c 2011 Faith A. Morrison, all rights reserved. 1 August 30, 2011

Transcription

1 c 20 Faith A. Moison, all ights eseved. August 30, 20

2 2 c 20 Faith A. Moison, all ights eseved.

3 An Intoduction to Fluid Mechanics: Supplemental Web Appendices Faith A. Moison Associate Pofesso of Chemical Engineeing Michigan Technological Univesity August 30, 20

4 2 c 20 Faith A. Moison, all ights eseved. /Uses/fmoiso/Desktop/untitled folde

5 Pat I Appendices

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7 Appendix A Mathematics Appendix A. Multidimensional Deivatives In section?? we eviewed the basics of the deivative of single-vaiable functions. The same concepts may be applied to multivaiable functions, leading to the definition of the patial deivative. Conside the multivaiable function fx,y. An example of such a function would be elevations above sea level of a geogaphic egion o the concentation of a chemical on a flat suface. To quantify how this function changes with position, we conside two neaby points, fx,y and fx+ x,y + y Figue A.. We will also efe to these two points as f x,y f evaluated at the point x,y and f x+ x,y+ y. In a two-dimensional function, the ate of change is a moe complex concept than in a one-dimensional function. Fo a one-dimensional function, the ate of change of the function f with espect to the vaiable x was identified with the change in f divided by the change in x, quantified in the deivative, df/dx see Figue??. Fo a two-dimensional function, when speaking of the ate of change, we must also specify the diection in which we ae inteested. Fo example, if the function we ae consideing is elevation and we ae standing nea the edge of a cliff, the ate of change of the elevation in the diection ove the cliff is steep, while the ate of change of the elevation in the opposite diection is much moe gadual. To quantify the change in the function fx,y in an abitay diection, we define the diffeential df. Conside two neaby points x,y and x+ x,y + y and the associated

8 6 c 20 Faith A. Moison, all ights eseved. Figue A.: A function of two vaiables fx,y can be sketched as a suface. To quantify how the value of the function changes with position, we conside two neaby points, f x,y and f x+ x,y+ y. values of the function f at these two points. The diffeential df is defined as df lim x 0 y 0 f A. Diffeential defined df lim x 0 y 0 fx+ x,y + y fx,y A.2 We can futhe expess df in tems of ates of change by beaking up the path fom x,y to x+ x,y+ y into two steps. Beginning at x,y we hold y constant and move a distance x in the x-diection. The function f changes as Change in f holding y constant fx+ x,y fx,y A.3 Fom this stating location, we now move a distance y in the y-diection, holding x constant. The function f changes as Change in f holding x+ x constant fx+ x,y + y fx+ x,y A.4

9 c 20 Faith A. Moison, all ights eseved. 7 The total change in f is just the sum of these two incemental changes. Change in f Change in f Total change holding + holding in f y constant x+ x constant f fx+ x,y + y fx,y [fx+ x,y fx,y] A.5 +[fx+ x,y + y fx+ x,y] A.6 The two incemental changes that make up the total change in f can be elated to deivatives in planes of constant y and x. Fo the fist step, changing x at constant y, the situation is sketched in Figue A.2. This situation, with y held constant, evets to the Figue A.2: The total change in f may be boken into fist a change in x and subsequently a change in y. Fo the fist of these two steps the function f changes as shown in this figue. classical situation of a single-vaiable function. The ate of change of f can be elated to a deivative. Because we must specify that y is held constant, we define a new deivative, the patial deivative. slope ise un f x Patial deivative defined y ise un f x y held constant y fx+ x,y fx,y lim x 0 x A.7 A.8 A.9

10 8 c 20 Faith A. Moison, all ights eseved. Fo the second step, changing y at constant x + x, the situation is sketched in Figue A.3. This situation, with x+ x held constant, allows us to wite the patial deivative Figue A.3: The second pat of the total change in f is a change in y at constant x+ x, as shown in this figue. with espect to y using an expession analogous to equation A.9. f y f y x+ x x+ x ise un x + x held constant lim y 0 fx+ x,y + y fx+ x,y y A.0 A. The two expessions fo patial deivative that we have developed, equation A.9 and equation A., may now be combined with the equation fo the total diffeential df equation A.6. The final step in ou development is to take the limits as x and y go to zeo. f fx+ x,y + y fx,y [fx+ x,y fx,y] f f x +[fx+ x,y + y fx+ x,y] A.2 f x+ A.3 y y y x+ x Taking the limit as both x and y go to zeo we obtain the final expession fo the total

11 c 20 Faith A. Moison, all ights eseved. 9 diffeential df. df lim x 0 y 0 f df f f dx+ y x y y x A.4 A.5 The esult in equation A.5 may be extended to functions of moe than two vaiables. Fo example, fo the fou-dimensional function fx,x 2,x 3,t, df becomes f f f f df dx + dx 2 + dx 3 + dt A.6 x x 2,x 3,t x 2 x,x 3,t x 3 x,x 2,t t x,x 2,x 3 This is often witten moe compactly as shown below, with the undestanding that the appopiate vaiables ae held constant duing each patial diffeentiation. df f x dx + f x 2 dx 2 + f x 3 dx 3 + f t dt A.7 A.2 Multidimensional Integals In section?? we eviewed the basics of the integal. In this appendix we expand this backgound mateial into two and thee dimensions. A.2. Double Integals Double integals ae defined as summations ove two dimensions, fo example, x and y. These types of integals can be useful in calculating suface aeas, volumes, and flow ates though sufaces, as well as othe quantities. A.2.. Enclosed Aea One use of the single integal of section?? was to calculate aea, specifically aea unde a cuve, but the single-integal technique is not appopiate fo calculating evey type of aea. Conside, fo example, the aea in a plane bounded by a closed cuve Figue A.4. Looking back at equation??, in the single-integal method we calculated aea unde a cuve by summing the aeas of ectangles constucted fom values of fx i multiplied by the inteval x. In the case of the aea bounded by a closed cuve Figue A.4, the poduct fx i x does not give a piece of the desied aea. Thus the single-integal method does not give us the quantity we seek.

12 0 c 20 Faith A. Moison, all ights eseved. y f x i x i x x Figue A.4: One intepetation of the single integal is as aea unde a cuve. This intepetation is not helpful in finding enclosed aea such as is shown hee. The aea elements that ae summed to give aea unde a cuve single integal do not coespond to a meaningful contibution to an enclosed aea. We can define a new type of integal that is suitable fo this new poblem. The aea bounded by a closed cuve can be appoximated by the sum of the aeas x y, whee x and y ae small intevals in the Catesian coodinates of the plane Figue A.5. The sizes of x and y ae abitay, just as the size of x was abitay in equation??. The key featue is that as x and y ae made smalle, the aea appoximation becomes bette, and in the limit x 0, y 0, we obtain the aea we seek. We now set out to define the double integal. Conside R, an aea in a plane bounded by a closed cuve Figue A.5. We fist constuct a gid by choosing initial sizes x and y and divide up the plane into ectangles of aea x y. The gid we constuct poduces ectangles that ae wholly within the closed cuve, ectangles that ae wholly outside of the closed cuve, and some ectangles that intesect the bounday. We will only count the ectangles that ae wholly inside the bounding cuve. We now sum up the aeas of the ectangles x y that ae wholly within the closed cuve. Aea N [ x y] i i N A i i A.8 A.9 whee N is the numbe of ectangles within the bounded egion, and A i [ x y] i. We define the double integal to be the limit of this sum as the dimensions x and y go to

