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Fundamental concepts of fluid mechanics
1.
INTRODUCTION TO FLUID MECHANICS SCOPE OF FLUID MECHANICS Fluid mechanics is the study of fluids at rest or in motion. It has traditionally been applied in areas as the design of canal and dam systems; the design of pumps, compressors, and piping and ducting used in the water and air conditioning systems of homes and businesses, piping systems needed in chemical plants; the aerodynamics of automobiles and sub and supersonic airplanes; and the development of many different flow measurement devices such as gas pump meters. Many exciting areas have developed in the last quartercentury. Some examples include environmental and energy issues (e.g., containing oil slicks, largescale wind turbines, energy generation from ocean waves, the aerodynamics of large buildings, and the fluid mechanics of the atmosphere and ocean and of phenomena such as tornadoes, hurricanes, and tsunamis, biomechanics (e.g., artificial hearts and valves and other organs, sports, “smart fluids” (e.g., in automobile suspension systems to optimize motion under all terrain conditions, military uniforms containing a fluid layer that is “thin” until combat, when it can be “stiffened” to give the soldier strength and protection, and fluid lenses with humanlike properties for use in cameras and cell phones); and microfluids (e.g., for extremely precise administration of medications). DE FI NITION OF FLUID AND BASIC CONCEPTS A fluid is a substance that deforms continuously under the application of a shear (tangential) stress no matter how small the shear stress may be. Fluids tend to flow when we interact with them while solids tend to deform or bend. We can also define a fluid as any substance that cannot sustain a shear stress when at rest.
2.
Difference In Behavior Of Fluid And Solid A fluid in contact with a solid surface does not slip—it has the same velocity as that of surface because of the noslip condition, an experimental fact. The amount of deformation of the solid depends on the solid’s modulus of rigidity. The rate of deformation of the fluid depends on the fluid’s viscosity μ. We refer to solids as being elastic and fluids as being viscous. Substances which exhibits both springiness and friction; they are viscoelastic. Many biological tissues are viscoelastic. Basic Equations Analysis of any problem in fluid mechanics includes statement of the basic laws governing the fluid motion. The basic laws, which are applicable to any fluid, are 1. The conservation of mass 2. Newton’s second law of motion 3. The principle of angular momentum 4. The first law of thermodynamics 5. The second law of thermodynamics In many problems it is necessary to bring into the analysis additional relations that describe the behavior of physical properties of fluids under given conditions
3.
Differential Versus Integral Approach The basic laws that we apply in our study of fluid mechanics can be formulated in terms of infinitesimal or finite systems and control volumes. In the first case the resulting equations are differential equations; solution of the differential equations of motion provides a means of determining the detailed behavior of the flow. We often are interested in the gross behavior of a device; in such cases it is more appropriate to use integral formulations of the basic laws. Lagrangian Versus Eulerian
Approach Method of description that follows the particle is referred to as the Lagrangian method of description. In Lagrangian approach we analyze a fluid flow by assuming the fluid to be composed of a very large number of particles whose motion must be described. In control volume analyses, it is convenient to use the field, or Eulerian, method of description, which focuses attention on the properties of a flow at a given point in space as a function of time. In the Eulerian method of description, the properties of a flow field are described as functions of space coordinates and time. Fluid Properties Density is mass per unit volume. Density is measured simultaneously at an infinite number of points in the fluid, we would obtain an expression for the density distribution as a function of the space coordinates, at the given instant. The density at a point may also vary with time (as a result of work done on or by the fluid and/or heat transfer to the fluid). The complete representation of density (the field representation) is given by Specific gravity is density of a substance (solid or fluid) compared to an accepted reference value, typically the maximum density of water. Specific weight is defined as the weight of a substance per unit volume.
4.
Velocity Field Velocity is a vector quantity. The velocity vector can be written in terms of its three scalar components. The velocity vector, V indicates the velocity of a fluid particle that is passing through the point x, y, z at time instant t, in the Eulerian sense. The point x, y, z is not the ongoing position of an individual particle, but a point we choose to look at. Hence x, y, and z are independent variables. The term uniform flow field (as opposed to uniform flow at a cross section) is used to describe a flow in which the velocity is constant, i.e., independent of all space coordinates, throughout the entire flow field. In a flow that is uniform at a given cross section, the velocity is constant across any section normal to the flow, the velocity field is a function of x alone, and thus the flow model is onedimensional Steady flowIf properties at every point in a flow field do not change with time, the flow is termed steady Timelines, Pathlines, Streaklines, And Streamlines If a number of adjacent fluid particles in a flow field are marked at a given instant, they form a line in the fluid at that instant; this line is called a timeline. Timelines were introduced to demonstrate the deformation of a fluid at successive instants. Timelines are created by marking a line in a flow and watching how it evolves over time. A pathline is the path or trajectory traced out by a moving fluid particle. They show, over time, the paths individual particles take. If we focus our attention on a fixed location in space and identify, by the use of dye or smoke, all fluid particles passing through this point. After a short period of time we would have a number of identifiable fluid particles in the flow, all of
5.
