- 1. FLUID MECHANICS BY ENGR. ALVIN M. ANTE
- 2. References: Fluid Mechanics 3rd Ed. By Frank M. White Fluid Mechanics and Hydraulics 4th Ed. By Gillesania Other related books, PDF, etc.
- 3. INTRODUCTION
- 4. What is Fluid Mechanics? It is the science that deals with the behavior of fluids at rest or in motion, and the interaction of fluids with solids or other fluids at the boundaries.
- 5. Focus on rate of flow and the various pressures that inhibit them
- 6. What is a Fluid? A fluid is a substance that deforms when subjected to force. Liquids occupy definite volumes; gases expand to occupy any containing vessel
- 7. The Concept of a Fluid A solid can resist a shear stress by a static deformation; a fluid cannot. The fluid moves and deforms continuously as long as the shear stress is applied.
- 8. LIQUIDS GASES incompressible compressible Compressibility is defined as the change of specific weight with pressure
- 11. Dimensions and Units A dimension is the measure by which a physical variable is expressed quantitatively. A unit is a particular way of attaching a number to the quantitative dimension.
- 12. Primary Dimensions In fluid mechanics there are only four primary dimensions from which all other dimensions can be derived: mass, length, time, and temperature.
- 16. FLUID PROPERTIES
- 17. FLUID PROPERTIES The Velocity Field Foremost among the properties of a flow is the velocity field V(x, y, z, t). In fact, determining the velocity is often tantamount to solving a flow problem, since other properties follow directly from the velocity field.
- 18. FLUID PROPERTIES Kinematic Properties of the Velocity Field
- 20. FLUID PROPERTIES Unit weight / Specific Weight ◦ Weight per unit volume of a fluid γ = 𝒘 𝑽 Where: γ = unit wt. (N/m3, lbf/ft3) w = weight of fluid (N, lbf) V = volume (m3, ft3)
- 21. FLUID PROPERTIES Mass Density ◦ Mass per unit volume of a fluid ρ = 𝒎 𝑽 ; ρair = 1.205 kg/m3; ρwater = 1000 kg/m3 Where: ρ = density (kg/m3, lb/ft3) m = mass of fluid (kg, lb) V = volume (m3, ft3)
- 22. FLUID PROPERTIES Density of Gases ρ = 𝒑 𝑹𝑻 Where: ρ = density (kg/m3) p = absolute pressure of gas (kPa) R = gas constant (J/kg-K) T = absolute temperature (K)
- 23. FLUID PROPERTIES Specific Gravity ◦ Unit wt. of fluid divided by the unit wt. of water s = 𝜸(𝒇) 𝜸(𝒘) Where: s = specific gravity (dmls) 𝜸(𝒇) = unit wt. of fluid (N/m3) 𝜸(𝒘) = unit wt. of water (N/m3)
- 24. FLUID PROPERTIES Viscosity – resistance to deformation which causes flow ◦ Kinematic Viscosity ν = 𝝁 𝝆 Where: ν = (nu) k. viscosity (m2/s or stoke) 𝝁 = dynamic/absolute viscosity (Pa*s) 𝝆 = density (kg/m3) 1 stoke = 1 cm2/s
- 25. FLUID PROPERTIES Viscosity ◦ Dynamic Viscosity 𝝁 = 𝝉 𝒅𝑽/𝒅𝒚 = 𝝉 𝜸 Where: 𝝁 = dynamic/absolute viscosity (Pa*s) or prop. const. τ = shear stress (Pa) dV/dy = γ = velocity gradient/shear rate (1/s) 1 centipoise (cP) = 10-3 Pa*s
- 27. FLUID PROPERTIES Viscosity is a proportionality constant suggesting a linear relationship between shear stress and velocity gradient.
- 28. FLUID PROPERTIES Surface Tension ◦ The surface tension of a fluid is the work that must be done to bring enough molecules from inside the liquid to the surface to form a new unit area of that surface in ft-lb/ft2 or N-m/m2.
