Application of polymeric coatings to fibers can be accomplished by drawing a fiber through a bath of molten polymer: see schematic below. R Wire 2kR Molten Polymer →V Let's assume steady, isothermal, and incompressible flow. i. Using Navier-Stokes equations in a cylindrical coordinate system along with appropriate boundary conditions, derive an expression for the z velocity as a function of the radial coordinate r. ii. Show that the amount of polymer coated per unit time, say M, is given by M =p¹Vk²(1-2Ink)-1] where p denotes the density of the molten (liquid) polymer. iii. In good engineering design it is important to be able to control the product parameters (coating thickness in the present case) by varying process variables (draw speed). From this point of view, can the proposed design be improved? If so, how?

Application of polymeric coatings to fibers can be accomplished by drawing a fiber through a bath of molten polymer: see schematic below. R Wire 2kR Molten Polymer →V Let's assume steady, isothermal, and incompressible flow. i. Using Navier-Stokes equations in a cylindrical coordinate system along with appropriate boundary conditions, derive an expression for the z velocity as a function of the radial coordinate r. ii. Show that the amount of polymer coated per unit time, say M, is given by M =p¹Vk²(1-2Ink)-1] where p denotes the density of the molten (liquid) polymer. iii. In good engineering design it is important to be able to control the product parameters (coating thickness in the present case) by varying process variables (draw speed). From this point of view, can the proposed design be improved? If so, how?

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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Application of polymeric coatings to fibers can be accomplished by drawing a fiber through a bath of molten polymer: see schematic below. Wire2kR Molten Polymer →V Let's assume steady, isothermal, and incompressible flow. i. Using Navier-Stokes equations in a cylindrical coordinate system along with appropriate boundary conditions, derive an expression for the z velocity as a function of the radial coordinate r.
A wind tunnel has a cross-section of 1.0m at its inlet by 1.0m and length 10m. Wind at uniform velocity 15m/s enters the tunnel at 20°C. Assume, velocity distribution in turbulent 1/5 y boundary layer to follow the law U (Kinematic viscosity= 1.53x10°m² /s) Assuming the boundary layer to be turbulent from the beginning, then the value of 8(represents in law) is calculated as
The following rheological data were collected on a new "Thick 'n Spicy" brand of tomato catsup at 20°C and 90 kPa. rate of shear (s") shear stress (Pa) 10.0 20.4 20.0 21.5 40.0 23.0 60.0 25.0 80.0 26.5 (a) Plot these measured values on graph-paper. Which class of fluid does the catsup belong to?
The following rheological data were collected on a new "Thick 'n Spicy" brand of tomato catsup at 20°C and 90 kPa. rate of shear (s') shear stress (Pa) 10.0 20.4 20.0 21.5 40.0 23.0 60.0 25.0 80.0 26.5 (a) Plot these measured values on graph-paper. Which class of fluid does the catsup belong to? (b) Write the appropriate model for describing the rheological behavior of "Thick 'n Spicy" catsup. Specify the numerical value(s) and the units of the associated constant(s). (c) What shear stress is expected when the rate of shear, under vigorous shaking of a bottle of "Thick 'n Spicy", is 150 s?
Ex. The force required to tow a 1:30scale model of a motor boat in a lake at a speed of 2m/s is 0.5 N, Assuming that the viscosity resistance due to water and air is negligible in comparison with the wave resistance, calculate the corresponding speed of the prototype for dynamically similar conditions. What would be the force required to propel the prptype at that velocity in the same lake? Ans.: 10.95m/s, 13500N
Find the equation of motion (Navier Stokes) for a viscous fluid between two rotating concentric cylinders (axle and shaft). The inner cylinder has the radius ro and rotates at angular speed wo. The outer cylinder has the radius R and is stationary. Write down each vector component of the equation in a separate line and use reasonable assumptions to simplify the equation, especially the derivatives. Be sure to use cylindrical coordinates for the convective operator and the other derivatives.
By using the expression for the shear stress derived in class (and in BSL), show that the shear force on asphere spinning at a constant angular velocity in a Stokes’ flow, is zero.This means that a neutrally buoyant sphere (weight equal buoyancy force) that is made to spin in aStokes’ flow, will neither rise nor fall, nor translate in any preferential direction in the (x-y) plane. expressions for velocity are: v_r (r,θ)= U_∞ [1-3R/2r+R^3/(2r^3 )] cos⁡θ v_θ (r,θ)= -U_∞ [1-3R/4r-R^3/(4r^3 )] sin⁡θ Where v_r and v_θ are the radial and angle velocity, U_∞ is the velocity of fluid coming to sphere which very faar away from the sphere. And R is the radius of sphere.
A jet engine on a test stand directs a stream of hot exhaust gasses against a vemcal wall. All of the exhaust gas leaving the wall after impact is in the y-z plane (ie no "x" direction velocity). The mass rate is 200 kg/s and the velocity is 400 m/s. (Note that the density and viscosity are not relevant) What is the force on the wall (include direction)?
Fluid Mechanics II Sheet NO (04) Damietta University Faculty of Engineering Mechanical Engineering Dept. FACULTY OF ENGINEERING DAMIETTA UNIVERSIm bluas daola-dunigil us 3- Fluid flow between two horizontal infinite parallel plates with a spacing of H mm shown in figure 3. Determine the volume flow rate per width and the shear stress on upper and lower plates. U2 , draw the velocity profile distribution. U1 Moving plate H Moving plate U2 Figure 3 , a tornado can be approximated As illustrated in Fig. 4 by a free vortex of strength I' for r > Re, where R̟ is the radius of the core. Velocity measurements at points A and B indicate that V = 125 ft/s and V3 = 60 ft/s. Determine the distance from point A to the center of the tornado. Why can the free vortex model not be used to approximate the tornado throughout the flow field (r > 0)? 4- B +100 ft- Figure 4
(b) In two dimensional boundary layer, shear stress was changed linearly from the solid surface toward y-axis until it reach the value of zero at y = 6. Based on Table 2 and setting given to you; () Derive the equation of displacement thickness and momentum thickness using Von Karman Approximation Method ; and (ii) Determine the accuracy of this method in determining the value of displacement thickness and momentum thickness. Table 2 : Equation of Velocity Profile Setting Equation wU = 3(y/8)/2 – (y/8j?/2
(b) In two-dimensional boundary layer, shear stress was changed linearly from the solid surface toward y-axis until it reaches the value of zero at y = 8. Based on Table 2 and setting given to you; (i) Derive the equation of displacement thickness and momentum thickness using Von Karman Approximation Method; and (ii) Determine the accuracy of this method in determining the value of displacement thickness and momentum thickness. Table 2: Equation of Velocity Profile Equation u/U = 3(y/8)/2 – (y/8)³/2 Setting 2
2. Consider the boundary-layer momentum equation for steady incompressible flow in two dimen- sions: ди dpe ду du du + po ay dx Assume the viscosity, u, to be constant. What does this equation simplifies to at a stationary solid surface? Based on this result, is a boundary layer more likely to separate in a favourable gradient (pressure decreasing) or an adverse pressure gradient? Sketch the velocity profile of a boundary layer that is attached and one that is separated. pressure