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Consider the following steady, three-dimensional velocity field in Cartesian coordinates:
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Fluid Mechanics: Fundamentals and Applications
- 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_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_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
- (d) Consider the following steady, three dimensional velocity field in Cartesian coordinates. V = (axy² – b)i – (2cy)³j +(dxy)k where a, b, c and d are constants. Under what conditions is this flow field incompressible?arrow_forwardAn 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_forward(a) Given the following steady, two-dimensional velocity field. [Diberi medan halaju yang mantap dan dua dimensi.] V = (u, v) = (8x + 6)ï + (-8y – 4)j (i) Is this flow field an incompressible flow? Prove your answer. (ii) Is this flow field irrotational? Prove your answer. (iii) Generate an expression for the velocity potential function if applicable.arrow_forward
- A steady, incompressible, two-dimensional velocity field is given by the following components in the xy-plane: u = 0.205 + 0.97x + 0.851y ? = −0.509 + 0.953x − 0.97y Calculate the acceleration field (find expressions for acceleration components ax and ay) and calculate the acceleration at the point (x, y) = (2, 1.5).arrow_forward4. A steady, incompressible, and two-dimensional velocity field is given by the following components in the xy-plane: Vxu = 2.65 + 3.12x + 5.46y = Vy= =v=0.8+ 5.89x² + 1.48y = Calculate the acceleration field (find expressions for acceleration components ax and ay and calculate the acceleration at the point (x,y) = (-1,3).arrow_forwardA steady, incompressible, two-dimensional velocity field is given by the following components in the xy-plane: u = 1.85 + 2.05x + 0.656y ? = 0.754 − 2.18x − 2.05yCalculate the acceleration field (find expressions for acceleration components ax and ay), and calculate the acceleration at the point (x, y) = (−1, 3).arrow_forward
- 4. 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_forwardHelp me pleasearrow_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
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