Welcome to MECH 280 Ian A. Frigaard Department of Mechanical Engineering, University of British Columbia
Lectures 1 & 2: Learning goals/concepts: What is a fluid Apply continuum hypothesis Stress and viscosity Understand stress strain rate relationship for a fluid Flow examples & classifications Review of fluid properties Surface tension and its molecular origins Problem solving: Compute simple examples with viscosity Simple calculations involving: Surface tension Capillary rise Reading: chapter 1 of any introductory text on fluid mechanics
Dimensions and Units Any physical quantity can be characterized by dimensions The magnitudes assigned to dimensions are called units Primary dimensions include: mass m, length L, time t, and temperature T. Sometimes force replaces mass Secondary dimensions can be expressed in terms of primary dimensions, e.g. velocity V, energy E Dimensional homogeneity: valuable in checking for errors Each term in an equation should have the same units (hence dimensions) Two common unit systems: English system (USCS): length (ft), force (lbf), time (s) 1 lbf = 32.174 lbm ft/s 2, (where lbm is the unit of mass) Metric SI (International System): length (m), mass (kg), time (s)
Properties of a fluid Any characteristic of a system is called a property Familiar: pressure P, temperature T, volume V, and mass m Less familiar: viscosity, thermal conductivity, vapor pressure, modulus of elasticity, thermal expansion coefficient, surface tension Intensive properties: independent of mass of the system, e.g. temperature, pressure, density Extensive properties: value depends on the size of the system, e.g. total mass, total volume, total momentum Extensive properties per unit mass are called specific properties, e.g. specific volume v = V/m and specific total energy e=e/m.
Physically, what is a fluid? Solid Liquid Gas Relative motion of the molecules? Intermolecular forces?
Continuum hypothesis A fluid is hypothesized to be continuous, homogeneous matter with no holes, i.e. a continuum. Both liquids & gases Allows us to treat physical properties of substance as smoothly varying quantities, e.g. we can differentiate Continuum hypothesis is reasonable for much of classical fluid mechanics Atoms are widely spaced in gas phase. Want to disregard atomic nature of a substance Continuum hypothesis is valid as long as averaging volume δu is large in comparison to distance between molecules, but small in comparison to physical changes in system
What is a fluid? Definition of a fluid: a continuum that is unable to resist a shear stress while at rest Consider a surface, with normal vector n: Stress is defined as the force per unit area The stress vector is also sometimes called the traction Note: stress vector depends on n Normal component: normal stress In a fluid at rest, the normal stress are independent of n, called the pressure Tangential components: shear stresses
Distinction between solids and fluids? Suppose a material fills the space between 2 plates. A shear stress τ=f/a is applied to the upper plate. Solid resists by a constant deformation: τ ε/h : (ε/h is the strain) Fluid resists by a continuous deformation: τ V/h Strain in the fluid: ε /h = Vt/h constant Constant of proportionality is the viscosity: V/h is the strain rate Notes: For solids, relationship between shear stress and strain is important For fluids, relationship between shear stress and strain rate is important
Conceptually Viscosity is a fluid property that represents internal resistance of fluid to motion: Under same stresses, more viscous fluid moves slower The force that a flowing fluid exerts on a body, in the flow direction is called the drag force Magnitude of drag force depends, in part, on viscosity How does the moving body generate viscous stresses? How are viscous stresses exerted on the body?
