Question 1:



5130: Viscous Flow Name: _________________

Final Exam

December 13, 2006

Question 1:

Molten solder is used to attach two brass blocks, as shown in the figure below:

[pic]

Side View:

[pic]

The solder is poured from a large reservoir into a very thin, approximately 2D gap, which has a constant thickness, h. The flow is initiated at time t=0 by opening the gate in the reservoir. The elevation of the solder in the reservoir is maintained at a constant value H, and the gap is considered thin enough that:

L1, L2, H >>> h

The liquid solder has a known density, ρ, and viscosity, μ.

We are interested in how fast the solder front, s(t) (see above), travels down the gap as a function of time, assuming that the gap is so thin that the flow in it is highly viscous and the flow is locally fully developed. (The locally fully developed flow assumptions do not necessarily apply within a few gap thicknesses upstream of the advancing solder front, but since h > 1, where L is the pan length. A fan blows a steady stream of air over the liquid surface as shown in the figure. The air exerts a shear force, τair, along the free surface of the liquid, which has a viscosity, μ. This tangential drag force (per unit area) on the surface produces a steady laminar flow of the liquid causing the liquid height h to vary along the pan length (the direction x in the figure below). It may be assumed that the inclination of the free surface is small and that the flow is essentially two-dimensional and approximately in the x-direction (except near the bounding walls on the side). It may also be assumed that along the free surface τair is constant and known, and the pressure is Pa (atmospheric). Moreover, the liquid layer is thin enough that inertial effects may be neglected.

[pic]

Part 1: Neglecting the end walls, derive an equation for the velocity profile at any station x, expressed in terms of z, μ, ρ, g, τair, and the local value of h and its derivatives with respect to x.

Part 2: Obtain an expression for the x-direction volume flow rate per unit depth (into the page) at any location x.

Part 3: Suppose we now consider the case of a final steady-state with walls at the left and right end of the pan. Sketch the velocity profile (as quantitatively as possible) in the liquid at a station x away from the end walls of the pan (be sure to make clear the profile shape at the air-liquid interface). If the liquid elevation at the center of the pan (x=0) is h0, obtain an explicit solution for h(x).

Question 3:

In this course we have concerned ourselves largely with boundary layers for engineering applications and components (hydrofoils, turbine blades, flame arrestors, paper coating process, etc.) that are on the order of cm-to-m as the primary length scale. In this problem consider the boundary layer of the entire earth. Measurements indicate that atmospheric boundary layers are very thick, but follow formulas similar to those of flat plate theory.

1. What is an appropriate Reynolds number for the boundary layer of the earth? The radius of the earth is about 6,378 km. Is flat plate theory justified?

2. Consider wind blowing at 10 m/s at a height of 100 m above a smooth beach. Estimate the wind shear stress, in N/m2, on the beach at standard sea-level conditions.

3. What will the wind velocity be at eye-level if you are standing up?

4. What will be the wind velocity if you are lying down on a blanket?

5. Comment on this results from questions (4) and (5).

Question 4:

Consider a flat plate with uniform suction along the plate and an uniform freestream velocity, U∞, as shown in the picture below (or refer to Figure 4-21 of White’s Viscous Flow):

[pic]

Section 4-5.2 does not give much information on how to solve this problem, so consider the following approach for arriving at a solution, which is quite similar to the approach that we have used numerous times in class. To make things even easier we will not deal with the stream function in this particular exercise.

Step 1: Make order of magnitude estimates using the Boundary Layer approximations of the Navier-Stokes Equations and Continuity to arrive at an expression for the boundary layer thickness, δ, as a function of only ν and v0.

Step 2: Use this value to suggest an appropriate ‘similarity’ variable, η.

Step 3: Put this similarity variable into the N-S approximation and obtain a second order differential equation which is only a function of f(η). Again, instead of dealing with a stream function, simply let the function f(η)=u/U∞.

Step 4: Solve this equation using a solution of the form f(η)=Cerη, where C and r are constants to be determined. Utilizing appropriate boundary conditions, obtain an expression for u/U∞.

