Fluid Mechanics Is the G.O.A.T. Subject | Introduction to Fluid Mechanics | Part 1

This is going to be one of the most comprehensive videos on fluid mechanics. The first question that comes to our mind is, what is a fluid? A fluid is anything that can flow, when you apply even a very small force to it. Fluids include both liquids like water, oil, or juice, and gases like air or helium in a balloon. Now let us discuss the nature of fluids. Note that, from now on, unless I mention otherwise, the type of fluid we will be studying will be an ideal fluid. An ideal fluid is a simplified model of a fluid that we imagine in order to understand fluid motion easily, even though such a fluid does not exist perfectly in real life. We assume that an ideal fluid has no viscosity, which means there is no internal friction, and the fluid can flow without any resistance. For example, water is considered a low viscosity fluid, or nearly non-viscous, when compared to thick liquids like honey or ketchup, whose viscosity is very high, because water flows easily and quickly, demonstrating low internal resistance to flow. We then assume that the ideal fluid is incompressible. Imagine you have 10 liters of water in a container, and you press it very hard from all sides. Even after applying this force, the water still occupies 10 liters, and does not shrink to 9 liters or 8 liters. So the idea that the volume remains the same, even when pressure is applied to it, is what we mean when we say a fluid is incompressible. Then the last assumption is that there is no energy loss during its motion. This means the total energy the fluid has at one point remains the same as it moves from one place to another. These assumptions may not be 100% true for real fluids like water or air, but they help us focus on the fundamental ideas of fluid flow, and make the analysis simple, clear, and easy to extend to real situations later. The assumptions we listed earlier describe the nature of the fluid itself, but they do not say anything about how the fluid actually moves. So we now make one final and very important assumption. The motion of the fluid is a streamline flow. Streamline flow is an assumption about the motion of the fluid, not about its material properties. It means that fluid particles move along smooth and well-defined paths, without crossing each other, mixing randomly, or showing chaotic motion. The flow pattern is stable and predictable. Also, at any specific point, the velocity of a fluid particle is always tangential to the streamline curve or path, and the velocity remains constant over time. In real situations, such orderly motion is usually observed when the fluid moves slowly, and its resistance is small, but this does not mean the fluid has to be ideal. Even non-ideal fluids can follow a streamline flow. Now, when studying fluids, we usually divide the subject into three types, fluid statics, fluid kinematics, and fluid dynamics. Statics comes from the word stationary, or standing still, so fluid statics studies fluids at rest, focusing on pressure or buoyancy, for example, the water sitting in a dam or a tank. Fluid kinematics describes how fluids move, its velocity, acceleration, and flow patterns, without including forces. Then fluid dynamics specifically studies forces, momentum, and energy using principles like Bernoulli's theorem, and at a deeper level, Navier-Stokes equations, for example, air moving over an airplane wing or water flowing through a pipe. Now you can also sometimes hear fluid mechanics. Mechanics is the broad study of how fluids behave, whether at rest or in motion, so fluid mechanics covers all of them. Awesome! With these ideas in mind, we are now ready to study pressure in fluid statics. When the fluid is static or not moving, they exert pressure on whatever touches them, like the walls of a container or an object inside them. We all know pressure is simply how much force is applied per unit area. Now in fluids, pressure behaves in a very special way. It always acts perpendicular to any surface it touches. This is a very important property. Whether the surface is horizontal, vertical, or slanted, the pressure force from the fluid always acts normal to that surface. This normal direction means at right angle to the surface. Let us now talk about how pressure exists at a point inside a fluid. Pick any tiny point inside a fluid that is completely at rest. Pressure at that point is the same in all directions. This property is called isotropy of pressure, which simply means pressure has the same value in all directions at a point. Imagine a very small solid cube being gently released inside a container filled with a fluid. The fluid is initially at rest. Because the cube is extremely small, all its faces are almost at the same depth inside the fluid. So, the pressure on all faces of the cube are equal in magnitude. On the top face of the cube, the fluid pressure acts perpendicular to the surface, pushing straight downward. On the bottom face of the cube, the fluid pressure also acts perpendicular to the surface, but it pushes straight upward. On the left face of the cube, the pressure force acts horizontally toward the right, directly normal to that face. On the right face, the pressure acts horizontally toward the