OCEN 678-600 Fluid Dynamics for Ocean and Environmental Engineering Homework #2 Viscous Flow Date distributed : 9.18.2005 Date due : 9.29.2005 at 5:00 pm Return your solution either in class or in my mail box (WERC Rm. 235F) by the date shown above. Please show all your work and follow the rules outlined in the course syllabus. 1 Viscosity Meter A viscosity meter consists of two concentric cylinders, where the outer cylinder is fixed and the inner cylinder is allowed to rotate. Fluid is poured into the gap between the cylinders and the torque Ω required to rotate the inner cylinder at constant velocity is measured. In the following analysis, assume that the extent of the viscosity meter in the z-direction is great enough that boundary effects at the top and bottom can be ignored. R 2 R 1 z θ r Figure 1: Diagram of a viscosity meter showing the cylindrical-polar coordinate system and the device dimensions. CIVIL 3136 TAMU College Station, Texas 77843-3136 (979) 845-4517 FAX (979) 862-8162
1. Show that the exact equations for this idealized case are u θ θ = 0 p r 0 = 1 p ρr θ + ν 0 = 1 p ρ z u2 θ r = 1 ρ ( 1 r r ( r u ) θ u ) θ r r 2 (1) 2. Assuming a narrow gap between cylinders h/r 1, where h = R 2 R 1, and slow flow, we can find the approximate solution for the velocity profile between the gap as u θ (R) = 1 ( dp R R 2 ln (R/R ) 2) µ dθ ln (R 1 /R 2 ) (R 1 R 2 ) + ln(r/r 2) ln(r 1 /R 2 ) R θ 1 0 (2) where θ 0 is the steady rotation rate of the inner cylinder in rad/s. From this velocity profile, find a relationship between the viscosity of the fluid and the torque applied to the inner cylinder for steady motion of the cylinder under the case dp/dθ = 0. 3. Do you obtain the same results if the cylinder is oriented so that the z-axis is perpendicular to the gravity vector? Why or why not? 2 Shallow Flow down an Incline In class we derived the solution for shallow free-surface flow down a slight incline of constant slope. Use this model to evaluate the flow through the Cape Cod Channel. 1. Use the lubrication flow solution to find the surface velocity of water of depth 0.3 mm flowing down an inclined plane of angle 1 degree. Does this estimate seem reasonable? Why or why not? 2. Estimate the surface current in the Cape Cod Channel, where the depth is 10 m, the length is 10 km, and the maximum difference in elevation between the two ends is 3 m. Give several reasons why this estimate is greater than the measured maximum surface velocity of about 5 knots. An alternative model of the Cape Cod Channel is a flat channel driven by tidal forcing. Reformulate the problem of Cape Cod Channel by assuming that the motion is sinusoidal in time with a frequency ω corresponding to a tidal period of 12 hours and that the velocity is horizontal and a function only of the vertical coordinate, e.g., u = f(y) cos(ωt). Assume 2
the water has a uniform density ρ. Note that sinusoidal functions can be written in terms of the exponential function by using the real part operator Re using the relationship Re [ e iωt] = Re [cos(ωt) + i sin(ωt)] = cos(ωt) (3) 1. Show that the exact equation for the x-direction momentum conservation is u t = 1 p ρ x + u ν 2 (4) y 2 2. For tidal motion, the pressure gradient can be related to the maximum slope of the tidal wave dh/dx according to 1 [ p ρ x = Re g dh ] dx eiωt Substitute this expression and the assumed form of the velocity profile u = f(y)e iωt into the momentum equation to obtain an ordinary differential equation for the unknown function f given by (5) d 2 f dy iω 2 ν f = g dh ν dx (6) Notice that dh/dx is a known constant that is independent of y. 3. Equation (6) is an inhomogeneous ordinary differential equation. The general solution can be found by first solving the homogeneous equation and then finding a particular solution using the method of undetermined coefficients. Show that the following general solution for f satisfies Equation (6) [ f(y) = Re Ae ky + Be ky + ig ω ] dh dx if k satisfies the dispersion relation ( k 2 iω ) = 0 (8) ν (7) A and B are constants to be evaluated from the boundary conditions. 4. Use the no-slip boundary condition at the bed u(0,t) = 0 and the shallow flow stressfree surface boundary condition u(h, t)/ y = 0 to obtain values for the unknown coefficients A and B in Equation (7). You can leave your solution in terms of the complex exponential functions. 3
