Fluid mechanics and my struggle to learn it

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About the Thumbnail: This is my solution sheet for determining pressure difference in a manometer example, exiting velocity for pipes, entrance region and fluid closest described in external flow given via reynolds number.

Preview: This is my personal experience. All my notes here are purely written which is also heavily advocated by the part-time instructor when taking reading assignments. exam is online and was open-book and resources, so we are tempted to check solutions online. :”)

OVERVIEW / RANT: One of the ongoing challenges faced by both educators and students is how to effectively teach and learn within the hybrid education setup—a blend of physical and online learning modalities. Personally, I find this arrangement… quite frustrating. It’s just incentivize both parties to be more and more dependent with online tools and it loses the effort of teaching AND studying. In a world that information is easily bought and retrieved but with questionable origins and reasoning, I make it a point to make my learnings strictly within the confines of what is being taught by instructor and without the help of others. (I choose to learn it the hard way, not sounding cringey pramis). nfortunately, there are moments when instructors may fail to empathize with students like me—those genuinely eager to engage critically with the lesson. A clear example of this was during my Fluid Mechanics lecture class in my final term as a major student, where this tension between curiosity and structure became especially apparent.

Below first I show what is being discussed via my handwritten notes that I have scanned :))

Overview and Resources, No Gatekeeping

As stated by many books and authorities about this topic, Fluid Mechanics is the branch of physics that studies the behavior of fluids — liquids and gases — at rest and in motion.

It forms the foundation for applications in mechanical, influencing everything from hydraulic systems and pumps to airplane aerodynamics and blood flow. Which is needed for my course on the mechanical side of any system.

At its core, fluid mechanics connects pressure, density, velocity, and viscosity through mathematical relationships derived from Newton’s laws and the conservation of mass and energy.

One of the godsend youtube series that I also highly recommend is by Enginerds that you can check here. Legit, I watched this before I sleep for the term.

To add also here is the fact that an OG book author for the fundamentals of fluid mechanics has made an entire YT series discussing his book and lessons here 🙏🙏

Of course if you are book-devoted, you need to have the solution manual of the textbook problems your are solving, you can check the solutions here in Litsolutions. But this may only help you or not given the expectations of the instructor and in my case, it definitely did not help me passing this course comfortably. Below are the lecture notes provided.

I give here also a brief rundown of what the contents are of our term discussions.

Hydrostatic Pressure

Hydrostatics deals with fluids at rest, where pressure varies only with depth.

The hydrostatic pressure at a given depth ( h ) is:

\[P = P_0 + \rho g h\]

Where:

• ( P ): absolute pressure at depth
• ( P_0 ): reference or surface pressure (often atmospheric)
• ( \rho ): fluid density (kg/m³)
• ( g ): acceleration due to gravity (9.81 m/s²)
• ( h ): depth below the free surface (m)

Pressure increases linearly with depth — an essential principle for submerged surfaces, dams, and fluid columns.

Hydrostatic Pressure in Layered Fluids

For fluids of different densities stacked in layers (e.g., oil over water), total pressure at the bottom is the sum of pressures from each layer:

\[P = P_0 + \rho_1 g h_1 + \rho_2 g h_2 + \rho_3 g h_3 + \dots\]

Each term corresponds to a fluid layer’s contribution based on its density and thickness.
This principle is critical in geotechnical engineering and fluid column experiments.

Manometers

A manometer is a U-shaped tube filled with fluid, used to measure pressure differences.

For a differential manometer between points A and B:

\[P_A - P_B = \rho g \Delta h\]

If multiple fluids are involved (e.g., mercury and water):

\[P_A - P_B = \rho_m g h_m - \rho_w g h_w\]

Where ( \rho_m ) and ( \rho_w ) are the densities of mercury and water, respectively.
This principle is widely used in laboratory pressure measurements and calibration.

Ideal Gas Law

For compressible gases, pressure, volume, and temperature relate through:

\[P V = n R T\]

or in mass-based form:

\[P = \rho R T\]

Where:

• ( P ): pressure (Pa)
• ( V ): volume (m³)
• ( n ): number of moles
• ( R ): universal gas constant (8.314 J/mol·K)
• ( T ): absolute temperature (K)
• ( \rho ): gas density (kg/m³)

The ideal gas law connects thermodynamics with fluid mechanics — useful for compressible flow and air system design.

Bernoulli’s Equation

Derived from the conservation of energy, Bernoulli’s Equation applies to steady, incompressible, and inviscid flow along a streamline:

\[P + \frac{1}{2}\rho v^2 + \rho g z = \text{constant}\]

Where:

• ( P ): pressure energy
• ( \frac{1}{2}\rho v^2 ): kinetic energy per unit volume
• ( \rho g z ): potential energy per unit volume

This principle forms the basis of Venturi meters, airfoils, pumps, and hydraulic design.

Energy Equation for Real Flow

In real (viscous) flow, head losses due to friction are considered:

\[\frac{P_1}{\rho g} + \frac{v_1^2}{2g} + z_1 = \frac{P_2}{\rho g} + \frac{v_2^2}{2g} + z_2 + h_L\]

Where ( h_L ) is the head loss caused by pipe friction, bends, fittings, or other resistances.

