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Consider the following steady, two-dimensional velocity field:
Is there a stagnation point in this flow field? If so, where is it?
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Fluid Mechanics: Fundamentals and Applications
- 1. 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_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_forwardThe stream function o in a two-dimensional flow field is given as 9 = 4x – 3y + 7xy (a) Prove that this flow field is irrotational and that it satisfies the continuity equation. Find the potential flow function P(x, y) for this flow field with boundary condition 0 = 0 at x = 2, y = 1. (b)arrow_forward
- Consider the following steady, two-dimensional velocity field: V = (u, v) = (0.5 + 1.2x)i + (-2.0 – 1.2y)j. Is there a stagnation point in this flow field? If so, where is it?arrow_forwardTwo velocity components of a steady, incompressible flow field are known: u = 2ax + bxy + cy2 and ? = axz − byz2, where a, b, and c are constants. Velocity component w is missing. Generate an expression for w as a function of x, y, and z.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) + Â(x + y + 3) (b) ▼ = î(xy) + ĵ(yz) + Â(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
- A fluid has a velocity field defined by u = x + 2y and v = 4 -y. In the domain where x and y vary from -10 to 10, where is there a stagnation point? Units for u and v are in meters/second, and x and y are in meters. Ox = 2 m. y = 1 m x = 2 m, y = 0 No stagnation point exists x = -8 m, y = 4 m Ox = 1 m, y = -1 m QUESTION 6 A one-dimensional flow through a nozzle has a velocity field of u = 3x + 2. What is the acceleration of a fluid particle through the nozzle? Assume u, x and the acceleration are all in consistent units. O 3 du/dt 9x + 6 1.5 x2 + 2x O Oarrow_forwardConsider the steady, two-dimensional velocity field given by V = (1.3 + 2.8x)i+ (1.5 – 2.8y)j. 6. Verify that this flow field is incompressible.arrow_forwardThe velocity field of incompressible flow in a Cartesian system is represented by V = 2 (x? – y?) i+ vj+ 3k Which one of the following expressions for v is valid?arrow_forward
- 2. Consider the two-dimensional time-dependent velocity field u(x, t) = (sint, cost, 0), in the basis of Cartesian coordinates. a) Determine the streamlines passing through the point x = 0 at the times t = 0, π/2, π and 3π/2. b) Determine the paths of fluid particles passing through the point x = 0 at the same times, to = 0, π/2, 7 and 37/2. Hence, describe their motion. ㅠ c) Find the streakline produced by tracer particles continuously released at the point xo = 0 and find its position at t = 0, π/2, π and 37/2. Hence describe its motion.arrow_forwardConsider the velocity field, ▼ — (x – 2y)i — (2x + y)j. What is the value of the velocity potential function at = the point (2,3)? Answer:arrow_forward2- For a certain incompressible flow field it is suggested that the velocity components are given by the equations u = 2xy v = –x²y w = 0 Is this a physically possible flow field? Explain.arrow_forward
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