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- Question : 1E - Many centuries ago, a mariner poured 100 cm3 of water into the ocean. As time passed, the action of currents, tides, and weather mixed the liquid uniformly throughout the earth
- Question : 2E - 1 An adult human expels approximately 500 mL of air with each breath during ordinary breathing. Imagining that two people exchanged greetings (one breath each) many centuries ago and that their breath subsequently has been mixed uniformly throughout the atmosphere, estimate the probability that the next breath you take will contain at least one air molecule from that age-old verbal exchange. Assess your chances of ever getting a truly fresh breath of air. For this problem, assume that air is composed of identical molecules having Mw
- Question : 3E - In Cartesian coordinates, the Maxwell probability distribution, f(u)
- Question : 4E - Using the Maxwell molecular velocity distribution given in Exercise 1.3 with U
- Question : 5E - By considering the volume swept out by a moving molecule, estimate how the mean-free path, l, depends on the average molecular cross section dimension d and the molecular number density n~ for nominally spherical molecules. Find a formula for ln~1=3 (the ratio of the mean-free path to the mean intermolecular spacing) in terms of the molecular volume (d 3 ) and the available volume per molecule (1=n~). Is this ratio typically bigger or smaller than one?
- Question : 6E - In a gas, the molecular momentum flux (MFij) in the j-coordinate direction that crosses a flat surface of unit area with coordinate normal direction i is: MFij
- Question : 7E - Consider the viscous flow in a channel of width 2b. The channel is aligned in the x-direction, and the velocity u in the x-direction at a distance y from the channel centerline is given by the parabolic distribution u
- Question : 8E - Estimate the height to which water at 20C will rise in a capillary glass tube 3 mm in diameter that is exposed to the atmosphere. For water in contact with glass the wetting angle is nearly 90. At 20C, the surface tension of a water-air interface is s
- Question : 9E - A manometer is a U-shaped tube containing mercury of density rm. Manometers are used as pressure-measuring devices. If the fluid in tank A has a pressure p and density r, then show that the gauge pressure in the tank is: p patm
- Question : 10E - Prove that if e(T, y)
- Question : 11E - Starting from the property relationships (1.18) prove (1.25) and (1.26) for a reversible adiabatic process when the specific heats Cp and Cv are constant.
- Question : 12E - A cylinder contains 2 kg of air at 50C and a pressure of 3 bars. The air is compressed until its pressure rises to 8 bars. What is the initial volume? Find the final volume for both isothermal compression and isentropic compression
- Question : 13E - Derive (1.29) starting from the arguments provided at the beginning of Section 1.10 and Figure 1.8
- Question : 14E - Starting with the hydrostatic pressure law (1.8), prove (1.30) without using perfect gas relationships
- Question : 15E - Assume that the temperature of the atmosphere varies with height z as T
- Question : 16E - Suppose the atmospheric temperature varies according to: T
- Question : 17E - Consider the case of a pure gas planet where the hydrostatic law is: dp=dz
- Question : 18E - Consider a heat-insulated enclosure that is separated into two compartments of volumes V1 and V2, containing perfect gases with pressures of p1 and p2, and temperatures of T1 and T2, respectively. The compartments are separated by an impermeable membrane that conducts heat (but not mass). Calculate the final steadystate temperature assuming each gas has constant specific heats.
- Question : 19E - Consider the initial state of an enclosure with two compartments as described in Exercise 1.18. At t
- Question : 20E - A heavy piston of weight W is dropped onto a thermally insulated cylinder of crosssectional area A containing a perfect gas of constant specific heats, and initially having the external pressure p1, temperature T1, and volume V1. After some oscillations, the piston reaches an equilibrium position L meters below the equilibrium position of a weightless piston. Find L. Is there an entropy increase?
- Question : 21E - A gas of noninteracting particles of mass m at temperature T has density r, and internal energy per unit volume 3. a) Using dimensional analysis, determine how 3 must depend on r, T, and m. In your formulation use kB
- Question : 22E - Many flying and swimming animalsdas well as human-engineered vehiclesdrely on some type of repetitive motion for propulsion through air or water. For this problem, assume the average travel speed U depends on the repetition frequency f, the characteristic length scale of the animal or vehicle L, the acceleration of gravity g, the density of the animal or vehicle ro, the density of the fluid r, and the viscosity of the fluid m. a) Formulate a dimensionless scaling law for U involving all the other parameters. b) Simplify your answer for part a) for turbulent flow where m is no longer a parameter. c) Fish and animals that swim at or near a water surface generate waves that move and propagate because of gravity, so g clearly plays a role in determining U. However, if fluctuations in the propulsive thrust are small, then f may not be important. Thus, eliminate f from your answer for part b) while retaining L, and determine how U depends on L. Are successful competitive human swimmers likely to be shorter or taller than the average person? d) When the propulsive fluctuations of a surface swimmer are large, the characteristic length scale may be U/f instead of L. Therefore, drop L from your answer for part b). In this case, will higher speeds be achieved at lower or higher frequencies? e) While traveling submerged, fish, marine mammals, and submarines are usually neutrally buoyant (ro z r) or very nearly so. Thus, simplify your answer for part b) so that g drops out. For this situation, how does the speed U depend on the repetition frequency f ? f) Although fully submerged, aircraft and birds are far from neutrally buoyant in air, so their travel speed is predominately set by balancing lift and weight. Ignoring frequency and viscosity, use the remaining parameters to construct dimensionally accurate surrogates for lift and weight to determine how U depends on ro/r, L, and g
- Question : 23E - The acoustic power W generated by a large industrial blower depends on its volume flow rate Q, the pressure rise DP it works against, the air density r, and the speed of sound c. If hired as an acoustic consultant to quiet this blower by changing its operating conditions, what is your first suggestion?