which had, at some time, passed through one fixed location in space. The line joining these fluid particles is defined as a streakline. A streakline is the line produced in a flow when all particles moving through a fixed point are marked in some way (e.g., using smoke). Streamlines are lines drawn in the flow field so that at a given instant they are tangent to the direction of flow at every point in the flow field. Since the streamlines are tangent to the velocity vector at every point in the flow field, there can be no flow across a streamline. In steady flow, the velocity at each point in the flow field remains constant with time and, consequently, the streamline shapes do not vary from one instant to the next. This implies that a particle located on a given streamline will always move along the same streamline. Furthermore, consecutive particles passing through a fixed point in space will be on the same streamline and, subsequently, will remain on this streamline. Thus in a steady flow, pathlines, streaklines, and streamlines are identical lines in the flow field. For unsteady flow, streaklines, streamlines, and pathlines will in general have differing shapes. Stress Field Each fluid particle can experience: surface forces (pressure, friction) that are generated by contact with other particles or a solid surface; and body forces (such as gravity and electromagnetic) that are experienced throughout the particle. Surface forces on a fluid particle lead to stresses. The concept of stress is useful for describing how forces acting on the boundaries of a medium (fluid or solid) are transmitted throughout the medium. When a body moves through a fluid, stresses are developed within the fluid. The difference between a fluid and a solid is that stresses in a fluid are mostly generated by motion rather than by deflection.
6.
Viscosity And Fluid Types Newton’s law of viscosity is given for onedimensional flow as μ =absolute (or dynamic) viscosity Coefficient of dynamic viscosityIt is shear stress required to drag one layer of fluid with unit velocity over other layer unit distance apart. Coefficient of kinematic viscosity The ratio of absolute viscosity, μ, to mass density, ρ, is given the name kinematic viscosity and is represented by the symbol ν. Units of viscosity [1 poise =1 g/(cm .s)]; In the SI system the units of viscosity are kg/(m . s) In the Absolute Metric system of units, the unit for ν is a stoke (1 stoke = 1 cm ^2 /s). NonNewtonian Fluids Fluids in which shear stress is not directly proportional to deformation rate. (Apparent viscosity)
7.
The difference in viscosity and apparent viscosity is that while μ is constant (except for temperature effects), η depends on the shear rate. Pseudoplastic Fluids in which the apparent viscosity decreases with increasing deformation rate. (n <1) are called pseudoplastic (or shear thinning) fluids. Most nonNewtonian fluids fall into this group; examples include polymer solutions, colloidal suspensions, and paper pulp in water. Dilatant If the apparent viscosity increases with increasing deformation rate (n >1) the fluid is termed dilatant (or shear thickening). Suspensions of starch and of sand are examples of dilatant fluids. Bingham plastic A “fluid” that behaves as a solid until a minimum yield stress is exceeded and subsequently exhibits a linear relation between stress and rate of deformation is referred to as an ideal or Bingham plastic. Thixotropic fluidsshow a decrease in η with time under a constant applied shear stress; many paints are thixotropic. Rheopectic fluids show an increase in η with time. Viscoelastic After deformation some fluids partially return to their original shape when the applied stress is released; such fluids are called viscoelastic. Surface Tension Whenever a liquid is in contact with other liquids or gases, an interface develops that acts like a stretched elastic membrane, creating surface tension. We define a liquid as “wetting” a surface when the contact angle θ <90. Factors that affect the contact angle include the cleanliness of the surface and the purity of the liquid. A force balance on a segment of interface shows that there is a pressure jump across the imagined elastic membrane whenever the interface is curved.
8.
Capiliarity Surface tension also leads to the phenomena of capillary, most important effect of surface tension is the creation of a curved meniscus that appears in manometers or barometers, leading to a (usually unwanted) capillary rise (or depression).this is due to difference in adhesive and cohesive forces. Capillary rise is inversely proportional to the radius of the tube. Therefore, the thinner the tube is, the greater the rise (or fall) of the liquid in the tube. Manometer and barometer readings should be made at the level of the middle of the meniscus. This is away from the maximum effects of surface tension and thus nearest to the proper liquid level.