left. Similarly, on the front and back faces, the pressure acts perpendicular to each face, pushing inward. Great. Now, we will move to one of the most important ideas, how pressure changes with depth. Imagine a cylindrical plastic water bottle like this. If you use a needle to poke five holes at the exact same height around the circumference of bottle, the water won't just leak out of one side. It will spray out of every hole with the exact same force. This shows that at a specific depth, the pressure is identical in every horizontal direction. But now, imagine you are inside a swimming pool. As you slowly go deeper into the pool, your ears start feeling heavy, blocked, or slightly painful. This is because the water outside your ears is pressing on your eardrums, and this pressure keeps increasing as you go deeper. This happens because pressure in a fluid increases with depth. The deeper you go, the more water lies above your head, and the weight of this water creates additional pressure on your eardrums. Near the surface, the pressure difference is small, so your ears feel normal. But as depth increases, the pressure rises, and your eardrums clearly sense this change. This increase in pressure with depth depends on three factors. First is the density of the fluid. Represented using rho sub w, a denser fluid has more mass packed into the same space, so its weight is greater. Because of this greater weight, a denser fluid creates more pressure at the same depth, compared to a lighter fluid. Second is gravity, represented by g. Third is the vertical depth below the surface, which is the height of the water column above the point, and is represented by h. As you go deeper in the pool, the value of h increases, so the pressure keeps increasing. Because of this, the extra pressure created by the water itself is equal to rho w times g times h. So if we have pressure p1 here and p2 here, then p2 minus p1 will be equal to this extra pressure delta p. This extra pressure delta p that arises purely because of the weight of a fluid at rest is called hydrostatic pressure. The word hydrostatic simply means a fluid that is not moving. A very important fact is that hydrostatic pressure depends only on the vertical depth and not on the shape of the container. Whether the pool is wide, narrow, circular, or irregular in shape, the pressure at the same depth in the same fluid is always the same everywhere. This is exactly why structures like dams are built much thicker at the bottom than at the top, because the pressure at greater depths is much higher. Now let us introduce atmospheric pressure, which is written like this. We live at the bottom of a huge ocean of air called the atmosphere. Air has weight, and because of that, it exerts pressure on everything around us. This pressure is called atmospheric pressure. At sea level, which means at the average height of the earth's surface, where the land meets the ocean, its value is about 101 kilopascal. By the way, Pa is the unit of pressure called pascal, which is nothing but newton per meter square. We usually do not feel it because our body is adapted to it, and the pressure inside our body balances it. Now using the same concept of hydrostatic pressure, tell me in the comments whether the atmospheric pressure decreases or increases as we go above the sea level. Based on atmospheric pressure, we define two important types of pressure. One is absolute pressure. Absolute pressure is the actual pressure at a point, measured from a perfect vacuum as a reference. Vacuum means complete absence of matter and pressure. So the pressure in a vacuum is zero, and therefore absolute pressure represents the true total pressure at that point. Atmospheric pressure, whose value is this, is an absolute pressure. This is because the atmospheric pressure is measured with respect to a perfect vacuum, as the reference, and not relative to any other surrounding pressure. The other type of pressure is gauge pressure. Gauge pressure is the pressure measured relative to atmospheric pressure. So if the absolute pressure at this point is p, then the gauge pressure at that point is p, minus the p atmospheric. Most instruments we use in daily life show gauge pressure, not absolute pressure. If gauge pressure is zero, it does not mean there is no pressure. It means the pressure is equal to atmospheric pressure. To measure pressure, we use devices like manometers and barometers. A manometer uses a liquid column to compare pressures. The difference in heights of the liquid columns tells us the pressure difference. Manometers primarily use mercury for high pressure measurements due to its high density, but water and alcohol are common for lower pressures because they offer greater sensitivity and visibility. Then a barometer is a special instrument used to measure atmospheric pressure. In a barometer, you fill a liquid inside a test tube up to the brim, say mercury, and then invert the tube carefully into a container filled with the same liquid such that no air is trapped inside it. After some time, you will notice that the mercury level inside the tube falls slightly and then becomes steady at a certain height, and this height is a direct measure of the atmospheric pressure acting on the liquid in the container. This height