13 c 20 Faith A. Moison, all ights eseved. y R x y x x y smalle y y x x enclosed aea R lim N x y 0 i [ x y] i Figue A.5: The aea inside a closed cuve may be calculated by dividing up coodinate space and adding up the aeas of enclosed ectangles x y. As x y goes to zeo, the coect aea is obtained. zeo. The double integal is equal to the bounded aea. Enclosed aea in a plane o Double Integal vesion defined I R [ N ] da lim A i A 0 i A.20 The techniques used to evaluate double integals may be found in standad calculus texts[0]. Note that when we defined the single integal, one use was to calculate aea aea unde a cuve and now we see that the double integal is used to calculate aea as well enclosed aea. We will see that to calculate volume, we must sometimes use a double integal section A.2..2 and sometimes a tiple integal section A The facto that detemines what type of integal to use is the definition of the quantity of inteest. We always begin with the definition, and subsequently we develop a limit of a summation that leads us to the appopiate type of integal. We can illustate the use of equation A.20 with an example. EXAMPLE A. What is the aea enclosed by a cicle of adius R? SOLUTION This is a poblem fom elementay geomety, and the solution

14 2 c 20 Faith A. Moison, all ights eseved. is πr 2. This esult can be calculated in cylindical coodinates as follows. Aea of R 2π ds ddθ A.2 cicle S 0 0 2π 2 R 2 πr 2 A.22 0 A.23 This calculation was staightfowad because we could easily wite ds in the cylindical coodinate system. Fo shapes that ae iegula o unusual, we must apply moe thought to the pocess. We will not use the cylindical coodinate system; we will cay out ou integation in the catesian systems shown in Figue A.6. z R R z z y y -R R y Figue A.6: We can calculate the aea of a cicle using geneal methods and a catesian coodinate system as descibed in the text. To calculate the aea of a cicle, we will divide it in half, pefom a double integal ove the half cicle, and double the esult. We begin with equation A.20. Aea enclosed da A.24 by half cicle The diffeential da in ou catesian system is dydz. The only emaining step is to detemine the limits of the integation, which ae elated to the equation fo the cicle. Equation R 2 y 2 +z 2 A.25 fo cicle R

15 c 20 Faith A. Moison, all ights eseved. 3 The vaiable z goes fom zeo to a maximum at z R. As z vaies, the limits between which y vaies change. If we daw a hoizontal line at an abitay value of z, we see that the limits on the coodinate y ae fom y R 2 z 2 to y + R 2 z 2. Thus the integal becomes Aea enclosed by half cicle da R R + R 2 z 2 Caying out this integal we obtain the final esult. 0 A.26 R 2 z 2 dydz A.27 Aea enclosed by half cicle 2 R + R 2 z 2 0 R 0 R 0 [ 2 πr2 2 dydz A.28 R 2 z 2 + R 2 z 2 y dz A.29 R 2 z 2 2 R 2 z 2 dz A.30 z R 2 z 2 +R 2 sin z ] R R 0 A.3 A.32 Thus, the aea fo a complete cicle is twice this esult o πr 2, as expected. No special coodinate system was equied fo this calculation; we needed to know only the equations of the bounday, which detemine the limits of the integation. A.2..2 Volume Thee ae many othe quantities that ae calculated as limits of two-dimensional sums simila to equation A.20. One example is the calculation of volume, specifically the volume V of the solid egion V bodeed by a function z fx,y and the domain in the xy plane ove which fx,y is defined. Conside the solid shown in Figue A.7. The base of the solid V is the suface R in the xy plane. Fist we divide R into sub-aeas x y A as befoe. Fo the i th aea A i, fx i,y i is a value of the function fx,y evaluated at any point x i,y i within A i. Then the quantity fx i,y i A i epesents the volume of a vetical ectangula pism that

16 4 c 20 Faith A. Moison, all ights eseved. z z f x, y y V R x volume f x, y A i i i Figue A.7: A double integal may be used to calculate the volume of the solid egion bodeed by a function z fx,y and the domain in the xy plane ove which fx,y is defined. appoximates the volume of the potion of V that stands diectly above A i. Volume between the uppe suface fx,y and A i in the xy plane fx i,y i A i A.33 The total volume of the solid V is then appoximated by the sum of these contibutions, counting only the aeas A i that ae wholly within R. Volume between the uppe suface fx,y and R in the xy plane N fx i,y i A i i A.34 The tue volume of the solid is given by the limit of this summation as x and y go to zeo. Volume between the uppe suface fx,y and R in the xy plane lim A 0 [ N ] fx i,y i A i i A.35 whee as befoe A i [ x y] i. The limit of the sum in equation A.35 is defined as the

17 c 20 Faith A. Moison, all ights eseved. 5 double integal of the function fx,y ove R. Double Integal of a function geneal vesion I R [ N ] fx,y da lim fx i,y i A i A 0 i A.36 Volume between the uppe suface fx,y and R in the xy plane R fx,y da A.37 Note that the definition of double integal above equation A.36 educes to the definition of double integal we made ealie equation A.20 when the function fx,y is taken to be fx,y. A.2..3 Mass Flow Rate The double integal is a quantity that can have meanings othe than aea equation A.20 o volume equation A.37. Fo example, in fluid mechanics we often need to calculate the total mass flow ate though a flat suface Figue A.8. Conside the flow though the flat v x i, yi z y nˆ v nˆ x R A i x i, yi Figue A.8: The double integal allows us to calculate the mass flow ate though a flat suface. In locations whee the velocity is not pependicula to the suface, only the component of velocity pependicula to the suface contibutes to mass flow though the suface.

18 6 c 20 Faith A. Moison, all ights eseved. suface R shown in Figue A.8. The fluid velocity vx,y is a vecto function of position, and the fluid density ρx,y is a scala function of position. We seek to calculate the mass flow ate though the suface R. If we divide R into ectangles A as befoe, we can appoximate the mass flow ate though R as the sum of the mass flow ates though the individual aeas A i. In the limit that A goes to zeo, the eo in this appoximation goes to zeo, and this sum becomes the total mass flow ate though R. Note that in the cuent case of a flat suface R, the unit nomal to the suface ˆn ê z is independent of position. Mass flow ate though the i th ectangle A i mass volume flow volume time A.38 vxi,y i ˆn A i A.39 ρx i,y i The quantity v ˆn is used in the expession fo the volume flow pe unit time above athe than v since only the component of velocity pependicula to the suface contibutes to the flow acoss the suface Figue A.8. Wenowsumoveallectangles A i thataefullycontainedwithinrandsubsequently take the limit as A becomes small. Mass flow ate though R Mass flow ate though R N ρx i,y i vx i,y i ˆn A i i lim A 0 [ N ρx i,y i vx i,y i ˆn ] A i i A.40 A.4 Compaing equation A.4 to the definition of double integal in equation A.36 we see that the total mass flow though R is given by a double integal ove the function fx,y ρv ˆn. Mass flow ate though flat suface R ρv ˆn da R A.42 In fluid mechanics we ae also inteested in flows thought cuved sufacesfigue A.9. With some adjustments, ou pevious stategy fo calculating mass flow though a flat suface should wok on this new poblem: divide up the suface, wite the mass flow fo each piece of suface, sum up these contibutions to obtain the total mass flow ate. In the case of a cuved suface, dividing up the suface is ticky, since the suface has a complex shape in geneal. We will need to be systematic. Ou appoach will be to poject S, the thee-dimensional suface aea of inteest, onto a plane we will call the xy plane Figue A.0. The aea of the pojection will be R. Since R is in the xy plane, the unit nomal to R is ê z. We divide the pojection R the way we did in the pevious calculation, into aeas A x y and seek to wite the mass flow ate in

