph.D 35 Universidad Tecnológica de Panamá = ( ) EOS for ideal gases = For air: = Specific heat at constant Volume = (1) Internal energy: Enthalpy is defined as: = + = + = + (3) = Specific heat at constant Pressure Specific heat at constant Volume Specific heat at constant Pressure = . (2) Finally, = ( not chemically reacting) = = = = + Gibbs Relations Prepared by: Dr. Vasco O. Duke-Walker Ph.D 34 Universidad Tecnológica de Panamá Prepared by: Dr. Vasco O. Duke-Walker Ph.D 39 Universidad Tecnológica de Panamá Prepared by: Dr. Vasco O. Duke-Walker Ph.D 40 Universidad Tecnológica de Panamá Prepared by: Dr. Vasco O.

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