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This relation is also valid for nonwetting liquids. Simple analysis of the capillary effect and gives reasonable results only for tube diameters less than 0.1 in. (2.54 mm) experimental data for the capillary rise with a waterair interface are correlated by the empirical expression. Excess Pressure In Droplet And Bubble For a water droplet in air, pressure in the water is higher than ambient; the same is true for a gas bubble in liquid. For a soap bubble in air, surface tension acts on both inside and outside interfaces between the soap film and air along the curved bubble surface.
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FLUID FLOW Viscous And Inviscid Flows Consider a ball flying through the air, the ball will experiences the aerodynamic drag of the air .What is the nature of the drag force of the air on the ball? Maybe it’s due to friction of the air as it flows over the ball but air has such a low viscosity, friction might not contribute much to the drag, and the drag might be due to the pressure buildup in front of the ball as it pushes the air out of the way. We can predict ahead of time the relative importance of the viscous force, and force due to the pressure buildup in front of the ball, we can estimate whether or not viscous forces, as opposed to pressure forces, are negligible by simply computing the Reynolds number. Here ρ and μ are the fluid density and viscosity, respectively, and V and L are the typical or “characteristic” velocity and size scale of the flow (in this example the ball velocity and diameter). If the Reynolds number is “large,” viscous effects will be negligible (but will still have important consequences), if the Reynolds number is small, viscous effects will be dominant, if the Reynolds number is neither large nor small, no general conclusions can be drawn.
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In inviscid flow the streamlines are symmetric fronttoback. Because the mass flow between any two streamlines is constant, wherever streamlines open up, the velocity must decrease, and vice versa. Hence we can see that the velocity in the vicinity of points A and C must be relatively low; at point B it will be high. In fact, the air comes to rest at points A and C they are stagnation points. The pressure in this flow is high wherever the velocity is low, and vice versa hence, points A and C have relatively large (and equal) pressures; point B will be a point of low pressure. In fact, the pressure distribution on the sphere is symmetric fronttoback, and there is no net drag force due to pressure. Because we’re assuming inviscid flow, there can be no drag due to friction either. Hence we have d’Alembert’s paradox of 1752: The ball experiences no drag! The noslip condition requires that the velocity everywhere on the surface of the sphere be zero (in sphere coordinates), but inviscid theory states that it’s high at point B. Prandtl suggested that even though friction is negligible in general for high Reynolds number flows, there will always be a thin boundary layer, in which friction is significant and across the width of which the velocity increases rapidly from zero (at the surface) to the value inviscid flow theory predicts (on the outer edge of the boundary layer) this boundary layer has another important consequence: It often leads to bodies having a wake, as shown in Fig from point D onwards. Point D is a separation point, where fluid particles are pushed off the object and cause a wake to develop. Consider once again the original inviscid flow. As a particle moves along the surface from point B to C, it moves from low to high pressure. This adverse pressure gradient (a pressure change opposing fluid motion) causes the particles to slow down as they move along the rear of the sphere. If we now add to this the fact that the particles are moving in a boundary layer with friction that also slows down the fluid, the particles will eventually be brought to rest and then pushed off the sphere by the following particles, forming the wake. This is generally very bad news: It turns out that the wake will always be relatively low pressure, but the front of the sphere will still have relatively high pressure. Hence, the sphere will now have a quite large pressure drag (or form drag—so called because it’s due to the shape of the object) It’s interesting to note that although the boundary layer is necessary to explain the drag on the sphere, the drag is actually due mostly to the asymmetric pressure distribution created by the boundary layer separation drag directly due to friction is still negligible!
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How Streamlining Of A Body Works? The drag force in most aerodynamics is due to the lowpressure wake: If we can reduce or eliminate the wake, drag will be greatly reduced. Consider once again why the separation occurred. Boundary layer friction slowed down the particles, but so did the adverse pressure gradient. The pressure increased very rapidly across the back half of the sphere in because the streamlines opened up so rapidly. If we make the sphere teardrop shaped, the streamlines will open up gradually, and hence the pressure will increase slowly, to such an extent that fluid particles are not forced to separate from the object until they almost reach the end of the object, as shown. The wake is much smaller (and it turns out the pressure will not be as low as before), leading to much less pressure drag. The only negative aspect of this streamlining is that the total surface area on which friction occurs is larger, so drag due to friction will increase a little. Laminar And Turbulent Flows Consider water flowing out of pipe. At a very low flow rate the water will flow very smoothly—almost “glasslike.” If we increase the flow rate, the water will flow in a churnedup, chaotic manner. These are examples of how a viscous flow can be laminar or turbulent, respectively. A laminar flow is one in which the fluid particles move in smooth layers, or laminas; a turbulent flow is one in which the fluid particles rapidly mix as they move along due to random three dimensional velocity fluctuations.