19 c 20 Faith A. Moison, all ights eseved. 7 z y S x Figue A.9: The flow ate though a cuved suface is calculated with a suface integal. diffeent egions of S associated with the pojections A i. By focusing on R and equal-sized divisions of R athe than dividing S diectly, we can aive at the appopiate integal expession. Figue A.0 shows the aea S and its pojection R in the xy plane. The aea R has been divided into ectangles of aea A i, and we will only conside the A i that ae wholly contained within the boundaies of R. Fo each A i in the xy plane we choose a point within A i, and we call this point x i,y i,0. The point x i,y i,z i is located diectly above x i,y i,0 on the suface S. If we daw a plane tangent to S though x i,y i,z i, we can constuct an aea S i that is a potion of the tangent plane whose pojection onto the xy plane is A i Figue A.0. We will soon take a limit as A i becomes infinitesimally small, and theefoe it is not impotant which point x i,y i,0 is chosen so long as it is in A i. Each tangent-plane aea S i appoximates a potion of the suface S, and thus we can wite the mass flow though S as a sum of the mass flows though all the egions S i. The mass flow though S i may be witten as Mass flow ate though the i th tangent plane S i mass volume flow volume time A.43 ρx i,y i,z i vx i,y i,z i ˆn i Si A.44 As befoe, the quantity v ˆn is used in the expession fo the volume flow pe unit time above since only the component of velocity pependicula to S i contibutes to the flow cossing

20 8 c 20 Faith A. Moison, all ights eseved. z R S x i, yi, zi y S i v xi, yi, zi nˆ i v ˆ i n i x A i A i nˆ i eˆ z Si Figue A.0: Fo a suface that is not flat, we fist poject the suface onto a plane called the xy plane. We then divide up the pojection and poceed to wite and sum up the mass flow ate though each small piece. The suface diffeential S can be elated to A, its pojection onto the xy plane, by S A/ˆn ê z. S i. Fo the cuent case of a cuved suface, the diection of the unit nomal ˆn i will vay with position. We now sum ove all tangent-planes S i that ae associated with those pojections A i that ae fully contained within R. Subsequently we take the limit as A becomes small. Mass flow ate though S Mass flow ate though S N ρx i,y i,z i vx i,y i,z i ˆn i Si i [ N lim ρx i,y i,z i ] vx i,y i,z i ˆn i Si A 0 i A.45 A.46 The ight-hand-side of equation A.46 is simila to equation A.36, the definition of the double integal, but it is not quite the same, since S i appeas athe than A i. We can elate the tangent-plane aea S i and the pojected aea A i though geomety see Appendix A.6. The elationship is A i ˆn i ê z S i A.47

21 c 20 Faith A. Moison, all ights eseved. 9 Substituting equation A.47 into equation A.46 we obtain [ Mass flow ate N ρx i,y i,z i ] vx i,y i,z i ˆn i lim A though S i A 0 ˆn i ê z i A.48 Compaison with equation A.36 shows that this expession may be witten as a double integal ove the pojected aea R. Mass flow ate ρ v ˆn da A.49 though S ˆn ê z If we define ds da/ˆn ê z then equation A.49 becomes Mass flow ate though abitay suface S R S ρv ˆn ds A.50 This final esult is witten as an integal ove the suface S, which is how mass flow ate though an abitay suface is usually expessed. The pevious vesion of this esult in equation A.49 may be moe convenient duing actual calculations, howeve[0]. A.2..4 Suface Aea Anothe quantity of inteest is the suface aea of an abitay suface. Conside the suface S discussed in the last section Figue A.0. If we poject the suface onto the divided xy plane as befoe and constuct the tangent planes S i, the suface aea S may be witten as the limit of a sum as follows. Suface aea of S Suface aea of S N S i i [ N ] lim S i A 0 i [ N ] A i lim A 0 ˆn i ê z i A.5 A.52 A.53 Compaing this expession with the definition of the double integal equation A.36 andagainwitingds da/ˆn ê z, weobtaintheequationfothesufaceaeaofanabitay suface. Suface aea of S R da ˆn ê z S We can illustate the use of equation A.54 with an example. ds A.54

22 20 c 20 Faith A. Moison, all ights eseved. EXAMPLE A.2 What is the total suface aea of a cylinde of adius R and length L? is SOLUTION This is a poblem fom elementay geomety, and the solution Total suface aea of cylinde 2πRL+2πR 2 A.55 This esult can be calculated in cylindical coodinates as follows. R 2π Aeas of top/bottom S ds ddθ A π 2 R 2 πr 2 A.57 0 Aea of L 2π ds R dθdz 2πRL A.58 sides Total suface aea of cylinde S 0 0 2πRL+2πR 2 A.59 This calculation was staightfowad because we could easily wite ds in the cylindical coodinate system. Fo shapes that ae iegula o unusual, we must use the vesion of equation A.54 witten in tems of an integal ove da. Suface aea of S R da ˆn ê z S ds A.60 We will not use the cylindical coodinate system in this case; we will cay out ou integation in the catesian systems shown in Figue A.. Weshowed insectiona.2.. howtocalculatetheaeasofthetopandbottom of the cylinde in a geneal coodinate system. To calculate the aea of the cylindical sides, we will divide the cylinde in half lengthwise, poject it onto a suface R in the xy-plane, calculate the aea using equation A.54, and multiply by 2 to get the total aea of the sides of the cylinde. We begin with equation A.54, which is an integation ove the ectangula pojection R. Fo ou chosen coodinate system, da dydx, and the limits of x and y that span R ae 0 x L and R y +R. Suface aea of half cylinde da ˆn ê z R L +R 0 R ˆn ê z dydx A.6 A.62

23 c 20 Faith A. Moison, all ights eseved. 2 sin θ nˆ cos θ θ R y R z z x L y R R Figue A.: The methods of this section can be used to fomally calculate the suface aea of any volume; one example whee we can check ou final esults is the suface aea of a ight cicula cylinde. The unit nomal ˆn to the suface is a function of position. Fom Figue A. we see that at an abitay point x,y,z, n can be witten as 0 ˆn cosθê y +sinθê z cosθ A.63 sinθ cosθ y R sinθ z R R2 y 2 R Theefoe ˆn ê z sinθ R 2 y 2 /R, and the est of the solution follows. xyz A.64 A.65 Suface aea of half cylinde L +R 0 R L +R 0 LR R +R R ˆn ê z dydx R R2 y 2 dydx R2 y 2 dy A.66 A.67 A.68 LR sin y R [ π LR R R 2 π ] 2 A.69 πrl A.70

