Chapter 7: Fluids and solids (C2971675)

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1 Fluids

Fluids continually deform. Deformation is the process of changing shape as a result of applied shear stress. Shear stress is a sliding force, which is parallel but not in the same line, thereby causing sliding. The ability to deform is a result of bonds that can be easily broken and reformed. Water only resists stress to a small extent, known as viscosity, a measurement of friction between fluid molecules. Viscosity in common language is known as “thickness”. For example, honey is more viscous than water. The viscosity is the shear modulus, which is shear stress rate divided by the shear strain rate [or in other words, the slope of the shear stress vs. shear strain graph] (see  for more information on shear modulus, stress and strain). Whereas drag related to the friction, fluid causes on an object; viscosity relates to the friction, fluid causes on itself. Evidently, if fluid has no viscosity, there is also no drag. Alternatively, it can be said that fluids can only resist forces normal to its surface, which from , was defined as perpendicular to its surface. Ability to resist normal forces explains why fluids do take on the shape of its container.

Density is mass per unit volume, and is defined as $\rho =\dfrac{m}{V}$, and hence has the SI units $kg/m^3$. Because density isn’t an intuitive measurement, specific gravity may be used instead, which is the ratio of the density of a substance, to the density of water (which has a specific gravity of 1), or alternatively, $SG=\dfrac{\rho}{\rho_{water}}$. This means a specific gravity of 1 has the same weight as water, if greater than 1 is heavier than water, or if less than 1 is lighter than water. The density of water is $1,000 kg/m^3$, or alternatively, $1g/cm^3$. The density of air is $1kg/m^3$.

Pressure is force of particle collisions per unit area, or alternatively, $p=\dfrac{F}{A}$. It has the SI unit Pascal ($Pa$), and is a scalar measurement. Other units of pressure include atmosphere ($atm$), Torr, and millimeter of mercury ($mmHg$). An atmosphere is the pressure at sea level. An atmosphere is equivalent to $760 mmHg=760 Torr=101,000 Pa$. A bar is equal to $100,000 Pa$, and for all practical purposes, replaces the atmosphere (which is slightly more). Liquid pressure is directly related to the depth of the liquid, because the increased weight of the fluid increases force. Or alternatively, since $F=mg$, in-substituting, $p=\dfrac{mg}{A}$. To show the heaviness of air, if lifting a utility hole that is $1m^2$, with air above it at a pressure of $1 atm=101,000 Pa$, the mass of the utility hole is $m=\dfrac{P.A}{g}=\dfrac{100,000\times 1}{10}=10,000 kg$. Because air molecules are both beneath and above the utility hole however, it is only until all the air from within the utility hole is sucked out, before this weight is actualized.

For fluids, since density is used instead of mass, the formula can be rewritten as $p=\rho gy$, where $y$ is the vertical displacement (depth) below the fluid to a chosen point. For example, the density of water at 100m, is $1,000\times 10\times 100=1,000,000 Pa\approx 10 atm$. Note that apart from volume, pressure is also unrelated to shape. This fluid pressure measurement is a gauge pressure, meaning it is absolute pressure, not including atmospheric pressure. Absolute pressure measures pressure relative to a vacuum of 0 Pa. Therefore, since there is an additional atmospheric pressure, the absolute pressure of water in the example, is in fact $10+1=11 atm$. Although absolute pressure cannot be negative, gauge pressure can be negative, as it means that the pressure is less than atmospheric pressure.

Pascal’s law is that pressure exerted anywhere in an enclosed fluid is transmitted undiminished to all points in the fluid. Hydraulic machinery is the use of fluid to do work. It involves the use of two pistons pushing against an enclosed fluid, such that when a force $F_{1}$ pushes against an area $A_{1}$, the change in pressure is $p=\dfrac{F}{A}=\dfrac{F_{1}}{A_{1}}$. Per Pascal’s law, since $p_{a}=p_{b}$, we thus know $\dfrac{F_{1}}{A_{1}}=\dfrac{F_{2}}{A_{2}}$. Remember from  that machines do not change work ($W=F\cdot d$), meaning $W_{1}=W_{2}$, or in-substituting, $F_{1}d_{1}=F_{2}d_{2}$.

