Fluid Mechanics: Momentum, Heat & Mass Transfer

• Lecture No. One relates to Transport Phenomena, Second Edition, by Dr. Mohammed Atiya Transport Phenomena - Iraq, 2019

Momentum Transport

• Chapter 1 discusses Viscosity and the Mechanisms of Momentum Transport, which is Part One
• Viscous fluids, especially those with low molecular weight, exhibit resistance to flow characterized by viscosity

Newton's Law of Viscosity

• Large parallel plates, each with area A and separated by distance Y, containing a fluid (gas or liquid), will have varying degrees of viscosity
• The buildup to the steady, laminar velocity profile for a fluid will contained between the 2 plates
• Laminar flow occurs because adjacent fluid layers slide past one another in an orderly fashion.
• Constant force 'F' is required to sustain the motion of the lower plate once attained.
• F/A = µ(V/Y) represents the force, proportional to area and velocity, but inversely proportional to distance, and its constant proportionality defines viscosity
• F/A is replaced by Tyx, symbolizing force in the x-direction acting on a unit area perpendicular to the y-direction and dvx/dy replaces V/Y. Then:
• τyx = -μ dvx/dy shearing force per unit area is proportional to the velocity gradient, termed Newton’s law of viscosity.
• All gases and liquids of molecular weight less than 5000 are described by the equation above, and are called Newtonian fluids
• Polymeric liquids, suspensions, pastes, and slurries that don't adhere to it, are called non-Newtonian fluids. Polymeric liquids are in Chapter 8.
• Kinematic viscosity denoted symbol v to represent the viscosity divided by density (mass per unit volume) by fluid dynamicists
• v = μ/ρ defines kinematic viscosity.
• In the SI system, Tyx has units of N/m² = Pa, vx is in m/s, and y is in m.
• μ = -τyx/(dvx/dy) has units of (Pa)/((m/s)/(m))⁻¹ = Pa·s

Summary of Units for Quantities

• The following units of quantities summary is related to equation 1.1-2
• In the SI system: pressure = Pa; velocity = m/s; length = m; viscosity = Pa * s; kinematic viscosity = m²/s
• In the c.g.s system: pressure = dyn/cm²; velocity = cm/s; length = cm; viscosity = gm/cm * s = poise; kinematic viscosity = cm²/s
• In the British system: pressure = poundals/ft²; velocity = ft/s; length = ft; viscosity = lb/ft * s; kinematic viscosity = ft²/s
• Note: Pa is the same as N/m², and the Newton N, is the same as kg * m / s ². The abbreviation for centipoise is “cp.”

Viscosity of Water and Air at 1 atm Pressure

• Water at 0°C, the viscosity in μ (mPa·s)= 1.787, and the kinematic viscosity νcm²/s = 0.01787
• Water at 20°C, the viscosity in μ (mPa·s)= 1.0019, and the kinematic viscosity ν cm²/s = 0.010037
• Water at 40°C, the viscosity in μ (mPa·s)= 0.6530, and the kinematic viscosity ν cm²/s = 0.006581
• Water at 60°C, the viscosity in μ (mPa·s)= 0.4665, and the kinematic viscosity ν cm²/s = 0.004744
• Water at 80°C, the viscosity in μ (mPa·s)= 0.3548, and the kinematic viscosity ν cm²/s = 0.003651
• Water at 100°C, the viscosity in μ (mPa·s)= 0.2821 and kinematic viscosity ν cm²/s = 0.002944
• Air at 0°C, the viscosity in μ (mPa·s) = 0.01716 and the kinematic viscosity v (cm²/s) = 0.1327
• Air at 20°C, the viscosity in μ (mPa·s) = 0.01813 and the kinematic viscosity v (cm²/s) = 0.1505
• Air at 40°C, the viscosity in μ (mPa·s) = 0.01908 and the kinematic viscosity v (cm²/s) = 0.1692
• Air at 60°C, the viscosity in μ (mPa·s) = 0.01999 and the kinematic viscosity v (cm²/s) = 0.1886
• Air at 80°C, the viscosity in μ (mPa·s) = 0.02087 and the kinematic viscosity v (cm²/s) = 0.2088
• Air at 100°C, the viscosity in μ (mPa·s) = 0.02173 and the kinematic viscosity v (cm²/s) = 0.2298

Momentum Flux

• The steady-state momentum flux 1.46 × 10⁻² lb/ft² when the lower plate velocity in Fig. 1.1-1 given as 1 ft/s in the positive x direction where the plate separation Y given as 0.001 ft, and the fluid viscosity μ given as 0.7 cp.
• The velocity profile is linear so that dvx/dy and is equal to -1.0 ft/s/0.001 ft, giving -1000s⁻¹

Generalizing newton's Law of viscosity

• Previous section defines viscosity simply, where vₓ is a function of y and vᵧ and v_z were zero
• Interested in more complicated flows in which the 3 velocity components depend on x, y, and z coordinates, and possibly time where an expression greater than eq 1.1-2 is needed
• Full generalization is complex, requiring more than century and half.
• Consider general flow in which the fluid velocity may vary in direction and depends on time, the velocity components include v = vₓ(x, y, z, t); vy = vᵧ(x, y, z, t); v₂ = v₂(x, y, z, t)
• General conditions have nine stress components τij( i and j are x, y, and z), rather than the component τyx

