Four prototypical flow fields are expressed below in terms of Cartesian coordinates. In theseexpressions, y is a constant.i) Planar elongation:ii) Uniaxial extension:iii) Biaxial extension:iv) Pure rotation:For each of these flows, determine:v = jxex - ¡ye, +0e₂v=2jxex - jy e, - ize₂v = jxẹx + jy ẹ, - 2jze₂v = jye - jxe, +0e₂a) the rate of strain tensor =e [Vy + Vy¹] and2b) the deviatoric stress tensor = [Vy - Vy¹].Based only on your answers to parts (a) and (b), determine:c) whether the flow is incompressible (i.e., is V. y = 0?)d) whether the flow is irrotational. (i.e., is V × v = 0 ?)

Answered: Image /qna-images/answer/990f64f1-0906-4d4a-8bab-727e852fb71b.jpg

Four prototypical flow fields are expressed below in terms of Cartesian coordinates. In these expressions, y is a constant. i) Planar elongation: ii) Uniaxial extension: iii) Biaxial extension: iv) Pure rotation: For each of these flows, determine: v=ixe - jye, +0e₂ v=2jxe - jy e, - ize, v = jxe + jy e, -2jze, v = iye - jxe, +0e₂ a) the rate of strain tensor € = ½ [Vy + Vy²] and b) the deviatoric stress tensor ( = ½ [Vy – Vy7]. - Based only on your answers to parts (a) and (b), determine: c) whether the flow is incompressible (i.e., is V. y = 0?) d) whether the flow is irrotational. (i.e., is V × v = 0 ?)

Four prototypical flow fields are expressed below in terms of Cartesian coordinates. In these expressions, y is a constant. i) Planar elongation: ii) Uniaxial extension: iii) Biaxial extension: iv) Pure rotation: For each of these flows, determine: v=ixe - jye, +0e₂ v=2jxe - jy e, - ize, v = jxe + jy e, -2jze, v = iye - jxe, +0e₂ a) the rate of strain tensor € = ½ [Vy + Vy²] and b) the deviatoric stress tensor ( = ½ [Vy – Vy7]. - Based only on your answers to parts (a) and (b), determine: c) whether the flow is incompressible (i.e., is V. y = 0?) d) whether the flow is irrotational. (i.e., is V × v = 0 ?)

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

See similar textbooks

Similar questions

1. Stagnation Points A steady incompressible three dimensional velocity field is given by: V = (2 – 3x + x²) î + (y² – 8y + 5)j + (5z² + 20z + 32)k Where the x-, y- and z- coordinates are in [m] and the magnitude of velocity is in [m/s]. a) Determine coordinates of possible stagnation points in the flow. b) Specify a region in the velocity flied containing at least one stagnation point. c) Find the magnitude and direction of the local velocity field at 4- different points that located at equal- distance from your specified stagnation point.
4. Consider steady flow of water through an axisymmetric garden hose nozzle (see Fig). Along the center line of the nozzle, the water speed increases from uentrance to Uexit as sketched. Measurements reveal that the centerline water speed increases parabolically through the nozzle. Write an equation for centerline speed u(x), based on the parameters given here, from x=0 to x=L. Dentrance Dexit Hentrance Hexit u(x) x=L
1. Find the stream function for a parallel flow of uniform velocity V0 making an angle α with the x-axis. 2. A certain flow field is described by the stream function ψ = xy. (a) Sketch the flow field. (b) Find the x and y velocity components at [0, 0], [1, 1], [∞, 0], and [4, 1]. (c) Find the volume flow rate per unit width flowing between the streamlines passing through points [0, 0] and [1, 1], and points [1, 2] and [5, 3].
4. The velocity vectors of three flow fileds are given as V, = axĩ + bx(1+1)}+ tk , V, = axyi + bx(1+t)j , and V3 = axyi – bzy(1+t)k where coefficients a and b have constant values. Is it correct to say that flow field 1 is one-, flow filed 2 is two-, and flow filed 3 is three-dimensional? Are these flow fields steady or unsteady?
A fluid flow is described (in Cartesian coordinates) by u = x2, v = 4xz. (a) Is this flow two-dimensional or three-dimensional? (b) Is this flow field steady or unsteady? (c) Find the simplest form of the z-component of velocity if the flow is incompressible.
1. If u- 3x'yr and v = -6x'y'r answer the following questions giving reasons, Is this flow or fluid: (a) Real (Satisfies Continuity Principle). (b) Steady or unsteady. (c) Uniform or non-uniform. (d) One, two, or three dimensional. (e) Compressible or incompressible. Also, Find the acceleration at point (1,1). %3D
Velocity Field Assignment 4 2 -5 -4 2 -1 N -4 W- E Consider the steady, two-dimensional velocity field of wind as: V= (u, v)= (8 – 0.5x)i + (0.5 - 5y)j where x- an y- coordinates are in m, time in s, and the magnitude of the velocity is in m/s. Determine: (a) A stagnation point, if existed. (b) Sketch the velocity vector for the given coordinate on the map. (c) Sketch the relevant streamlines on a different graph. (d) Verify if the flow is rotational or irrotational flow. (e) Looking at the velocity vector, which section of the country will receive the most rain if the wind brings rainy season from the south-china sea?
Q.5 A stream function is given by Y = (x² – y2). The Velocity potential function (b) of the flow will be A 2xy + f(x) B -2xy + constant © 2(x2 -y2) D 2xy + f(y)
A Fluid Mechanics, Third Edition - Free PDF Reader E3 Thumbnails 138 FLUID KINEMATICS Fluid Mechanies Fundamenteis and Applicationu acceleration); this term can be nonzero even for steady flows. It accounts for the effect of the fluid particle moving (advecting or convecting) to a new location in the flow, where the velocity field is different. For example, nunan A Çengel | John M. Cinbala consider steady flow of water through a garden hose nozzle (Fig. 4-8). We define steady in the Eulerian frame of reference to be when properties at any point in the flow field do not change with respect to time. Since the velocity at the exit of the nozzle is larger than that at the nozzle entrance, fluid particles clearly accelerate, even though the flow is steady. The accel- eration is nonzero because of the advective acceleration terms in Eq. 4-9. FLUID MECHANICS FIGURE 4-8 Flow of water through the nozzle of a garden hose illustrates that fluid par- Note that while the flow is steady from the…
Problem 1 Given a steady flow, where the velocity is described by: u = 3 cos(x) + 2ry v = 3 sin(y) + 2?y !! !! a) Find the stream function if it exists. b) Find the potential function if it exists. c) For a square with opposite diagonal corners at (0,0) and (47, 27), evaluate the circu- lation I = - f V.ds where c is a closed path around the square. d) Calculate the substantial derivative of velocity at the center of the same box.
Flow over a sphere is given by the superposition of uniform flow and 3D doublet flow. In cylindrical coordinates the flow resembles that pictured below. See Lecture Pack 3 for more information on vector operations in polar and cylindrical coordinates. Uniform flow in the z direction is o = Uz. Doublet flow is U„R³z + z2 where R is the radius of the sphere. 1. Calculate ur, uo, and uz as functions of r and z for superposed uniform and doublet flow. 2. Show thatr 0, z ±R are stagnation points.
Consider the following three-dimensional velocity vector: V = 4xy² i + fj- z²y² k a) Find the appropriate form of the function f such that the velocity vector represents a physical incompressible flow. b) What is the stream function for this flow? c) Write down the expression for the velocity potential along the plane z = 0.