24 22 c 20 Faith A. Moison, all ights eseved. Thus, the suface aea of the sides of a half-cylinde is πrl and of the fullcylinde is 2πRL. The total suface aea of the cylinde is the aea of the sides plus the aeas of the top and bottom. Total suface aea of cylinde 2πRL+2πR 2 A.7 No special coodinate system was equied fo this calculation; we needed to know only the equations of the bounday, which allowed us to calculate ˆn and ˆn ê z. A.2.2 Tiple Integals Integals in thee dimensions x, y, z ae called tiple integals. Like the double integal and the single integal, the tiple integal is a limit of a sum. The simplest application of the tiple integal is to calculate the volume of an abitay solid. A.2.2. Volume Conside a solid of abitay shape with a volume V Figue A.2. To calculate V we constuct the thee-dimensional vesion of a gid by dawing bounding planes paallel to the x, y, and z axes at intevals x, y, and z, espectively. These planes ceate a gid of volumes V x y z that fill all of space. Some of these volumes ae wholly enclosed within V shown in Figue A.2, some ae cut by the oute suface of V, and some ae outside of V not shown. We will estimate the volume of V by summing only those volumes that ae wholly within V. V Volume of solid N V i i A.72 In the limit that x, y, and z go to zeo, the appoximation in equation A.72 becomes the exact volume of the solid. [ V Volume N ] lim V of solid i A.73 V 0 i We define the tiple integal to be this limit of the sum. Volume of a solid o Tiple Integal vesion defined I V [ N ] dv lim V i V 0 i A.74

25 c 20 Faith A. Moison, all ights eseved. 23 V z y x z y x x y z smalle z y x Figue A.2: To calculate the volume V of an abitay solid, we divide space into ectangula paallelepipeds ectangula boxes of volume x y z. In the limit that the volume x y z goes to zeo, the total volume of the solid is equal to the sum of the volumes of all the paallelepipeds that ae located within the solid. Only the wholly enclosed V i ae shown above. A Mass Closely elated to volume is the concept of mass. We can use the idea of the tiple integal to calculate the mass of an abitay solid. Conside once again the solid of abitay shape in Figue A.2. Let ρx,y,z be the density of the solid as a function of position. We can appoximate the mass of the solid by dividing up the solid as we did when calculating its volume. The mass of the solid is appoximately equal to the sum of the masses of those volumes V i x i y i z i located inside the solid. Mass of a solid of volume V N ρx i,y i,z i V i i A.75

26 24 c 20 Faith A. Moison, all ights eseved. In the limit that x, y, and z go to zeo, this appoximation becomes the exact mass of the solid. [ N ] Mass of a solid lim ρx of volume V i,y i,z i V i A.76 V 0 We define the tiple integal of the function ρ to be this limit of the sum. Mass of a solid o Tiple Integal of a function defined I V i [ N ] ρx,y,zdv lim ρx i,y i,z i V i A.77 V 0 This definition is valid fo the tiple integal of any function fx,y,z with fx,y,z substituting fo ρx, y, z in equation A.77. The techniques used to evaluate tiple integals may be found in standad calculus texts[0]. i

27 c 20 Faith A. Moison, all ights eseved. 25 A.3 Diffeential Opeations on Vectos and Tensos To calculate the deivative of a vecto we fist must expess the vecto with espect to a basis. Diffeentiation is then caied out by having a diffeential opeato, e.g. / y, act on each tem, including the basis vectos. Fo example, if the chosen basis is the abitay basis ẽ,ẽ 2,ẽ 3 not necessaily othonomal o constant in space, we can expess a vecto v as v ṽ ẽ +ṽ 2 ẽ 2 +ṽ 3 ẽ 3 A.78 The y-deivative of v is thus, v y y +ṽ 2ẽ 2 +ṽ 3 ẽ 3 y ṽ ẽ + y ṽ 2ẽ 2 + y ṽ 3ẽ 3 ṽ ẽ y +ẽ ṽ y +ṽ ẽ 2 2 y +ẽ ṽ 2 2 y +ṽ ẽ 3 3 y +ẽ ṽ 3 3 y A.79 A.80 A.8 Note that we used the poduct ule of diffeentiation in obtaining equation A.8. This complex situation is simplified if fo the basis vectos ẽ i we choose to use the Catesian coodinate system, ê x, ê y, ê z. In the Catesian coodinate system, the basis vectos ae constant in length and fixed in diection, and with this choice the tems in equation A.8 involving diffeentiation of the basis vectos ae zeo; thus half of the tems disappea. Note that since vecto quantities ae independent of coodinate system, any vecto quantity deived in Catesian coodinates is valid when popely expessed in any othe coodinate system. Thus, when deiving geneal expessions, it is convenient to epesent vectos in Catesian coodinates. Thee ae times when coodinate systems othe than the spatially homogeneous Catesian system ae useful, and we will discuss two such coodinate systems cylindical and spheical in the next section. Remembe that the choice of coodinate system is one of convenience, since vecto expessions ae independent of coodinate system. In Catesian coodinates x x, y x 2, z x 3, the spatial diffeentiation opeato is defined as ê +ê 2 +ê 3 A.82 x x 2 x 3 Del is a vecto opeato, not a vecto. This means that it has some of the same chaacte as a vecto, but it cannot stand alone. We cannot sketch it on a set of axes, and it does not have a magnitude in the usual sense. Note also that although has vecto chaacte, common convention omits the undeba fom this symbol. Since is an opeato, fo it to have meaning it must opeate on something. Del may

28 26 c 20 Faith A. Moison, all ights eseved. opeate on scalas o vectos. When opeates on a scala, it poduces a vecto. α ê +ê 2 +ê 3 α x x 2 x 3 α α α ê +ê 2 +ê 3 x x 2 x 3 The vecto it poduces, α, is called the gadient of the scala quantity α. α x α x 2 α x 3 23 A.83 A.84 A.85 We pause hee to claify two tems we have used, scalas and constants. Scalas ae quantities that ae of ode zeomoe on the concept of ode late. They convey magnitude only. Scalas may be vaiables, such as the distance xt between two moving objects o the tempeatue Tx,y,z at vaious positions in a oom with a fieplace. Multiplication by scalas follows the ules outlined ealie: it is commutative, associative, and distibutive. When combined with a opeato, howeve, the position of a scala is quite impotant. If the position of a scala vaiable is moved with espect to the opeato, the meaning of the expession has changed. We can summaize some of this by pointing out the following ules with espect to opeating on scalas α and ζ: Laws of algeba fo del opeating on scalas NOT commutative α α NOT associative ζα ζα distibutive ζ +α ζ + α The fist limitation, that is not commutative, elates to the fact that is an opeato: α is a vecto while α is an opeato, and they cannot be equal. The second limitation above eflects the ule that the diffeentiation opeato / x acts on all quantities to its ight until a plus, minus, equals sign o backet,{},[] is eached. Thus, is not associative, and expession of the type ζα/ x must be expanded using the usual poduct ule of diffeentiation: ζα x ζ α x +α ζ A.86 x The tem constant is sometimes confused with the wod scala. Constant is a wod that descibes a quantity that does not change. Scalas may be constant as in the speed of light, c m/s o the numbe of cas sold last yea woldwide, and vectos may be constant asin the Catesian coodinate basis vectos, ê x, ê y, and ê z. The issue of constancy only comes up now because we ae dealing with the change opeato,. Constants may be positioned abitaily with espect to a diffeential opeato since they do not change. Anothe thing to notice about the opeato is that it changes the chaacte of the expession onwhich it acts. We saw abovethat when opeates onascala, avecto esults.

