COURSE NUMBER: ME 321 Fluid Mechanics I. Fluid: Concept and Properties

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1 COURSE NUMBER: ME 321 Fluid Mechanics I Fluid: Concept and Properties Course teacher Dr. M. Mahbubur Razzaque Professor Department of Mechanical Engineering BUET 1

2 What is Fluid Mechanics? Fluid mechanics is a branch of mechanics. It is the study of fluids either in motion (fluid dynamics) or at rest (fluid statics) and the subsequent effects of the fluid upon the boundaries, which may be either solid surfaces or interfaces with other fluids. Both gases and liquids are classified as fluids, and the number of fluids engineering applications is enormous: breathing, blood flow, swimming, pumps, fans, turbines, airplanes, ships, rivers, windmills, pipes, missiles, icebergs, engines, filters, jets, and sprinklers, etc. Everything on this planet either is a fluid or moves within or near a fluid. 2

3 What is Fluid? From the point of view of fluid mechanics, all matter consists of only two states, fluid and solid. The technical distinction lies with the reaction of the two to an applied shear or tangential stress. A solid can resist a shear stress by a static deformation; a fluid cannot. Any shear stress applied to a fluid, no matter how small, will result in motion of that fluid. The fluid moves and deforms continuously as long as the shear stress is applied. As a corollary, we can say that a fluid at rest must be in a state of zero shear stress, a state often called the hydrostatic stress condition in analysis. 3

4 What is Fluid? There are two classes of fluids: liquids and gases. The distinction technically lies in the effect of cohesive forces. A liquid, being composed of relatively close-packed molecules with strong cohesive forces, tends to retain its volume and will form a free surface in a gravitational field if unconfined from above. Gas molecules are widely spaced with negligible cohesive force, a gas is free to expand until it encounters confining walls. A gas has no definite volume and when left to itself without confinement, a gas forms an atmosphere which is essentially hydrostatic. Gases cannot form a free surface. 4

5 What is Fluid? 5

6 Model of Fluids Fluids are aggregations of molecules, widely spaced for a gas, closely spaced for a liquid. The distance between molecules is very large compared with the molecular diameter. The molecules are not fixed in a lattice but move about freely relative to each other. Thus fluid density or mass per unit volume, has no precise meaning because the number of molecules occupying a given volume continually changes. This effect becomes unimportant if the unit volume is large compared with the molecular spacing, when the number of molecules within the volume will remain nearly constant in spite of the enormous interchange of particles across the boundaries. If, however, the chosen unit volume is too large, there could be a noticeable variation in the bulk aggregation of the particles. 6

7 Fig. The limit definition of fluid density: (a) an elemental volume in a fluid region of variable density; (b) calculated density versus size of the elemental volume. The limiting volume δυ* is about 10-9 mm 3 for all liquids and for gases at atmospheric pressure mm 3 of air at standard conditions contains approximately 3x10 7 molecules, which is sufficient to define a nearly constant density. 7

8 Fluid as a Continuum Most engineering problems are concerned with physical dimensions much larger than this limiting volume, so that fluid density is essentially a point function and fluid properties can be thought of as varying continually in space. Such a substance is called a continuum, which simply is a mathematical idealization of fluids. Although any matter is composed of several molecules, the concept of continuum assumes a continuous distribution of mass within the matter or system with no empty space and the properties of the matter are continuous functions of space variables. It means that the variation in properties is so smooth that the differential calculus can be used to analyze the substance. This approximation is invalid for low pressure gases where the molecular spacing and mean free path are comparable to, or larger than, the physical size of the system. Classical fluid mechanics is not applicable in such cases. 8

9 Density and Specific Weight Fluid Properties Fluid density is defined as mass per unit volume. The units of density are Kg/m 3 or slug/ft 3. r = m/v A fluid property directly related to density is the specific weight. Specific weight is defined as the weight per unit volume. g = W/V = mg/v = (m/v)g = rg Where g is the local gravitational acceleration. The units of specific weight are N/m 3 or lb/ft 3. 9

10 Fluid Properties Specific gravity The specific gravity is used to determine the specific weight or density of a fluid (usually a liquid). It is defined as the ratio of the density of a substance to that of water at a reference temperature of 4 o C. For example, the specific gravity of mercury is 13.6, a dimensionless number; means the mass of mercury is 13.6 times that of water for the same volume. 10