A Greek king wanted to determine if a gold crown was pure gold, or if some silver has been substituted by the dishonest goldsmith. Archimedes was asked to determine this without damaging the crown. Noting that the water in the bathtub rose as he dipped in, he realized this effect could be used to determine the volume of the crown. As Archimedes knew the density of gold and the mass of the crown, using  he could figure out volume of displacement, supposing the crown was pure gold. Then figuring out actual volume of displacement, he submerged the crown in water, measuring how much water spilt. The volume of fluid displaced is equal to the volume of (the submerged portion of) a submerged object. Because other metals are not as dense as gold, if the volume of water spilt is less than the hypothetical volume of displacement, impure metals have been added. Note therefore that as an object is submerged, the (displaced) fluid pushes upward against the object, a force known as buoyancy. Archimedes principle is that buoyancy, is equal to the weight of the displaced fluid. The weight of the displaced fluid is , or since fluid is measured in density, , or in-substituting, , where  is the buoyancy force (per Archimedes principle, equal to the weight of the displaced fluid),  is the density of the displaced fluid,  is the volume of the displaced fluid, and  is the gravitational constant. Keep in mind that the weight of the displaced fluid (by Archimedes principle, equivalent to buoyancy) is not necessarily the same as the weight of the object. If the object weight is greater than buoyancy, it will sink. If the object weight is less than buoyancy, it will float. Keep in mind that , meaning the statements can be restated as, if the density of the object is greater than the density of the fluid, it will sink; and if the density of the object is less than the density of the fluid, it will float. Thus, if the ratio of the density of the object to the fluid is greater than 1, will sink; and if less than 1, will float. As the density of the object decreases, the volume of the fluid displaced decreases proportionally. Note from  that the volume of the submerged portion of an object is equal to the volume of the fluid displaced, meaning that the sentence can be restated as, as the density of the object decreases, the volume of the submerged portion of an object decreases proportionally. This means that the ratio of the density of an object to the fluid, is the ratio of the object that is submerged. From , since specific gravity is a density ratio to water, it is thus the proportion of an object that will be submerged if floated in water. For example, if the specific gravity of ice is 0.9, it means that 90% of an iceberg will be underwater.

An ideal fluid has non-viscous, non-compressible, non-turbulent and non-rotational flow. Turbulent flow is flow that is chaotic, and increases with increasing velocity. Laminar flow is flow at lower velocities, before the onset of turbulent flow. Rotational flow is the rotation of fluid particles, as it moves.

Ideal fluids obey the continuity equation, which are equations that describe the conservation of mass, in the case of a fluid, the conservation of fluid running through a tube, and is $Q=Av$, where $Q$ is the volume flow rate, $A$ is the cross-sectional area of flow, and $v$ is the fluid velocity. Note that velocity relates to uniform translational motion, and not random translational motion of the fluid. Random motion is the movement of particles suspended in a fluid. Uniform motion is the movement shared by all particles. Alternatively, since $\rho =\dfrac{m}{V}$, and $Q$ relates to volume, to find the mass equivalent, $\dfrac{m}{\rho}$ can in-substitute for the mass equivalent flow rate. So in-substituting, $\dfrac{m}{\rho}=Av$, and since it is mass equivalent flow rate that is required [and not mass per se], assigning mass flow rate as $I$, therefore, $\dfrac{I}{\rho}=Av$, or reshuffling, $I=\rho Av$, where $\rho$ is density. Note that by the equation $Q=Av$, that flow rate is constant (the same) throughout the entire fluid (i.e. it is not dependent on location). Evidently, as velocity relates to time, it, and therefore flow rate, can change over time. Another formula for volumetric flow rate is $\Delta P=QR$, which is the fluid equivalent of the voltage equation $V=IR$, where $\Delta P$ is the different in pressure between two points, $Q$ is the volume flow rate, and $R$ is resistance to flow, although it assumes a horizontal pipe, with constant cross-sectional area. This formula states that volumetric flow rate is directly proportional to the difference in pressure between the two ends of the pipe, and inversely proportional to the resistance to the flow. Another formula for volumetric flow rate is Poiseuille's law, which states that $\Delta P=\dfrac{8\mu LQ}{\pi r^4}$, where $r$ is the radius of the pipe, $\mu$ is the viscosity, and $L$ is the length of the pipe.

