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The introductory Python tutorials are available here: Introduction to Python and NumPy I - Introduction to Python and NumPy II

Table of Contents
Purpose
In this exercise you will apply the Newtonian (linear) and non-Newtonian (nonlinear) viscous flow equations to make plots of horizontal and vertical velocity profiles in glaciers. It is based on a MATLAB exercise provided by Prof. Todd Ehlers at the University of Tübingen, Germany.

Overview


As we’ve seen previously, most landforms on Earth are sculpted by the flow of fluids across the landscape. Erosion by glaciers can significantly change the Earth’s surface, moving vast quantities of material from high elevations to lower elevations. Glacial landscapes are unique and clearly reflect erosion by processes that differ greatly from those found in fluvial systems. To understand how material is moved within glacial systems, it is important to first understand the dominant controls on glacier velocities.

Non-Newtonian flow in open channels


Background

Figure 1. Schematic diagram of pressure-driven flow in a channel.

The viscosity of ice depends on both temperature and stress. For simplicity, we’ll consider the case where viscosity is only a function of shear stress. Consider a channel of width h centered about y = 0, with lateral boundaries at y = ±h/2 (Figure 1). A pressure gradient Δpp1 - p0 drives flow within the channel of length L.  The shear stress in the fluid depends on the applied pressure gradient and the distance from the channel boundaries

\[ \begin{align} \frac{d \tau}{dy} = \frac{dp}{dx} = -\frac{p_{1} - p_{0}}{L} \end{align} \]

where τ is the shear stress. We know that for Newtonian fluids

\[ \begin{align} \tau = \eta \frac{du}{dy} \end{align} \]

where η is the dynamic viscosity of the fluid. Substituting the right side of Equation 2 into the left side of Equation 1 we find

\[ \begin{align} \eta \frac{d^{2}u}{dy^{2}} = \frac{dp}{dx} \end{align} \]

Integrating Equation 3 twice we get

\[ \begin{align} \int \frac{d^{2}u}{dy^{2}} &= \frac{1}{\eta} \int \frac{dp}{dx} && \text{Integrate once} \nonumber\\ \frac{du}{dy} &= \frac{1}{\eta} \frac{dp}{dx}y + c_{1}\\ \int \frac{du}{dy} &= \frac{1}{\eta} \int \frac{dp}{dx}y + c_{1} && \text{Integrate again} \nonumber\\ u(y) &= \frac{1}{2 \eta} \frac{dp}{dx} y^{2} + c_{1}y + c_{2} \end{align} \]

Since we know du/dy = 0 at y = 0, we find c1 = 0, and because we assume zero slip at the boundaries of the channel (u = 0 at y = ±h/2), c2 = (1/2η)(dp/dx)(h/2)2. Thus,

\[ \begin{align} u(y) = \frac{1}{2 \eta} \frac{dp}{dx} \left[ y^{2} + \left( \frac{h}{2} \right)^{2} \right] \end{align} \]

This is the velocity profile for a Newtonian fluid, but as you will recall, we learned that ice is a non-Newtonian (nonlinear viscous) fluid. As such, the behavior should be modeled using a power-law, where the strain rate is a related to the shear stress as follows

\[ \begin{align} \frac{du}{dy} = a \tau^{n} \end{align} \]

where a is a function of the viscosity and temperature (we will ignore temperature for now), and n is the power-law exponent. If we solve Equation 7 in terms of τ, we can substitute the result into Equation 1 to get

\[ \begin{align} a^{-1/n} \frac{d}{dy} \left( \frac{du^{1/n}}{dy} \right) = -\frac{p_{1} - p_{0}}{L} \end{align} \]

Assuming symmetry about the center line, y = 0, and integrating we see

\[ \begin{align} \int \frac{d}{dy} \left( \frac{du^{1/n}}{dy} \right) &= -a^{1/n} \int \frac{p_{1} - p_{0}}{L} \nonumber && \text{Integrate}\\ \frac{du^{1/n}}{dy} &= -a^{1/n} \frac{p_{1} - p_{0}}{L} y + c_{1} \nonumber && \text{Raise both sides to the power } n\\ \frac{du}{dy} &= -a \left( \frac{p_{1} - p_{0}}{L} \right)^{n} y^{n} + c_{1}^{n} \nonumber && \text{Apply boundary condition } \left. \frac{du}{dy} = 0 \right|_{y=0}\\ \frac{du}{dy} &= -a \left( \frac{p_{1} - p_{0}}{L} \right)^{n} y^{n} \end{align} \]

Now integrate Equation 9 and assume u = 0 at y = ±h/2

\[ \begin{align} \int \frac{du}{dy} &= -a \left( \frac{p_{1} - p_{0}}{L} \right)^{n} \int y^{n} \nonumber\\ u(y) &= -\frac{a}{n + 1} \left( \frac{p_{1} - p_{0}}{L} \right)^{n} y^{n+1} + c_{2} && \text{Apply boundary condition } u = 0 \text{ at } y = \pm\frac{h}{2} \nonumber\\ u(y) &= \frac{a}{n + 1} \left( \frac{p_{1} - p_{0}}{L} \right)^{n} \left[ \left( \frac{h}{2} \right)^{n+1} - y^{n+1} \right] \end{align} \]