If a pump or turbine is included:

\[\frac{P_1}{\rho g} + \frac{v_1^2}{2g} + z_1 + h_P = \frac{P_2}{\rho g} + \frac{v_2^2}{2g} + z_2 + h_T + h_L\]

Where:

• ( h_P ): head added by pump
• ( h_T ): head extracted by turbine

Viscous Flows in Ducts

In real fluids, viscosity causes internal friction that resists motion.

Newton’s law of viscosity:

\[\tau = \mu \frac{du}{dy}\]

Where:

• ( \tau ): shear stress (Pa)
• ( \mu ): dynamic viscosity (Pa·s)
• ( \frac{du}{dy} ): velocity gradient perpendicular to flow

Viscosity defines whether a fluid behaves as ideal (no resistance) or real (resists deformation).

Laminar and Turbulent Flow

The flow regime depends on the Reynolds number (Re), a dimensionless parameter representing the ratio of inertial to viscous forces:

\[Re = \frac{\rho v D}{\mu}\]

Where:

• ( \rho ): fluid density (kg/m³)
• ( v ): mean velocity (m/s)
• ( D ): pipe diameter (m)
• ( \mu ): dynamic viscosity (Pa·s)

Flow types:

• Laminar: ( Re < 2000 ) — smooth, orderly layers (e.g., oil flow)
• Transitional: ( 2000 < Re < 4000 )
• Turbulent: ( Re > 4000 ) — chaotic, mixed motion (e.g., water pipelines)

Frictional Head Loss in Pipes

For steady, fully developed flow in a circular pipe, frictional head loss is given by the Darcy–Weisbach equation:

\[h_f = f \frac{L}{D} \frac{v^2}{2g}\]

Where:

• ( h_f ): head loss (m)
• ( f ): friction factor (dimensionless)
• ( L ): pipe length (m)
• ( D ): pipe diameter (m)
• ( v ): average velocity (m/s)

The friction factor depends on the flow regime:

• Laminar flow:
  \(f = \frac{64}{Re}\)
• Turbulent flow (smooth pipes, approximate):
  \(f = 0.3164 Re^{-0.25}\)

• Turbulent flow (rough pipes):
  use the Colebrook-White equation:

  \[\frac{1}{\sqrt{f}} = -2.0 \log_{10} \left( \frac{\varepsilon/D}{3.7} + \frac{2.51}{Re\sqrt{f}} \right)\]

Where ( \varepsilon ) is the pipe roughness (m).
This is solved iteratively or with the Moody diagram.

Below, I give a sample of the answer sheets but I dont have the given problem. Still this provides the overview of the thought process on how to solve such fluid mechanics problems.

REFLECTIONS / RANT: This class was already introduced to us way way back when we were first years (ENGPHYS) and on our 2nd year (MFTHSCI) and now in the most critical time of our study, they bring the theoreticals back to us as like a slap in the face since the information has been scratched. (Its like ENGMEC all over again!!). No excuses though, I was able to deliver the core competencies being asked from this course but it left a salty feeling in my mouth that I cannot process until I write it down here.

Admittedly, I cannot follow to the discussion and this left me not being confident to solve the problems without consultation with my notes. I was not also in a good place given thesis. No biggie, there are many resources out there. Expecting that I can catch up by watching YT videos of PH chem engineers (the resource i provided above!), I silently and barely pass each exam. Then a big bomb dropped on me. 50% of the grade was on the final exam. 💀💀 haha amazing. I was barely passing then my grade was dictated on one assessment and on my final term no less. Like the one who made the course felt like we can pass this easily on our final year. Ang ending I got tempted to check the solutions online on that final exam T_T. To just pass the exam, I relied on the solutions of others like diko alam ung subject matter :< My harmartia being allergic to cross reference my solution with the solution online and I felt like I sinned doing that (/hs).

I kept thinking if my degree had a PRC, was this a topic that we have to prepare? What is the point of teaching the topic and making us do reading assignments if the topic were discussed were it was not being followed… It felt wrong and of course, I see my coursemates acting like it was smooth sailing but on my end I was lowkey panicking. I dont follow others as much as possible since I really wanted to learn it on my own way Kahit diko magagamit in the future, it was a matter of principles for me (Of course, Im not perfect, Im not a latin honor standing so this is just a rant of a normal student)

Recently too, recruiters I got to discuss with asked about this exact thing of having a course without a PRC regulation joining their workforce. I brushed it off, saying we dont need it given our line of work, but this experience made me think heavily back how lax the teachings can be if there is no expectations that the education will not be part of a bigger picture. That bigger picture is the secret handshake of an ABET accredited program to stakeholders in the industry looking up to the college to be the stewards of excellence.

Of course, this is just an isolated case. This is my personal experience of what online exam did to corrupt the essence of learning which I cannot put a word to it but it is slowly creeping towards the mindset of both students and educators. Respect goes both ways and I respect the instructor and their experience (heck there are doctorates!) but I hope they will also respect the time and effort put into learning the subject too. I’ll probably look back to this soon to see if I was not alone in this reflection I have made! Yon lang

Let me know your thoughts on my reflections rants above/ :”> 😅😅

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PS. Nasusukot ako sa english ko, sinamahan ko na nang konting filipino sa reflection haha