# 1. Introduction, fluid properties (1.1, 2.8, 4.1, and handouts)

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1 1. Introduction, fluid properties (1.1, 2.8, 4.1, and handouts) Introduction, general information Course overview Fluids as a continuum Density Compressibility Viscosity Exercises: A1

2 Fluid mechanics Fluid properties (2) Hydrostatics (3) Basic equations (6) Pipe flow (5) Flow around submerged bodies (1) Channel flow (3) Repetition (2)

3 Applications of fluid mechanics Transport of gas and fluids in pipes and channels Societal supply of safe energy and water Energy production (oil, hydropower, nuclear energy, natural gas) Environmental engineering and water treatment (channels, basins, filtering) Process technology (relationship temperature, pressure, and energy) Protection against climate extremes/catastrophes (flooding, harbours, wind forces)

4 FLUID AS A CONTINUUM A fluid is considered to be a continuum in which there are no holes or voids velocity, pressure, and temperature fields are continuous. Validity criteria: Smallest length scale in a flow >> average spacing between molecules composing the fluid.

5

6 DENSITY (ρ) Mass/ unit volume (kg/m 3 ) Density decreases normally with increasing temperature ρ water = ρ(t,s,p) i.e., dependent on - Temperature - Salt content (ρ S, S in per mille; S = 3.5% in ocean ρ= 1026 kg/m 3 ) - Pressure (but only a small variability)

7 OTHER DEFINITIONS Weight = mass gravity acceleration (W = mg, [N = kg m/s 2 ]) (Eqn. 1.4) Weight density (or specific weight)= density gravity acceleration (w = ρg, [N/m 3 = kg /(m 2 s 2 )]) (Eqn. 1.6) (Note w = γ in exercises) Specific volume ν = reciprocal of density (ν = 1/ρ, [m 3 /kg]) Relative density (or specific gravity), s, is the density normalized with the density of water at a specific temperature and pressure (normally 4 C and atmospheric pressure): s = R.d. = ρ/ρ water (often = ρ/1000) (Eqn. 1.7) Power P [W = J/s = kg m 2 /s 3 = Nm/s]; P = T ω (T = torque, ω = angular velocity [rad/s, 360 o = 2πrad]

8 Example density. The specific weight of water at ordinary temperature and pressure is 9.81 kn/m 3. The specific gravity of mercury is Compute the density of water and the specific weight and density of mercury.

9 COMPRESSIBILITY All fluids can be compressed by application of pressure elastic energy being stored Modulus of elasticity ( elastitetsmodul ) describes the compressibility properties of the fluid and is defined on the basis of volume

10 Modulus of elasticity: E=-dp/(dV/V 1 ) [Pa] For liquids, region of engineering interest is when V/V 1 1 ΔV Δp V E E water ~ Pa (function of temperature)

11 A1 What pressure must be applied to water to reduce its volume 1 %?

12 Example compressibility. At a depth of 8 km in the ocean the pressure is 81.8 MPa. Assume that the specific weight of sea water at the surface is kn/m 3 and that the average volume modulus of elasticity is 2.34*10 9 N/m 2 for the pressure range. A) What will be the change in specific volume between that at the surface and at that depth? B) What will be the specific volume at that depth? C) What will be the specific weight at that depth

13 IDEAL FLUID A fluid in which there is no friction REAL FLUID A fluid in which shearing forces always exist whenever motion takes place due to the fluid s inner friction viscosity.

14 VISCOSITY Viscosity is a measure of a fluid s inner friction or resistance to shear stress. It arises from the interaction and cohesion of fluid molecules. All fluids posses viscosity, but to a varying degree. For instance, syrup has a considerably higher viscosity than water.

15 DEFINITION OF DYNAMIC VISCOSITY - μ y Shearing of thin fluid film between two plates. The upper plate has an area A. Experiments have shown that for a large number of fluids: F ~ AV/h (if V and h not too large) Linear velocity profile V/h = dv/dy

16 Introduction of the proportionality constant μ, named dynamic viscosity, gives Newton s viscosity law shear force: F V dv (Eqn ) N/m τ = = μ = μ 2 A h dy μ [Pa s or kg/ms] ν = μ/ρ [m 2 /s] - Kinematic viscosity No-slip condition water particles adjacent to solid boundary has zero velocity (observational fact)

17 μ (Pa s) VVR 120 Fluid Mechanics

18 Implication of viscosity: a fluid cannot sustain a shear stress without deformation

19 Implications of Newton s law: τ, μ independent of pressure (in contrast to solids) no velocity gradient no shear stress Restriction of Newton s law: law only valid if the fluid flow is laminar in which viscous action is strong

20 Laminar flow: smooth, orderly motion in which fluid elements appears to slide over each other in layers (little exchange between layers). Turbulent flow: random or chaotic motion of individual fluid particles, and rapid mixing and exchange of these particles through the flow Turbulent flow is most common in nature.

21 Newtonian non-newtonian fluids Examples non-newtonian fluids: Plastics, blood, suspensions, paints, foods VVR 120 Fluid Mechanics Shear vs. rate of strain relations for non-newtonian fluids: Bingham plastic du τ τ = μ, τ > τ i dy i n>1: Shear-thickening fluid, n<1: Shear-thinning fluid du n τ = μ( ) dy

22 Example viscosity. A space, 3 cm wide, between two plane horizontal surfaces is filled with SAE 30 Western lubricating oil at 20 C. What force is required to drag a very thin plate of 1 m 2 area through the oil at a velocity of 0.1 m/s if the plate is 1 cm from one surface?

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