FLUID PROPERTIES Independent variables SCALARS VECTORS TENSORS

, u, w, v REFERENCE FRAME

SCALARS Need a single number to represent them: P, T, ρ besttofind.com Temperature May vary in any dimension x, y, z, t

VECTORS Have length and direction Need three numbers to represent them:

Unit vector = vector whose length equals 1 x y z

VECTORS In terms of the unit vector:

CONCEPTS RELATED TO VECTORS Nabla operator: Denotes spatial variability Dot Product:

CONCEPTS RELATED TO VECTORS CrossProduct:

INDICIAL or TENSOR NOTATION Vectoror First Order Tensor Vector Dot Product Matrix or Second Order Tensor

INDICIAL or TENSOR NOTATION Gradient of Scalar Gradient of Vector Second Order Tensor Special operator – Kronecker Delta

TENSORS Need nine numbers to represent them:

For a fluid at rest: Normal (perpendicular) forces caused by pressure

MATERIAL (or SUBSTANTIAL or PARTICLE) DERIVATIVE

Fluids Deform more easily than solids Have no preferred shape

Deformation, or motion, is produced by a shear stress z x u μ = molecular dynamic viscosity [Pa·s = kg/(m·s)]

Continuum Approximation Even though matter is made of discrete particles, we can assume that matter is distributed continuously. This is because distance between molecules << scales of variation ψ (any property) varies continuously as a function of space and time space and time are the independent variables In the Continuum description, need to allow for relevant molecular processes – Diffusive Fluxes

Diffusive Fluxes e.g. Fourier Heat Conduction law: z x t = 0 t = 1 t = 2 Continuum representation of molecular interactions This is for a scalar (heat flux – a vector itself) but it also applies to a vector (momentum flux)

Shear stress has units of kg m -1 s -1 m s -1 m -1 = kg m -1 s -2 Shear stress is proportional to the rate of shear normal to which the stress is exerted at molecular scales µ is the molecular dynamic viscosity = kg m -1 s -1 for water is a property of the fluid or force per unit area or pressure: kg m s -2 m -2 = kg m -1 s -2 Diffusive Fluxes (of momentum)

Net momentum flux by u

Diffusive Fluxes (of momentum) For a vector (momentum), the diffusion law can be written as (for an incompressible fluid): Shear stress linearly proportional to strain rate – Newtonian Fluid (viscosity is constant)

Boundary Conditions Zero Flux No-Slip [u (z = 0) = 0] z x u

The Hydrostatic Equation Hydrostatics - The Hydrostatic Equation z g A z = z 0 z = z 0 + dz dz p p + (∂p/∂z ) dz Integrating in z:

the Hydrostatic Equation - 1 Example – Application of the Hydrostatic Equation - 1 z H Find h Downward Force? Weight of the cylinder = W Upward Force? Pressure on the cylinder = F Same result as with Archimedes’ principle (volume displaced = h A c ) so the buoyant force is the same as F Pressure on the cylinder = F = W ACAC h

the Hydrostatic Equation - 2 Example – Application of the Hydrostatic Equation - 2 z W D Find force on bottom and sides of tank On bottom? On vertical sides? Same force on the other side x L A T = L W dF x Integrating over depth (bottom to surface)