11 Fluid Properties The density, specific weight and specific gravity of air and water at standard conditions are given in Table

12 12

13 Fluid Properties Viscosity Viscosity is the most important fluid property in the study of fluid flows. -It can be thought of as the internal stickiness of a fluid. -It is one of the properties that controls the fluid flow rate in a pipeline. -It accounts for the energy losses associated with the transport of fluids in ducts, channels, and pipes. -It plays a primary role in the generation of turbulence. -The rate of deformation of a fluid is directly linked to the viscosity of the fluid. For a given stress, a highly viscous fluid deforms at a slower rate rhan d fluid with a low viscosity. 13

14 Viscosity Consider a flow in which the fluid particles move in the x-direction at different speeds, so that particle velocities u vary with the y- coordinate. The Figure shows two particle positions at different times. For such a simple flow field, in which u = u(y), we can define the viscosity m of the fluid by the relationship where t is the shear stress and u is the velocity in the x-direction. The units of are N/m 2 or Pa, and of m are N.s/m 2. The quantity du/dy is a velocity gradient and can be interpreted as a strain rate. This equation is known as the Newton s Law of Viscosity. 14

15 Example Consider a fluid within the small gap between two concentric cylinders. A torque is necessary to rotate the inner cylinder at constant speed while the outer cylinder remains stationary. This resistance to the rotation of the cylinder is due to viscosity. The shear tress that resists the applied torque for this simple flow depends directly on the velocity gradient in the fluid film in the gap between the cylinders, i.e. 15

16 Example For a small gap h<<r, this gradient can be approximated by assuming a linear velocity distribution in the gap. Thus using the Newton s Law of viscosity, the shear stress on the surface of the inner cylinder may be written as 16

17 Example We can then relate the applied torque T to the viscosity and other parameters by the equation Here the shearing stress acting on the ends of the cylinder is neglected; L represents the length of the rotating cylinder. Note that the torque depends directly on the viscosity, thus the cylinders could be used as a viscometer, a device that measures the viscosity of a fluid. 17

18 Fluids which follow the linear pattern of the Newton s law of viscosity are called Newtonian fluids. There are many non- Newtonian fluids and they are treated in rheology. Figure: Rheological behavior of various materials Stress vs. Strain The figure compares four examples of non- Newtonian fluids with the behaviour of Newtonian fluids. 18

19 A dilatant (or shear-thickening) fluid increases resistance with increasing applied stress. A pseudoplastic (or shearthinning) fluid decreases resistance with increasing stress. If the thinning effect is very strong (the dashed curve) the fluid is termed plastic. The limiting case of a plastic substance is one which requires a finite yield stress before it begins to flow. The linear-flow Bingham plastic idealization is shown. The flow behavior after yield may also be nonlinear. An ex ample of a yielding fluid is toothpaste, which will not flow out of the tube until a finite stress is applied by squeezing. 19

20 A further complication of non-newtonian behavior is the transient effect shown in the following figure. Figure: Rheological behavior of various materials Effect of time on applied stress Some fluids require a gradually increasing shear stress to maintain a constant strain rate and are called rheopectic. The opposite case of a fluid which thins out with time and requires decreasing stress is termed thixotropic. 20

21 Example: A 60-cm-wide belt moves as shown. Calculate the horsepower requirement assuming a linear velocity profile in the 10 o C water. Sol: du/dy = 10*1000/2 = 5000 s -1 m for 10 o C water = x 10-3 N.s/m 2 t = m. du/dy = 5000*1.308 x 10-3 N/m 2 F = t.a = 5000*1.308 x 10-3 *4*0.6 N Power = F.U = 5000*1.308 x 10-3 *4*0.6*10 Nm/s = 0.21 Hp 21

22 Example: A 1.2 m long, 2 cm diameter shaft rotates inside an equally long cylinder that is 2.06 cm in diameter. Calculate the torque required to rotate the inner shaft at 2000 rpm if SAE-30 oil at 20 o C fills the gap. Also, calculate the horsepower required. Assume symmetric motion. Sol: N = 2000 rpm w = 2*3.142*N/60 = rad/s du = wr - 0 = * 0.01 = m/s dr = 0.06/2 = 0.03 cm = 0.03 x 10-2 m du/dr = s -1 m for SAE-30 oil at 20 o C = 0.4 N.s/m 2 t = m. du/dr = 0.4* = N/m 2 F = t.a = *3.142*2 x 10-2 *1.2 = N T = r. F = 0.01* = Nm Power = T. w = * Nm/s = watt = 0.6 Hp 22