turns out to be 760 millimeters or 0.76 meters. The space above the mercury column inside the tube has almost no air in it, so a vacuum is created there. This vacuum is called the Torricelli vacuum, and it shows that the mercury column is supported only by atmospheric pressure. The density of mercury is 13,600 kilograms per meter cube. Use gravity as 9.8 and substitute all of them here to get difference in pressure as this. This point has same pressure as atmospheric pressure because there is no depth here, and as mentioned before, the pressure is the same everywhere horizontally. So, vacuum pressure plus this difference equals atmospheric pressure. Vacuum pressure is zero, and hence atmospheric pressure equals this. If atmospheric pressure increases, the liquid column rises higher, and if atmospheric pressure decreases, the liquid column falls. Now, the next important principle in fluid statics is Pascal's law. Pascal's law says that when pressure is applied to a confined fluid, that pressure is transmitted equally and undiminished in all directions throughout the fluid. This means, if you push a fluid at one place, the pressure increase is felt everywhere inside the fluid. This principle is used in hydraulic machines like hydraulic lifts and hydraulic brakes, where a small force applied at one place can produce a large force at another place. Consider a simple solved example. Imagine a completely closed container filled with oil, and inside it, there are two pistons, one small and one large. The small piston has an area of 10 square units, and the large piston has an area of 100 square units. Now, suppose you apply a force of 10 units on the small piston. This force creates a pressure inside the confined fluid, whose value will be 10 over 10, or one unit. According to Pascal's law, this same pressure is transmitted equally to every part of the fluid, and reaches the large piston without any reduction. When this pressure acts on the large piston, which has 10 times more area, it produces a force of one unit of pressure times 100 square units of area, or 100 units. So, a force of 10 units applied at one place results in a force of 100 units at another place. Isn't this amazing? Next up, we have buoyancy and Archimedes' principle that naturally come after understanding pressure in fluids. When an object is placed inside a fluid, the fluid exerts pressure on all sides of the object. Since pressure increases with depth, the pressure on the lower part of the object is more than the pressure on the upper part. Because of this difference in pressure, there is a net upward force acting on the object. This upward force exerted by a fluid on a body immersed in it is called buoyant force. Buoyant force always acts vertically upward, and it always exists whether the object sinks, floats, or remains fully submerged. Now the Archimedes' principle explains the magnitude of this buoyant force. It states that when a body is wholly or partially immersed in a fluid, it experiences an upward buoyant force equal to the weight of the fluid displaced by the body. So, buoyant force, Fb, is equal to density of the fluid, represented by rho sub w, multiplied by acceleration due to gravity, g, multiplied by volume of fluid, V, displaced by the object. Now, many of us get confused with this V. Is this volume of fluid, V, displaced by the object the same as the volume of the object? For that, look here. When there is no object inside the fluid, its volume is 100 liters, and there is no buoyant force acting on the object. Now, when I slowly submerge a cube, whose volume is 10 liters into the fluid, say 40 percent or 4 liters submerged inside like this, we have this red upward arrow representing buoyancy, which is 39.2 newtons right now. Also, the total volume of the fluid, plus only the submerged part, is therefore 104 liters. The density of this fluid is 1 kilogram per liter. So, substituting the values in this formula, we get this. Solving for V, we get V as 4 liters. This means the V in the buoyancy formula is not the total volume of the object, but only the volume of fluid displaced, which is exactly equal to the submerged volume of the object. Now, as the cube is pushed further down, a larger part gets submerged, more fluid is displaced, and the buoyant force increases. When the cube is completely submerged, then and only then, the displaced volume becomes equal to the total volume of the object, and at that point, buoyant force does not increase further. This leads us to the idea of floating and sinking. If the buoyant force acting on an object is less than the weight of the object, the object sinks. If the buoyant force is equal to the weight of the object, the object floats in equilibrium. If the buoyant force is greater than the weight of the object, the object rises upward until balance is achieved. With this, we end our journey through fluid statics. I am super tired right now. If this video gets 7,000 likes, then in the next video, we will move from fluid statics to fluid kinematics and dynamics, where fluids are in motion and cover topics like equations of continuity and Bernoulli's theorem. We will then study applications of Bernoulli's theorem, viscosity, and surface tension with their key ideas and real-life applications. So good!