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The stream function for steady, incompressible, two-dimensional flow over a circular cylinder of radius a and free-stream velocity
FIGURE P10-70
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Fluid Mechanics: Fundamentals and Applications
- A common flow encountered in practice is the crossflow of a fluid approaching a long cylinder of radius R at a free stream speed of U∞. For incompressible inviscid flow, the velocity field of the flow is given as in fig. Show that the velocity field satisfies the continuity equation, and determine the stream function corresponding to this velocity field.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_forwardPlease answer botharrow_forward
- 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. 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_forwardAy 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
- Pravin bhaiarrow_forwardProblem 1 Given a steady flow, where the velocity is described by: u = 3 cos(x) + 2ry v = 3 sin(y) + 2?y !! !! a) Find the stream function if it exists. b) Find the potential function if it exists. c) For a square with opposite diagonal corners at (0,0) and (47, 27), evaluate the circu- lation I = - f V.ds where c is a closed path around the square. d) Calculate the substantial derivative of velocity at the center of the same box.arrow_forwardConsider a two-dimensional flow which varies in time and is defined by the velocity field, u = 1 and v = 2yt. Compute the convective derivative of each velocity component: Du/Dt and Dv/Dt.arrow_forward
- Consider 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(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_forwardplease answer quicklyarrow_forward
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