29 c 20 Faith A. Moison, all ights eseved. 27 When opeates on a vecto, it yields something of even geate complexity, a 2nd-ode tenso[8,, 4, 7]. A complete discussion of tensos is beyond the scope of this text; late, howeve, we will be giving some impotant equations in vecto notation, and these equations contain del opeating on a vecto fo example w. Fo completeness, theefoe, we show the meaning of this expession below. Note that in caying out the distibution ule in the expessions below the unit vectos emain in the same ode in the final expession as shown in equation A.88 below as they appeaed in the oiginal opeation. w ê +ê 2 +ê 3 w ê +w 2 ê 2 +w 3 ê 3 A.87 x x 2 x 3 w x ê ê + w 2 x ê 2 ê + w 3 x ê 3 ê + w x 2 ê ê 2 + w 2 x 2 ê 2 ê 2 + w 3 ê 3 ê 2 + w ê ê 3 + w 2 ê 2 ê 3 + w 3 ê 3 ê 3 x 2 x 3 x 3 x w k ê p ê k x p p k w x w 2 x w 3 x w x 2 w 2 x 2 w 3 x 2 w x 3 w 2 x 3 w 3 x 3 23 A.88 A.89 a tenso A.90 whee the matix holds the coefficients of the expessions ê ê, ê ê 2, and so on. These expessions ê i ê j ae called indeteminate vecto poducts and ae themselves simple secondode tensos[7]. The ules of algeba fo del opeating on non-constant scalas is NOT commutative, NOT associative, but is distibutive, also hold fo non-constant vectos as outlined below. Laws of algeba fo del opeating on vectos NOT commutative w w NOT associative a b a b a b a b distibutive w +b w + b A second type of diffeential opeation is pefomed when del is dot-multiplied with a vecto o a tenso. This opeato, the divegence, educes the ode of the quantity acted upon. The following opeations ae defined: Scalas, vectos, and tensos can all be classified as tensos of diffeent odes. Scalas ae zeo-ode tensos, vectos ae fist-ode tensos, and the usual tensos encounteed in fluid mechanics ae 2nd-ode tensos. What is changing when del opeates on a scala o vecto is the ode of the quantity upon which it acts [8,, 4, 7].

30 28 c 20 Faith A. Moison, all ights eseved. The divegence of a vecto. w ê +ê 2 +ê 3 w ê +w 2 ê 2 +w 3 ê 3 x x 2 x 3 The esult is a scala. w x + w 2 x 2 + w 3 x 3 The divegence of a tenso appeas in the momentum balance. Fo the tenso B: B ê +ê 2 +ê 3 x x 2 x B mn ê ê m ê n + x m n B mn ê 3 ê m ê n x m n 3 B + B 2 + B 3 ê + x x 2 x 3 B3 + + B 23 + B 33 ê 3 x 3 x 2 x 3 B x + B 2 x 2 + B 3 x 3 B 2 x 2 + B 22 x 2 + B 32 x 3 B 3 x 3 + B 23 x 2 + B 33 x m n 3 m n 3 B mn ê m ê n 3 x 2 B mn ê 2 ê m ê n B2 + B 22 + B 32 ê 2 x 2 x 2 x 3 A.9 A.92 A.93 A.94 A.95 A.96 The esult is a vecto examine equation A.95. The ules of algeba fo the opeation of the divegence,, on vectos and tensos can be deduced by witing the expession of inteest in Catesian coodinates and following the ules of algeba fo the opeation of the diffeentiation opeato / x p on scalas and vectos. One final diffeential opeation is the Laplacian, o 2. This opeation leaves unchanged the ode of the quantity acted upon, and thus we may take the Laplacian of scalas, vectos, and tensos. The action of 2 on scalas and vectos is shown below: The Laplacian of a scala. α ê +ê 2 +ê 3 x x 2 x 3 2 a + 2 a + 2 a x 2 x 2 2 x 2 3 The esult is a scala. ê +ê 2 +ê 3 α x x 2 x 3 A.97 A.98

31 c 20 Faith A. Moison, all ights eseved. 29 The Laplacian of a vecto. w ê +ê 2 +ê 3 x x 2 x 3 ê +ê 2 +ê 3 w ê +w 2 ê 2 +w 3 ê 3 A.99 x x 2 x 3 2 w + 2 w + 2 w 2 w 2 ê x 2 x 2 2 x w w 2 ê 3 x 2 x 2 2 x w w w 3 ê x 2 x 2 2 x 2 3 A w x 2 2 w 2 x 2 2 w 3 x w x w 2 x w 3 x w x w 2 x w 3 x A.0 The esult is a vecto. Coectly identifying the quantities on which del opeates is an impotant issue, and the ules ae woth epeating. The diffeentiation opeato / x i acts on all quantities to its ight until a plus, minus, equals sign, o backet,{},[] is eached. To show how this popety affects tems in a vecto expession, we now do an example. EXAMPLE A.3 What is α b? SOLUTION We begin by witing α b in a Catesian coodinate system: α b ê +ê 2 +ê 3 αb ê +b 2 ê 2 +b 3 ê 3 A.02 x x 2 x 3 ê +ê 2 +ê 3 αb ê +αb 2 ê 2 +αb 3 ê 3 A.03 x x 2 x 3 Since both α and the coefficients of b ae to the ight of the del opeato, they ae all acted upon by its diffeentiation action. The Catesian unit vectos ae also affected, but these ae constant. Now we cay out the dot poduct, using the distibutive law. Since the basis vectos ae othogonal and of unit length,

32 30 c 20 Faith A. Moison, all ights eseved. most of the dot poducts ae zeo. α b ê x αb ê +αb 2 ê 2 +αb 3 ê 3 +ê 2 x 2 αb ê +αb 2 ê 2 +αb 3 ê 3 +ê 3 x 3 αb ê +αb 2 ê 2 +αb 3 ê 3 ê ê αb x +ê ê 2 αb 2 x +ê ê 3 αb 3 x +ê 2 ê αb x 2 +ê 2 ê 2 αb 2 x 2 +ê 2 ê 3 αb 3 x 2 +ê 3 ê αb x 3 +ê 3 ê 2 αb 2 x 3 +ê 3 ê 3 αb 3 x 3 αb x + αb 2 x 2 + αb 3 x 3 A.04 A.05 A.06 To futhe expand this expession, we use the poduct ule of diffeentiation on the quantities in paentheses. α b α b α +b +α b 2 α +b 2 +α b 3 α +b 3 A.07 x x x 2 x 2 x 3 x 3 b α + b 2 + b 3 α α α + b +b 2 +b 3 A.08 x x 2 x 3 x x 2 x 3 This is as fa as we can go. It is possible to wite this final esult in vecto also called Gibbs notation. α b α b+b α A.09 The equivalency of equations A.08 and A.09 may be veified by witing out the tems in equation A.09 and caying out the dot poducts. If the diffeentiation of the poduct had not been caied out coectly, the second tem on the ighthand side would have been omitted. A summay of vecto identities involving the opeato ae given in Table A..