- 29. FLUID PROPERTIES Capillarity ◦ Rise or fall of fluid in for example a tube which is caused by surface tension and depends on the relative magnitudes of the cohesion of the fluid and its adhesion to the walls. Rise – cohesion < adhesion Fall – cohesion > adhesion
- 35. Ideal Fluids Assumed to have no viscosity (no resistance to shear) Incompressible Have no uniform velocity when flowing No friction b/w moving layers of fluid No eddy currents or turbulence
- 36. Real Fluids Exhibit infinite viscosities Non-uniform velocity distribution when flowing Compressible Experience friction and turbulence in flow
- 37. Newtonian Fluids ◦Newtonian fluids show linear relationship between shear stress (τ) and velocity gradient (du/dy) – that is the viscosity remains constant irrespective of changes to shear stress and velocity gradient
- 38. Non-Newtonian Fluids ◦The ratio between shear stress (τ) and velocity gradient (du/dy) or shear rate is not constant but depends on the shear force exerted on the fluid. ◦Non-Newtonian fluids doesn’t show linearity between shear stress and shearing rate (velocity gradient).
- 39. Non-Newtonian Fluids Bingham Fluids - has true shear rate (damp clay, concentrated slurries, cheese) Pseudoplastics – viscosity decreases as shear rate increases (paint, mayonnaise, blood, heavy slurries) Dilatent fluids – viscosity increases as shear rate increases (most honeys, sand) Thixiotropic – become more fluid (viscosity decreases) when being more stirred with time Rheopetic – become less fluid with time, opposite of thixiotropic
- 40. Most fluid problems assume real fluids with Newtonian characteristics for convenience. This assumption is appropriate for: water air gases steam simple fluids: alcohol, gasoline, acid solutions
- 42. Unit Pressure or Pressure (p) Pressure - is the force per unit area exerted by a liquid or gad on a body or surface, with the force acting at right angles to the surface uniformly in all directions. Where: p = pressure, N/m2 or Pa (psf, psi) F = force, N or lbf A = area, m2 or ft2 or in2 𝑝 = 𝐹 𝐴
- 43. Pascal’s Law States that the pressure entity at a point in a fluid at rest is the same in all directions P1 = P2 =P3
- 45. Fluid-Static Law States that the pressure in a fluid increases with increasing depth P = ρgh
- 46. Hydrostatic Pressure Distribution Pressure in a continuously distributed uniform static fluid varies only with vertical distance and is independent of the shape of the container. The pressure is the same at all points on a given horizontal plane in the fluid. The pressure increases with depth in the fluid
- 48. Absolute and Gage Pressures Absolute pressure is the pressure above absolute zero (vacuum - no air). It is the lowest possible pressure attainable. It can never be negative value. Gage pressure is pressure above or below the atmosphere and can be measured by pressure gauges or manometers. The smallest gage pressure is equal to the negative of ambient atmospheric pressure. Atmospheric pressure is the pressure at any one point on the earth’s surface from the weight of the air above it.
- 49. Absolute and Gage Pressures 𝑝𝑎𝑏𝑠 = 𝑝𝑔𝑎𝑔𝑒 + 𝑝𝑎𝑡𝑚
- 63. Steps in Solving Manometer Problems 1. Decide on the fluid in feet or meter, of which the heads are to be expressed. 2. Starting from an end point, number in order, the interface of different fluids. 3. Identify points of equal pressure. Label these points with the same number. 4. Proceed from level to level, adding (if going down) or subtracting (if going up) pressure heads as the elevation decreases or increases, respectively with due regard for the specific gravity of the fluids.