z x h(z) h(x) R 0 (z) Air Icicle Water Water γ γ R y (a) (b) Figure 2: Schematic of the sheet flow down an icicle far from the tip. 5. Use Matlab to plot the velocity profile in Cape Cod Channel at times 0, 3, 6, 9, and 12 hours (beware of your units and note that you can use the Matlab commandreal to obtain the real part of your complex solution. From the figure, determine the value of dh/dx that is needed to obtain a maximum velocity of 5 knots. Is this value consistent with a tidal wave with amplitude about 1 m and wave length about 450 km? Why or why not? Adapted from Newman, Marine Hydrodynamics, The MIT Press, Cambridge, MA, 1977. 3 Liquid Water Flow along an Icicle Short et al. (Short, Baygents, & Goldstein, A free-boundary theory for the shape of the ideal dripping icicle, Physics of Fluids 18, 083101, 2006) analyze the flow of water down an icicle in order to obtain a non-dimensional relationship for the geometry of icicles. The following paragraph is taken from their paper with a slight change in the names of the variables so that it is more consistent with our notation: We first consider the water layer flowing down the surface of a growing icicle to set some initial scales. The volumetric flow rate Q over icicles is typically on the order of tens of milliliters per hour ( 0.01 cm 3 /s), and icicle radii are usually in the range of 1 10 cm. To understand the essential features of the flow, consider a cylindrical icicle of radius R 0, over the surface of which flows an aqueous film of thickness h (refer to Figure 2(a)). Since h R 0 over nearly the entire icicle surface, the velocity profile in the layer may be determined as that flowing on a flat surface. Furthermore, we expect the Reynolds number to be low enough that the Stokes approximation is valid [e.g., Re O(1) or less]. If y is a coordinate normal to the surface and γ is the angle that the icicle surface makes 4
with respect to the horizontal (see Figure 2(b)), then the Stokes equation for gravity-driven flow is νd 2 u/dy 2 = g sin γ, where g is the gravitational acceleration and ν = 0.01 cm 2 /s is the kinematic viscosity of water. Enforcing no-slip and stress-free boundary conditions at the solid-liquid and liquid-air interfaces, the thickness is ( ) 1/3 3Qν h =. (9) 2πgR 0 sin γ Using typical flow rates and radii, we deduce a layer thickness that is tens of microns and surface velocities u s (gh 2 /2ν) sin γ below several mm/s, consistent with known values, yielding Re = 0.01 0.1, well in the laminar regime as anticipated. At distances from the icicle tip comparable to the capillary length (several millimeters), the complex physics of pendant drop detachment takes over and the thickness law Equation (9) ceases to hold. Verify their analysis by working through the following steps. 1. Write the exact equations for steady, gravity-driven, sheet flow down a cylinder using cylindrical coordinates. Simplify the equations for steady flow and the given geometry. Gravity is the only driving force; for now, leave this body force as f = (f R, 0,f z ). 2. Use the assumption of shallow sheet flow (h R 0 ) to show that the equations from step one can be written in the form for two-dimensional flow over a flat plate. That is, show that the exact equations may be approximated by u R R + u z z = 0 (10) ) u R p u R u R R + u z u z u R R + u u z z ( 2 z = 1 ρ R + ν u R R + 2 u z + f 2 z 2 R (11) ( ) z = 1 p 2 ρ z + ν u z R + 2 u z + f 2 z 2 z (12) 3. Use the coordinate system in Figure 2(b) to convert the equations in step two to a set of equations in the Cartesian coordinate system x y for the flow over a flat plate. Substitute the correct form of the gravity terms based on the geometry in the figure. 4. Use scale analysis and the assumptions of shallow flow (h L), Stokes approximation (Re h 1), and a shallow slope (( π γ) 1) to obtain the following set of approximate 2 5
governing equations u x + v y = 0 (13) 1 p = g cos γ ρ y (14) ν 2 u = g sin γ y2 (15) 5. Solve these simplified equations for the u-component velocity to obtain u(y) = g sin γ ( ) y 2 ν 2 hy (16) by applying a no slip boundary condition at the icicle wall (u(0) = 0) and a shallow-flow stress-free boundary condition at the free surface (du(h)/dy = 0). 6. To obtain an equation for the net flux Q over the icicle under the assumption h R 0, we may use the solution for flow over a flat plate just obtained as follows Q(x) = 2πR 0 (x) h 0 udy (17) Solve this equation for the thickness h to obtain the expression in Equation (9). 7. From real icicles, we measure the net flux to be Q 0.01 cm 3 /s, a typical radius to be R 0 1 10 cm, and a typical wall angle as γ 5. Using these data, verify the assumptions in this derivation by showing that h R 0 and Re h 1. 6