33 c 20 Faith A. Moison, all ights eseved. 3 ap a p + p a A-. v f f v + v f A-.2 ρv v ρ+ρ v A-.3 A v A T : v +v A A-.4 v A A : v +v A T A-.5 pi p A-.6 v 2 v A-.7 v T v A-.8 ρv f ρv f+f ρv A-.9 Table A.: Vecto identities involving the opeato. A.4 Diffeential Opeations in Rectangula and Cuvilinea Coodinates A A xx A xy A xz A yx A yy A yz A.2-7 A zx A zy A zz xyz w w x x w x y w x z w y x w y y w y z w z x w z y w z z xyz A w 2 w x + 2 w x + 2 w x x 2 y 2 z 2 2 w y + 2 w y + 2 w y x 2 y 2 z 2 A w z + 2 w z + 2 w z x 2 y 2 z 2 xyz

34 32 c 20 Faith A. Moison, all ights eseved. Table A.2 Table of diffeential opeations in the ectangula coodinate system x, y, z Continued w w x w y A.2- w z xyz ê x x +ê y y +ê z z A.2-2 a a x a y a z xyz A.2-3 a 2 a 2 a + 2 a + 2 a x 2 y 2 z 2 A.2-4 w wx + wy + wz x y z A.2-5 Table A.2: Table of diffeential opeations in the ectangula coodinate system x, y, z Continues.

35 c 20 Faith A. Moison, all ights eseved. 33 w w w θ A.3- w z θz ê +ê θ θ +ê z z A.3-2 a a a θ a z θz A.3-3 a 2 a a + 2 a + 2 a 2 θ 2 z 2 A.3-4 w w + w θ + wz θ z A.3-5 w w z w θ θ z w wz z w θ w θ θz A.3-6 Table A.3: Table of diffeential opeations in the cylindical coodinate system, θ, z Continues.

36 34 c 20 Faith A. Moison, all ights eseved. A A A θ A z A θ A θθ A θz A.3-7 A z A zθ A zz θz w w w w θ θ w θ w θ + w θ w z w z θ A.3-8 w z w θ z w z z θz 2 w w w θ w z + 2 w + 2 w 2 w θ 2 θ 2 z 2 2 θ + 2 w θ + 2 w θ + 2 w 2 θ 2 z 2 2 θ + 2 w z + 2 w z 2 θ 2 z 2 θz A.3-9 A A + A θ + Az A θθ θ z 2 2 A θ + A θθ + A zθ + A θ A θ θ z A z+ A θz + Azz θ z θz A.3-0 u w u w w +uθ w θ w +uz θ z u wθ w +uθ θ + w wθ +uz θ z u wz w +uθ z wz +uz θ z θz A.3- Table A.3 Table of diffeential opeations in the cylindical coodinate system, θ, z Continued

37 c 20 Faith A. Moison, all ights eseved. 35 w w w θ A.4- w φ θφ ê +ê θ +ê θ φsinθ φ A.4-2 a a a θ a sinθ φ θφ A.4-3 a 2 a 2 2 a + 2 sinθ θ sinθ a θ + 2 a 2 sin 2 θ φ 2 A.4-4 w 2 2 w + w sinθ θ θsinθ+ w φ sinθ φ A.4-5 w sinθ w θ φsinθ w θ sinθ φ w sinθ φ w θ w φ w θ θφ A.4-6 A A A θ A φ A θ A θθ A θφ A.4-7 A φ A φθ A φφ θφ Table A.4: Table of diffeential opeations in the spheical coodinate system, θ, φ Continues.

38 36 c 20 Faith A. Moison, all ights eseved. w w w w θ θ w w φ sinθ φ w θ w θ + w θ w φ w φ θ w θ w φ cotθ w φ + w φ + w θ sinθ φ sinθ φ cotθ θφ A w w + 2 sinθ 2 w θ 2 w φ 2 2 sinθ + 2 θ θ sinθ w θ + θ w θsinθ 2 2 sinθ sinθ w θ 2cotθ 2 sinθ + 2 θ sinθ w φ φ 2 sin 2 θ w θ θsinθ + 2 sin 2 θ w φ φ + 2 w + 2cotθ 2 sinθ φ 2 sinθ w θ φsinθ + 2 sin 2 θ w θ φ 2 w φ 2 2 w θ φ 2 2 w φ φ 2 θφ A.4-9 A 2 2 A + A sinθ θ θsinθ+ A φ A θθ+a φφ sinθ φ 3 3 A θ + A sinθ θ θθsinθ+ A φθ + A θ A θ A φφ cotθ sinθ φ 3 3 A φ + sinθ A θ θφsinθ+ A φφ + A φ A φ +A φθ cotθ sinθ φ θφ A.4-0 u w u w w +uθ w θ +uφ θ w +uθ θ + w +uφ θ u wθ u wφ +u θ w φ θ +u φ sinθ sinθ sinθ w w φ φ w θ φ w φ cotθ w φ φ + w + w θ cotθ θφ A.4- Table A.4 Table of diffeential opeations in the spheical coodinate system, θ, φ Continued

39 c 20 Faith A. Moison, all ights eseved. 37 Catesian coodinates ρ t + ρ v x x +v y ρ y +v ρ z z vx +ρ x + v y y + v z 0 A.5- z Cylindical coodinates ρ t + ρv + ρv θ + ρv z θ z 0 A.5-2 Spheical coodinates ρ t + ρ 2 v + ρv θ sinθ + ρv φ 2 sinθ θ sinθ φ 0 A.5-3 Table A.5: The continuity equation in thee coodinate systems.

40 38 c 20 Faith A. Moison, all ights eseved. Catesian coodinates vx ρ t +v v x x x +v v x y y +v v x z p z x + τxx x + τ yx y + τ zx +ρg x z vy ρ t +v v y x x +v v y y y +v v y z p z y + τxy x + τ yy y + τ zy +ρg y z vz ρ t +v v z x x +v v z y y +v v z z p z z + τxz x + τ yz y + τ zz +ρg z z A.6- A.6-2 A.6-3 Cylindical coodinates v ρ t +v v + v θ v θ v2 θ +v v z p z + τ + τ θ θ τ θθ + τ z +ρg z vθ ρ t +v v θ + v θ v θ θ + v θv v θ +v z p z θ + 2 τ θ + τ θθ 2 θ + τ θz +ρg θ z vz ρ t +v v z + v θ v z θ +v v z z p z z + τ z + τ θz θ + τ zz +ρg z z A.6-4 A.6-5 A.6-6 Spheical coodinates v ρ t +v v p + vθ ρ t +v v θ + v θ vφ ρ t +v v φ + v θ v θ + v φ v sinθ 2 τ + 2 sinθ + v θ v θ θ + v φ sinθ p θ + sinθ 2 τ θ + 2 sinθ p φ + φ v2 θ +v2 φ τ θ sinθ θ v θ φ + v v θ v φ θ + v φ sinθ 2 τ φ + 2 τ θθ sinθ θ v φ φ + v v φ + τ φ sinθ φ τ θθ +τ φφ +ρg v2 φ cotθ + sinθ + v φv θ cotθ τ θφ θ + sinθ τ θφ φ + τ θ cotθτ φφ +ρg θ τ φφ φ + τ φ + 2cotθτ θφ +ρg φ A.6-7 A.6-8 A.6-9