23 Example: A 25-cm-diameter horizontal disk rotates a distance of 2 mm above a solid surface. Water at 10 o C fills the gap. Estimate the torque required to rotate the disk at 400 rpm. Sol: N = 400 rpm, h = m w = 2*3.142*N/60 = 41.9 rad/s m for 10 o C water = x 10-3 N.s/m 2 du = wr - 0 = wr; du/dy = wr /h t = m. du/dy = mwr /h df = t.da = (mwr /h)*2prdr = 2pmwr 2 dr/h dt = r*df = 2pmwr 3 dr/h T R 0 2pm r h 3 dr 2pm r pm 4 R h 4 R 4 0 2h = 3.142*1.308 x 10-3 *41.9/(2*2 x 10-3 )* = Nm Power = T. w = 0.44 watt 23 da w r

24 Viscosity Examples of Newtonian fluids: air, water, and oil, etc. Examples of non-newtonian fluids: liquid plastics, blood, slurries, paints, and toothpaste. An important effect of viscosity is to cause the fluid to adhere to the surface; this is known as the no-slip condition. This was assumed in the previous examples. The viscosity is very dependent on temperature in liquids. Viscosity of liquids decreases with increased temperature. For a gas, the viscosity increases as the temperature increases. The CGS physical unit for viscosity or dynamic viscosity is the poise (P), named after Jean Leonard Marie Poiseuille. It is more commonly expressed, as centipoise (cp). Water at 20 C has a viscosity of cp. 1 P = 0.1 Pa s, 1 cp = 1 mpa s = Pa s = N s/m 2. 24

25 Kinematic Viscosity Viscosity is often divided by the density in the derivation of equations, it has become useful and customary to define kinematic viscosity to be n = m/r Where the units of n are m 2 /s (ft 2 /sec). Note that for a gas, the kinematic viscosity will also depend on the pressure since the density is pressure sensitive. The SI unit of kinematic viscosity is m 2 /s. The CGS physical unit for kinematic viscosity is the stokes (St), named after George Gabriel Stokes. It is sometimes expressed in terms of centistokes (cst). 1 St = 1 cm 2 s 1 = 10 4 m 2 s 1. 1 cst = 1 mm 2 s 1 = 10 6 m 2 s 1. Water at 20 C has a kinematic viscosity of about 1 cst. 25

26 EXAMPLE: A viscometer is constructed with two 30-cm-long concentric cylinders, one 20.0 cm in diameter and the other 20.2 cm in diameter. A torque of 0.13 N-m is required to rotate the inner cylinder at 400 rpm (revolutions per minute). Calculate the viscosity. Ans: Ns/m 2 EXAMPLE: Express the above result in cp and cst. Ans: 1 cp = 1 mpa s = 1cSt Ns/m 2 = 1.65 mpa s = 1.65 cp = 1.65 cst 26

27 Compressibility In the preceding section we discussed the deformation of fluids that results from shear stresses. In this section, we discuss the deformation that results from pressure changes. All fluids compress if the pressure increases, resulting in an increase in density. A common way to describe the compressibility of a fluid is by the following definition of the bulk modulus of elasticity B: In words, the bulk modulus is defined as the ratio of the change in pressure (Dp) to relative change in density (Dr/r) while the temperature remains constant. The bulk modulus obviously has the same units as pressure. 27

28 Compressibility The bulk modulus for water at standard conditions is approximately 2100 MPa (310,000 psi), or times the atmospheric pressure. For air at standard conditions, B is equal to 1 atm. In general, B for a gas is equal to the pressure of the gas. To cause a 1% change in the density of water a pressure of 21 MPa (210 atm) is required. This is an extremely large pressure needed to cause such a small change; thus liquids are often assumed to be incompressible. For gases, if significant changes in density occur, say 4%, they should be considered as compressible; for small density changes they may also be treated as incompressible. 28

29 Compressibility Small density changes in liquids can be very significant when large pressure changes are present. For example, they account for "water hammer," which can be heard shortly after the sudden closing of a valve in a pipeline. When the valve is closed an internal pressure wave propagates down the pipe, producing a hammering sound due to pipe motion when the wave reflects from the closed valve. The bulk modulus can also be used to calculate the speed of sound in a liquid; it is given by This yields approximately 1450 m/s (4800 ft/sec) for the speed of sound in water at standard conditions. 29