Component Forces

• Viscous forces come into play only with velocity gradients.
• Forces aren't perpendicular to a surface element and are angled to the surface.
• Force per unit area τ₂ is exerted on the shaded area, force per unit area τy force per unit area τ2.
• Each force (vector) has a composite scalars of Txz, Try, Txz.
• Molecular stresses:πij = − pδij + τij where i and j may be x, y, or z and δij is the Kronecker delta, which is 1 if i = j and zero if i ≠ j
• (and also the πij ) signify aspects such as
  • force in j direction on a unit area perpendicular to the i direction, fluid of lesser x, is exerting force on fluid of greater xi and
  • the flux of j-momentum in the positive i direction from a point of lesser xito a point of greater xi.
• πxx = p + τxx, πyy = p + τyy, πzz = p + τzz are called normal stresses, whereas the remaining quantities τxy = τyx, τyz = τzy are shear stresses
• Viscous stresses are linear combinations of all velocity gradients: τij = −Σ_k Σ_l μijkl (∂v_k/∂x_l)where i, j, k, and l may be 1, 2, 3 where "viscosity coefficients." are included
• Quantities x₁, x₂, x₃ in derivatives denote the Cartesian coordinates x, y, and z, and quantities v₁, v₂, v₃ are same as vx, vy, vz.
• Expression should exclude time derivatives or time integrals.
• Viscous forces are not expected to be in pure rotation, where this necessitates a symmetric combination of velocity gradients and combinations of velocity gradients are determined to be scalars.

Molecular Stress Tensor Component Summary

Following entities are assumed that the expression components is the "molecular momentum flux tensor" as is assoc. with the molecular motions are discussed in §, 1.4 and appendix D and the “convective momentum flux tensor” associated with the bulk movement of the fluids are discussed in § 1.7

• direction components like x, y, and z define the expression as the normal Vector force, and force per unit area components like: x-component, y-component, and z-component

Viscosity Coefficients

• The "viscosity coefficients" reduced from 81 to 2
  • the equation simplifies to for an existing formula
  • the scalar must exist as a negative of the viscosity
  • a set scalar constant equals the other scalar constants to zero for monatomic gases The generalization for Newton's law of viscosity is 9 relations of Newton's law of viscosity with defined equations applicable for 1, 2, and 3

Relations that can be written more concisely with the Lennard-Jones parameters

τ = −μ[∇v + (∇v)†] + (⅔ μ − κ) (∇ · v)δ δ is the unit tensor with components δij., ∇v vector contains multiple components to a vector Shear stresses can demonstrate to be easy to visualize with the flow

Pressure, Temperature Dependence/Viscosity

• Most accurate viscosity data of pure gases and liquids has been researched in various science and engineering handbooks
• When experimental data is lacking and there is not time, viscosity can be estimate data-sets for correlation from trends of viscosity of liquids
• Global graph can describe how dependency on temperature
  • reduced quantity is a dimensionless by dividing the quantity - limits as the pressure of the gas, pressure can be nearly attained at 1 atm - the viscosity of gas increases at low temperatures Experimental has been seldom available and can either estimate viscosity based on pVT or use empirical relations

Molecular Theory of Viscosity

• Molecular momentum transport is an appreciation to the point of view of kinetic energy
• Mass given by sphere molecules and number density of volumes is presumed the rate that the number of gasses is limited
  • the average momentum is given by one or more

Constant and Frequency Molecular

• The average distance given between collisions and path by molecule
• average plane will have has its average at a high rate
• the viscosity of a gass gradient describes how all rates are relative and a momentum represents the rate by which assumptions may be further simplified
• The molecular given is equal to the Boltzmann equation
• Maxwell derived an equation for the viscosity in the 1860's using the collision cross section.
• Accurate depictions of the momentum and temperature makes the term better at the existing state that they will be known from the sphere model
• Also in the approximation, empirical approximations of relations of the transport of molecules undergoes in liquid form given in an equation.

Liquids at rest

• Molecules constantly move between vibrations, given an energy state and undergoes re-arrangements that the molecules exists in each other.
• Coordinate directions of the jumps in lengths are in equation by the rate and the Plank constants is derived to a set of formulas when the is accounted for. A further equation of approximation helps defines a relationship by how all units of matter has specific set of limits, and derives an estimate viscosity by it.

Suspension Models and Emulsions

• Suspension should involve liquid discussion of phases
• the models are more related to known symbols that does not relate to all the relationships
• Two modifications with Newton laws of viscosity and effective components is involved, for steady-state flows of viscosity that are inappropriate
• Einstein considered solid of volume or mediums that describes the equation and is the volume fraction of the spheres

Molecular Transport components

The dielectric of the fluid should have relations to know specific forces Convective transport and bulk flows involves a flux can lead to the total momentum three vectors describes momentum on three axis

Explore the principles of momentum, heat, and mass transfer in fluids. Understand driving forces, transport phenomena, and applications. Investigate Newtonian and non-Newtonian fluids and their properties.