41 c 20 Faith A. Moison, all ights eseved. 39 A.5 Diffeential Equations Enginees and scientists study the solutions of diffeential equations in depth; hee we give a bief eview of the most common solution techniques fo the types of equations encounteed in the intoductoy study of fluid mechanics. In outlining mathematical techniques that ae used to solve diffeential equations, it is helpful to be systematic. To this end we divide diffeential equations into those that depend on a single independent vaiable and those that depend on seveal independent vaiables. The fist goup of equations deal only with odinay deivatives df/dx, fo example, and this goup is called odinay diffeential equations ODEs. The second goup concens patial deivatives f/ x, f/ y, fo example, and these ae known as patial diffeential equations PDEs. The ode of an odinay diffeential equation is the ode of the highest deivative that appeas in the equation. Thus dy +2xy +6 0 dx A.0 is a fist-ode ODE fo the vaiable y fx because it only contains fist deivatives of y. An example of a highe-ode ODE is d 2 u d du 2 d 0 which is a second-ode ODE fo the vaiable u f. A. A diffeential equation is thus chaacteized by the numbe of independent vaiables in the equation one vaiable fo ODEs, two o moe vaiables fo PDEs, and its ode, that is, how high the deivatives ae in the equation. The ode of the diffeential equation tells us how many bounday conditions ae needed to fully solve the equation. Because integation intoduces abitay constants into the solution fo the function, each integation necessitates a bounday condition. Bounday conditions ae values of the function at known values of the independent vaiables. Second-ode diffeential equations equie two bounday conditions while fist-ode diffeential equations equie only one bounday condition. Also quite significant is whethe the equation is linea o nonlinea[5]. A diffeential equation is linea if it is a linea function of the vaiable and of all its deivatives. The geneal linea odinay diffeential equation of ode n is a 0 x dn y dx +a x dn y n dx + +a n x dy n dx +a nxy gx A.2 Note that in equation A.2 each tem has only one deivative in it, multiplied by a function of the independent vaiable x. A nonlinea ODE has tems whee, fo example, the function and its fist deivative ae multiplied togethe. d 2 v dx 2 +vhxdv dx gx A.3

42 40 c 20 Faith A. Moison, all ights eseved. Catesian coodinates vx ρ t +v v x x x +v v x y y +v v x z p 2 z x +µ v x x + 2 v x 2 y + 2 v x +ρg 2 z 2 x vy ρ t +v v y x x +v v y y y +v v y z p 2 z y +µ v y x + 2 v y 2 y + 2 v y +ρg 2 z 2 y vz ρ t +v v z x x +v v z y y +v v z z p 2 z z +µ v z x + 2 v z 2 y + 2 v z +ρg 2 z 2 z A.7- A.7-2 A.7-3 Cylindical coodinates v ρ t +v v + v θ v p vθ ρ t +v v θ + v θ vz ρ t +v v z + v θ θ v2 θ +v v z z +µ v + 2 v 2 θ 2 v θ 2 2 θ + 2 v +ρg z 2 v θ θ + v v θ v θ +v z z p θ +µ v θ + 2 v θ 2 θ + 2 v 2 2 θ + 2 v θ +ρg z 2 θ v z θ +v v z z p z z +µ v z + 2 v z 2 θ + 2 v z 2 z 2 +ρg z A.7-4 A.7-5 A.7-6 Spheical coodinates v ρ t +v v + v θ v p +µ vθ ρ t +v v θ + v θ p vφ ρ t +v v φ + v θ v φ v2 θ +v2 φ θ + v φ sinθ 2 v 2v 2 v θ 2 2 θ 2v θcotθ 2 2 v θ v2 φ cotθ θ + v φ sinθ v θ φ + v v θ 2 v θ v θ +µ v φ θ + v φ sinθ p sinθ φ +µ v θ 2 sinθ θ 2 sin 2 θ 2cosθ 2 sin 2 θ v φ φ + v v φ + v φv θ cotθ 2 v φ v φ 2 sinθ sinθ v φ +ρg φ v φ +ρg θ φ v φ + 2cosθ 2 sin 2 θ v θ +ρg φ φ A.7-7 A.7-8 A

43 c 20 Faith A. Moison, all ights eseved. 4 Catesian coodinates τ xx τ xy τ xz τ yx τ yy τ yz τ zx τ zy τ zz xyz µ 2 vx x v y + vx x y v z + vx x z v x + vy y x 2 vy y v z + vy y z v x + vz z x v y + vz z y 2 vz z xyz A.8- Cylindical coodinates τ τ θ τ z τ θ τ θθ τ θz τ z τ zθ τ zz θz µ 2 v vθ + v + vz z vθ v θ + v 2 v θ + v θ θ v z + v θ θ z v + vz z v z + v θ θ z 2 vz z θz A.8-2 Spheical coodinates τ τ θ τ φ τ θ τ θθ τ θφ τ φ τ φθ τ φφ θφ µ sinθ 2 v vθ + v φ + v vφ vθ + v θ 2 v θ + v θ θ sinθ vφ θ sinθ + sinθ v θ φ 2 v + vφ sinθ φ sinθ vφ θ sinθ sinθ + sinθ v θ φ v φ + v + v θ cotθ φ θφ A.8-3 Table A.8: The Newtonian constitutive equation fo incompessible fluids in ectangula, cylindical, and spheical coodinates. These expessions ae geneal and ae applicable to thee-dimensional flows. Fo unidiectional flows they educe to Newton s law of viscosity as discussed in Chapte??.

44 42 c 20 Faith A. Moison, all ights eseved. Catesian coodinates T t + T v x x +v T y y +v T z k [ ] 2 T z ρĉp x + 2 T 2 y + 2 T + S 2 z 2 ρĉp A.9. Cylindical coodinates T t + T v + v θ T θ +v T z z k ρĉp [ T + ] 2 T 2 θ + 2 T + S 2 z 2 ρĉp A.9.2 Spheical coodinates T v + v θ T θ + v φ T sinθ φ k [ 2 T + ρĉp 2 2 sinθ T t + sinθ T + θ θ ] 2 T 2 sin 2 + S θ φ 2 ρĉp A.9.3 Table A.9: The micoscopic enegy equation in ectangula, cylindical, and spheical coodinates.

45 c 20 Faith A. Moison, all ights eseved. 43 Catesian coodinates τ xx τ xy τ xz τ yx τ yy τ yz τ zx τ zy τ zz xyz η γ A.0- η m γ n m 2 sum of squaes of each tem in γ n 2 m p j γ 2 pj n 2 A.0-2 γ 2 vx x v y + vx x y v z + vx x z v x + vy y x 2 vy y v z + vy y z v x + vz z x v y + vz z y 2 vz z xyz A.0-3 Cylindical coodinates τ τ θ τ z τ θ τ θθ τ θz τ z τ zθ τ zz θz η γ A.0-4 η m γ n m 2 sum of squaes of each tem in γ n 2 m p j γ 2 pj n 2 A.0-5 γ 2 v vθ + v + vz z vθ v θ + v 2 v θ + v θ θ v z + v θ θ z v + vz z v z + v θ θ z 2 vz z θz A.0-6 Table A.0: The powe-law, genealized Newtonian constitutive equation fo incompessible fluids in ectangula, cylindical, and spheical coodinates. These expessions ae geneal and ae applicable to thee-dimensional flows. Fo unidiectional flows they educe to the simple powe-law expession discussed in Chapte?? Continues.

