The ratio of inertial to viscous forces is commonly used to scale fluid flow, and is called the Reynolds number, given as:


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1 LAB 3C: STOKES FLOW DUE: WEDNESDAY, MARCH 9 Lab Overview and Background The viscosity of a fluid describes its resistance to deformation. Water has a very low viscosity; the force of gravity causes it to flow immediately. Honey (or karo syrup) has a higher viscosity. The viscosities of Earth materials largely control mantle convection and therefore volcanic activity, heat loss from the interior, and plate tectonics; how volcanoes erupt and how far lava travels; and the shapes of landforms created through weathering, to give a few examples. To give you a feeling for both what viscosity values are and how they are measured in the lab, we will measure honey (or karo syrup) viscosity using Stokes law. Stokes law, derived by George Gabriel Stokes around 1850, describes the velocity of a sphere sinking or rising through a viscous fluid under the influence of gravity. The ratio of inertial to viscous forces is commonly used to scale fluid flow, and is called the Reynolds number, given as: ρνl (1) Re = η where ρ is the density of the fluid, v is the fluid velocity, L is a characteristic length scale, and η is fluid viscosity. The solid flow of the mantle has very low inertia compared to viscosity, so its Reynolds number is vanishingly small. Stokes law pertains well to mantle flow. In the mantle the flow is entirely laminar and very far from turbulence. kg Viscosity in SI is measured in units of Pa s, or Pascalseconds, which are m sec. If a fluid with a viscosity of one Pa s is placed between two plates, and one plate is moved laterally with a shear stress of one pascal for the time interval of one second, the plate moves a distance equal to the thickness of the fluid layer. The cgs unit for viscosity is the poise, which equals 0.1*Pa s. Thus ten poise make a pascalsecond. Viscosity can be also measured using the Stokes flow law. The original formulation of Stokes law simply described the force needed to drive a sphere through a quiescent, continuous, viscous fluid: (2) F = 6πrv S where a sphere of radius r is moving with velocity v S. This law can be rewritten as 2(ρsolid ρ (3) v S = liquid ) gr 2 9η to describe how a sphere moves under the influence of gravity. The density of the steel ball ElkinsTanton, Blackburn and Jordan (2011) 1
2 bearing with radius r is 7,800 kg/m 3. Equation 3 can be solved for viscosity: 2(ρsolid (4) η = ρ 2 liquid ) gr 9v S Our spheres are not falling through a continuous fluid, though, and nor do they have an infinite distance to fall. Stokes velocity can be tempered for edge effects (critical in all fluid tank experiments). Experimentalists have found that the best correction for edge effects is called the Faxon correction, as follows: 2gr 2 ( ρsolid ρliquid ) (5) η = r r 3 r r rc r 9v S 1+ c r c hc where r c is the radius of the container; h c is the distance the sphere falls; and η is in units of poise. Note: this equation is written for cgs units, NOT SI units. You will have to convert to Pascals. By measuring the fall of the sphere using a ruler and a stopwatch, this equation can be used to calculate the viscosity of the fluid. Experimental Procedure There will be a series of experiments measuring honey (or karo syrup) viscosity using Stokes flow, equation 5. You should work in small teams to complete the experiment. 1. Make a table for data. First column: trial number. Second column: radius of the ball bearing. Third column: temperature of the fluid. Fourth column: distance between the marks on the graduated cylinder. Fifth column: time the ball takes to sink between the markings on the graduated cylinder. Later columns: calculations. 2. Measure the radii of several ball bearings (use various sizes). 3. Record the temperature of the fluid. 4. Drop the ball into the fluid and time its fall over a measured distance. 5. Calculate viscosity separately for each ball experiment using equations 3, 4, and 5. Each team should hand in a very neat sheet showing their measurements, calculations, and a graphical representation of temperature and viscosity. Include every team member s name on your final writeup. Each student must complete and hand in the following table and questions separately. You may collaborate with other students, but your work must be your own. ElkinsTanton, Blackburn and Jordan (2011) 2
3 Data (m) (K or C) (m) (sec) (g/cm 3 ) (g/cm 3 ) Stokes r T x t ρ solid ρ liquid This table is for your own personal use during the lab. You must turn in a sheet with your measurements, calculations, and graph (depicting the temperature vs. viscosity relationship) separately. Lab Questions (answer on separate sheet) 1. Does the viscosity of the fluid (honey or karo syrup) exhibit temperature dependence? If so, what is the relationship? 2. In addition to temperature dependence of viscosity, the effective viscosity of the mantle is dependent upon additional parameters, including strain rate, water fugacity, and grain size. The power law equation, 1 n (6) η eff = ε n f r n H 2 O m d n Ae RT 1 n H can be used to quantify the relationship between effective mantle viscosity (η eff ), strain rate (ε), water fugacity (ƒ H2O ), and grain size (d). There are, however, significant exponential dependence for grain size (m) and strain rate (n) that vary depending on the dominant deformation regime. The dislocation regime is dependent upon strain rate (n=3.5) while the diffusion regime is insensitive to it (n=1). Similarly, the diffusion regime is strongly dependent on grain size (m= ) while the dislocation regime is not (m=0). Qualitatively state how these variables affect mantle viscosity. For example, if we increase strain rate in the mantle how will this affect the effective viscosity? ElkinsTanton, Blackburn and Jordan (2011) 3
4 This image has been removed due to copyright restrictions. Figures taken from Global Tectonics, Kearey, Klepeis, and Vine, WileyBlackwell publishing. 3. Viscosity experiments like you ve done in lab can be used to investigate the viscosity of Earth materials and further applied towards understanding the deformation and heat transfer regimes in the Earth. The Rayleigh number is a dimensionless parameter that quantified the mechanisms of heat transfer in the Earth s mantle and is highly sensitive to viscosity. If the viscosity is too high, conductive heat transfer dominates the system; if the viscosity is low and the Rayleigh number is above what we call the critical Rayleigh number, heat is transferred by convection. The Rayleigh number is defined as: (7) where ρ is fluid density (kg/m 3 ), g is gravity (m/s 2 ), α is thermal expansivity (K 1 ), ΔT is the temperature difference across the region in question, h is the height of the region (m), η is viscosity (Pa s), and κ is thermal diffusivity (m 2 /s). Given parameters for the Earth s mantle in Table 1, complete the following: a. Calculate if the whole mantle is capable of convecting. b. What is the minimum viscosity that will permit convection? Table 1 ρ (kg/m 3 ) 4000 g (m/s 2 ) 10 α (1/K) ΔT (K) 3500 h (m) κ (m 2 /s) 106 Critical Ra η (Pa s) ElkinsTanton, Blackburn and Jordan (2011) 4
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