- 68. Buoyancy Principle which determines whether the body will sink, rise or float
- 70. Buoyancy Upthrust (F) or Buoyant force F = Wair – Wliquid = ρ g V V = volume of immersed object (or immersed part if it floats)
- 76. Hydrostatic Forces on Surfaces 1. Hydrostatic forces on plane surfaces 2. Hydrostatic forces on curved surfaces 3. Hydrostatic forces in layered surfaces
- 77. Hydrostatic Forces on Plane Surfaces If the pressure over a plane area is uniform, as in the case of a horizontal surface submerged in a liquid, the total hydrostatic force/pressure is given by: F= 𝑝𝐴
- 78. Hydrostatic Forces on Plane Surfaces For an inclined or vertical plane submerged in a liquid, the total pressure is given by: Lifted from Fluid Mech. & Hydraulics 4th Ed. By Gillesania
- 79. Lifted from Fluid Mech. & Hydraulics 4th Ed. By Gillesania
- 80. Hydrostatic Forces on Plane Surfaces The hydrostatic problem is reduced to simple formulas involving centroid and moments of inertia of the plate cross- sectional area. Lifted from Fluid Mechanics 3rd Ed. by White
- 82. Sample Problems
- 89. Hydrostatic Forces on Curved Surfaces The resultant force on a curved surface is most easily computed by separating it into horizontal and vertical components. Lifted from Fluid Mechanics 3rd Ed. by White
- 90. Hydrostatic Forces on Curved Surfaces The horizontal component of force on a curved surface equals the force on the plane area formed by the projection of the curved surface onto a vertical plane normal to the component. The vertical component of pressure force on a curved surface equals in magnitude and direction the weight of the entire column of fluid, both liquid and atmosphere, above the curved surface.
- 91. Sample Problems
- 98. Hydrostatic Forces on Layered Surfaces If the fluid is layered with different densities, a single formula cannot solve the problem because the slope of the linear pressure distribution changes between layers. However, the formulas apply separately to each layer, and thus the appropriate remedy is to compute and sum the separate layer forces and moments. Lifted from Fluid Mechanics 3rd Ed. by White
- 99. Sample Problems
- 105. Governing Eqns. Of Flow BASIC EQUATIONS 1. continuity equation 2. energy equation 3. momentum equation
- 106. CONTINUITY EQUATON 2 1 m m 2 2 2 1 1 1 t t 2 2 1 1 Q Q 2 1 Q Q 2 2 1 1 V A V A
- 107. ENERGY EQUATION (BERNOULLI’S EQUATION) 2 1 E E g v z p g v z p 2 2 2 2 2 2 2 1 1 1
- 109. Sample Problems: Conservation Laws involving Fluids
- 113. Open Channel Flow The flow of liquids in a pipe is partially filled with the liquid and there is a free surface
- 114. PRISMATIC VS. NON-PRISMATIC A prismatic channel is a channel built with constant cross-section and constant bottom slope Otherwise, it is a non-prismatic channel, characterized by non-uniform cross section and slope.
- 115. Artificial channels such as rectangular, trapezoid, parabola and circle are the most commonly used prismatic channels. Natural channels are usually non-prismatic.
- 116. FLOW CLASSIFICATION 1. TIME AS THE CRITERION 2. SPACE AS THE CRITERION 3. BASED ON FLOW REGIMES STEADY FLOW/UNSTEADY FLOW UNIFORM FLOW/NON-UNIFORM FLOW INERTIAL FORCES VISCOUS FORCES GRAVITATIONAL FORCES
- 118. Incompressible vs. Compressible •Incompressible fluids maintain a nearly constant density at given temperatures and pressures. Density changes should be under 5%. •Compressible fluids have significantly varying densities during the flow
- 119. While all flows are compressible, flows are usually treated as being incompressible when the Mach number (the ratio of the speed of the flow to the speed of sound) is less than 0.3 (30%) Compressibility is NOT a fluid property but a flow property.
- 120. Viscous vs. Inviscid Flow Flows in which the effects of viscosity are significant are called viscous flows. Idealized flows of zero-viscosity fluids are called frictionless or inviscid flows. The effects of viscosity is very small.
- 121. Laminar vs. Turbulent Flow The highly ordered fluid motion characterized by smooth streamlines is called laminar. The highly disordered fluid motion that typically occurs at high velocities characterized by velocity fluctuations is called turbulent.
