HW_6 - 210 CHAPTER 5 I IIqunucnON TD DIFFERENTIAL ANALYSIS...

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Unformatted text preview: 210 CHAPTER 5 I IIqunucnON TD DIFFERENTIAL ANALYSIS OF FLUID MOTION $5.26 Does the velocity field of Problem 5.22 represent a possible 5.37 COHSidef the flow fiEId given by ‘7 =xy2i — h“; + xyl. D0- incompressible llow case? If so. evaluate and sketch the stream lemme (a) the number OfdiI-‘neflSiOUS Oflhfi fi0W. (b) if“ is apossr function for the flow. If not. evaluate the rate of change of density ible incompreSSibIC flow. and (C) lhe acceleration Ufa flUid panic in the flow field. at point (x.y.: = (l. 2. 3). $5.27 Consider a flow with velocity components a = 0. U = 5-33 confide" fhe flow field given by V :a-‘ll’iHbl’j'i'rzz ‘ y(_v2 — 3:2), and w = 5 a: — 3f) wherea = I mi‘vsLEJ: : 3 s' I. and c = 2 m_1-s’l. Determ’ a, [5 [his a 0113., mm, or mree_dimensionat flow? (a) the number of dimensions of the flow, (b) if it is a possible in- b. Demonstrate whether this is an incompressible or com— compressible flow- all“ (C) [he acceleration 0f a mild Panic“? =' pressible flow. Win! (313’. Z} = (3. l. 2). c. If possible. deriveastream function for this flow. 5.39 The velocity field within a laminar boundary layer r “$.28 An incompressible frictionless flow field is specified by the aPPmXimale‘l by the expressmn stream function If) =-2Ax - My. where A = 1 mix. and x and y .. g AUy . AUy2 5 are coordinates in meters. Sketch the streamlines ti; = l) and If: = V — rm; f 4“ 4x3]; J 5. Indicate the direction of the velocity vector at the point (0. 0) on the sketch. Determine the magnitude of the flow rate between the streamlines passing through the points (2, 2) and (4, 1). In this expression. A = 141 m' “2. and U = 0.240 m/s is the free- stream velocity. Show that this velocity field represents a possib I. incompressible flow. Calculate the acceleration of a fluid panic at point (.r. y) = (0.5 m. 5 mm). Determine the slope of u- streamline through the point. $5.29 In a parallel one-dimensional flow in the positive I direc— tion. the velocity varies linearly from zero at y = O to 30 m/s at y = LS Tn. Determine an expression for the stream function. I11. Also determine the y coordinate above which the volume flow rate is halfthe total between _v I 0 and _v = 1.5 m. 5.40 The x component of velocity in a steady. incompressi -- flow field in the xy plane is u = A(.r5~10x3y2+5xy4), where A . _‘ . _ ‘ I 2 m'4 -s ' and x is measured in meters. Find the simplest y c n “5-30 A “"63" “3"le Pmfilc was “sad [0 "1056] flow 1" a “mum” ponent of velocity for this flow field. Evaluate the acceleration ii incompressible boundary layer in Problem 5.”). Derive the stream a fluid Daniele at point (x. v) : (1, 3)_ function for this flow held. Locate streamlines at one-quarter and 5.“ Consider the velocity field V = mm; +y2)f+Ayl(r, "H2" one—half the total volume flow rate in the boundary layer. . i 2 . In the Ajv plane. where A — l0 m Is, and x and y are measured _ $5.31 A parabolic velocity profile was used to model flow in *1 meters. Is this an incompressible flow field? Derive an expressi laminar incompressible boundary layer in Problem SJ 1. Derive for [ha fluid acceleration Evajuate the velocity and accelerati the stream function for this flow field. Locate streamlines at one— along the x axis. the y axis. and along a fine defined by y = 11-. quarter and one-half the total volume flow rate in the boundary can you conclude abouuhis flow field? layer‘ 5.42 They component of velocity in a two-dimensional. inco I *5.32 Derive the stream function that represents the sinusoidal pressible flow field is given by v = —Axy. where U is in m/SJa-fld approximation used to model. the 1: component of velocity for the are in meterS. and A is a dimensional constant. There is no veloci a: boundary layer Of Problem 5.12. Locate streamlines at one-quarter component or variation in the z direction, Determine the dime‘ and one-half the total volume flow rate in the boundary layer. sions of the constant. A. Find the simplest .r component of veloci . 