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 10, Problem 67P
To determine
The Laplace equation is valid in an irrotational region of flow.
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Define vorticity in terms of the vector velocity field. How is the “rotation” of a fluid particle related to its vorticity? In Cartesian coordinates, write down the general form of the z-component of vorticity. What is the condition on the vector velocity field such that a flow is irrotational?
Write out the three components of the Navier–Stokes equation in Cartesian coordinates in terms of modified pressure. Insert the definition of modified pressure and show that the x-, y-, and z-components are identical to those in terms of regular pressure. What is the advantage of using modified pressure?
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a) Start with Navier-Stokes equations and determine the velocity at the point x = 4 cm and y= 0.58 cm. The value of the velocity is ________
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Chapter 10 Solutions
Fluid Mechanics: Fundamentals and Applications
Ch. 10 - Discuss how nondimensalizsionalization of the...Ch. 10 - Prob. 2CPCh. 10 - Expalain the difference between an “exact”...Ch. 10 - Prob. 4CPCh. 10 - Prob. 5CPCh. 10 - Prob. 6CPCh. 10 - Prob. 7CPCh. 10 - A box fan sits on the floor of a very large room...Ch. 10 - Prob. 9PCh. 10 - Prob. 10P
Ch. 10 - Prob. 11PCh. 10 - In Example 9-18 we solved the Navier-Stekes...Ch. 10 - Prob. 13PCh. 10 - A flow field is simulated by a computational fluid...Ch. 10 - In Chap. 9(Example 9-15), we generated an “exact”...Ch. 10 - Prob. 16CPCh. 10 - Prob. 17CPCh. 10 - A person drops 3 aluminum balls of diameters 2 mm,...Ch. 10 - Prob. 19PCh. 10 - Prob. 20PCh. 10 - Prob. 21PCh. 10 - Prob. 22PCh. 10 - Prob. 23PCh. 10 - Prob. 24PCh. 10 - Prob. 25PCh. 10 - Prob. 26PCh. 10 - Prob. 27PCh. 10 - Consider again the slipper-pad bearing of Prob....Ch. 10 - Consider again the slipper the slipper-pad bearing...Ch. 10 - Prob. 30PCh. 10 - Prob. 31PCh. 10 - Prob. 32PCh. 10 - Prob. 33PCh. 10 - Prob. 34EPCh. 10 - Discuss what happens when oil temperature...Ch. 10 - Prob. 36PCh. 10 - Prob. 38PCh. 10 - Prob. 39CPCh. 10 - Prob. 40CPCh. 10 - Prob. 41PCh. 10 - Prob. 42PCh. 10 - Prob. 43PCh. 10 - Prob. 44PCh. 10 - Prob. 45PCh. 10 - Prob. 46PCh. 10 - Prob. 47PCh. 10 - Prob. 48PCh. 10 -
Ch. 10 - Prob. 50CPCh. 10 - Consider the flow field produced by a hair dayer...Ch. 10 - In an irrotational region of flow, the velocity...Ch. 10 -
Ch. 10 - Prob. 54CPCh. 10 - Prob. 55PCh. 10 - Prob. 56PCh. 10 - Consider the following steady, two-dimensional,...Ch. 10 - Prob. 58PCh. 10 - Consider the following steady, two-dimensional,...Ch. 10 - Prob. 60PCh. 10 - Consider a steady, two-dimensional,...Ch. 10 -
Ch. 10 - Prob. 63PCh. 10 - Prob. 64PCh. 10 - Prob. 65PCh. 10 - In an irrotational region of flow, we wtite the...Ch. 10 - Prob. 67PCh. 10 - Prob. 68PCh. 10 - Water at atmospheric pressure and temperature...Ch. 10 - The stream function for steady, incompressible,...Ch. 10 -
Ch. 10 - We usually think of boundary layers as occurring...Ch. 10 - Prob. 73CPCh. 10 - Prob. 74CPCh. 10 - Prob. 75CPCh. 10 - Prob. 76CPCh. 10 - Prob. 77CPCh. 10 - Prob. 78CPCh. 10 - Prob. 79CPCh. 10 - Prob. 80CPCh. 10 - Prob. 81CPCh. 10 -
