I was always curious about the implication of math or specifically calculus in our real lives. People used to tell me that math was a great part of our lives, simple as to grocery shopping to complicated engineering equations. But how exactly is math really used in our daily lives? This type of question has been frequently asked within myself and I was determined to write something that could relate to many people as well as my own area of interest. I have to say that my interest was promoted as a kid. The suspenseful alcohol wipe. The initial squeeze. The slight pressure and the feeling as a slender metal rod injects mysterious, foreign liquids into your body. To my sister, this was D-Day; the Reckoning. To me, an exhilarating adventure. I stared down with awe at the long and sharp needle entering my body as it fed what seemed to be magic healing liquids to my 5-year-old self. I’m proud to say I accumulated some fame around the hospital as my doctor used to call me the ‘brave little girl’ because I never cried or complained about getting shots ever since I was a baby. The nurses even ended up giving me the privilege of sanitizing my own arm before a shot. These nurses were not a necessity for me because I would calmly clean the spot for the shot and patiently wait for the doctor to give me the medications. People said that I was brave, but that wasn’t really it. Rather, I didn’t cry because my curiosity trumped the pain. Growing up, these shots and needles rather developed my interest in the field of human biology. I was interested in exploring the medical field which derived me to choose a topic that correlates with the human body. Searching for possible topics intrigued me to write a paper that examines the relationship between calculus and the rate of the intravenous fluid entering in our human body. As I was looking around for an area of research, I discovered the Hagen Poiseuille’s Law. The equations involved with this law show the factors that influence the rate of flow in a tube. This investigation aims to find the factors that affect the rate by examining the Hagen Poiseuille’s Law and how this equation shows the the relationship between the resistance of a tube and its radius, length, and the fluid’s viscosity. This paper will also discuss how this relates to every practice embedded in the medical field. I also hope the readers will learn the significance usage of calculus in our daily lives.Background:Interestingly,  Gotthilf Heinrich Ludwig Hagen in 1839 and  Jean Léonard Marie Poiseuille in 1938 experimentally derived this equation independently. Hagen Poiseuille’s Law, also known as Hagen Poiseuille’s equation, provides the velocity of a flow through a pipe as the function of the radius, =f(r). This paper will utilize this basic equation to derive the Hagen Poiseuille’s Law. Assumptions: Before working with the Hagen Poiseuille’s equation, we must appreciate several assumptions when working with this equation:The liquid flows through only circular tubes. The fluid is laminar. Laminar means that the flow is in a parallel pathway with no disruptions.The fluid itself is a newtonian fluid. Newtonian fluids are defined as those forces that are proportional to the rates of change of the fluid’s velocity vector as one moves away from the point in question in various directions. For example, like water and air are types of newtonian fluid. Even though blood is not a newtonian liquid, its physical characteristics are sufficiently similar to water, so blood is assumed to be a newtonian liquid. Understanding Viscosity:One of the aims involves the relationship of the resistance of the tube and the fluid’s viscosity. So the question is, what is viscosity? First, imagine a tube filled with water moving from left to right. The individual molecules in water do not flow or move in a smooth way, instead, it varies intramolecular forces which causes some pulling and some pushing, resulting in the internal friction. Viscosity is a measure of a fluid’s resistance to flow and deformation by stress due to internal friction. It approximates the common notion of “thickness”. Viscosity is dependent on temperature but independent on the pressure. This is evident when we are cooking. We notice that the substance decreases in viscosity as it is heated longer. However, in some cases, such as sauces, gets thicker when it’s cooked longer due to the increase in molecular interchange and evaporation.  Derivation of Hagen Poiseuille’s Law:=velocity r= the distance from the center of the tube to the edge of the tube. Beginning with the equation, =f(r)=?, the diagram below (figure 1) shows how the velocity will be closer to zero as it gets closer to the edge and how the velocity gets faster as it moves to the center.Figure 1Figure 2 Since we know that Pressure=ForceArea, then the Force=PA . So what is the force that is driving the fluid into the tube? Figure 2  shows the basic diagram with pressure, radius and length labeled.on the tube. Looking at this figure, we can determine the driving force equation by using, Force=PA. Assuming that P1 has a higher pressure and the fluid flowing to the direction of the arrow, the pressure in this case would be P1-P2. Next, pressure is multiplied by the area of the cross section of the fluid, r2. We conclude that, FD=(P1-P2)r2. This equation is the force that is driving the fluid through the tube and we will call that FD.Now we have to figure out the retarding force which is what is holding the fluid traveling through this tube. This involves viscosity and how the size of the portion of the fluid that comes through that is being retarded by the friction. This is basically the friction force of viscosity of the fluid. The retarding force is given to us as, FR=-Advdr. This force keeps the fluid from flowing through the pipe. represents the coefficient of friction and area is the surface area of the fluid which is circumference of the cross section multiplied by the length of the tube FR=-2rLdvdr. Now, why is there a negative sign in front of the equation? This is because as the liquid goes further away from the center, the velocity decreases so it is a negative change in the velocity as the function of r. *An important thing to notice in these equations is during the constant travel of the fluid, FD=FR. If the driving force is larger than the retarding force, than the fluid is accelerating and vice versa. FD=FR(P1-P2)r2=-2rLdvdr. We are trying to find the velocity as a function of rso for somehow we have to solve this equation for velocity. Notice that there are things that we can cancel on both sides. After, we are going to isolate dvdron one side.