30 Vapor Pressure When a small quantity of liquid is placed in a closed container, a certain fraction of the liquid will vaporize. Vaporization will terminate when equilibrium is reached between the liquid and gaseous states of the substance in the container - in other words, when the number of molecules escaping from the water surface is equal to the number of incoming molecules. The pressure resulting from molecules in the gaseous state is the vapor pressure. The vapor pressure is different from one liquid to another. For example, the vapor pressure of water at standard conditions (15 o C, kpa) is 1.70 kpa absolute and for ammonia it is 33.8 kpa absolute. 30

31 Vapor Pressure The vapor pressure is highly dependent on pressure and temperature; it increases significantly when the temperature increases. For example, the vapor pressure of water increases to kpa (14.7 psi) if the temperature reaches 100 o C. In general, a transition from the liquid state to the gaseous state occurs if the local absolute pressure is less than the vapor pressure of the liquid. In liquid flows, conditions can be created that lead to a pressure below the vapor pressure of the liquid. When this happens, bubbles are formed locally. This phenomenon is called cavitation. Cavitation in a flow can be very damaging when bubbles are transported by the flow to high pressure regions and collapse. It has the potential of damaging a pipe wall or a ship s propeller. 31

32 Surface Tension Suface tension is a property that results from the attractive forces between molecules. As such, it manifests itself only in liquids. The forces between molecules in the bulk of a liquid are equal in all directions, and as a result, no net force is exerted on the molecules. However, at the surface, the molecules exert a force that has a resultant in the surface layer. This force holds a drop of water suspended on a rod and limits the size of the drop that may be held. It also causes the small drops from a sprayer or atomizer to assume spherical shapes. Surface tension has units of force per unit length, N/m (lb/ft). The force due to surface tension results from a length multiplied by the surface tension; the length to use is the length of fluid in contact with a solid, or the circumference in the case of a bubble. 32

33 Surface Tension A surface tension effect can be illustrated by considering the free body diagrams of half a droplet and half a bubble as shown in Fig The droplet has one surface and the bubble is composed of a thin film of liquid with an inside surface and an outside surface. The pressure inside the droplet and bubble can now be calculated. The pressure force ppr 2 in the droplet balances the surface tension force around the circumference. Hence 33

34 Surface Tension Similarly, the pressure force in the bubble is balanced by the surface tension forces on the two circumferences. Therefore, So, we can conclude that the internal pressure in a bubble is twice as large as that in a droplet of the same size. Figure 1.12 shows the rise of a liquid in a clean glass capillary tube due to surface tension. The liquid makes a contact angle b with the glass tube. 34

35 Surface Tension Experiments have shown that this angle for water and most liquids is zero. There are also cases for which this angle is greater than 90 o (e.g. mercury); such liquids have a capillary drop. If h is the capillary rise, D the diameter, and r the density, s can be determined from equating the surface tension force to the weight of the liquid column. 35

36 EXAMPLE A 2-mm-diameter clean glass tube is inserted, as shown, in water at l5 o C. Determine the height that the water will climb up the tube. The water makes a contact angle of 0 o with the clean glass. Solution A free-body diagram of the water shows that the upward surfacetension force is equal and opposite to the weight. Writing the surface-tension force as surface tension times distance, we have => 36

37 EXAMPLE Solving for h, we get, The numerical values for s and r were obtained from Table of water properties. Note that the nominal value used for the density of water is 1000 kg/m 3. 37

Contact Angle b b Another important surface effect is the contact angle b which appears when a liquid interface intersects with a solid surface, as in the above Fig.

38 Contact Angle b b Another important surface effect is the contact angle b which appears when a liquid interface intersects with a solid surface, as in the above Fig. The force balance would then involve both s and b. If the contact angle is less than 90 o, the liquid is said to wet the solid; if b > 90 o, the liquid is termed nonwetting. For example, water wets soap but does not wet wax. Water is extremely wetting to a clean glass surface, with b = 0 o. Like s, the contact angle b is sensitive to the actual physico-chemical conditions of the solid-liquid interface. For a clean mercury-air-glass interface, b = 130 o. 38

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