The mean velocity in the fluid is

\[ \begin{align} \bar{u} &= \frac{2}{h} \int_{0}^{h/2} u dy \nonumber\\ &= \frac{a}{n+2} \left( \frac{p_{1} - p_{0}}{L} \right)^{n} \left( \frac{h}{2} \right)^{n+1} \end{align} \]

We can calculate a non-dimensional velocity now by dividing the velocity at any point in the channel by the mean velocity

\[ \begin{align} \frac{u}{\bar{u}} = \left( \frac{n+2}{n+1} \right) \left[ 1 - \left( \frac{2y}{h} \right)^{n+1} \right] \end{align} \]
Exercise 1 - Planform flow velocity across a glacier

Files for this exercise

The goal of this exercise is to calculate horizontal velocity profiles across a glacier for Newtonian and non-Newtonian fluid flow resulting from a pressure gradient.

  1. Modify the Python script file open_channel_ex1.py to plot the non-dimensional velocity (u/) of a fluid as a function of non-dimensional distance (y/h) across the channel of width h (Equation 12). Assume the flow is symmetrical about = 0 and the velocity is zero at the boundaries of the channel. In your Python script you should
    1. Solve for the non-dimensional velocity across the channel for a Newtonian fluid and for non-Newtonian fluid with power-law exponents of n = 2, 3, 4, 5

      The flow is symmetric about y = 0 and for negative y values the equation does not work properly. You can solve one half of the flow (y ≥ 0) and use symmetry to plot the other half of the flow.

    2. Create one plot of the non-dimensional velocity of all of the fluids as a function of non-dimensional distance.
      1. Be sure to label your axes and add a title
      2. Also include text labels beside each velocity profile listing the power-law exponent
      3. How sensitive is the velocity to the power-law exponent? What is the maximum percent difference in velocity between n = 2 and n = 4?
    3. Include clear comments that explain what each section of your code does
  2. The Saskatchewan glacier near Banff in Alberta, Canada (Figures 2-4) is 1400 m wide and part of a large ice field known as the Columbia Icefield.


    Figure 2. Map of western Canada with the location of the Saskatchewan glacier in Banff, Alberta indicated with a filled black circle.  
    (Source: Microsoft MapPoint 2004)


    Figure 3. Topographic map of Saskatchewan glacier.  
    (Source: http://earth.esa.int/workshops/fringe_1996/cumming/loadmap.htm)


    Figure 4. Saskatchewan glacier from Parker Ridge.  
    (Source: http://www.giorgiozanetti.ca/rockies_2002/banff_large.html)

     Velocity 
    Distance from center line [m][m/a][m/s]
    -660123.80E-07
    -640227.00E-07
    -570421.33E-06
    -460632.00E-06
    -220742.35E-06
    40762.41E-06
    180742.35E-06
    260722.28E-06
    500511.62E-06

    Table 1. Velocity measurements across the Saskatchewan glacier.

    The file sask_glacier_velo.txt contains a series of surface velocities measured at various locations across the glacier (Table 1). Modify the Python script sask_glacier_ex1.py to plot the measured velocities as a function of distance from the center line of the glacier, and also plot predicted velocity profiles for several non-Newtonian fluids (Equation 10). Assume a = 5 × 10-24 Pa-3 s-1. In your program you should

    1. Solve for the velocity profile of a non-Newtonian fluid with power-law exponents of n = 2, 3, 4, 5

      The term (p1 -p0)/L in Equation 10 is an unknown. However, you can solve for it using the condition that the maximum flow velocity (~2.41 × 10-6 m/s) occurs at y = 0. Once you have solved for that term, you can insert it back into Equation 10 to solve for the velocities elsewhere.

      Again, the flow is symmetric about y = 0 and for negative y-values Equation 10 does not work properly. Use the symmetry of the solution to make your plot.

    2. Load the observed velocity data
    3. Plot the measured velocities along with the 4 predicted velocity profiles. Be sure to label your axes and include a title. Also include text labels beside each velocity profile listing the power-law exponent.
    4. Include clear comments explaining what each section of the code does.

For exercise 1 you should submit

  1.  2 Python plots
    1. Non-dimensional velocity profiles for Newtonian and non-Newtonian fluids (Part 1 above)
    2. Velocity profile across the Saskatchewan glacier with points showing the measured velocities (Part 2 above)
  2. Copies of your modified Python scripts that produce the plots
  3. Figure captions for each plot describing the plot as if it were in a scientific journal article.
    1. For the second plot, list the most likely power-law exponent n for the Saskatchewan glacier in the caption text

Non-Newtonian flow down an inclined plane


Background

Figure 5. Schematic diagram of a viscous fluid flowing down an inclined plane.