46 44 c 20 Faith A. Moison, all ights eseved. Spheical coodinates τ τ θ τ φ τ θ τ θθ τ θφ τ φ τ φθ τ φφ θφ η γ A.0-7 η m γ n m 2 sum of squaes of each tem in γ n 2 m p j γ 2 pj n 2 A.0-8 γ sinθ 2 v vθ + v φ + v vφ vθ v θ + 2 v θ + v θ θ sinθ vφ θ sinθ + sinθ v θ φ 2 v + vφ sinθ φ sinθ vφ θ sinθ sinθ + sinθ v θ φ v φ + v + v θ cotθ φ θφ A.0-9 The function v fx appeas in the second tem multiplied by its fist deivative; this nonlinea tem complicates the equation consideably. 2 The diffeential equations we solve to completion in this text ae linea diffeential equations, but the equations of fluid mechanics ae in geneal nonlinea. In the sections that follow we eview some intoductoy techniques fo solving two types of ODE and one type of PDE. Fo moe infomation, see the mathematics liteatue[5]. Note that although moden calculatos ae helpful in pefoming many integations, it is often up to us to educe the poblem to a solvable fom befoe the calculato solution is helpful. A.5. Sepaable Equations ODEs Odinay diffeential equations ODEs ae functions of a single vaiable, fo example y fx. An odinay, fist-ode diffeential equation fist-ode ODE may be witten as dy dx φx,y dy φx, ydx A.4 A.5 2 Equation A.3 appeas in fluid mechanics as pat of the momentum balance fo compessible fluids in one paticula flow.

47 c 20 Faith A. Moison, all ights eseved. 45 The function φx,y is a multivaiable function. If equation A.5 can be witten in the following fom, dy hx gy dx gydy hxdx A.6 A.7 then the equation is called sepaable, since all the tems explicitly containing y ae on the left of equation A.7 and all the tems containing x explicitly ae on the ight. With this eaangement we have simplified the multivaiable poblem to two single-vaiable poblems. We can now integate the two sides of the equation sepaately. gydy hxdx + C A.8 whee C is an abitay constant of integation that depends on the bounday conditions. EXAMPLE A.4 Solve fo y fx. dy dx 6y A.9 SOLUTION By algebaic eaangements we can wite equation A.9 as follows. dy y 6dx A.20 This can now be integated diectly and solved. y dy 6dx A.2 lny 6x+C e lny e 6x+C e 6x e C A.22 A.23 y fx Ce 6x A.24 whee C e C is an abitay constant of integation that must be detemined by a bounday condition.

48 46 c 20 Faith A. Moison, all ights eseved. EXAMPLE A.5 Solve fo yx: y dy dx x 3x3 y A.25 By algebaic eaangements, we can wite equation A.25 as follows. dy y 3x 2 + dx x A.26 Integating both sides we obtain, lny x 3 +lnx+c A.27 y ln x x 3 +C A.28 y x ex3 +C e x3 +e C A.29 y fx Cxe x3 A.30 whee C e C is an unknown constant of integation. Substituting equation A.30 into equation A.25 confims the esult. A.5.2 Integating Functions ODEs We can integate odinay diffeential equations y fx of the following type, 3 dy +y ax+bx 0 dx A.3 by using an integating function, ux, whee ux is defined as: ux e ax dx A.32 To solve equation A.3, we fist multiply though by ux. ux dy +uxaxyx bx ux dx A.33 3 This type of diffeential equation is classified as a linea equation of the fist ode[6].

49 c 20 Faith A. Moison, all ights eseved. 47 We seem to have complicated the equation, but actually it has become simple. The choice of ux in equation A.32 makes it now possible to wite the left-hand side of equation A.33 as the x-deivative of the combination uxy. To check this, conside the x-deivative of uxy. Using the poduct ule we obtain We calculate du/dx fom equation A.32. d uxyx udy dx dx +ydu dx A.34 Theefoe u e axdx lnu axdx du udx ax du dx axux d uxyx udy dx dx +axuxy which is the left-hand side of equation A.33, and the factoization checks out. We can theefoe wite equation A.33 as d ux yx dx A.35 A.36 A.37 A.38 A.39 bx ux A.40 The final solution to the diffeential equation comes fom integating equation A.40 to obtain uxyx and then solving fo yx. d u y bx ux A.4 dx du y bx ux dx A.42 u y bx ux dx +C A.43 { } y fx bx ux dx +C A.44 ux whee C is an abitay constant of integation, and x is a dummy vaiable of integation. We intoduce this notation to avoid any confusion between the x on the outside of the integal and the x within the integal.

50 48 c 20 Faith A. Moison, all ights eseved. EXAMPLE A.6 Solve fo y fx: dy +2xy 3x dx A.45 This equation may be solved using the integating function method. Reaanging equation A.45 we obtain, dy +2xy 3x 0 dx A.46 and compaing to equation A.3 we ecognize that ax 2x and bx 3x. We calculate ux fom 2x dx x 2 A.47 ux e ax dx e x 2 Multiplying equation A.45 by the integating function ux we obtain, dy dx ex2 +2xe x2 y 3xe x2 A.48 A.49 By the factoization outlined in equation A.39 this becomes d e x2 y 3xe x2 A.50 dx We can veify the factoization above by caying out the diffeentiation on the left-hand side. Integating equation A.50 we obtain, e x2 y 3x e x 2 dx +C A.5 whee C is an unknown constant of integation. Caying out the integation on the ight side using a calculato, if desied, we obtain the final esult. e x2 y 3x e x 2 dx +C A.52 3 e x2 2xdx+C A ex2 +C A.54 y fx 3 2 +Cex 2 A.55 Substituting equation A.55 into equation A.45 confims the esult.

51 c 20 Faith A. Moison, all ights eseved. 49 A.5.3 Sepaable Equations PDEs Patial diffeential equations PDEs ae functions of a two independent vaiables, such as z fx,y. In multivaiable poblems such as ae encounteed in fluid mechanics, we ae called upon to solve patial diffeential equations PDEs. The simplest type of patial diffeential equation to solve is the sepaable PDE. A geneal fist-ode PDE fo the vaiable z fx,y may be witten as, ax,y z x +bx,y z y A.56 It is a challenge to solve complex equations of this type. It may wok out, howeve, that the solution fo zx, y may be sepaated into the poduct of two single-vaiable functions, fo example gx and hy. zx, y gxhy A.57 If zx, y may be witten this way, this function is temed sepaable. Substituting equation A.57 into the geneal equation fo the PDE equation A.56, we obtain, z x gxhy hydg x dx z y gxhy gxdh y dy ax,y dg dx hy+bx,ydh dy gx ax,y dg dxgx +bx,ydh dy hy gyhx A.58 A.59 A.60 A.6 If the equations coopeate which they ae not doing so fa because we have not specified ax,y o bx,y, then the PDE sepaates into two expessions that ae equal to each othe. Xx Yy A.62 If we ae able to obtain a sepaated esult in the fom of equation A.62, the left side is a function of x only, and the ight side is a function of y only. This is a vey paticula cicumstance, because both x and y ae independent vaiables. As independent vaiables, x and y may be chosen to be any values whatsoeve, completely independent of one anothe. Fo the functions Xx and Yy in equation A.62 to be equal to each othe fo all possible choices of x and y, the two functions must individually be equal to the same constant. We call that constant λ. Xx Yy λ constant A.63