3. Understanding of Convective and Diffusive Fluxes. A solution containing species A in water moves to the right with a mass average velocity of 0.06cm/min. Species A obeys Fick's law with diffusion coefficient of 10-5 cm²/s. The mass concentration of species A vs. position is shown in the figure. Find the diffusion flux of species A at the position x=2cm. Is the total flux of A greater or less than the convective flux of A? Justify your answer. PA (g/ml) 0.004- Flow 0.003 0.002 0.001 0 0 1 2 3 x (cm)

Fluid runs through a drainage pipe with a 10-cm radius and a length of 10 m (1000 cm). The velocity of the fluid gradually decreases from the center of the pipe toward the edges as a result of friction with the walls of the pipe. For the data shown, v (x) is the velocity of the fluid (in cm/sec) and X represents the distance (in cm) from the center of the pipe toward the edge. 1 5 8 9 2 3 4 6 7 v (x) 195.9 195.2 194.2 192.7 190.8 188.4 185.6 182.3 178.6 174.5 Part: 0/ 4 Part 1 of 4 (a) The pipe is 10 m long (1000 cm). Determine how long it will take fluid to run the length of the pipe through the center of the pipe. Round to 1 decimal place. It will take fluid approximately 5.104 sec to run the length of the pipe through the center of the pipe. Part: 1/4 Part 2 of 4 (b) Determine how long it will take fluid at a point 9 cm from the center of the pipe to run the length of the pipe. Round to 1 decimal place. It will take fluid approximately sec to run the length of the pipe at a point 9…

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2. Sketch the streamlines for the flow v = -ay, w = 0, u = ax, where a is a positive constant. Next, let the concentration of some pollutant in the fluid be: c(x, y, t) = ßx²yet, for y>0, where ß is a constant. Does the pollutant concentration for any particular fluid element change with time? An alternative way of describing any flow is to specify the position x of each fluid element at time t in terms of the position X of that element at t= 0. For the above flow this Lagrangian' description is x = Xeat, y = Yeat z= Z. Verify by direct calculation that du = u, ốt Du Dt in this particular case. Why are these results true in general?

9. WATER DRAINING FROM A BOTTLE When water flows out of a bottle, the potential energy of the water (Ep = mgh = pAh, where p is the density of water, A is the cross-sectional area of the bottle and h is the height of the water in the bottle) is turned into kinetic energy (Ek = mv² = pa(d)2, where a is the cross-sectional area of the hole out of which the water drains). Energy is conserved, so these quantities must be equal: Ek = Ep 1 dh pa()² = pAh dt we can simplify this equation to (dh)² = dt = 2=h a Let's define the ratio A/a as the constant c, which we could measure. Then, taking the square root of both sides, we get the equation dh √2ch dt where we negative sign arises because we know the height of the water in the bottle decreases with time. a.) Solve the differential equation for h(t) assuming the initial height is h(0) = 1. Don't worry about units, and your answer will involve the known constant c. b.) Of course, the height of water in the bucket should decrease and,…

Fourier's Law of heat transfer (or heat conduction) states that the heat flow vector F at a point is proportional to the negative gradient of the temperature: that is, F = − kVT, which means that heat energy flows from hot regions to cold regions. The constant k is called the conductivity, which has metric units of J/m-s-K or W/m-K. A temperature function T for a region D is given below. Find the SSF FondSk -KSS VT n dS across the boundary S of D. It may be easier to use the Divergence Theorem and evaluate a triple integral. Assume that k = 1. S S T(x,y,z) = 65e¯x² - y² − z²; net outward heat flux D is the sphere of radius a centered at the origin.

A one quarter scale model aircraft is tested at high incidence in a compressed air wind tunnel where the density is 10 times standard air density. It is found that the wings shed vortices at a frequency of 25 Hz. For conditions of dynamic similarity, what will be the equivalent vortex shedding frequency for the full-scale aircraft? You may assume that the air dynamic viscosity remains constant in model and full-scale conditions.

Fourier's Law of heat transfer (or heat conduction) states that the heat flow vector F at a point is proportional to the negative gradient of the temperature; that is, F = -KVT, which means that heat energy flows from hot regions to cold regions. The constant k is called the conductivity, which has metric units SS S of J/m-s-K or W/m-K. A temperature function T for a region D is given below. Find the net outward heat flux boundary S of D. It may be easier to use the Divergence Theorem and evaluate a triple integral. Assume that k = 1. T(x,y,z) = 100 - 5x+ 5y +z; D = {(x,y,z): 0≤x≤5, 0≤y≤4, 0≤z≤ 1} The net outward heat flux across the boundary is (Type an exact answer, using as needed.) -KSS S F.ndS = -k VT n dS across the

1. Jumping on a trampoline can be modelled by a mass-spring system. A person is jumping on a trampoline vertically and lightly so that his/her feet do not leave the trampoline. off The jumper has a mass of m = 50 kg. The natural height of the trampoline is 1.5 m above ground level and the trampoline has a spring constant k = 1000 Nm¯¹. At equilibrium the trampoline is compressed by a distance s metres. Natural height System at equilibrium off 1.5m It T Air resistance acts on the jumper with a damping constant 3 = 50 N sm-¹. Assume that the gravitational constant is g = 10 ms ². Sm 1.5m Let y(t) be the distance in metres of the jumper below the equilibrium position at time t seconds after the start of the jump, and let h(t) be the height in metres of the jumper above ground level at time t. At time t = 0, the jumper starts at height 0.6 m above the ground and the velocity of the jumper is 0. You may assume that the equation of motion for the system is 50y + 50y + 1000y = 0. Please turn…

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