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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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Fluid Mechanics: Fundamentals and Applications
- Homework! Parts A-C and label them Question: A two-dimensional incompressible flow has the velocity components u = 5y and v = 4x. Part A - Check continuity equation is satisfied. Part B - Are the Navier-Stokes equations valid? Part C - If so, determine p(x,y) if the pressure at the origin is po. HINT *** : u = 5y and v = 4x.arrow_forwardIn your own words,a. Describe what is viscosity and explain why viscosity of fluid is related to Navier-Stokes equation.b. Describe what is vorticity and explain its relationship with circulation.c. Explain why the stream function ψ is restricted to 2-D flows and you have to use the velocity potential ζ to define 3-D flows.arrow_forwardInvestigate the complex potential f(z) = A cosh [π(z/a)],and plot the streamlines inside the region shown in Fig.What hyphenated word (originally French) mightdescribe such a flow pattern?arrow_forward
- The 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_forwardDuring the course, we have learned about the Bernoulli equation as well as rotational and irrotational flows. Answer the following problemsc. Give explicit one example of velocity field (v = … i + … j meter/second) for rotational flow and one example for irrotational flow.d. Is Bernoulli equation valid only for rotational flows or it is also valid for irrotational flow? Explain your answer.arrow_forwardA velocity field is specified as shown, the coordinates are measured in meters. Is the flow field one-, two-, or three-dimensional? Why? Calculate the velocity components at the points (2, 1/2). Develop an equation for the streamline passing through this point. Plot several streamlines in the first quadrant including the one that passes through the point (2, 1/2)arrow_forward
- For 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_forwardConsider fully developed Couette flow—flow between two infinite parallel plates separated by distance h, with the top plate moving and the bottom plate stationary as illustrated in Fig. The flow is steady, incompressible, and wo-dimensional in the xy-plane. The velocity field is given by V-› = (u, ?) = (Vy/h) i-› + 0 j-›. Generate an expression for stream function ? along the vertical dashed line in Fig. For convenience, let ? = 0 along the bottom wall of the channel.calculate the volume flow rate per unit width into the page of Fig. from first principles (integration of the velocity field). Compare your result to that obtained directly from the stream function. Discuss.arrow_forwardIn a certain two‐dimensional flow field, the velocity is constant with components u = –4 ft/s and v = –2 ft/s.Determine the corresponding stream function and velocity potential for this flow field. Sketch theequipotential line φ = 0 which passes through the origin of the coordinate system. Could you answer and explain every step pleasearrow_forward
- The velocity components are ur = 0, u? = Г/(2?r), and uz = 0. Compute the viscous term of the ?-component of the Navier–Stokes equation, and discuss. Verify that this velocity field is indeed irrotational by computing the z-component of vorticity.arrow_forwardFor each statement, choose whether the statement is true or false, and discuss your answer briefly. (a) The velocity potential function can be defined for threedimensional flows. (b) The vorticity must be zero in order for the stream function to be defined. (c) The vorticity must be zero in order for the velocity potential function to be defined. (d) The stream function can be defined only for two-dimensional flow fields.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
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