Transport Phen. 3 - J.Chew Flashcards
What are the 3 types on non-Newtonian fluid (+ examples)?
1) Purely viscous fluids (inelastic or plastic)
E.g.
- pseudo-plastic fluids (shear thinning e.g. yoghurt)
- dilatant fluids (shears thickening e.g. paint)
- Bingham (plastic) fluid (solid until yield stress is reached e.g.toothpaste)
2) Time dependent fluids
E.g.
- Thixotropic fluids (thins with time of shearing e.g. gels)
- Rheopectic fluids (thicken with time of shearing e.g. lubricants)
3) Viscoelastic fluids
Viscous and elastic behaviour e.g. paste extrusion
How do pseudoplastics respond to shearing?
Viscosity decreases as shear rate is increased.
What law is used to describe pseudoplastic behaviour with relation to shear rate?
The power law.
Tau = k*gamma^n
Shear stress = consistency index * shear rate ^ flow behaviour index
What is the value of the flow behaviour index, n, for a pseudplastic (shear thinning)?
0 < n < 1
If 1, the fluid would be Newtonian and k would be equal to viscosity.
What is the value of the flow behaviour index, n, for a dilatant fluid (shear thickening)?
n > 1
Log apparent viscosity vs log shear rate graph would have a positive gradient.
How do pseudoplastics behave with shearing?
They become thinner with increased shearing
How do dilatant fluids behave with shear?
They become thicker
How to Bingham (plastic) fluids behave?
They are solid until their yield stress is reached, and will then behave as a (Newtonian) fluid.
If 𝜏(y,x) is less than 𝜏(y) (shear stress < yield stress), shear rate of a Bingham plastic is zero. This does not mean that the fluid is not flowing, this just means that the fluid is not being deformed.
Note that μB (coefficient of rigidity) is not equal to apparent
viscosity.
How do thixotropic fluids behave with shear?
They get thinner with time of shearing.
How do rheopectic fluids behave?
They get thicker with time of shearing.
What fluids is the power law model used for?
Pseudoplastic and dilatant fluids
Pseudoplastics: 0 < n < 1
Dilatant: n > 1
Newtonian: n = 1
Where n is the flow index
When is a fluid a Herschel Buckley fluid?
When a Bingham plastic fluid begins to flow, but is either shear thinning or thickening, and not Newtonian, then it is a Herschel Buckley fluid.
If tau(y,x) is less than tau(y) (shear stress < yield stress), shear rate of a Bingham plastic is zero. This does not mean that the fluid is not flowing, this just means that the fluid is not being deformed.
What is apparent viscosity?
Apparent viscosity is the shear stress applied to a fluid divided by the shear rate.
For a Newtonian fluid, the apparent viscosity is constant, and equal to the Newtonian viscosity of the fluid, but for non-Newtonian fluids, the apparent viscosity depends on the shear rate
Why do we study the development of velocity profiles and volumetric flowrate expressions?
From these, we can determine pressure drop.
E.g. From the u and Q expressions for laminar flow through a pipe, we can find the Hagen-Poiseulle equation for pressure drop
In order to express laminar pressure drop (via Hagen-Poiseulle equation), what do we need to find?
How are centreline and mean velocity found?
Firstly, determine the velocity profile (via a force balance and equating said force balance to the constitutive equation)
Then, determine the relationship for the volumetric flowrate.
Q = Au (so integrate the product of A and velocity which you have just found)
From the Q and dp/dz relationship, an expression for dP may be found.
Centreline velocity: velocity at the centre of the channel e.g. when r = 0 for pipe flow. Substitute r = 0 into the velocity profile. This is also likely to be u max.
Mean velocity: u = Q/A. Divide the volumetric flowrate by the total cross sectional area.
How may apparent viscosity and shear rate be graphed to determine the consistency index, k, and the flow behaviour index, n, for a pseudoplastic / power law fluid?
