Major ideas
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1. The flow between two independently rotating co-axial cylinders has an exact solution to the equation of motion for moving fluids subject to "no-slip boundary conditions" (the flow velocity much match the velocity of any walls with which it is in contact at the locations of the walls). Here the equation of motion, Newton's second law for fluids, is written as the Navier-Stokes equation: it expresses how the time variation of the flow velocity field is related to spatial derivatives of the velocity field and to forces imposed on the fluid (including viscous stresses). In describing fluid motion, this equation must be accompanied by an equation of local conservation of mass (the "continuity equation") which for incompressible fluids is simply that the velocity field has zero divergence. (The form of the exact solution is found in the first prelab exercise.)
2. The exact solution described above can become unstable to small perturbations to the velocity field. This happens when, for a given rotational speed Omega_2 of the outer cylinder, the rotational of the inner cylinder Omega_1 reaches a critical value. This critical value is typically expressed in terms of a critical value of the Re_c of the corresponding "Reynolds number" (Re) that is a dimensionless parameter defining the conditions imposed on the fluid. (The definition of Reynolds number is explored in the second prelab exercise.) When instability occurs, a new flow state evolves with a pattern that has a spatial periodicity in the axial direction. This periodicity is typically expressed in terms of a "critical wavenumber" k_c that appears in mathematical representations of the spatial periodicity of the perturbation to which the original flow state is most sensitive. We say that the original simple flow state has undergone a "pattern forming instability."
Major equipment
1. Micro-stepping stepper motors are used to rotate the cylinders. These motors are controlled by computer commands through a serial-communications interface between the stepper-motor "driver" and the host computer running the experiment.
2. The flow pattern is made visible by a technique of "flow visualization". This technique uses a suspension of small particles that are neutrally buoyant in the flow and which align with the flow like tiny mirrors that show the local shear pattern in the flow. Time variations in the flow pattern can be recorded and analyzed by measuring the intensity of light reflecting off of the flow visualization material. One way to do this is to shine a laser onto a location in the flow and measure the intensity of the light scattered into a detector.
Other features to know about:
3. The working fluid is a mixture of water and glycerol designed to have a viscosity about 10 times greater than the viscosity of water. This results in rotation speeds and time-scales of the experiment that are "reasonable". (The speeds and time scales are too slow using just water.)
4. Careful experiments require cylinders that have been precision machined to high tolerance. Ideally, the gap between the cylinders should be uniform to 1 part in a thousand, even as the cylinders rotate (there should be no "wobble" in the gap).
5. The experiment must also be controlled to prevent undue influence of temperature variations in the fluid, since these affect density and viscosity. Ideally, these variations in time and space should be well below 0.1 degree Celsius. Such temperature control can be achieved by various means, usually by immersing the whole cylinder system in a temperature-regulated water bath. (This is not the case for our set-up at present so we must try other measures to prevent non-uniform temperatures.)
6. A major step up in instrumentation for measuring the flow is to make local measurements of the actual velocity at a point using a technique called laser-Doppler velocimetry (LDV). (This is not currently available in our system; we intend to eventually implement this for the experiment.)
Data analysis
1. Plot critical "inner cylinder" Reynolds number versus "outer cylinder" Reynolds number and compare to predicted values.
2. Plot critical wavenumber versus "outer cylinder" Reynolds number and compare to predicted values.
(For this lab, the prelab may be done after the first lab day and before the second lab. This will allow us time to discuss key concepts such as the Naiver-Stokes equation during the first lab meeting. However, to prepare for this discussion, you are strongly advised to start looking up some of the terminology and concepts used in the prelab problems before the first lab.)
(1) Look up the form of the Navier-Stokes equation in cylindrical polar coordinates. Simplify it assuming axial and azimuthal symmetry: that is, eliminate any terms containing derivatives with respect to the axial coordinate (z) and azimuthal coordinate (phi). Show that a solution to the equation is the following:
v_r = 0
v_phi = A r + B / r
v_z = 0
Find expressions for the coefficients A and B by matching "no-slip boundary" conditions:
v_phi = Omega_1 * r_1 at r = r_1
v_phi = Omega_2 * r_2 at r = r_2
(2) Look up the Navier-Stokes equation in vector form (containing del and grad operators). Assume there is typical distance scale d and a typical velocity scale V_0, giving a typical time scale t_0 = d / V_0. Write down the Navier-Stokes equation in dimensionless form by:
-- writing spatial coordinates as x = x_tilda * d
-- writing time coordinates as t = t_tillda * t_0 = t_tilda * d / V_0
-- writing velocity components as v = v_tilda * v_0
Show that the dimensionless equation can be written in a form that contains only one dimensionless parameter called the Reynolds number
Re = V_0 * d / nu
where nu is the kinematic viscosity (units m^2/s) which is related to the dynamic viscosity mu (units Pa-s) and density rho (units Kg/m^3) by
nu = mu / rho
Python code for controlling the motor speeds:
Download Jupyter notebook for control
Python code for aquiring and analyzing images
Download Jupyter notebook for imaging and Fourier transforms
Books
Faber, T. E. (1995), Fluid Dynamics For Physicists (Cambridge).
Guyon, E., J. Hulin, L. Petit, and C. D. Mitescu (2001), Physical Hydrodynamics (Oxford Univ. Press).
Tritton, D. J. (1988), Physical Fluid Dynamics, 2nd ed. (Oxford Univ. Press).
Journal articles
Langford, W.F.; Tagg, R.; Kostelich, E.J.; Swinney, H.L.; Golubitsky, M., "Primary instabilities and bicriticality in flow between counter-rotating cylinders," Physics of Fluids, v 31, n 4, p 776-85, April 1988
R. Tagg, “The Taylor Couette Problem” Nonlinear Science Today, Vol. 4, 1994, pp. 1-25.