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In this chapter, we discuss the line vortex (Fig. 10-109) as an example of an irrotational flow field. The velocity components are
FIGURE P10-109
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EBK FLUID MECHANICS: FUNDAMENTALS AND A
- In a steady, two-dimensional flow field in the xyplane, the x-component of velocity is u = ax + by + cx2 where a, b, and c are constants with appropriate dimensions. Generate a general expression for velocity component ? such that the flow field is incompressible.arrow_forwardA 2-D flow field has velocity components along X-axis and y-axis given by u = x't and v = -2 xyt respectively, here, t is time. The equation of streamline for the given velocity field is : (а) ху — сonstant (с) ху' — сonstant (b) x´y = constant (d) x + y constantarrow_forwardConsider irrotational flow past a stationary sphere of radius R located at the origin. In the limit r→∞, the velocity field v = U2, as in Fig. 8-6 in the book. (a) Calculate the velocity field v assuming potential flow given by v = Vo(r, 0, 0), where the potential can be assumed to be independent of the azimuthal coordinate and vo= 0. Here, since ə rde Ə Ər for large r/R, look for solutions of the form = f(r) cos 0. Assume a no-penetration boundary condition at the surface of the sphere. (b) Calculate the pressure P and the drag force due to pressure. Vr = U cos 0 and Vo = -U sin 0arrow_forward
- Write down the continuity equation and the Navier-Stokes equations in the x-, y-, and z-directions for an incompressible, three-dimensional flow. There should be a total of fourequations. If we make the assumptions that the flow is steady and inviscid, what do thesefour equations simplify to? Note: this is notvan assignment question and not a grade questionarrow_forward(2) Consider the following fluid velocity fields: F(x,y) = (x,y), F(x,y)=(-x, y), F(x,y) = (y, 0). (a) Plot the three fields as glyphs. Which of these vector fields represent an expansion, a compression and a shear flow? (b) Calculate the divergence of the three fields V F. Can you relate the value of the divergence with the nature (compression, expansion or shear of the flow)? (c) Calculate the circulation V x F and relate it with the nature of the flow.arrow_forwardThe velocity potential function (0) is given by an expression xy' x'y x* + 3 3 (i) Find the velocity components in x and y direction. (ii) Show that o represents a possible case of flow.arrow_forward
- Consider the following steady, two-dimensional, incompressible velocity field: V-› = (u, ? ) = ( 1/2ay2 + b) i-› + (axy2 + c) j-›. Is this flow field irrotational? If so, generate an expression for the velocity potential function.arrow_forwardA incompressible, steady, velocity field is given by the following components in the x-y plane: u = 0.205 + 0.97x + 0.851y ; v = v0 + 0.5953x - 0.97y How would I calculated the acceleration field (ax and ay), and the acceleration at the point, v0= -1.050 ? Any help would be greatly appreciated :)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
- he velocity at apoint in aflued for one-dimensional Plow wmay be aiven in The Eutkerian coordinater by U=Ax+ Bt, Show That X Coordinates Canbe obtained from The Eulerian system. The intial position by Xo and The intial time to zo man be assumeal · 1. x = foxo, yo) in The Lagrange of The fluid parficle is designatedarrow_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_forwardThe velocity components of an incompressible, two-dimensional velocity field are given by the equations Show that the flow satisfies continuity. (b) Determine the corresponding stream function for this flow field. (c) Determine if the flow is irrotational.arrow_forward
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