The velocity of a fluid flowing down an inclined plane can be modelled using some basic physical relationships. First, recall that the basal shear stress for a fluid flowing down an inclined plane is due to the down slope component of the weight of the overlying fluid

\[ \begin{align} \tau = \rho g h \sin(\alpha) \end{align} \]

where ρ is the fluid density, g is the acceleration due to gravity, h is the thickness of the fluid perpendicular to the inclined plane and α is the angle of the plane with respect to horizontal. At any distance z above the bed

\[ \begin{align} \tau &= \rho g (h - z) \sin(\alpha)\nonumber\\ &= \gamma_{x} (h - z) \end{align} \]

where γx ρgsin(α) is the downslope component of the gravitational force. Combining Equation 2 with Equation 14 we find the constitutive equation for a Newtonian fluid is

\[ \begin{align} \frac{du}{dz} = \frac{\gamma_{x}(h - z)}{\eta} \end{align} \]

Integrating Equation 15 yields

\[ \begin{align} \int \frac{du}{dz} &= \int \frac{\gamma_{x}(h - z)}{\eta} \nonumber\\ u(z) &= \frac{\gamma_{x}}{\eta} \left( zh - \frac{z^{2}}{2}\right) + c_{1} \end{align} \]

where c1 = 0 from the boundary condition u = 0 at z = 0. Equation 16 can be rewritten as

\[ \begin{align} u(z) = \frac{1}{2} \left( \frac{\gamma_{x}}{\eta} \right) \left[ h^{2} - (h-z)^{2} \right] \end{align} \]

For a non-Newtonian fluid, Equation 14 can be modified to account for the case where the strain rate varies as a power of the shear stress (Equation 7)

\[ \begin{align} \frac{du}{dz} = a\left[ \gamma_{x} (h - z) \right]^{n} \end{align} \]

To determine the velocity profile, we need to integrate Equation 18. If we use the boundary condition that the basal sliding velocity is equal to ub rather than zero, we get

\[ \begin{align} \int \frac{du}{dz} &= a \int \left[ \gamma_{x} (h - z) \right]^{n} \nonumber\\ u(z) &= u_{\mathrm{b}} + \frac{a}{n+1}\gamma_{x}^{n} \left[ h^{n+1} - (h - z)^{n+1} \right] \end{align} \]
Exercise 2 - Vertical velocity profile in a glacier

Files for this exercise

The goal of this exercise is to calculate vertical velocity profiles across a glacier for Newtonian and non-Newtonian fluid flow resulting from a gravitational forces on the glacier.

  1. The Athabasca glacier (Figures 6, 7) is another glacier in the Columbia Icefield (see Figures 2, 3 for location).


    Figure 6. Athabasca glacier landscape.
    (Source: http://www.pbase.com/image/27382295)


    Figure 7. Athabasca glacier landscape.
    (Source: http://www.canadianheritage.org/reproductions/23026.htm)

    Depth from surfaceHeight from base zHorizontal velocity
    [m][m][m/a][m/s]
    020928.69.10E-07
    1519528.59.00E-07
    3018028.59.00E-07
    4516528.49.00E-07
    6015028.28.90E-07
    7513528.08.90E-07
    9012027.78.80E-07
    10510527.28.60E-07
    1209026.58.40E-07
    1357525.58.10E-07
    1506024.07.60E-07
    1654521.56.80E-07
    1803017.55.50E-07
    19515103.20E-07
    209041.30E-07

    Table 2. Velocities measured from a vertical profile through Athabasca glacier.

    A vertical velocity profile for the Athabasca glacier has been measured and the measurements are in the file atha_glacier_velo.txt (Table 2). Modify the provided Python script atha_glacier_ex2.py to plot the velocity measurements (x-axis) versus the height (z-axis) from the bed. On the same plot, include Newtonian and non-Newtonian fluid flow profiles as well (Equation 19). Assume = 5 × 10-24 Pa-3 s-1. In your code, you should

    1. Solve for the velocity profile for a Newtonian fluid. Your profile should have a no-slip basal boundary condition (ub = 0) and honor the observed surface velocity.

      Again, in this problem there is an unknown term γx in Equation 19. However, you can solve for it using the condition that the maximum flow velocity (~0.91 × 10-6 m/s) occurs at z = h = 209 m. Once you have solved for that term, you can insert it back into Equation 19 to solve for the velocities elsewhere.

    2. Solve for the velocity profiles of non-Newtonian fluids with power-law exponents of n = 2, 3, 4, 5 (Equation 19). You should set ub equal to the observed value (Table 2) and your Python program should honour the observed surface velocity (see tip above)
    3. Load in the observed velocity profile data
    4. Generate a plot of the observed velocity values, the predicted Newtonian velocity profile and the 4 non-Newtonian velocity profiles
    5. Include clear comments that explain what each section of the program does

For Exercise 2 you should submit

  1. One Python plot of the Newtonian and non-Newtonian velocity profiles across the Athabasca glacier with data points showing the measured velocities
  2. A copy of your Python script that produces the plot
  3. A figure caption for the plot describing the plot as if it were in a scientific journal article.
    1. In the caption you should list the most likely value for the power-law n exponent for the Athabasca glacier.

1 Comment

    • Students somewhat confused about where to start. Maybe it would help to give a bit more of an introduction.
    • Values of h and y are confusing. Maybe just list a value for h to get started.