An important parameter in fluid flow problems involving thin films is the Weber number (We) which can be expressed in equation form as We=[pv^2L/(omega)] where p is the density of the fluid, v is a velocity, L is a length, and (omega) is the surface tension of the fluid. If the Weber number is dimensionless, what are the dimensions of the surface tension (omega)?

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Which of the following dimensionless parameters is the correct arrangement of the given parameters? (select all that O a. Pi = (Dp)/(Rho*V) Dp: Pressure change Rho: Density V: Velocity O b. Pi = (V*L*(Rho))/(nu) V: Velocity L: length Rho: Density nu: Dynamic viscosity O. Pi = Q/((A^1.5)*w) Q: Volumetric discharge rate A: Area w: Angular velocity O d. Pi = (a)(t^2)/(L) L: length a: acceleration t time

Pressure drop AP depends on four variables: the diameter of the pipe D, the density of the fluid p, the kinematic viscosity of the fluid v and the mean velocity v. Applying Buckinham's Pi Theorem, a dimensionless pi-group can be identified using the following power product: [AP] = [D]ª[p]b[v][v]ª. Which of the following statements in term of [T] is correct? Select one: O [T]: -2 = -c-d [T]: -1 = -c-d [T]: -2 = -c-2*d [T]: -2 = -2*c-d [T]: -2=-c+d

**Problem 1.15 Suppose you wanted to describe an unstable particle, that spon- taneously disintegrates with a "lifetime" t. In that case the total probability of finding the particle somewhere should not be constant, but should decrease at (say) an exponential rate: too P(t) = | V(x,1)1²dx = e=1/*. -0- A crude way of achieving this result is as follows. In Equation 1.24 we tacitly assumed that V (the potential energy) is real. That is certainly reasonable, but it leads to the "conservation of probability" enshrined in Equation 1.27. What if we assign to V an imaginary part: V = Vo – ir, where Vo is the true potential energy and r is a positive real constant? (a) Show that (in place of Equation 1.27) we now get dP 21 = --P. dt (b) Solve for P(1), and find the lifetime of the particle in terms of r.

2. The apparatus shown below is designed to measure the density of an unknown fluid (p2₂). The two sides of the device are separated by a movable, frictionless partition. The partition is attached to the immobile sidewalls of the device via springs (different spring constants) on either side. Before pouring fluid into the device, both springs are unstretched. The device has a rectangular cross-section and extends a width w into the page. Derive an expression for the unknown density p2 = f(p1, h₁, h₂, k₁, k2, Ax, g), where Ar is the displacement of the partition relative to its equilibrium location before the fluids are poured into the apparatus. h₁ P1 k₁ 5 P2 ли Ax k₂ h₂

What is the temperature profile for an atmosphere with uniform lapse rate y, [where y = -dT/dz]? Find the corresponding pressure p(z) and density p(z) profiles. Assume the atmosphere is an ideal gas in hydrostatic balance, with sea level temperature and pressure given by To and po, respectively. [~Holton 1.18]

An elastic foundation is often used to support heavy machinery that vibrates in such a way to cause problems for the rest of the structure. An elastic foundation with two equal springs each of stiffness k = 600 lbs/in is used to support the equipment of weight W = 1000 lbs shown. The equipment vibrates in such a way that it causes a vertical forcing function P(t) = 100 sin(20t) [lbs]. %3D (a) Draw the degree of freedom for the motion of the machinery. (b) Determine the equation of motion. What type of motion is this? (c) Determine the natural frequency and the displacement of the machinery. (d) What is the maximum amount of force that will be felt by the rest of the structure? (Think about what causes force to the rest of the structure) Rest of structure Hww