Ideal fluids also obey Bernoulli’s principle/equation, which is $\dfrac{1}{2}\rho v^2 + \rho gz + p = constant$, where $p$ is the pressure at a chosen point, and $z$ is the height of the chosen point above a reference plane (known as the elevation head), and $\dfrac{1}{2}\rho v^2$ is the velocity head. Note that whereas $=\rho gy$, instead of using $y$, that $z$ is used, because whereas $y$ increases in the downward direction, that $z$ increases in the upward direction. Note therefore that elevation head is not pressure. From Bernoulli’s principle, as the elevation head or velocity head increases, pressure decreases. Keeping in mind from  that pressure is particle collisions per unit area, it thus relates to random translational motion (also useful since pressure per se is present in Bernoulli’s principle). From , also note that the fluid velocity in the equation relates to uniform translational motion. Note therefore that from Bernoulli’s principle, random translational motion and uniform translational motion are inversely proportional to each other. , it will be explained that temperature depends on kinetic energy (which is caused by increased random translational motion), meaning that uniform translational motion is also inversely proportional with temperature. Since $\rho =\dfrac{m}{V}$, Bernoulli’s principle can be restated as $\dfrac{1}{2}mv^2.\dfrac{1}{V}+mgz.\dfrac{1}{V}+p=constant$. Also since $p=\rho gy$, $\dfrac{1}{2}mv^2.\dfrac{1}{V}+mgz.\dfrac{1}{V}+\rho gy=constant$, or in-substituting, $\dfrac{1}{2}mv^2.\dfrac{1}{V}+mgz.\dfrac{1}{V}+mgy.\dfrac{1}{V}=constant$. Note that $\dfrac{1}{2}mv^2$ is kinetic energy of molecules due to uniform translational motion, $mgz$ is the gravitational potential energy, and $mgy$ is the work done by molecules due to random translational motion (since this is determinant of pressure). Bernoulli’s equation is thus a conservation of energy formula. In fact, the modification of the uniform acceleration motion equation $v^2 = 2gh$ for circumstances where there is no initial velocity (meaning all potential energy is converted into kinetic energy), can also be derived from Bernoulli’s principle, known as Torricelli’s theorem. By the law of conservation of energy, as the pressures should be same before and after, $\dfrac{1}{2}\rho v_{1}^2 + \rho gz_{1} + p_{1} = \dfrac{1}{2}\rho v_{2}^2 + \rho gz_{2} + p_{2}$. Note that as water is has no initial velocity, $v_{1}=0$. Also, as $z$ is the height above the reference plane, $z_{2}=0$. Thus, the equation becomes $\rho gz_{1} + p_{1} = \dfrac{1}{2}\rho v_{2}^2 + p_{2}$. Since Pascal’s law states that pressure is transmitted undiminished, therefore $p_{1}=p_{2}$, meaning the equation becomes $\rho gz_{1}=\dfrac{1}{2}\rho v_{2}^2$. Thus, solving, $v_{2}^2 = 2gz_{1}$, quod erat demonstrandum.

Note that in the mass flow rate formula $Q=Av$, if a pipe has the cross-sectional area of a circle $A=\pi r^2$, the formula becomes $Q=\pi r^2 v$, meaning that radius and velocity are inversely related. This means that in ideal fluids, as radius is decreased, velocity increases. Real fluids are non-ideal fluids, which have reduced flow. Deviation from ideal behavior (reduction of flow) increases in pipes with smaller diameter.

