Volume Flow Rate & Equation of Continuity

Fluids in Motion Volume Flow Rate & Equation of Continuity: Laminar Flow is when a fluid has no resistance and is not viscous, meaning it flows freely (low velocity or when fluid is very viscous) Equation of Continuity states that the same amount of volume coming into the pipe must leave the pipe as well → Aivit = Afvft (time can be taken out since it is the same on both sides of the equation) The volume per second coming in is the same as coming out which is Volume in per unit time (also known as flux) Eddies are the swirling of a fluid and the reverse current created when the fluid is in a turbulent flow regime Bernoulli’s Equation Derivation: You can derive this by starting with the idea of conservation of energy (Wi + PEi + KEi = Wf + PEf + KEf) Bernoulli’s Equation Pi + 𝜌ghi + 0.5𝜌vi2 = Pf + 𝜌ghf + 0.5𝜌vf2 Mass is not present because it can be canceled out on both sides since mass in is the same as mass out P = 0 if it is done in a vacuum (air above the fluid has nothing in it) - Viscosity & Poiseuille Flow: The viscous force (FV) only has to do with the area of the lid in contact with the fluid (kind of like friction but with objects being dragged on a liquid) and also the velocity by which you drag it along the fluid, inversely proportional to the depth of the fluid, and ɳ (coefficient of viscosity of the fluid in units of Pa*s → Ns/m2) Sometimes people is Poise which is equal to 1/10 Pa*s The coefficient of viscosity is very dependent on temperature Gases have viscosities even less than that of liquids ! FV = ɳ𝐴 " Newtonian fluids are those that have a constant coefficient of viscosity, regardless of the speed ! Poiseuille’s Law is # = 𝑄 = - $%&' 4 8ɳ( Δp refers to the difference in pressures at the start of the tube and at the end (pressure must be greater at start in order for it to flow) r is simply the radius L is the length of the tube Turbulence at High Velocities & Reynold’s Number: Once you get to a certain speed, fluids start exhibiting turbulent flow which is highly unpredictable and very difficult to describe Reynold’s Number gives you a way to predict what the critical speed is that you cross when the fluid begins to show turbulence -ɳ (𝑉)'*#*)+, = 2.') - The R in this case is Reynold’s Number NOT RADIUS and is a unitless, dimensional quantity Surface Tension & Adhesion: Floating only occurs when the object is LESS dense than the liquid If it does float when the object is MORE dense than the liquid, then surface tension is counteracting this action Water molecules have attractions to each other due to H bonding and this is known as cohesion (same molecules attract each other) Water molecules at the surface are much more able to form interaction because they don’t have any liquid above them to pull them in every single direction Adhesion is when one set of molecules are attracted to another set (such as water molecules attracted to the container) and can lead to the formation of a meniscus (which is the curved upper surface of a liquid in a tube) Capillary action describes the fact that if you place a tube into the surface of a liquid, the liquid will rise due to the adhesion forces between the tube and the liquid Venturi Effect & Pitot Tubes: The flow rate through every section of the pipe must remain constant ( - 𝑉1 𝑡1 = 𝑉2 𝑡2 ) This means, the velocity of the fluid will likely mean that the fluid flows faster through a constricted portion of tubing If the velocity of the liquid increases, then the pressure must decrease according to Bernoulli’s equation Bernoullie’s Principle states what was said above, that when the pressure decreases within a fluid, that the fluid therefore increases its speed Two Circulations in the Body: Red Blood Cells don’t have a nucleus, mitochondria, etc. and therefore don’t really use oxygen (just carry them to a location) The heart does use blood that it pumps out (one of the first vessels that come from the heart) The lungs also has its own blood vessels in order to get oxygen Arteries vs. Veins - What’s the Difference?: Arteries carry blood away from the heart and veins carry blood towards the heart Capillaries are the very small vessels where there are a lot of exchanges of gases between the vessels and their surroundings Pulmonary Artery carries deoxygenated blood to the lungs for refuel and returns it through the Pulmonary Veins Arteries have high pressure/low volume and therefore would spurt out a lot more blood compared to a low pressure/high volume vein Resistance in a Tube: Why is it more difficult to blow through a straw with a smaller radius (same length)? 8(ɳ Poiseuille’s Equation 2.0? → 𝑅𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒 (𝑅) = &' 4 Arterioles are covered in smooth muscle that can either be contracted (vaso-constriction) or relaxed (vaso-dilated) in order to generate different resistances based on what the body needs at the specific time - Putting it all Together: Pressure, Flow, & Resistance: 𝛥𝑃 = 𝑄𝑅 where Q = blood flow rate (volume/min) and R is resistance Another way to find Q is finding the stroke volume (which is the volume of blood per one beat) and multiple by the heart rate (beats per minute) and an example of this is shown to the right Power Output is simply Pressure multiplied by the flow rate (Power = PQ)