The Fluxes and the Equations of Change

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1 Appendix 15 The Fluxes nd the Equtions of Chnge B.l B. B.3 B.4 B.5 B.6 B.7 B.8 B.9 Newton's lw of viscosity Fourier's lw of het conduction Fick's (first) lw of binry diffusion The eqution of continuity The eqution of motion in terms of т The eqution of motion for Newtonin fluid with constnt p nd fx The dissiption function Ф^ for Newtonin fluids The eqution of energy in terms of q The eqution of energy for pure Newtonin fluids with constnt p nd к B.1O The eqution of continuity for species in terms of ) B.ll The eqution of continuity for species A in terms of (o A for constnt ряь АВ B.l NEWTON'S LAW OF VISCOSITY [T = - - K)(V v)5] Crte coordintes (x, y, z): дх (f/x - K)(V v) (f/x, - K)(V v) (В.1-Г dz (l fc)(v v) dx + (B.l-4) in which - + dy ~dv x dz ~d~z (B.l-5) (B.l-6) dy dz (B.l-7) а When the fluid is ssumed to hve constnt density, the term contining (V v) my be omitted. For montomic gses t low density, the dilttionl viscosity к is zero. 843

2 844 Appendix B Fluxes nd the Equtions of Chnge B.l NEWTON'S LAW OF VISCOSITY (continued) Cylindricl coordintes (г, в, z): v) (B.I-9)" (В.1-10Г I T(ir / = T,/, = i дгг ) 1 ^ ^ Г дв dz г дв (B.l-ll) (B.l-1) in which dv z ~dz + ~dr (B.l-13) (B.l-14) When the fluid is ssumed to hve constnt density, the term contining (V v) my be omitted. For montomic gses t low density, the dilttionl viscosity к is zero. Т п = -fl Too = Тфф V 1 ^Ф, Pr + Pfl COt g] 1 : - -ГТ- + r I + (з/х, - fc)(v v) Г Sin 0 о>ф /J (В.1-17Г TrO = T 0r = " (B.l-18) т вф Т ФО ~ дв r (B.l-19) in which Тфг = Туф = r в дф + dr г (B.l-0) ~ (B.l-1) а When the fluid is ssumed to hve constnt density, the term contining (V v) my be omitted. For montomic gses t low density, the dilttionl viscosity к is zero.

3 B. Fourier's Lw of Het Conduction 845 B. FOURIER'S LAW OF HEAT CONDUCTION" Crte coordintes (x, y, z): [q = -fcvt] (B.-3) Cylindricl coordintes (г, в, z): (В.-4) (В.-5) (B.-6) (B.-7) (B.-8) (B - " 9) For mixtures, the term X (H /M Q )] must be dded to q (see Eq ).

4 846 Appendix B Fluxes nd the Equtions of Chnge B.3 FICK'S (FIRST) LAW OF BINARY DIFFUSION" Crte coordintes (x, y, z): (B.3-) (B3-3) Cylindricl coordintes (г, в, z): ^ (B.3-4) J~ (B.3-5) ^ (В.З-6) (B.3-7) (B.3-8) ( B 3 " 9 ) To get the molr fluxes with respect to the molr verge velocity, replce A, p, nd ш А by J% c, nd x A. B.4 THE EQUATION OF CONTINUITY" [dp/dt + (V pv) = 0J Crte coordintes (x, у,z): Cylindricl coordintes (r, 0, z): dp, (pv dt 6 x ) + "Г" (pi )X r x dy V* 0 (B.4-1)? (. 1 д ipv 0 ) = 0 (B.4-) Sphericl coordintes (г, в, #9 + д dt Гд1 Ф): - (pr v r ) + 1 д r в дв PVe 0) +- r 1 д ( ) 0 дф ) (В.4-3) 3 When the fluid is ssumed to hve constnt mss density p, the eqution simplifies to (V v) = 0.

