Fluid Mechanics: Fundamentals and Applications
4th Edition
ISBN: 9781259696534
Author: Yunus A. Cengel Dr., John M. Cimbala
Publisher: McGraw-Hill Education
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Chapter 9, Problem 54P
Flow separates at a shap corner along a wall and froms a recirculating and a recirculating separtion bubble as sketched in Fig. 9-54 (streamlines are shown). The value of the stream function at the wall is zero, and that of the uppermost streamline is positive value
FIGURE P9-54
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Chapter 9 Solutions
Fluid Mechanics: Fundamentals and Applications
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- Consider dimensionless velocity distribution in Couette flow (which is also called generalized Couette flow) with an applied pressure gradient which is obtained in the following form as attached, where u, V, ∂P/∂x, and h represent fluid velocity, upper plate velocity, pressure gradient, and distance between parallel plates, respectively. Also, u* , y* , and P* represent dimensionless velocity, dimensionless distance between the plates, and dimensionless pressure gradient, respectively. (a) Explain why the velocity distribution is a superposition of Couette flow with a linear velocity distribution and Poiseuille flow with a parabolic velocity distribution. (b) Show that if P*>2, backflow begins at the lower wall and it never occurs at the upper wall. Plot u* versus y* for this situation. (c) Find the position and magnitude of maximum dimensionless velocity.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_forwardHow does the Navier-Stokes equation encapsulate the complexities of fluid behavior, and what challenges arise when attempting to solve it in the context of mechanical engineering's fluid mechanicsarrow_forward
- Consider 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_forwardConsider a planar irrotational region of flow in the r?-plane. Show that stream function ? satisfies the Laplace equation in cylindrical coordinates.arrow_forwardWhat is the definition of a streakline? How do streaklines differ from streamlines?arrow_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_forwardIs the Lagrangian method of fluid flow analysis more similar to study of a system or a control volume? Explain.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_forward
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