- Question : 24E - A machine that fills peanut-butter jars must be reset to accommodate larger jars. The new jars are twice as large as the old ones but they must be filled in the same amount of time by the same machine. Fortunately, the viscosity of peanut butter decreases with increasing temperature, and this property of peanut butter can be exploited to achieve the desired results since the existing machine allows for temperature control. a) Write a dimensionless law for the jar-filling time tf based on: the density of peanut butter r, the jar volume V, the viscosity of peanut butter m, the driving pressure that forces peanut butter out of the machine P, and the diameter of the peanut buttere delivery tube d. b) Assuming that the peanut butter flow is dominated by viscous forces, modify the relationship you have written for part a) to eliminate the effects of fluid inertia. c) Make a reasonable assumption concerning the relationship between tf and V when the other variables are fixed so that you can determine the viscosity ratio mnew/mold necessary for proper operation of the old machine with the new jars.
- Question : 25E - As an idealization of fuel injection in a diesel engine, consider a stream of high-speed fluid (called a jet) that emerges into a quiescent air reservoir at t
- Question : 26E - 3 One of the simplest types of gasoline carburetors is a tube with a small port for transverse injection of fuel. It is desirable to have the fuel uniformly mixed in the passing airstream as quickly as possible. A prediction of the mixing length L is sought. The parameters of this problem are: r
- Question : 27E - Consider dune formation in a large horizontal desert of deep sand. a) Develop a scaling relationship that describes how the height h of the dunes depends on the average wind speed U, the length of time the wind has been blowing Dt, the average weight and diameter of a sand grain w and d, and the air
- Question : 28E - An isolated nominally spherical bubble with radius R undergoes shape oscillations at frequency f. It is filled with air having density ra and resides in water with density rw and surface tension s. What frequency ratio should be expected between two isolated bubbles with 2 cm and 4 cm diameters undergoing geometrically similar shape oscillations? If a soluble surfactant is added to the water that lowers s by a factor of two, by what factor should air bubble oscillation frequencies increase or decrease?
- Question : 29E - In general, boundary layer skin friction, sw, depends on the fluid velocity U above the boundary layer, the fluid density r, the fluid viscosity m, the nominal boundary layer thickness d, and the surface roughness length scale 3. a) Generate a dimensionless scaling law for boundary layer skin friction. b) For laminar boundary layers, the skin friction is proportional to m. When this is true, how must sw depend on U and r? c) For turbulent boundary layers, the dominant mechanisms for momentum exchange within the flow do not directly involve the viscosity m. Reformulate your dimensional analysis without it. How must sw depend on U and r when m is not a parameter? d) For turbulent boundary layers on smooth surfaces, the skin friction on a solid wall occurs in a viscous sublayer that is very thin compared to d. In fact, because the boundary layer provides a buffer between the outer flow and this viscous sub-layer, the viscous sublayer thickness lv does not depend directly on U or d. Determine how lv depends on the remaining parameters. e) Now consider nontrivial roughness. When 3 is larger than lv a surface can no longer be considered fluid-dynamically smooth. Thus, based on the results from parts a) through d) and anything you may know about the relative friction levels in laminar and turbulent boundary layers, are high- or low-speed boundary layer flows more likely to be influenced by surface roughness?
- Question : 30E - Turbulent boundary layer skin friction is one of the fluid phenomena that limit the travel speed of aircraft and ships. One means for reducing the skin friction of liquid boundary layers is to inject a gas (typically air) from the surface on which the boundary layer forms. The shear stress, sw, that is felt a distance L downstream of such an air injector depends on: the volumetric gas flux per unit span q (in m2 /s), the free stream flow speed U, the liquid density r, the liquid viscosity m, the surface tension s, and the gravitational acceleration g. a) Formulate a dimensionless law for sw in terms of the other parameters. b) Experimental studies of air injection into liquid turbulent boundary layers on flat plates has found that the bubbles may coalesce to form an air film that provides near perfect lubrication, sw/0 for L > 0, when q is high enough and gravity tends to push the injected gas toward the plate surface. Reformulate your answer to part a) by dropping sw and L to determine a dimensionless law for the minimum air injection rate, qc, necessary to form an air layer. c) Simplify the result of part b) when surface tension can be neglected. d) Experimental studies (Elbing et al., 2008) find that qc is proportional to U2 . Using this information, determine a scaling law for qc involving the other parameters. Would an increase in g cause qc to increase or decrease?

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