- 122. Laminar vs. Turbulent Flow Laminar: high-viscosity fluid such as oil at low velocity Turbulent: low viscosity fluid such as air at high velocity
- 123. Laminar vs. Turbulent Flow
- 124. Laminar vs. Turbulent Flow
- 125. Internal vs. External Flow ◦A fluid flow is classified as being internal and external, depending on whether the fluid is forced to flow in a confined channel or over a surface.
- 126. Internal Flow The flow in a pipe or duct in which the fluid is completely bounded by solid surfaces.
- 127. External Flow
- 128. External Flow
- 129. External Flow – The Boundary Layer is the flow of an unbounded fluid over a surface
- 130. Velocity Boundary Layer The x-coordinate is measured along the plate surface from the leading edge of the plate in the direction of the flow, and y is measured from the surface in the normal direction. The velocity of the particles in the first fluid layer adjacent to the plate becomes zero because of the no-slip condition. Thus, the presence of the plate is felt up to some normal distance from the plate beyond which the free-stream velocity u remains essentially unchanged.
- 131. Natural vs. Forced Flow A fluid flow is said to be natural or forced, depending on how the fluid motion is initiated. In forced flow, a fluid is forced to flow over a surface or in a pipe by external means such as a pump or a fan. In natural flows, any fluid motion is due to a natural means such as the buoyancy effect, which manifests itself as the rise of the warmer (and thus lighter) fluid and the fall of cooler (and thus denser) fluid.
- 132. Steady vs. Unsteady (Transient) Flow The term steady implies no change with time. The opposite of steady is unsteady, or transient. Many devices such as turbines, compressors, boilers, condensers, and heat exchangers operate for long periods of time under the same conditions, and they are classified as steady-flow devices.
- 133. INERTIAL FORCES VISCOUS FORCES GRAVITATIONAL FORCES
- 134. Reynold’s Number The transition from laminar to turbulent flow depends on the surface geometry, surface roughness, free-stream velocity, surface temperature, and type of fluid, among other things. After exhaustive experiments in the 1880s, Osborn Reynolds discovered that the flow regime depends mainly on the ratio of the inertia forces to viscous forces in the fluid.
- 135. REYNOLDS NO. LAMINAR FLOW ◦Re < 500 TURBULENT FLOW ◦Re > 1,000 TRANSITION FLOW ◦500 < Re < 1,000 VL Re English Units
- 136. LAMINAR FLOW Re < 2130 TURBULENT FLOW Re > 4000 TRANSITION FLOW 2130 < Re < 4000 S.I. Units
- 137. Reynold’s Number At large Reynolds numbers, the inertia forces, which are proportional to the density and the velocity of the fluid, are large relative to the viscous forces, and thus the viscous forces cannot prevent the random and rapid fluctuations of the fluid. At small Reynolds numbers, however, the viscous forces are large enough to overcome the inertia forces and to keep the fluid “in line.” The Reynolds number at which the flow becomes turbulent is called the critical Reynolds number.
- 138. Reynold’s Number For flow over flat plate: where xcr is the distance from the leading edge of the plate at which transition from laminar to turbulent flow occurs
- 140. Streamlined vs. Turbulent Flow Streamlined flow is when fluid moves in parallel elements, depicted by streamlines. The velocity of any element is constant but not necessarily the same as that of an adjacent element. Turbulent flow is when the fluid moves in elemental swirls or eddies. Both velocity and direction of each element change over time, thus a violent mixing results.
- 141. FROUDE NO. CRITICAL FLOW ◦ Fr = 1 SUBCRITICAL FLOW ◦ Fr < 1 SUPERCRITICAL FLOW ◦ Fr > 1 gD V Fr
- 143. Darcy-Weisbach Equation relates the head loss, or pressure loss, due to friction along a given length of pipe to the average velocity of the fluid flow for an incompressible fluid hf = 𝟒𝒇𝑳 𝑫 𝒙 (𝑽𝟐) 𝟐𝒈
- 144. Fluid Power
- 146. Cavitation the vaporization that may occur at locations where the pressure drops below the vapor pressure Cavities are known as “bubbles” or “voids”
- 147. Cavitation
- 148. Practice Solving :D
- 149. Practice Problems