The following differential equation describes the momentum conservation in the streamwise (x) direction in a region near a surface with fluid flowing over it - this region is called a boundary layer. ди pu მე + pv. = Ju др მყ მთ a² u + fl. მყ2 where ρ is density, u is velocity in the x-direction, v is velocity in the y-direction, p is pressure, and μ is the total viscosity. Using only this equation (meaning, don't just look up the answer), find the primary dimensions and units (both SI and British) for viscosity. Show your work. Also show that the other terms in the equation are dimensionally consistent with each other.

A laboratory study models the flow around the pier of an off-shore wind turbine with a scaled model of Lr = 20. Determine the model velocity (VM) past the pier, if the water travels with V = 2.2 km/h past the prototype pier. You can assume that water with the same fluid properties is used in both model and prototype with density of Pw = 1005 kg/m³ and dynamic viscosity of nw = 0.001 Pas. HINT: the flow around the pier is described by the Reynolds number. VM m/s (Provide your answer to 3 decimal places)

(b) One form of fluid movement is rotation and deform angularly. Figure Q1(b) shows the rotation and angular deformation caused by velocity variation about z-axis. Based on Table 1 and setting given to you, derive an equation of rotation. ди Sy St ây > B' ĉu B B ôy dy A' ↑ Sa v+. ôx A ôx Figure Q1(b) : Rotation and Angular Deformation Table 1: Axis of Rotation Setting Axis of Rotation 2 у-ахis

3. A circular cylinder of radius a is fitted with two pressure sensors to measure pressure at 0 = 180° and at 150°. The intent is to use this cylinder as a stream velocimeter, i.e. a device to determine the velocity of a stream by measuring the pressures at the two taps. The fluid is incompressible with a density of p. Figure for Part (a) U Figure for Part (b) 30 a) Using potential flow approximation, derive a formula for calculating U from the measured pressure difference at the two pressure taps. Note that for accurate measurement, the velocimeter must be aligned to have one of the taps exactly facing the stream as shown in the figure. (Ans: 2|Aptaps|/p ) b) Suppose the velocimeter has been misaligned by ổ degrees so that the two pressure taps are now at 180° + 8 and 150° + 8. Derive an expression for the percent error in stream velocity measurement. Then, calculate the error for 8 = 5°,10° and –10°. (Ans: [2/(sin2(150 + 8) – sin²(180 + 8) )– 1] × 100 )

For the flow of a viscous fluid, with the velocity V = f(x)g(y)h(z)i (where f, g, h are arbitrary functions), the following conditions are given: . The flow is adiabatic. • The quantities v = 2 and 3 = $ are constants. • The velocity circulation is conserved for the flow, irrespective of the values of vand 3. What is the general solution for the functions f, g, h?

Important variables in a particular water machine are: Fluid density (ρ), impeller angular velocity (ω), impeller diameter (D), volume flow rate (), fluid viscosity (µ), head (gH). In fact, (gH) shows the energy that the fluid unit mass receives from or gives to the machine, independent of gravitational acceleration (g). Therefore, gH value is used instead of H. Important sizes in terms of the operation of the water machine are power (P), (gH) and efficiency ( h). Derive dimensionless numbers by typing these two parameters based on the above variables and applying Buckingham's p method (repeating variables method).

#4 1.11 For a small particle of styrofoam (1 lbm/ft) (spherical, with diameter d = 0.3 mm) falling in standard air at speed V, the drag is given by FD-3mVd, where is the air viscosity. Find the maximum speed starting from rest, and the time it takes to reach 95 percent of this speed. Plot the speed as a function of time. s) Answer: (Vmax=0.0435",t=0.0133 S

In this question, assume the "additional displacement" is in the positive u direction. A mass weighing 16 lbs stretches a spring 8 inches. The mass is in a medium that exerts a viscous resistance of 1 lbs when the mass has a velocity of 2 ft/sec. Suppose the object is displaced an additional 5 inches and released. Find an equation for the object's displacement, u(t), in feet after t seconds. u(t) =

Fluid Mechanics. Fluid kinematics. The figure below represents the upper view of pillars, with circular sections of 0.5 m, which support a bridge. The figure also shows the river speeds before and after reaching the pillars. The width of the bridge is 2d. *Consider that in the symmetry section of the bridge there is no flow and the slightest force is exerted in the cross section. d Uniform speed (v) = 2 m/s d' fo) = 15 + V = = 1,5 +: Whereas d = 2 m and d' 1 m, determine: a) What possible control volumes could be used to resolve the issue? b) The flow rate per unit of length entering the plane of the image. c) The slightest force exerted by the river water in this pillar of the figure. d) Are there any generated momentum? If so, justify it.

p) A mass of fluid is in motion so that the lines of motion lie on the surface of coaxial cylinders. Show that the equation of continuity is partial differential ap 1 a 5(pvo) + əz - (ρυ.) 0 r d0 where (v,, vg, vz) are the components of velocity in cylindrical co-ordinates and p is the density.