Ch. 10 - On a hot day (T=30C) , a truck moves along the...Ch. 10 - A boat moves through water (T=40F) .18.0 mi/h. A...Ch. 10 - Air flows parallel to a speed limit sign along the...Ch. 10 - Air flows through the test section of a small wind...Ch. 10 - Prob. 87EPCh. 10 - Consider the Blasius solution for a laminar flat...Ch. 10 - Prob. 89PCh. 10 - A laminar flow wind tunnel has a test is 30cm in...Ch. 10 - Repeat the calculation of Prob. 10-90, except for...Ch. 10 - Prob. 92PCh. 10 - Prob. 93EPCh. 10 - Prob. 94EPCh. 10 - In order to avoid boundary laver interference,...Ch. 10 - The stramwise velocity component of steady,...Ch. 10 - For the linear approximation of Prob. 10-97, use...Ch. 10 - Prob. 99PCh. 10 - One dimension of a rectangular fiat place is twice...Ch. 10 - Prob. 101PCh. 10 - Prob. 102PCh. 10 - Prob. 103PCh. 10 - Static pressure P is measured at two locations...Ch. 10 - Prob. 105PCh. 10 - For each statement, choose whether the statement...Ch. 10 - Prob. 107PCh. 10 - Calculate the nine components of the viscous...Ch. 10 - In this chapter, we discuss the line vortex (Fig....Ch. 10 - Calculate the nine components of the viscous...Ch. 10 - Prob. 111PCh. 10 - The streamwise velocity component of a steady...Ch. 10 - For the sine wave approximation of Prob. 10-112,...Ch. 10 - Prob. 115PCh. 10 - Suppose the vertical pipe of prob. 10-115 is now...Ch. 10 - Which choice is not a scaling parameter used to o...Ch. 10 - Prob. 118PCh. 10 - Which dimensionless parameter does not appear m...Ch. 10 - Prob. 120PCh. 10 - Prob. 121PCh. 10 - Prob. 122PCh. 10 - Prob. 123PCh. 10 - Prob. 124PCh. 10 - Prob. 125PCh. 10 - Prob. 126PCh. 10 - Prob. 127PCh. 10 - Prob. 128PCh. 10 - Prob. 129PCh. 10 - Prob. 130PCh. 10 - Prob. 131PCh. 10 - Prob. 132PCh. 10 - Prob. 133PCh. 10 - Prob. 134PCh. 10 - Prob. 135PCh. 10 - Prob. 136PCh. 10 - Prob. 137PCh. 10 - Prob. 138P
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- Consider a three-dimensional, incompressible, irrotationalfl ow. Use the following two methods to prove that theviscous term in the Navier-Stokes equation is identicallyzero: (a) using vector notation; and (b) expanding out thescalar terms and substituting terms from the defi nition ofirrotationality.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_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
- 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_forwardIn deriving the vorticity equation, we have used the identity divergence x (divergence P) = 0 Show that this identity is valid for any scalar lamda by checking it in Cartesian and cylindrical coordinates.arrow_forwardConsider the following steady, incompressible, twodimensional velocity field: V-›= (u, ? ) = (4.35 + 0.656x) i-›+ (−1.22 − 0.656y) j-›Generate an analytical expression for the flow streamlines and draw several streamlines in the upper-right quadrant from x = 0 to 5 and y = 0 to 6.arrow_forward
- 3.3 Starting with a small fluid element of volume dx dy dz, derive the continuity equation (Eq. 3.4) in rectangular cartesian coordinates.arrow_forwardUx = -ax Uy = ay Using the streamline equations show the streamlines of the flow as well as stating the family of streamlines.arrow_forwardthe differential equation for conservation of mass, the continuity equation. In cylindrical coordinates, and for steady flow, 1/ r ∂(rur) /∂r + 1/ r ∂u? /∂? + ∂uz /∂z = 0 Write the primary dimensions of each additive term in the equation, and verify that the equation is dimensionally homogeneous. Show all your work.arrow_forward
- A subtle point, often missed by students of fluid mechanics (and even their professors!), is that an inviscid region of flow is not the same as an irrotational (potential) region of flow. Discuss the differences and similarities between these two approximations. Give an example of each.arrow_forwardWhat is the definition of a pathline? What do pathlines indicate?arrow_forwardQuestion 03: Consider a flow field represented by the stream function ψ = (4+ID) x3y - (6+ID) x2y2 + (4+ID) xy3. Is this a possible two-dimensional incompressible flow? Is the flow irrotational? Discuss the reasons against your findings.ID is the last two digits of student’s reg. no.arrow_forward
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