(P1-P2)r2=-2rLdvdr(P1-P2)r=-2Ldvdr-dvdr=(P1-P2)r2LBefore we integrate both sides, we want the differential drto be on the right side of the equation.-dv=(P1-P2)rdr2LNow we will start integrating. When integrating, keep in mind that (P1-P2)2Lis a constant and to include the c(constant) after the integration. -dv=(P1-P2)2Lrdr-v=(P1-P2)2Lr22+cAfter the integration, we want to solve for the c,constant. When r, the radius of the liquid, is equal to R, the diameter of the tube, then the velocity of the liquid will equal to 0. Let’s try and plug this information into the equation.0=(P1-P2)4LR2+cc= -(P1-P2)4LR2 (we will add this equation back into -v=(P1-P2)2Lr22+c)-v=(P1-P2)4Lr2-(P1-P2)4LR2Now we can combine and simplify the equation.-=(P1-P2)4L(r2-R2)Since there is a negative sign, we will move it to the right side which results in flipping the R and r.=(P1-P2)4L(R2-r2)Notice that v is is the function of the r2which makes it a parabolic equation. However, (R2-r2)shows that it is an inverted parabola instead of a regular parabola that is opening up. Figure 3 below shows the graph of the fluid’s velocity in respect to the radius. Figure 3Poiseuille’s Law and the Discharge Rate:Since we understand the derivation of the Poiseuille’s Law, we can now apply this concept to calculate the discharge rate of the fluid coming through a pipe, understanding that the velocity of the fluid forms a parabolic shape. We are going to say the discharge rate by definition is the amount of volume per unit that comes through the pipe is equal to the cross section area times the velocity. We are going to express the function as Q. If Q=dVdt=Av, then dQ is the discharge rate of a “sleeve” of fluid coming through the pipe. The sleeve is the position of a distance, r away from the pipe and it has a thickness of dr.Therefore we can say that,dQ=vdA Since we know the v, we can plug it into the equation. And the A, area, of course would equal to the circumference times the thickness (dr).dQ=(P1-P2)4L(R2-r2)2rdrThis equation would be the discharge rate of the small little amount. To find the discharge rate of the entire pipe, we will integrate it this from r=0 to R.Q=0RdQ=(P1-P2)24L0RR2rdr-0Rr3drIntegrating this function gives us,Q=(P1-P2)2LR2r22-r440RQ=(P1-P2)2LR2R22-R44Q=(P1-P2)2LR44Q=(P1-P2)8LR4 This equation calculates the amount of fluid that will flow through the pipe depending upon, the radius of the pipe, the length of the pipe, coefficient of viscosity of the fluid, and the difference of the pressure between the segment of the pipe. Medical Application:Hagen-Poiseuille’s Law can be applied into the medical field when it comes to determining the rate of the IV, also known as a intravenous fluid. Even though this equation only refers to Newtonian fluids and straight tubes, this equation clearly depicts the relationship in blood vessels with radius and viscosity. You can apply this to determine which one is the most suitable IV.Example:A hypotensive patient requires immediate infusion of intravenous saline. If the goal is to infuse the saline as fast as possible, which is the preferred route of administration: a standard triple lumen central line (length 20 cm, radius of each lumen 0.84mm), or a standard 16 gauge peripheral IV (length 4 cm, radius 1.2 mm)? We can start by using the Hagen Poiseuille’s Law Q=(P1-P2)8LR4 (P1-P2)=8LQr4And given that the (P1-P2)=QR, we can derive the corollary to Poiseuille’s Law to solve this problem.R=8Lr4We can compare the resistance to the central line and the resistance to the IV. If the ratio is greater than 1, than the resistance to the central line is greater so we must use the IV and vise versa. RCLRIV=8LCLrCL48LIVrIV4Simplifying and plugging in the given values, we get, RCLRIV=LCLrCL4LIVrIV4=(20cm)(0.84mm)4(4cm)(1.2mm)421This means that a central line port has about 21x the resistance as a 16 gauge IV. IV fluids will infuse 21x as fast through the IV compared to a central line.Conclusion: Hagen Poiseuille’s equation shows and calculates the flow rate of the fluid depending on its radius, length, and viscosity. Surprisingly, this implications are often used by many doctors and scientists when deciding upon the best medications for their patients. I also think that this derived equation can be used in many other fields such as physics and biology. By taking my time to explore and research on this Hagen Poiseuille’s equation, I became more fond in the field of biology and how this correlated to the length and viscosity of blood veins. Researching a little more than the equation, I found something that was really intriguing in the field of medicine. I learned that this equation can be used to find the discharge rate and how much the patient has improved from the intravenous fluids. However, I think I could have made some changes and improvements to my paper. I think I could have included more statistics to this paper rather than just focusing on the calculus part of the derivation. Also, I could have made the mathematics part shorter and included the implications of this equation in the world and how it affects the lives of individuals today. Also, instead of using graphs and pictures from other research professors, I could have include more of my own graphs. Despite all the possible changes to this paper, this exploration paper has strong grounds to how the Poiseuille’s equation came to life and how it relates to the discussion of length, radius and viscosity. After writing this paper, I am more determined about what I want to be when I grow up. I would like to further my studies in this field of biology and math in the future. Overall, I really enjoyed writing this paper.

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