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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
- Consider the steady, two-dimensional, incompressible velocity field, namely, V-›= (u, ?) = (ax + b) i-›+ (−ay + cx) j-›. Calculate the pressure as a function of x and y.arrow_forwardConsider a steady, two-dimensional flow field in the xy-plane whose x-component of velocity is given by u = a + b(x − c)2 where a, b, and c are constants with appropriate dimensions. Of what form does the y-component of velocity need to be in order for the flow field to be incompressible? In other words, generate an expression for ? as a function of x, y, and the constants of the given equation such that the flow is incompressiblearrow_forwardFor the velocity field that is linear in both spatial directions (x and y) is V-›= (u, ? ) = (U + a1x + b1y) i-›+ (V + a2x + b2y) j-›where U and V and the coefficients are constants, calculate the shear strain rate in the xy-plane.arrow_forward
- derive the equation of stream line in cylinderical coordinates in 2 dimensions.arrow_forwardConsider the steady, two-dimensional velocity field given by V-› = (u, ?) = (1.6 + 2.8x) i-› + (1.5 − 2.8y) j-›. Verify that this flow field is incompressible.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_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 a velocity field where the radial and tangential components ofvelocity are Vr = 0 and Vθ = cr, respectively, where c is a constant. Is the flow field given is irrotational? Prove your answer.arrow_forwardThe stream function of an unsteady two-dimensional flow field is given by ? = (4x/y2 )t Sketch a few streamlines for the given flow on the xy-plane, and derive expressions for the velocity components u(x, y, t) and v(x, y, t). Also determine the pathlines at t = 0.arrow_forward
- In cylindrical polar coordinates (r, theta, z), the velocity components of a flow field are given as vr=r2zcos(theta), vtheta=rzsin(theta), vz=z2t. Determine the local, convective and total acceleration of a fluid particule at r=0.5m, z=1m, theta = (pi)/4arrow_forwardThe velocity field of a flow is given by V axyi + by2j where a = 1 m-1s-1 and b = - 0.5 m-1s-1. Thecoordinates are in meters. Determine whether the flow field is three-, two-, or one-dimensional. Findthe equations of the streamlines and sketch several streamlines in the upper half plane (arrow_forwardIn a certain region of steady, two-dimensional, incompressible flow, the velocity field is given by V-› = (u, ? ) = (ax + b) i-› + (−ay + cx) j-›. Show that this region of flow can be considered inviscid.arrow_forward
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