Fluids Biomechanics Lecture 5 PDF

Fluids Biomechanics A part of mechanics studying fluids in resting or equilibrium condition is called fluids statistics, while the part studying fluids in process of movement is called fluids dynamics. Hydrostatics Fluid pressure Inside the immobile fluid pressure force is acting, which is caused...

Fluids Biomechanics A part of mechanics studying fluids in resting or equilibrium condition is called fluids statistics, while the part studying fluids in process of movement is called fluids dynamics. Hydrostatics Fluid pressure Inside the immobile fluid pressure force is acting, which is caused only by its weight. Pressure (the symbol: P) is the force per unit area applied in a direction perpendicular to the surface of an object. Liquid pressure is the force per unit area of hard substance surface from fluid layer or fluid itself.  Pascal’s Law Pascal’s principle, also called Pascal’s law, in fluid (gas or liquid) mechanics, statement that, in a fluid at rest in a closed container, a pressure change in one part is transmitted without loss to every portion of the fluid and to the walls of the container. For instance, squeezing a toothpaste container at the bottom will cause the toothpaste to spill (come out) of the container from the top hole. Two principles of Pascal’s law: 1. Pressure exerted anywhere in a confined incompressible fluid is transmitted equally in all directions throughout the fluid 2. The pressure ratio (initial difference) remains the same in all points of a horizontal plane. Despite the shape of a container, pressure on a given horizonal plane is the same since height (h) is the same.  Hydrostatic pressure Pascal's law or the Principle of transmission of fluid-pressure states that "pressure exerted anywhere in a confined incompressible fluid is transmitted equally in all directions throughout the fluid”. -the pressure ratio (initial difference) remains same in all points of horizontal plane Human organism is not a confined bag of fluid. Hence, pressure applied somewhere is not equally transmitted to every part of the body, but when pressure is applied to a given component of a confined system of the body, pressure is transmitted to the system of the body equally. For instance, it is not recommended to wrap waist with tight clothes or squeeze it in a different way during pregnancy because extra pressure will affect the fetus as well.  Hydrostatic Pressure Hydrostatic pressure is the pressure exerted by a fluid at equilibrium at a given point within the fluid, due to the force of gravity. Hydrostatic pressure increases in proportion to depth measured from the surface because of the increasing weight of fluid exerting downward force from above. So, when you dive depper and deeper in water, hydrostatic pressure increases since the water height increases above your head.  Fluid hydrostatic pressure at different points Let's calculate what pressure creates a static fluid column with height h on the surface on which it acts. Suppose we have a cylindrical container, the bottom area of which is S and is filled by H height fluid, with density. Thus, h height fluid column (static!) causes a pressure, which is proportional to column height. It is clear, that at h depth of static water (lake, sea) pressure: Fluid hydrostatic pressure is the pressure of non-moving (static) fluid. It is determined by exertion of gravity forces on fluid (fluid mass).  hydraulic press Two cylinder tubes (arms) have different diameter and they are interconnected. They are filled with fluid. When compressor or pistons pushes against the smaller diameter tube, according to Pascal’s principle, pressure is distributed equally, so pressures must be equal in both tubes, but since those two tubes have different areas due to different diameter, force generated will also be different. Hence, we have gain in force because of larger area in the second, bigger diameter tube. Displacement distance of fluid in smaller diameter tube will be more than the displacement distance in the bigger diameter tube though. In the bigger diameter tube more force is generated and this principle is widely used to gain a mechanical force for various purposes, like lifting heavy objects like cars. U-pipe principle If the U- tube is filled with two different fluids with different densities ρ 1 and ρ 2 , according to the Pascal’s low we will receive following equation If ρ 1 = ρ 2, then h 1 = h 2, thereby in U-type tube homogenous fluids are on the same level. The On the basis of Pascal's law many of technical equipment is built. Let us get to know some of them.  U-tube Principle Hydrailic press obeys U-tube principle. U-tube principle is used in medicine too. For instance, gauging different liquid and gas pressures through manometer. A Manometer is a device to measure pressures. A common simple manometer consists of a U shaped tube of glass filled with some liquid. Typically the liquid is mercury because of its high density. One arm is open-ended and is in contact with air – atmospheric pressure, the second Manometer arm is closed and is in contact with the fluid to be measured. If the fluid’s pressure is more than atmospeheric pressure, then mercury level (height) will be increased in the second arm (that is in contact with air) and vice versa. If the fluid’s pressure equals to atmospheric pressure, then mercury’s level will be balanced in both arms. Sphygmomanometer is used to mesaure blood pressure. Archimedes law An object is immersed in a fluid is buoyed up by a force equal to the weight of the fluid displaced by the object Archimedes' principle states that the upward buoyant force that is exerted on a body immersed in a fluid, whether fully or partially submerged, is equal to the weight of the fluid that the body displaces and acts in the upward direction at the center of mass of the displaced  Archimede’s Príncipe  When object is too light, it will float on the surface of fluid (water, for instance) so that it will not displace any amount of water – membrane of water due to surface tension will not be breached and zero buoyant force will act on the object.  