Concept explainers
Given the following steady axisymmetric stream function:,
valid in the region
Want to see the full answer?
Check out a sample textbook solutionChapter 4 Solutions
Fluid Mechanics
- .3 2. 1. If u = 3x*yt and v = -6x°y´t“ 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).arrow_forward1. For incompressible flows, their velocity field 2. In the case of axisymmetric 2D incompressible flows, where is Stokes' stream function, and u = VXS, S(r, z, t) = Uz = where {r, y, z} are the cylindrical coordinates in which the flow is independent on the coordinate and hence 1 Ꭷ r dr 1 dy r dz Show that in spherical coordinates {R, 0, 0} with the same z axis, this result reads Y(R, 0, t) R sin 0 S(R, 0, t) UR uo Y(r, z, t) r = = -eq, and Up = = 1 ay R2 sin Ꮎ ᎧᎾ 1 ƏY R sin Ꮎ ᎧR -eq 2 (1) (2) (3)arrow_forward1. For the following velocity fields, determine if they are possible for incompressible flows and if they are irrotational: (a) √ = î(x+y) + ĵ(x − y + z) + Ê(x + y +3) (b) ▼ = î(xy) + ĵ(yz) + k(yz + z²) (c) V = î[x(y +z)] + ĵ[y(x + z)] + k[−(x + y)z − z²] (d) V = î(xyzt) + ĵ(−xyzt²) + k[(z²/2) (xt² − yt)]arrow_forward
- An incompressible velocity field is given by u=a(x°y²-y), v unknown, w=bxyz where a and b are constants. (a)What is the form of the velocity component for that the flow conserves mass? (b) Write Navier- Stokes's equation in 2-dimensional space with x-y coordinate system.arrow_forward1. A Cartesian velocity field is defined by V = 2xi + 5yz2j − t3k. Find the divergence of the velocity field. Why is this an important quantity in fluid mechanics? 2. Is the flow field V = xi and ρ = x physically realizable? 3. For the flow field given in Cartesian coordinates by u = y2 , v = 2x, w = yt: (a) Is the flow one-, two-, or three-dimensional? (b) What is the x-component of the acceleration following a fluid particle? (c) What is the angle the streamline makes in the x-y plane at the point y = x = 1?arrow_forwardA velocity field is given by u = 5y2, v = 3x, w = 0. (a) Is this flow steady or unsteady? Is it two- or three- dimensional? (b) At (x,y,z) = (3,2,–3), compute the velocity vector. (c) At (x,y,z) = (3,2,–3), compute the local (i.e., unsteady part) of the acceleration vector. (d ) At (x,y,z) = (3,2,–3), compute the convective (or advective) part of the acceleration vector. (e) At (x,y,z) = (3,2,–3), compute the (total) acceleration vector.arrow_forward
- a. Given the velocity field u=(u,v,w) in Cartesian coordinates with u=2x+y, v=2zt, w=0. i. Find the equations of the corresponding streamlines (Eulerian concept) ii. Find the equations of the corresponding particle paths, i.e., the pathlines (Lagrangian concept). b. Show that the Vu=0 everywhere implies that volumes are conserved, i.e., the volume of red particles at t-0 is the same as at t=t. Hint: Write out what you must prove and use the theorems to get there.arrow_forward4. Consider a velocity field V = K(yi + ak) where K is a constant. The vorticity, z , is (A) -K (B) K (C) -K/2 (D) K/2arrow_forward1. For a velocity field described by V = 2x2i − zyk, is the flow two- or threedimensional? Incompressible? 2. For an Eulerian flow field described by u = 2xyt, v = y3x/3, w = 0, find the slope of the streamline passing through the point [2, 4] at t = 2. 3. Find the angle the streamline makes with the x-axis at the point [-1, 0.5] for the velocity field described by V = −xyi + 2y2jarrow_forward
- In plane stagnation flow, an incompressible fluid occupying the space y>0 has one velocity component given by Vx-x. The flow is two-dimensional and steady, such that V₂=0 and nothing depends on z or time t. (a) Use the continuity equation to determine Vy(x,y), given that Vy(x,0) =0. (This condition for Vy corresponds to the plane y=0 being an impenetrable boundary.) (b) is arbitrary, so you may set Y=0 at any convenient location.) Determine the stream function for this flow, (x,y). (The absolute value ofarrow_forwardA velocity field is described (in Cartesian coordinates) by u = 2−x3/3, v = x2y−zt, w = 0. (a) Write down the y-component of the acceleration of a fluid particle (in the Eulerian system) for this flow field. (b) Is this flow field incompressible?arrow_forward1. 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.arrow_forward
- Elements Of ElectromagneticsMechanical EngineeringISBN:9780190698614Author:Sadiku, Matthew N. O.Publisher:Oxford University PressMechanics of Materials (10th Edition)Mechanical EngineeringISBN:9780134319650Author:Russell C. HibbelerPublisher:PEARSONThermodynamics: An Engineering ApproachMechanical EngineeringISBN:9781259822674Author:Yunus A. Cengel Dr., Michael A. BolesPublisher:McGraw-Hill Education
- Control Systems EngineeringMechanical EngineeringISBN:9781118170519Author:Norman S. NisePublisher:WILEYMechanics of Materials (MindTap Course List)Mechanical EngineeringISBN:9781337093347Author:Barry J. Goodno, James M. GerePublisher:Cengage LearningEngineering Mechanics: StaticsMechanical EngineeringISBN:9781118807330Author:James L. Meriam, L. G. Kraige, J. N. BoltonPublisher:WILEY