215‘ 3 3 A cubic velocity profile was used to model flaw in a laminar in this flow field. Calculate the acceleration of a fluid particle incompressible boundary layer in Problem 5.13. Derive the stream mint (3" 3') = (1* 2)' function for this flow field. Locate streamlines at one-quarter and 5.43 An incompressible liquid with negligible viscosity ll one-half the total volume flow rate in the boundary layer. steadin through a horizontal pipe of constant diameter. In a n ous section of length L = 0.3 m. liquid is removed at a cons zu‘ rate per unit length. so the uniform axial velocity in the pipe I. nix) = Utl - .n'ZL). where U = 5 m/s. Develop an expressi for the acceleration of a fluid panicle along the centerline of I: porous section. i“:5..l4 A rigid-body motion was modeled in Example 5.6 by the velocity field l7 = mic”. Find the stream function for this flow. Evaluate the volume flow rate per unit depth between r. = 0. I0 m and r1 = 0.|2 m, if m = 0.5 radls. Sketch the velocity profile along a line of constant 0. Check the. flow rate calculated from the stream function by integrating the velocity profile along this line. \9‘ 5.44 An incompressible liquid with negligible viscosity steadily through a horizontal pipe. The pipe diameter linearly -: ies from a diameter of It} cm to a diameter of 2.5 cm over :1 Ian of 2 m. Develop an expression for the acceleration of a fluid =I= ticle along the pipe centerline. Plot the centerline velocity and : ' celeration versus position along the pipe. if the inlet centerl' velocity is 1 m/s. 35.3.5 Example 5.6 showed that the velocity field for a free vortex in the rt) plane is 17 = éuC/r. Find the stream function for this flow. Evaluate the volume [low rate per unit depth between r1 = 0.10 In and r2 = 0. l 2 m. if C = 0.5 mln’s. Sketch the velocity profile along a line of constant 0. Check the flow rate calculated from the stream function by integrating the velocity profile along this line. 536 Consider the velocity field I? : ADA _ 612).: + yit); + 5.45 Solve Problem 4.] 18 to show that the radial velocity in up. I narrow gap is V, = Q/anh. Derive an expression for the accele Al4xy" —4l.t3_v)yi in the xy plane. where A = 0.25 m'i-s . and | . tion ofa fluid pamcle In the gap. the coordinates are measured in meters. Is this a possible incom- pressible flow field? Calculate the acceleration of a fluid particle 5.46 Consider the low-speed flow of air between parallel di a! Will! (-l'n'l = {2. l). as shown. Assume that the flow is incompressible and invisc *These problems require material from sections that may be omitted without loss of continuity in the text material. 5’6 212 CHAPTER 5 I INTRODUCTION TO DIFFERENTIAL ANALYSIS OF FLUID MOTION 5.60 Air flows into the narrow gap. of height h. between closely spaced parallel disks through a porous surface as shown. Use a control volume. with outer surface located at position r. to show that the uniform velocity in the r direction is V : our/2h. Find an expression for the velocity component in the z direction (U0 < V). Evaluate the components of acceleration for a fluid particle in the gap. P560 5.61 The velocity field for steady inviscid flow from left to right over a circular cylinder, of radius R, is given by l7=Ucosfl [1— (5)]é,—Usin6 [1+ (5)]60 r 1' Obtain expressions for the acceleration of a fluid Particle moving along the stagnation streamline (H = 1t) and for the acceleration along the cylinder surface (r : R). Plot a, as a function of n'R for 9 : It, and as a function of 6 for r = R; plot an as a function of 6 for r = R. Comment on the plots. Determine the locations at which these accelerations reach maximum and minimum values. \96562 Consider the incompressible flow of a fluid through a noz— zle as shown. The area of the nozzle is given by A = Anll 7 bx) and the inlet velocity varies according to U = Unll — 97"”). where A0 = 0.5 mg. L = 5 m, b = 0.! m_[, t = 0.2 s“. and U0 : 5 mfs. Find and plot the acceleration on the centerline. with time as a parameter. P562 63‘563 Consider the one—dimensional. incompressible flow through the circular channel shown. The velocity at section (D is given by U = U0 + U] sin mt, where U.) : 20 m/s, Ul : 2 m/s. and w : 0.3 rad/s. The channel dimensions are L I l to. RI 2 0.2 m, and R2 = 0.1 m. Determine the particle acceleration at the channel exit. Plot the results as a function of time over a complete cycle. On the same plot. show the acceleration at the channel exit if the channel is constant area. rather than convergent. and explain the difference between the curves. 135.63 5.64 Consider again the steady. two-dimensional velocity field of Problem 5.53. Obtain expressions for the particle coordinates. xi, : fit!) and vi, = fitt). as functions of time and the initial par- ticle position. txn, Iv”) at t z 0. Determine the time required for a particle to travel from initial position. (x0. yo) 2 Q. 2) to positions (x, y) 2 (l. l) and (2, l). Compare the particle accelerations de- termined by differentiating fitt) and fit!) with those obtained in Problem 5.53. 