Surface tension is the tendency of the surface of a liquid to contract, caused by intermolecular forces towards the center of the fluid. This force which causes like molecules to stick together is the cohesive force. Cohesive force is responsible for the spherical shape of liquid droplets. In contrast, adhesive force is the force which causes dissimilar molecules to stick together. Adhesive force is responsible for water droplets sticking on the walls of a glass tube. Surface tension causes meniscus, which is the curve of the liquid surface at points close to the container. Meniscus is concave for water in a test tube, because the cohesive forces (between water to each other) are stronger than the adhesive forces (between water and the container). Meniscus is convex for mercury in a test tube, because of the vice versa reason. Concave is a curve that curves inward, and convex is a curve that bulges outward. Capillary action is where the adhesive forces in a tube with sufficient small diameter, will cause the liquid to lift (against the action of gravitational force).

Gas are like liquids, but in contrast, expands to fill the entire volume available. The ability to expand is a result of very weak bonds, which are equivalent to being nonexistent. Compressibility is the ability to vary density. Gases are far more compressible than fluids.

 Formative learning activity Maps to RK7.1 What are fluids?

2 Solids

The atoms of solids are tightly bound to each other, and vibrate about fixed mean positions. Due to this rigidity, there is [permanent] resistance to deformation, as very strong bonds need to be broken before deformation will occur. Although rigid, solids have a small degree of elasticity. Elasticity is the property of materials to return to their original shape after deformation.

Stress is the force applied to a solid, $\rho=\dfrac{F}{A}$, where $f$ is average force, $A$ is area. Average force is the average of the forces not applied [and not the addition], such that a compressive force of $10N$ on either side has an average force of $10N$, not $20N$. This is analogous to tension, where either end of the rope had a force $\dfrac{1}{2}.F$. Stress has the units $N/m^2$. Note that stress has the same formula as pressure, so can be thought of as the solid version of pressure, and has thus also has the SI units Pascal. Strain (epsilon, $\epsilon$) describes the deformation, that has resulted due to the stress applied, and is defined as the change of a length, with respect to the original length. As strain is a ratio, it has no units. If stress is plotted against strain, the slope of the graph is known as the modulus of elasticity, or alternatively, $\lambda=\dfrac{stress}{strain}$. For any particular metal, the modulus of elasticity is constant. There are different types of moduli of elasticity, including:

• Young’s modulus, which describes tensile elasticity. As defined , tension is forces acting on the same line. In the case of tensile elasticity, this is deformation due to opposing forces acting along a single line. $A$ is the cross-sectional area of the solid, and strain is $\epsilon=\dfrac{\Delta L}{L_{0}}$, where $\Delta L$ is the change in length, and $L_{0}$ is the original length
• Shear modulus, which describes tendency to shear. As defined , shear is caused by two parallel forces that are not acting in the same line, thereby causing sliding. $A$ is the area on which the force acts, and shear strain is $\epsilon =\dfrac{\Delta x}{l}$, where $\Delta x$ is the [transverse] displacement caused due to the rectangle being slid into the shape of a rhomboid, and $l$ is the initial length

[img]shear-strain.png[/img]

• Bulk modulus, which describes volumetric elasticity, which is tendency of an object to resist deformation (from all directions) when uniformly compressed (from all directions). It is a 3D version of Young’s modulus. Volumetric stress is the change in pressure, and volumetric strain is $\epsilon=\dfrac{\Delta V}{V_{0}}$, where $\Delta V$ is the change in volume, and $V_{0}$ is the original volume. For example, the bulk modulus of diamond is much greater than the bulk modulus of air

Thermal expansion is the tendency of solids to expand volume when heated. The change in linear dimension is defined as $\dfrac{\Delta L}{L}=\alpha \Delta T$, where $\Delta L$ is change in length, $L$ is the original length, $\alpha$ is the linear expansion coefficient, and $\Delta T$ is the change in temperature. The change in volume is defined as $\dfrac{\Delta V}{V}=\beta \Delta T$, where $\beta$ is the coefficient of volume expansion, defined as $\beta =3\alpha$.

 Formative learning activity Maps to RK7.2 What are solids?

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