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In Chap. 4, we defined the material acceleration, which is the acceleration following a fluid particle,
(a) That are the primary dimensions of the gradient operator
Answers: (a)
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Fluid Mechanics: Fundamentals and Applications
- 1.6 An incompressible Newtonian fluid flows in the z-direction in space between two par- allel plates that are separated by a distance 2B as shown in Figure 1.3(a). The length and the width of each plate are L and W, respectively. The velocity distribution under steady conditions is given by JAP|B² Vz = 2µL B a) For the coordinate system shown in Figure 1.3(b), show that the velocity distribution takes the form JAP|B? v, = 2μL Problems 11 - 2B --– €. (a) 2B (b) Figure 1.3. Flow between parallel plates. b) Calculate the volumetric flow rate by using the velocity distributions given above. What is your conclusion? 2|A P|B³W Answer: b) For both cases Q = 3µLarrow_forwardTwo-dimensional irrotational fluid flow is conveniently described by a complex poten- tial f(z) = u(x, v) + iv(x, y). We label the real part, u(x, y), the velocity potential, and the imaginary part, v(x, y), the stream function. The fluid velocity V is given by V = Vu. If f(z) is analytic: 11.2.11 (a) Show that df/dz= Vx – i Vy. (b) Show that V · V = 0 (no sources or sinks). (c) Show that V x V=0 (irrotational, nonturbulent flow).arrow_forwardIn a fluid flow, the density of the fluid is constant for incompressible flow Select one: True Falsearrow_forward
- #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 Sarrow_forwardA particle rotates with constant speed in a circle. Let be the net torque on the particle and F the net force on the particle. Then: a. tau > 0 and F > 0 O b. tau > 0 and F = 0 . tau = 0 and F = 0 Od. tau = 0 and F > 0arrow_forward1. (a) The motion of a floating vessel through the surrounding fluid results in a drag force D which is thought to depend upon the vessel's speed v, its length I, the density p and dynamic viscosity μ of the fluid and the acceleration due to gravity g. Show that:- D = pv²1² (1) (b) In order to predict the drag on a full scale 50m long ship traveling at 7m/s in sea water at 5°C of density 1027.7225 kg/m³ and viscosity 1.62 x 103 Pa.s, a model 3m long is tested in a liquid of density 805 kg/m³. What speed does the model need to be tested at and what is the required viscosity of the liquid?arrow_forward
- Q4: Answer the following 1) If for a flow a stream function exists and satisfies the Laplace equation, then which of the following is the correct statement? (a) The flow is rotational (b) The flow is rotational and incompressible (c) The flow is irrotational and compressible (d)The flow is irrotational and incompressible |2) The boundary layer thickness for flow over a flat plate (a) decreases with an increase in the free stream velocity (b) increases with an increase in the free stream velocity (c) decreases with an increase in the kinematic viscosity 3) In the Fanno flow ,if the flow is supersonic ,a shock appears in the duct when (b) L > Lmax 4) An automotive wing is a device whose intended design is to generate (a) L = Lmax (c) L< Lmax ----------as air passes around it. 5) -- is a unit less value denotes how much an object resists movement through a fluid |6)Fluid accelerate or decelerates at any point in a variable area duct depends on ------ and 7) To decrease drag force it is…arrow_forwardIN FLUID MECHANICS, THE DIVERGENCE OF VELOCITY EXPRESSES In fluid mechanics, the divergence of velocity expresses None of the above answers A volume dilatation rate A particular eigen value of the Navier-Stokes equation. An elongational rate in the direction of motion Viscous effectsarrow_forwardConsider fully developed flow between two infinite parallel plates separated by distance h, with the top plate moving and the bottom plate stationary. The flow is steady, incompressible, and two-dimensional in the xy-plane. a) Use the first principle (dimensional analysis) to generate a dimensionless relationship for the x-component of fluid velocity u as a function of fluid viscosity μ, top plate speed v, distance h, fluid density ρ, and distance y. b) Name the common dimensionless number formed in (a). Hint: modifying the dimensionless number if necessary.arrow_forward
- As shown in the following figure, vortices are shed from the rear of a bluff cylinder placed across a flow. The vortices alternately leave the top and bottom of the cylinder, causing an alternating force normal to the freestream velocity. The vortex shedding frequency, f, depends on the fluid density p, width of the cylinder d, freestream velocity V, and fluid viscosity u. (a) Use Buckingham Pi Theorem to develop a functional relationship for f. Use M, L, t as the primary dimensional. Use p, V, and d as the repeating parameters. (b) Vortex shedding occurs in standard air on two cylinders with a diameter ratio of 2. Determine the velocity ratio for dynamic similarity, and the ratio of vortex shedding frequencies. -Vortices Varrow_forwardConsider steady, incompressible, two-dimensional flow due to a line source at the origin. Fluid is created at the origin and spreads out radially in all directions in the xy-plane. The net volume flow rate of created fluid per unit width is V·/L (into the page of Fig), where L is the width of the line source into the page in Fig Since mass must be conserved everywhere except at the origin (a singular point), the volume flow rate per unit width through a circle of any radius r must also be V·/L. If we (arbitrarily) specify stream function ? to be zero along the positive x-axis (? = 0), what is the value of ? along the positive y-axis (? = 90°)? What is the value of ? along the negative x-axis (? = 180°)?arrow_forwardEXAMPLE Leaking Tank. Outflow of Water Through a Hole (Torricelli's Law) This is another prototype engineering problem that leads to an ODE. It concerns the outflow of water from a cylindrical tank with a hole at the bottom. You are asked to find the height of the water in the tank at any time if the tank has diameter 2 m, the hole has diameter 1 cm, and the initial height of the water when the hole is opened is 2.25 m. When will the tank be empty? 2.20 M Water level asime Outiine walls 200 200 30t .00- 50- D 10000 30000 tebe Revelion 50000arrow_forward
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