Generating stress inside a fluid? The body deforms the streamlines of the flow Viscous stresses result from velocity gradients Notes: Other stresses are also generated: inertial stresses, turbulent stresses, pressure field If fluid viscosity is low, parts of flow field without large velocity gradients may be considered inviscid Reynolds number, Re= ρud/µ,
Exerting viscous stresses on a body No-slip condition: A fluid in direct contact with a solid sticks to the surface due to viscous effects Responsible for generation of wall shear stress τ w, surface drag F D = τ w da, and development of the boundary layer The fluid property responsible for the no-slip condition is viscosity The no-slip condition is also very important in giving a boundary condition to the Navier-Stokes equations Used in analytical and computational fluid dynamics analyses Used in making simplifying approximations to the full Navier-Stokes equations, e.g. boundary layer analyses used in much of aerodynamics, ship dynamics, heat transfer
Viscosity Shear stress τ =F/A is applied to the upper plate. Using the no-slip condition, u(0) = 0 and u(h) = V, velocity profile and gradient are: u(y)= Vy/h du/dy=v/h Shear stress for a Newtonian fluid: τ = µdu/dy µ is the dynamic viscosity and has SI units of kg/m s = Pa s (= 10 poise) How viscous is? Air Water Engine oil Maple syrup Bitumen
Newtonian fluids A fluid which has a linear relationship between the shear stress and velocity gradient (rate of strain) The coefficient of proportionality is the coefficient of viscosity: µ Note that the assumption of linear relations to model transported quantities is very common in physics Fourier s law of heating Fick s law of diffusion
Example 1 Calculate the force (per unit area) required to pull the thin plate as illustrated if the fluid filling the gap between the walls is maple syrup (µ = 0.15 Pa s)
Example 2: Viscosity lubricates A block of dimension 50cm x 20cm x 20 cm moves steadily up an incline, as illustrated. Calculate the force required to push the block: a) Assuming dry friction with the base, c f = 0.27 b) Assuming the motion is lubricated by a thin oil layer (µ = 0.012 Pa s), of thickness 0.4mm a)
b)
Viscometry: How is viscosity measured? A rotating viscometer. Two concentric cylinders with a fluid in the small gap l. Inner cylinder is rotating, outer one is fixed. Relate viscosity to geometry and torque du F = τ A= µ A dy If l/r << 1, then gap between cylinders can be modeled as flat plates h = l Torque T = FR, and tangential velocity V=ωR Wetted surface area A=2πRL. Measure T and ω to compute µ
Non-Newtonian fluids Non-Newtonian fluids are those that do not follow the linear stress-strain rate law Generalised Newtonian fluids Generalised Newtonian fluids have a nonlinear relation between shear stress and strain rate Viscoelastic fluids have shear stress that depends on the shear history Thixotropic fluids have shear stress that depends on structure & time E.g. polymer solutions, slurries, pastes, muds, food products, cosmetics, biological fluids, Many engineering jobs
Classification of Flows Fluid flows are governed by Navier-Stokes equations NS-equations are too difficult to be solved analytically except in simple situations Range of flow phenomena observed is very broad Conservation of mass ρ + [ ρu] = 0 t Conservation of momentum U ρ + t 2 [ U ] U = P + ρg + µ U Both features have led engineers and scientists to classify flows, in an effort to simplify and understand
Viscous vs. Inviscid Regions of Flow Regions where frictional effects are significant are called viscous regions. They are usually close to solid surfaces. Regions where frictional forces are small compared to inertial or pressure forces are called inviscid For inviscid flows : ρ t U 2 + [ U ] U = P + ρg + µ U X
Internal vs. External Flow Internal flows dominated by influence of viscosity throughout the flow field External flows, viscous effects are limited to the boundary layer and wake.
Compressible vs. Incompressible Flow All fluids are compressible, but not all flows Flow is classified incompressible if the fluid density remains nearly constant for a given flow Liquid flows are typically incompressible Exceptions: high pressures, acoustics Gas flows often compressible, especially for high speeds. Mach number, Ma = V/c is a good indicator of whether or not compressibility effects are important Ma < 0.3 : Incompressible Ma < 1 : Subsonic Ma = 1 : Sonic Ma > 1 : Supersonic Ma >> 1 : Hypersonic
Steady vs. Unsteady Flow Steady implies no change at a point with time. Time derivative terms in Navier- Stokes equations are zero Unsteady is the opposite of steady. Transient usually describes a starting, or developing flow. Periodic refers to a flow which oscillates about a mean. Unsteady flows may appear steady if time-averaged
Laminar vs. Turbulent Flow Laminar: highly ordered fluid motion with smooth streamlines. Turbulent: highly disordered fluid motion characterized by velocity fluctuations and eddies. Transitional: a flow that contains both laminar and turbulent regions Reynolds number, Re= ρul/µ, is often the key parameter in determining whether or not a flow is laminar or turbulent.
One-, Two-, and Three-Dimensional Flows N-S equations are 3D vector equations. Velocity vector, U(x,y,z,t)= [U x (x,y,z,t),u y (x,y,z,t),u z (x,y,z,t)] Lower dimensional flows reduce complexity of analytical and computational solution Change in coordinate system (cylindrical, spherical, etc.) may facilitate reduction in order. Example: for fully-developed pipe flow, velocity V(r) is a function of radius r and pressure p(z) is a function of distance z along the pipe.