5.65 Expand (l7 - V)? in cylindrical coordinates by direct sub- stitution of the velocity vector to obtain the convective accelera- tion ot‘a fluid particle. (Recall the hint in footnote 1 on page 169.) Verify the results given in Eqs. 5.12. 5.66 Which. if any. of the flow fields of Problem 5.1 are irrotational? 5.67 A flow is represented by the velocity field 17 = (.r7 — 2119);2 + 35x52“ — 7m“)? + (Tray - 35x4y-l + 2113}!5 * yllj. Determine if the field is (a) a possible incompressible flow and (b) irrotational. 5.68 Consider again the sinusoidal velocity profile used to mode] the .r component of velocity for a boundary layer in Problem 5.12. Neglect the vertical component of velocity. Evaluate the circula- tion around the contour bounded by x = 0.4 m. x = 0.6 m. y I 0. and _v : 8 mm. What would be the results of this evaluation if it were performed 0.2 in further downstream? Assume U : 0.5 m/s. 5.69 Consider the velocity field for flow in a rectangular “cor- ner.“ V = Axi— ij, with A = 0.3 57L. as in Example 5.8. Evalu- ate the circulation about the unit square of Example 5.8. 5.70 Consider the two-dimensional flow field in which it : Axy and U : Brig. whereA= | m‘L - s". B=—%m 'l - s". and the coordinates are measured in meters. Show that the velocity field represents a possible incompressible flow. Determine the rotation at point (.r, y) = (I. I). Evaluate the circulation about the “curve” bounded by y = 0. .r = 1. y = l. and .r = 0. *5.71 Consider the flow field represented by the stream function if; 2 x6 * 1556i}:z + 15.3}:4 i )2“. Is this a possible two-dimen- sional, incompressible How? Is the flow irrotational? $5.72 Consider a flow field represented by the stream function If; = 3.t5_v e ltbr'ly3 + 3x35“. Is this a possible two-dimensional in- compressible flow? Is the flow irrotational? $5.73 Consider a flow field represented by the stream function ti} =7 A1203 + Uri). whereA : constant. Is this a possible two—di- mensional incompressible flow? ls the flow irrotational? *55'4 Consider a velocity field for motion parallel to the x axis with constant shear. The shear rate is did/(iv = A. where A = 0.1 s' '. Obtain an expression for the velocity field. l7. Calcu- late the rate of rotation. Evaluate the stream function for this flow field. \9‘4‘535 A flow field is represented by the stream function If; = X2 — yl. Find the corresponding velocity field. Show that this flow field is irrotational. Plot several streamlines and illustrate the velocity field. \C’fififl'fi Consider the velocity field given by V=Axyi+ 833}. where A 2 4 m‘ ' -s_ l. B = —2 m '-s". and the coordinates are measured in meters. Determine the fluid rotation. Evaluate the Cir- cttlation about the "curve" bounded by y I 0. x : 1.); : l. and .r *These problems require material from sections that may be omitted without loss ot’continuity in the text material. = U. Obtain an expression for the stream function. Plot several streamlines in the first quadrant. *577 Consider the flow field represented by the stream function tit = At)’ 4" Avg. where A I l s7]. Show that this represents a possible incompressible. flow field. Evaluate the rotation of the flow. Plot a few streamlines in the upper halfplane. i“5.78 Consider the [low represented by the velocity field i V=lAy+Blf+ij, where A 2 6 '. B = 3 m-s' '. and the coordinates are measured in meters. Obtain an expression for the stream function. Plot several streamlines (including the stagnation streamline] in the first quadrant. Evaluate the circulation] about the "curve" botmded by y : ft. x = l. _v I l. and ,r I t'). 5.79 Consider again the viscomctric flow of Example 5.7. Evaluate the average rate of rotation of a pair of perpendicular line segments oriented at :45 from the x axis. Show that this is the saute as in the example. *530 The velocity field near the core of a tornado can be approximated as q al+ K V=-- , an Ear do is this an irrotational flow field“? Obtain the stream function for this flow. 5.8] Consider the pressure~driven flow between stationary paral- lel plates separated by distance t'J. Coordinate _v is measured from the bottom plate. The velocity field is given by u = Utyl'blll — ty/btl. Obtain an expression for the circttlation about a closed con, tour of height h and length L. Evaluate when ft = [7/2 and when it: b. Show that the same result is obtained from the area integral ofthe Stokes Theorem {Eq. 5.18). 