μ.app = kγ^(n-1)
[ 𝜏 = kγ^n ]
μ.app = k*γ^(n-1)
log [μ.app] = log[k] + (n-1)*log[γ]
Plotting log [μ.app] vs log[γ],
Slope: n - 1
y-int: log [k]
What is the procedure for modelling a plastic fluid?
i) Plot 𝜏 vs γ on a linear graph
ii) If the intercept is 0, the fluid exhibits plastic behaviour
iii) If the slope is constant, the classical Bingham model can be used. Else, use the Herschel-Buckley model
iv) If the slope is not constant, plot (𝜏 - 𝜏y) vs γ on a log-log graph
log [𝜏 - 𝜏y] = log[k] + (n)*log[γ]
Slope: n
y-int: k
v) Quote all results and reference equations used
What would the basic force balance be for flow through a circular pipe?
Flow is in z direction
Radius of pipe is R
p is pressure
i) pπr^2 = (p + dp)πr^2 + 𝜏(rz)*2πrdz
where 𝜏(rz) is the shear stress applied by the cylinder of fluid on its surrounding at a distance r from the centre.
AND
𝜏(rz) = μ*du/dr
Equating and integrating, the velocity profile becomes:
u = [(R^2 - r^2)/(4μ)](-dp/dz)
[Integrating the product of u*A gives the volumetric flow rate. Note - Area used in this case is 2πrdr
How is mean velocity found?
By dividing the volumetric flowrate (derived from the force balance and velocity profile) by the total area.
What are tensors?
A mathematical object analogous to but more general than a vector, represented by an array of components that are functions of the coordinates of a space.
The tensors that we shall discuss are quantities which have a magnitude and two associated directions, e.g. a stress has two associated directions: the plane on which the stress acts and the direction of the stress.
What does the notation of a tensor tell you, e.g. for 𝜏 yx
For 𝜏 yx:
𝜏 is the shear stress
y is the plane upon which it acts
x is its direction
What is the definition of nabla / del (∇)?
A differential operator which, operating on a function of several variables, gives the sum of the partial derivatives of the function with respect to the three orthogonal spatial coordinates.
∇ ≡ ∂/∂xi + ∂/∂yj + ∂/∂z*k
What does the grad (gradient) operator show?
Applied to a scalar field, a vector function is obtained, called the gradient.
Grad Φ = ∇Φ = ∂Φ/∂xi + ∂Φ/∂yj + ∂Φ/∂z*k
What does the divergence operator show?
Divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field’s source at each point. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point.
div u ≡ ∇·u ≡ ∂u/∂x + ∂v/∂y + ∂w/∂z
What is the dot product of ∇·u where u = (u, v, w) [a vector]
div u ≡ ∇·u ≡ ∂u/∂x + ∂v/∂y + ∂w/∂z
≡ (∂/∂x, ∂/∂y, ∂/∂z)·(u, v, w)
What is the Laplacian operator?
The Laplacian is given by the sum of second partial derivatives of the function with respect to each independent variable.
∇·∇ or ∇^2
How is the scalar operator, u·∇Φ written?
(where u is a vector [u, v, w], · is the dot product - leading to summation, and Φ is the grad operator).
u·∇Φ ≡ u∂Φ/∂x + v∂Φ/∂y + w*∂Φ/∂z
Φ is a variable e.g. T, P etc.
What are the cartesian/rectangular, cylindrical/polar, and spherical coordinates?
Cartesian / rectangular: (x, y, z)
Cylindrical/polar: (r, θ, z)
Spherical coordinates: (r, θ, φ)
How may the total derivative be written in terms of vector operators?
D[…]/Dt = u·∇[…] + ∂[…]/∂t
How do Eulerian and Lagrangian flow descriptions differ?
Lagrangian - tracks diffusive/conductive effects only.
In the Lagrangian description of fluid flow, individual fluid particles are “marked,” and their positions, velocities, etc. are described as a function of time.
Eulerian - tracks convective and diffusive effects.
In the Eulerian description of fluid flow, individual fluid particles are not identified. Instead, a control volume is defined, as shown in the diagram. Pressure, velocity, acceleration, and all other flow properties are described as fields within the control volume.