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In a onedimensional laminar flow, the shear stress is related to the velocity gradient by the simple relation. For a turbulent flow in which the mean velocity field is onedimensional, no such simple relation is valid. Random 3d velocity fluctuations (u', v', and w') transport momentum across the mean flow streamlines, increasing the effective shear stress. Compressible And Incompressible Flows Flows in which variations in density are negligible are termed incompressible; when density variations within a flow are not negligible, the flow is called compressible. The most common example of compressible flow concerns the flow of gases, while the flow of liquids may frequently be treated as incompressible. For many liquids, density is only a weak function of temperature. At modest pressures, liquids may be considered incompressible. However, at high pressures compressibility effects in liquids can be important. Pressure and density changes in liquids are related by the bulk compressibility modulus, or modulus of elasticity If the bulk modulus is independent of temperature, then density is only a function of pressure the fluid is barotropic. Water hammer and cavitation are examples of the importance of compressibility effects in liquid flows. Water hammer is caused by acoustic waves propagating and reflecting in a confined liquid, for example, when a valve is closed abruptly. The resulting noise can be similar to “hammering” on the pipes, hence the term. Cavitation occurs when vapor pockets form in a liquid flow because of local reductions in pressure (for example at the tip of a boat’s propeller blades). Depending on the number and distribution of particles in the liquid to which very small pockets of undissolved gas or air may attach, the local pressure at the onset of cavitation may be at or below the vapor pressure of the liquid. These particles act as nucleation sites to initiate vaporization. Vapor pressure of a liquid is the partial pressure of the vapor in contact with the saturated liquid at a given temperature. When pressure in a liquid is reduced to less than the vapor pressure, the liquid may change phase suddenly and “flash” to vapor. The vapor pockets in a liquid flow may alter the geometry of the flow field substantially. When adjacent to a surface, the growth and collapse of vapor bubbles can cause serious
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damage by eroding the surface material. Very pure liquids can sustain large negative pressures—as much as 260 atmospheres for distilled water—before the liquid “ruptures” and vaporization occurs. Undissolved air is invariably present near the free surface of water or seawater, so cavitation occurs where the local total pressure is quite close to the vapor pressure. The ratio of the flow speed, V, to the local speed of sound, c, in the gas is defined as the Mach number, For M <0.3, the maximum density variation is less than 5 percent. Gas flows with M < 0.3 can be treated as incompressible; a value of M = 0.3 in air at standard conditions corresponds to a speed of approximately 100 m/s. Internal And External Flows Flows completely bounded by solid surfaces are called internal or duct flows. Flows over bodies immersed in an unbounded fluid are termed external flows. Both internal and external flows may be laminar or turbulent, compressible or incompressible. Flow will generally be laminar for Re <2300 and turbulent for larger values: Flow in a pipe of constant diameter will be entirely laminar or entirely turbulent, depending on the value of the velocity V. The internal flow of liquids in which the duct does not flow full—where there is a free surface subject to a constant pressure—is termed openchannel flow. Common examples of openchannel flow include flow in rivers, irrigation ditches, and aqueducts. Supersonic flows (M > 1) will behave very differently than subsonic flows (M < 1). For example, supersonic flows can experience oblique and normal shocks, and can also behave in a counterintuitive way—e.g., a supersonic nozzle (a device to accelerate a flow) must be divergent (i.e., it has increasing crosssectional area) in the direction of flow! We note here also that in a subsonic nozzle (which has a convergent crosssectional area), the pressure of the flow at the exit plane will always be the ambient pressure; for a sonic flow, the exit pressure can be higher than ambient; and for a supersonic flow the exit pressure can be greater than, equal to, or less than the ambient pressure!
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USEFUL EQUATIONS 1The variation with temperature of the viscosity of air is represented well by the empirical Sutherland correlation. 2The velocity distribution for laminar flow between parallel plates is given by
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