Flow examples: Inviscid flows: vortex rings https://www.youtube.com/watch?v=lthivhl1uey External flow: http://www.youtube.com/watch?v=q-wuellen9y&feature=related Internal flow: http://media.efluids.com/galleries/microfluidics?medium=583 Shedding vortices behind a cylinder: https://www.youtube.com/watch?v=ipbkr9csce0 Viscous fingering during displacement: http://www.youtube.com/watch?v=nzeb8tq3eom
Density and Specific Gravity Density is defined as the mass per unit volume ρ = m/v. Density has units of kg/m 3 For a gas, density depends on temperature and pressure Specific volume is defined as v = 1/ρ = V/m. Specific gravity, or relative density is defined as the ratio of the density of a substance to the density of some standard substance at a specified temperature Usually water at 4C is used SG = ρ/ρ H2O SG is a dimensionless quantity. Specific weight is defined as the weight per unit volume, γ s = ρg where g is the gravitational acceleration γ s has units of N/m3
Density of Ideal Gases Equation of State: equation for the relationship between pressure, temperature and density. The simplest and best-known equation of state is the ideal-gas equation. PV = nrt or P = ρr specific T Ideal-gas equation holds for most gases. Dense gases such as water vapor and refrigerant vapor should not be treated as ideal gases. Tables should be consulted for their properties.
Vapor Pressure and Cavitation Vapor Pressure P v is defined as the pressure exerted by its vapor in phase equilibrium with its liquid at a given temperature If P drops below P v, liquid is locally vaporized, creating cavities of vapor: cavitation Vapor cavities collapse when local P rises above P v. Collapse of cavities is a violent process which can damage machinery. Cavitation is noisy, and can cause structural vibrations.
Energy and Specific Heats Total energy E is comprised of numerous forms: thermal, mechanical, kinetic, potential, electrical, magnetic, chemical, and nuclear. Units of energy are joule (J) or British thermal unit (BTU). Microscopic energy Internal energy u is for a non-flowing fluid and arises is due to molecular activity. Enthalpy: h=u+p/ρ is for a flowing fluid and includes flow energy (P/ρ). Macroscopic energy Kinetic energy ke=v 2 /2 Potential energy pe=gz In the absence of electrical, magnetic, chemical, and nuclear energy, the total energy is e flowing =h+v 2 /2+gz Note that u, h, ke, pe are all specific properties. For total quantities, (U, H, KE, PE) we must multiply by ρ and integrate over system volume
Coefficient of Compressibility How does fluid volume change with P and T? Fluids expand as T or P Fluids contract as T or P Need fluid properties that relate volume changes to changes in P and T. Coefficient of compressibility P P κ = v = ρ v ρ T T Coefficient of volume expansion Combined effects of P and T can be written as 1 v 1 ρ β = = v T ρ T P P v v dv = dt + dp T P T P
Surface tension: molecular origins Liquid molecules experience attractive forces from other liquid molecules Molecule in the interior of liquid experiences forces from all directions symmetric in some statistical sense A molecule at the surface experiences attractive forces from the interior and weaker attractive forces from the gas Therefore, there is a net attractive force on each surface molecule, pulling inwards and trying to minimize the surface area Droplets behave like small spherical balloons filled with liquid The surface of the liquid appears to act like a stretched elastic membrane under tension. Magnitude of the surface force (per unit length) is called the surface tension σ s
Example 1: movable wire frame What is the surface tension in the fluid if F = 0.012N and b = 0.08m?
Example 2: How much larger is the pressure inside a spherical droplet than outside?
Units and typical values of σ S Units (SI) are N/m Frequently dyn/cm is used (cgs) 1 dyn/cm = 10-3 N/m English units lbf/ft From its molecular origin, σ S depends on both fluids More important with liquid-liquid Temperature dependency of σ S often strong Sensitive to impurities and surfactants
Example 3: A 35mm long needle rests on a water surface at 20C. What force is required to lift the needle?
Capillary Effect Capillary effect is the rise or fall of a liquid in a smalldiameter tube Curved free surface in tube is called the meniscus Water menisci curve up: water is a wetting fluid Mercury menisci curve downwards: mercury is a non-wetting fluid With no curved meniscus there is no surface tension Force balance gives an expression for magnitude of capillary rise Measuring capillary rise gives estimate of surface tension
Example 4: Capillary rise How high will a wetting fluid rise in a capillary tube exposed to the atmosphere?
What we covered Molecular origins of surface tension Units, typical values Simple calculations involving: Surface tension Capillary rise Visualisation of surface tension effects: Collapse of a soap bubble: http://media.efluids.com/galleries/interfacial?medium=722 Surfactant effects: the soap boat https://www.youtube.com/watch?v=miwldvohrse