When object is less dense than water, it will float on the surface but it will be partially immersed – it will displace some amount of water.  When object’s density equals to the density of water, object and water will be in equilibrium – object will be inside the bulk of water. The position depends on many factors – when object is dropped in water, gravitational potential energy will be converted to kinetic energy and the object will accelerate due to gravity.Then it will slide throught the bulk of water. At the same time, water will exert resistance force on the moving object. If the object is spherical (for simplicity), this will be in accordance with Stockes’ law - an expression describing the resisting force on a particle moving through a viscous fluid and showing that a maximum velocity is reached in such cases, e.g. for an object falling under gravity through a fluid.  If object’s density is more than water’s density, it will rest at the bottom. Eventually, the object will stop at some point and it will have some position in the bulk of water. Though, this position will not be permanent since a lot of factors will come into play. If the object is microscopic, it will undergo even browninan motion. When we put the object manually in another spot of the water, it will still be in equilibrium with the water (at least temporarily).  Archimede’s Principle Archimede’s principle has several applications in medicine. For instance, in CNS. This principle is used to determine the density changes of cerebrospinal fluid and/or the spinal cord based on buoyancy. Breast asymmetry is also determined by Archimede’s principle based on buoyancy after breast immersion in liquid, which yields some important data regarding breast tissue density change due to some pathological conditions and diseases. Body location depends on Archimedes upward force value 1. ρ body < ρ fluid – body is floating on fluid surface; 2.ρ body = ρ fluid – body is in equilibrium condition in fluid; 3. ρ body > ρ f luid – body is immersed)  Surface Tension Surface tension is the tendency of fluid surfaces to shrink into the minimum surface area possible. Surface tension allows insects, usually denser than water, to float and slide on a water surface. Surface tension is typically measured in dynes/cm, the force in dynes required to break a film of length 1 cm. Surface tension formula is: σ = F/l σ – surface tension, F – force, l – length Otherwise stated, surface tension is the energy, or work, required to increase the surface area of a liquid due to intermolecular forces. Another formula of surface tension: σ = A/S σ – surface tension, A – work done, S – area The second formula is derived from the first one and vice versa given the fact that A=Fl Surface tension Surface tension, the force of contraction acting across a line of unit length on the surface Surface tension has the dimension of force per unit length or of energy per unit area. The two are equivalent - but when referring to energy per unit of area, it is used the term surface energy, which is a more general term in the sense that it applies also to solids and not just liquids. In materials science, surface tension is used for either surface stress or surface free energy At fluid static equilibrium surface free energy is minimal. In case of deficiency of external forces minimum energy matches with minimum area. Minimization of fluid surface area means minimization of its surface energy. Fluid surface is certainly smooth. Fluid surface tension value is dependent not only on properties of fluid, but on those of neighboring area.  Surface Tension Between water surface and air there is surface tension. Inside water, water molecules interact with one another and these interaction forces cancel out one another in all directions. Hence, zero resultant force acting on water molecules inside the bulk of water. On the surface on water, water mlecules interact both with water molecules below and air molecules above. Since water-air interaction force is less than water- water interaction, there is a resultant force acting downwards that pulls water molecules on the surface downwards. Since water is a poorly compressible fluid, even though water molecules on the surface try to get into the bulk of the water, they are unable to achieve the goal and on the surface we get a tenion created by water molecules – a thin membrane (imagine the surface of oceans).  Surface Tension Surface tension examples are: Water droplets are spherical due to surface tension between water and air. In sperical shape the system has the lowest free energy and the least contact area with environmental medium, and it is a favorable condition from thermodynamics perspective. Planets are spherical for this reason as well. Eventually, when water droplets are formed from tap, they will elongate due to gravity and they will be  On the boundary of two surfaces (liquid and solid) tangent angle θ is formed, that depends on interaction force between fluid molecules and molecules of fluid and solid material (container wall). Fluid surface is called a meniscus The meniscus is concaved, if interaction between molecules of fluid and solid material is stronger than interaction between fluid molecules (θ < 900). This kind of fluid is called a wating fluid. The meniscus is convex, if interaction between molecules of fluid and solid material is weaker, than interaction between fluid molecules (θ < 900). This type of fluid is called a non- wating fluid.  