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Consider a spiraling line vortex/sink flow in the xy-plane as sketached in Fig. 9-26.The two-dimensional cylindrical velocity components
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Fluid Mechanics: Fundamentals and Applications
- Ay j. Is this a possible case of incompres- 3.9 A velocity field is given by V= Axyi -- %3D sible flow? If yes, obtain the stream function and find the value of constant A for which the flow rate between the streamlines passing through the points (3, 3) and (3, 4) is 18 units. Axy Ans: V = 12 + C, A 7 2arrow_forward(a) A two-dimensional flow field is given by u = 5a2 – 5y, v = -10xy (i) Find the streamfunction and velocity potential Ø. (ii) Find the equation for the streamline and potential line which passes through the point (1, 1).arrow_forwardIn the figure, consider the flow on a rotating plate (thick line). Which of the following do we see in the picture? In the picture shown, the radial lines are streamlines coming from the center while the circles are equipotential lines. It is the flow net of a a. b. What is the corresponding stream function for a flow field with velocity potential function, $ = (x² + xy - y²)? C. x² + y² 2 x² + y² y²-x² d. ²+² 2 a. Pathlines b. Streaklines c. Streamlines d. All of the above + 2xy xy a. sink b. vortex c. source d. cannot be determinedarrow_forward
- The stream function of a flow field is y = Ax3 – Bxy², where A = 1 m1s1 and B = 3 m-1s1. (a) Derive the velocity vector (b) Prove that the flow is irrotational (c) Derive the velocity potentialarrow_forwardconsider the 2 dimensional velocity field V= -Ayi +Axj where in this flow field does the speed equal to A? Where does the speed equal to 2A?arrow_forwardA steady, two-dimensional, incompressible flow field in the xy-plane has the following stream function: ? = ax2 + bxy + cy2, where a, b, and c are constants. (a) Obtain expressions for velocity components u and ?. (b) Verify that the flow field satisfies the incompressible continuity equation.arrow_forward
- Converging duct flow is modeled by the steady, two- dimensional velocity field V = (u, v) = (U₁ + bx) i-by. For the case in which Ug = 3.56 ft/s and b = 7.66 s¯¹, plot several streamlines from x = 0 ft to 5 ft and y=-2 ft to 2 ft. Be sure to show the direction of the streamlines. (Please upload you response/solution using the controls provided below.)arrow_forwardConsider a steady two-dimensional flow with the velocity field in the Cartesian coordinate system is given by u = -Ax and v = Ay, where A is a constant. Obtain the equation for a streamline and the equation for a streamfunction of the two-dimensional flow. What is the acceleration vector at (x.y) = (1,1)?arrow_forwardConsider fully developed Couette flow between two infinite parallel plates separated by distance h, with the top plate moving and the bottom plate stationary, as illustrated in the figure below. The flow is steady, incompressible, and two-dimensional in the XY plane. The velocity field is given by V }i = (u, v) = (v² )i +0j = V (a) Find out the acceleration field of this flow. (b) Is this flow steady? What are the u and v components of velocity? u= V² harrow_forward
- The velocity field for a line vortex in the r?-plane is given byur = 0 u? = K / rwhere K is the line vortex strength. For the case with K = 1.5 m/s2, plot a contour plot of velocity magnitude (speed). Specifically, draw curves of constant speed V = 0.5, 1.0, 1.5, 2.0, and 2.5 m/s. Be sure to label these speeds on your plot.arrow_forwardConsider steady, incompressible, two-dimensional flow through a converging duct (Figure below). Uo A simple approximate velocity field for this flow of the Converging duct flow is modeled by the steady, two- dimensional velocity field given by: V = (u, v) = (U, + bx)i – byj The pressure field is given by: P = P, – 2U,bx + b*(x² + y²) Where Po is the pressure at x = 0. Generate an expression for the rate of change of pressure following a fluid particle?arrow_forwardFor a two-dimensional incompressible flow field given by - Ajxi - vi , where A> 0, which one of the following statements is FALSE? A. It satisfies continuity equation B. It is unidirectional when x 0 and y → o. C. Its streamlines are given by x = y. D. It is irrotationalarrow_forward
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