Comment on these results, in particular, what are your thoughts on the so-called similarity variable that you selected in Step 2? Is this flow similar in nature, as the Blasius flat plate solution?

Step 5: Obtain an expression for the displacement thickness, δ*.

Step 6: Obtain an expression for the wall shear stress, τw.

Table of Useful Integrals

|Function |Indefinite Integral |Comments |

|xn |x(n+1)/(n+1) |n ≠ -1 |

|1/x |ln |x| | |

|1/(ax+b) |(1/a) ln |ax+b| | |

|x/(ax+b) |(1/a²)(ax + b - b ln|ax + b|) | |

|1/(a²+x²) |(1/a) arctan (x/a) | |

|1/(a²-x²) |(1/a) arctanh (x/a) | |

|1/sqrt(a²-x²|arcsin (x/a) |0 ≤ x ≤ a |

|) | | |

|1/sqrt(x²-a²|ln(x + sqrt(x²-a²)) |0 ≤ a ≤ x |

|) | | |

|1/sqrt(x²+a²|ln(x + sqrt(x²+a²)) | |

|) | | |

|x/sqrt(a²-x²|-sqrt(a²-x²) |0 ≤ x ≤ a |

|) | | |

|x/sqrt(x²-a²|sqrt(x²-a²) |0 ≤ a ≤ x |

|) | | |

|x/sqrt(x²+a²|sqrt(x²+a²) | |

|) | | |

|x |(-1/3)(a²-x²) sqrt(a²-x²) |0 ≤ x ≤ a |

|sqrt(a²-x²) | | |

|x |(1/3)(x²-a²) sqrt(x²-a²) |0 ≤ a ≤ x |

|sqrt(x²-a²) | | |

|x |(1/3)(x²+a²) sqrt(x²+a²) | |

|sqrt(x²+a²) | | |

|sin x |-cos x | |

|cos x |sin x | |

|sec² x |tan x | |

|csc² x |-cot x | |

|tan x |-ln cos x | |

|sec x |ln(sec x + tan x) | |

|csc x |ln(csc x - cot x) | |

|sin² x |x/2 - sin(2x)/4 | |

|cos² x |x/2 + sin(2x)/4 | |

|tan² x |tan(x) - x | |

|sin(ax)sin(b|(1/2)(sin(a-b)x/(a-b) - |a ≠ ±b |

|x) |sin(a+b)x/(a+b)) | |

|sin(ax)cos(b|-(1/2)(cos(a-b)x/(a-b) + |a ≠ ±b |

|x) |cos(a+b)x/(a+b)) | |

|cos(ax)cos(b|(1/2)(sin(a-b)x/(a-b) + |a ≠ ±b |

|x) |sin(a+b)x/(a+b)) | |

|x sin(ax) |(sin(ax) - ax cos(ax))/a² | |

|x cos(ax) |(cos(ax) + ax sin(ax))/a² | |

|arcsin x |x arcsin x + sqrt(1-x²) | |

|arccos x |x arccos x - sqrt(1-x²) | |

|arctan x |x arctan x - ln(1+x²)/2 | |

|eax+b |(1/a)(eax+b) | |

|xeax |((ax-1)/a²)(eax) | |

|eaxsin(bx) |(eax/(a²+b²)) (a sin(bx) - b | |

| |cos(bx)) | |

|eaxcos(bx) |(eax/(a²+b²)) (a cos(bx) + b | |

| |sin(bx)) | |

|sinh x |cosh x | |

|cosh x |sinh x | |

|tanh x |ln cosh x | |

|ln x |x ln x - x | |

|x ln x |x²((1/2)ln(x) - (1/4)) | |

|(ln x)/x |(ln²x)/2 | |

|1/(x ln x) |ln(ln(x)) | |

-----------------------

g

x

z

Pa

liquid (ρ, μ)

wind

wind

h

h0

τair

L/2

L

g

atmosphere at Pa

θ2

θ1

x

s(t)

L2

L1

molten solder

gate opens at time t=0

gap thickness = h

molten solder

H

width = W

U∞

v0

Suction

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