Wetting and non-wetting fluids Wetting fluids: 1. Meniscus is concave and liquid likes interacting with the walls of a capillary tube 2. Cos θ angle is less than 90 degrees. Non-wetting fluids: 3. Meniscus is convex and liquid tries to avoid contact with the walls of a capillary tube 4. Cos θ angle is more than 90 degrees. If meniscus is convex, (θ > 900), cosine value is negative, which means that fluid/solid material surface tension prevails (non-wetting fluid). Otherwise, (wetting fluid) the meniscus is concaved (θ < 900), cosine value is positive and fluid/solid material’s surface tension is very low Meniscus shape of the glass capillary with water (waring) (A) and mercury (non- wating) (B) liquid It should be noted, that in case of water/mercury contact, tangent angle equals 90°. If water-teflon is in contact, tangent angle equals 180°. In this last case, surface tension of fluid/solid body and fluid/gas equals each other. In the figure an in the table there are examples of fluid/solid body interaction.  Surface tension force is proportional to the length of surface limiting contour: F=σl σ is surface tension coefficient or it is simply called surface tension: This means, that surface tension is equivalent to energy per unit area of surface, that is, it defines fluid surface free energy. σ is dependent on temperature; by the increase in temperature it is decreased and at critical temperature σ = 0. In medicine in order to diagnose, determination of surface tension coefficient of various biological fluids is necessary. The simplest way to determine a coefficient, is drop detachment from capillary tube. The drop is detaching from capillary, when it’s waight equal to or greater than the surface tension force:  Surface Tension From a capillary tube droplet will be separated when the weight (mg) of the droplet is more than surface tension. Stalagmometer method is widely used in medicine to determine the surface tension of liquids including blood based on drop count. Insectes can move on the surface of the water without disruption a surface. The forme of their extrémités give them the possibility to distribute their mass equally (like in the case of skis), this ensures the formation of superficial tension force and minimum deformation of water surface. Superficial tension compensates for the force of Gravity and ensures that insects stay on the surface of the water  Capillary effect Capillary effect is related to fluid surface increase or decrease in thin open capillary in comparison with surrounding fluid level. Capillarity action is imployed in estimation of blood clotting time. In this method blood from pricked finger is filled in a free glass tube capillary by capillary action. Absorbent cotton or gauze used in surgical dressing also work on the principle capillarity. In addition, capillarity action is implode for a surgical drain. where h - is the height the liquid is lifted, ρ - is the density of the liquid, r - is the radius of the capillary, g - is the acceleration due to gravity, θ - is the angle of contact described above.  Capillary effect (action) Capillary action is the result of adhesion and surface tension. Adhesion of water to the walls of a vessel will cause an upward force on the liquid at the edges and result in a meniscus which turns upward. The surface tension acts to hold the surface intact, so instead of just the edges moving upward, the whole liquid surface is dragged upward. Capillary action occurs when the adhesion to the walls is stronger than the cohesive forces between the liquid molecules. The height to which capillary action will take water in a uniform circular tube is limited by surface tension. Acting around the circumference, the upward force is  Capillary Action Capillary effect has many applications in medicine like blood drain from a finger through a glass tube, blood clotting time determination etc. When liquid is wetting, height will be increased in a tube and the meniscus will be concave. When liquid is non-wetting, height will be decreased in a tube and the meniscus will be convex. Do not forget: height of liquid depends on the balance of two forces acting opposite each other – upward force determined by surface tension and Capillary effect depends not only on liquid type/nature, but also on radius of tube, and downward force due to the the substance of tube walls – due weight of the liquid. to different surface tensions Capillary action is the ability of a liquid to flow in narrow spaces without the assistance of, or even in opposition to, external forces like gravity. The effect can be seen in the drawing up of liquids between the hairs of a paint-brush, in a thin tube, in porous materials such as paper and plaster, in some non-porous materials such as sand and liquefied carbon fiber, or in a cell. It occurs because of intermolecular forces between the liquid and surrounding solid surfaces. If the diameter of the tube is sufficiently small, then the combination of surface tension (which is caused by cohesion within the liquid) and adhesive forces between the liquid and container wall act to lift the liquid.  Laplace’s Law Laplace studied water bubbles and came up with this law based on water bubbles, but since it turned out to be a universal law, it applies to many different settings including elastic materials like rubber. For simplicity we will discuss Laplace’s law on a baloon example. According to Laplace, under any non-flat (curved) surface there is an extra pressure acting opposite to surface tension. In this case tension acts downwards and extra pressure – upwards.  