5.82 The velocity profile for fully developed flow in a circular tube is l/E Z Vlmx[l * (HEEL Evaluate the rates of linear and angular deformation for this flow. Obtain an expression for the vorticity vector. 5.33 Consider the pressure-driven flow between stationary paral- lel plates separated by distance 21). Coordinate y is measured from the channel centerline. The velocity field is given by it = nmaxll * {Ir/8):]. Evaluate the rates of linear and angular deformation. Obtain an expression for the vorticity vector, C. Find the location where the vorticity is a maximum. 5.34 A linear velocity profile was used to model flow in a lami- nar incompressible boundary layer in Problem 5.10. Express the rotation of a fluid particle. Locate the maximum rate of rotation. Express the rate of angular deformation for a fluid particle. Locate the maximum rate of angular deformation. Express the rates of linear defonrtation for a lluid particle. Locate the maximum rates oflinear deformation. Express the shear force per unit volume in the .r direction. Locate the maximum shear force per unit volume: interpret this result. 5.85 The x component of velocity in a laminar boundary layer in water is approximated as it : U sin(rr'vt'2r5). where U : 3 oils and t5: 2 mm. The _v component ofvelocity is much smaller than it. Oh- 5-5 SUMMARY AND USEFUL EQUATIONS 213 5.86 Problem 4.3l gave. the velocity profile for fully developed laminar flow in a circular tube as it I ttnmll e (r/Rf]. Obtain an expression for the shear force per unit volume in the x direction for this flow. Evaluate its maximum value for the conditions of Problem 4.3 l . \06587 Use Ext-cl to generate the solution of Eq. 5.28 for m = 1 shown in Fig. 5.16. To do so, you need to learn how to perform linear algebra in Excel. For example. for N : 4 you will end up with the matrix equation of Eq. 5.34. To solve this equation for the n values. you will have to compute the inverse of the 4 X 4 matrix. and then multiply this inverse into the 4 X l matrix on the right of the equation. In Excel. to do array operations. you must use the following rules: Prerselect the cells that will contain the result; use the appropriate Et‘t't’f army firm-rim: (look at Excel's Help for def tails); press Ctrl-tShit‘t +Entct‘. not just Enter. For example, to invert the 4 X 4 matrix you would: Prerseleet a blank 4 X 4 array that will contain the inverse matrix: type : mim'r'rset [array containing matrix to be invertedl): press Ctrl-l-Shift-l—Enter. To multiply a 4 X 4 matrix into a 4 x | matrix you would: PreAselect a blank 4 X l array that will contain the result; type = mmut‘t([array containing 4 X 4 matrix]. [auray containing 4 X | matrixl); press CLrl+Shift+ Enter. 065.88 Following the steps to convert the differential equation Eq. 5.28 (for m I l) into a difference equation (for example. Eq. 5.34 for N : 4). solve (In 7+. :2s'n.‘ ()S.‘ dx u tlt) t It"\ I tth) =0 for N : 4. 8. and lo and compare to the exact solution Hem” = sin(x) — cos(x) + e" ‘ Hints: Follow the rules for Excel array operations as described in Problem 5.87. Only the right side of the difference equations will change. compared to the solution method of Eq. 5.28 (for example. only the right side of Eq. 5.34 needs modifying). SE51?) Following the steps to convert the differential equation Eq. 5.28 (for m : 1) into a difference equation (for example. Eq. 5.34 for N I 4). solve do 1 —+tt=,r“ 05x51 (it For N : 4. 8. and 16 and Compare to the extract solution 14(0) : 2 ‘1 Lima : .t‘ i 21 + 2 Hint: Follow the hints provided in Problem 5.88. \S‘aSSU A lU-cm cube of mass M = 5 kg is sliding across an oiled surface. The oil viscosity is ,u = 0.4 N - 57mg. and the thickness of the oil between the cube and surface is (5 2 0.25 mm. 1fthe initial speed of the block is a” 2 i this. use the numerical method that was applied to the linear form of Eq. 5.28 to predict the cube motion for the first second of motion. Use N i 4, 8. and 16 and compare to the exact solution 5 “mun : uni, [Alli/[Ml )t‘ where A is the area of contact. Hint: Follow the hints provided in Problem 5.87. tairt an expression for the net shear force per unit volume in the ,r di~ SE59] Use Excel to generate the solutions of Eq. 5.28 for m : 2 rection on a fluid element. Calculate its maximum value for this flow. showu in Fig. 5. l9. *These problems require material from sections that may be omitted without loss ofcontinuity in lbe text material. ...
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HW_6 - 210 CHAPTER 5 I IIqunucnON TD DIFFERENTIAL ANALYSIS...

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