Laplace’s Law According to Pascal’s principle, when we inflate a baloon, pressure is distributed evenly throughout the walls of a baloon. When we increase the pressure inside a baloon, it is inflated and eventually becomes stable (neither inflating nor deflating after we stop pumping air into it). That does not mean we have atmospheric pressure inside the baloon too since inflated baloon is in equilibrium with the environment. It simply implies that inside we have atmospheric pressure+ecess/extra pressure. Extra pressure is balanced by the tension of baloon walls. Once we stick a needle or otherwise disrupt the wholeness of the baloon, tension will not be present any more and it can’t cancel out the extra pressure inside. Hence, due to pressure gradient (atmospheric pressure+extra pressure>atmospheric pressure) air will be driven out of the baloon to environment very quickly – bursting. According to Laplace, extra pressure (delta P) for spherical objects/parts equals 2T/R, where T is tension of baloon walls and R is a curvature radius of a corresponding part of the baloon. For cylindrycal components of the baloon, extra pressure formula is T/R. Laplace’s law Let’s discuss a thin liquid membrane, the thickness of which can be ignored. In order to minimize its free energy, the membrane creates pressure difference between its both surfaces. The membrane is compressed until pressure inside liquid drop (or gas bubble) is higher than atmospheric pressure by so-called additional pressure. At any point of surface, additional pressure is dependent on surface mean curvature (K) at this point and is determined by Laplace’s law where R1 and R2 are main curvature radii (perpendicularly directed). Under liquid bended surface the pressure equals: Where P0 is atmospheric pressure, σ – is fluid surface tension coefficient. Pressure increase, caused by surface curvature equals:  If meniscus is convex, ΔP > 0; if it is concaved, ΔP < 0. For spherical surface R1 = R2 = R and In case of long round cylinder, R1 = R (R is cylinder radius), R2 = ∞ - cylinder surface curvature radius in longitudinal direction of axis and  Laplace’s Law According to the formulas given above, when curvature radius is more at a given tension, extra pressure will be less (inversely related). But this is against pascal’s law – all kind of pressures must be equal throughout the walls of the baloon. Hence, where we have more radius, tension must be more too to maintain the same delta P value throughout the walls. Indeed, when you press on the surface of a cylindrical part of the baloon where curvature radius is more, you will feel more tension of walls. Please remember: curvature radius is not the same as geometrical radius. In this case they just coincide. Curvature radius is inversely related to curvature. According to Laplace’s law, it is clear, that the artery walls with large radius, could be thicker, (have fibril belts), than those of thin arteries. Thin capillaries due to their small sizes (radius) are subject to much less tension, therefore their walls might be thin. That is, physiologically, vessels thickness is decreased disproportionately to change in radius. Thus, at hypertension or aneurism, vein could be damaged when vessel radius is increased even slightly. Certainly, in this case, FP = Prl > FW =σld and wall structure will be failed.  Laplace’s Law Laplace’s law has many applications in medicine. During dilatative cardiomyopathy, spherical heart becomes flattened – more cylindrical. That means the curvature of heart is decreased, but curvature radius in increased. Hence,according to Laplace’s formula, tension must also be increased to maintain the normal pressure of heart to let it function more or less normally. Tension increase means that heart walls become thinner (like stretching bubble gums). Thinned walls of heart is a huge risk for even normal pressure of heart to cause malfunctioning in pumping blood effectively. When there is aneurysm of aorta, the same thing happens – aortic walls flatten due to bulging and curvature radius in increased, aortic walls become thinner and tension is increased. In this condition even normal blood pressure can cause the rupture of aorta – life-threatening emergency condition. Laplace’s Law Gas embolism is a dangerous condition because when gas bubbles are stuck in blood vessels, they are not ruptured by pressing of blood and blood circulation is blocked. The reason why gas bubbles sometimes are not collapsed is Laplace’s law and extra pressure inside gas bubbles due to curved surface – they withstand the pressure exerted from blood and they are not collapsed. Pressure inside the sphere Water is said to "boil" when bubbles of water vapor grow without bound, bursting at the surface. For a vapor bubble to expand, the temperature must be high enough that the vapor pressure exceeds the ambient pressure – the atmospheric pressure, primarily. Below that temperature, a water vapor bubble will shrink and vanish. Superheating is an exception to this simple rule: a liquid is sometimes observed not to boil even though its vapor pressure does exceed the ambient pressure. The cause is an additional force, the surface tension, which suppresses the growth of bubbles. Surface tension makes the bubble act a bit like a rubber balloon. The pressure inside is raised slightly by the "skin" attempting to contract. For the bubble to expand — to boil — the temperature must be raised slightly above the boiling point to generate enough vapor pressure. Pressure inside the sphere What makes superheating so explosive is that a larger bubble is easier to inflate than a small one, just as when blowing up a balloon, the hardest part is getting it started. It turns out the excess pressure due to surface tension is inversely proportional to the diameter of the bubble. This means if the largest bubbles in a container are only a few microns in diameter, overcoming the