5 B.5 The Eqution of Motion in Terms of т 847 B.5 THE EQUATION OF MOTION IN TERMS OF r [pdv/dt = - Vp - [V т] + pg] Crte coordintes(x,y, z): (dv x P -dt +V dv. dv x I- v dy ' z dv x dz) = J ix~ dx xx + Pgx (B.5-1) (dv z i~dt +i x S A. _ + Vy >v z dv tj 1- v z dz) dz) = Л- dy dp d dx Txy ^ d ' dz Tzz + Pgy + Pgz (B.5-) (B.5-3) These equtions hve been written without mking the ssumption tht т is symmetric. This mens, for exmple, tht when the usul ssumption is mde tht the stress tensor is symmetric, r vl/ nd r yx my be interchnged. Cylindricl coordintes (г, в, z): b v o dp i д i д д т вв ] (B.5-4) dv 0 + Vr ^r + т дв dv + v + + v *>e Wo) W 1 *P _ fl l_, Y 1 d d m + )- Tie ydr {r Tde Toe T dp r or - т p g e (B.5-5) (В.5-6) b These equtions hve been written without mking the ssumption tht т is symmetric. This mens, for exmple, tht when the usul ssumption is mde tht the stress tensor is symmetric, т гв т в1. = 0. Sphericl coordintes (r, в, ф): с (dvr dv, VgdV, У dv V] + V _ dp t ф r p Vr r dt dr r r в d$ dr 1 d,... : ^ -T Твг Sin 0) 1 d + T "7 ^1, + P^r (B.5-7) dv 0 d dv 0 Щ d v r - vl cot 6) _ i dp r elkj> г дв (т в1. - т гв ) - т фф cot в (В.5-8) p [ dt dv (h V e dv ф COt +Vr dr + г r 1 r (T^r - Т гф) + Т фв COt (В.5-9) с These equtions hve been written without mking the ssumption tht т is symmetric. This mens, for exmple, tht when the usul ssumption is mde tht the stress tensor is symmetric, т гь - т вг = 0.

6 848 Appendix B Fluxes nd the Equtions of Chnge B.6 EQUATION OF MOTION FOR A NEWTONIAN FLUID WITH CONSTANT p AND /x [pdv/dt = -Vp + fiv v + pg] Crte coordintes (x, y, z): dv x dv x dv x dv x dp d V x d V x d V : dx dv u dv u dv x dy dv z dv z dv z dv z dp d V 7 d, 7, -i Pgz (B.6-3) Cylindricl coordintes (г, в, z): dv r dv r V o dv r ut С/Г ' Ow (dvi d V o dv e dz dp ( drrdr + r do dz d v r v _ г dp э ( d, Л, 1 <?Ч, 4S +PZe (B.6-4) (B.6-5) (dv z dv z V o dv z dp d ( 1 d v z d vj (В.6-6) ( У Ф 9 +Vr dt ~dr~ + T~d6 + r в 1ф ар dr л в dl (В.6-7У dv 0 dv 0 v Q d Уф dv 0 v r - v cot в dt r dr т SO r 0 dф r Idp r do Г Sin в dф Г d6 COt 0 (B.6-8) dt r V dr r SO r в dф cot в 1 dp ~ r в dф а The quntity in the brckets in Eq. B.6-7 is not wht one would expect from Eq. (M) for [V Vv] in Tble A.7-3, becuse we hve dded to Eq. (M) the expression for (/r)(v v), which is zero for fluids with constnt p. This gives much simpler eqution.

7 B.8 The Eqution of Energy in Terms of q 849 B.7 THE DISSIPATION FUNCTION ФУ FOR NEWTONIAN FLUIDS (SEE EQ ) Crte coordintes (x, y, z): [Д dx ) dy ) Щ Cylindricl coordintes (r, ( I z): 1 dv y Sv^V [_ dy dz J dv x dl ~ 3 [li" + ^ m H(") ' f [t (B.7-1) У + f + dr) r дв г 0У1 )] <? ( v» 1 ^.l, Г в д ( v+ 1 dv^v Г 1 ^р i_fm дг г ) г дв] _ г эв в) г в дф] [г в дф дг г -^ * ( -L_ W + г 0 (?0 г sm в (B.7-3) В.8 THE EQUATION OF ENERGY IN TERMS OF q [pc,,dt/dt = -(V q) - (d In p/d In T) p Dp/Dt - (T:VV)] Crte coordintes (x, y,,z): p, (dt, dt, pl dt +Vx dx v dt + dt dy dz /<?ln p Dp - (T:VV) d In r/ F Dt (В.8-1Г Cylindricl coordintes (r, 0, z): pel + v,. + - ^ot dr г с x (dt, dt ^v fl st pc[ + v r + - dt r ( ^ф) dq 0 di r do d zj V^lnT/ p Dp (B.8-) fl l i <^<Л / d In p Dp ) 1 ^ ( ^ + - (T:VV) ' r ^ du r в dф iв In T/p Dt (В.8-ЗУ The viscous dissiption term, -(T:VV), is given in Appendix A, Tbles A.7-1,,3. This term my usully be neglected, except for systems with very lrge velocity grdients. The term contining (d In p/d In T) p is zero for fluid with constnt p.

8 850 Appendix B Fluxes nd the Equtions of Chnge B.9 THE EQUATION OF ENERGY FOR PURE NEWTONIAN FLUIDS WITH CONSTANT" p AND к Crte coordintes (x, y, z): [pcpt/dt = kv T + Cylindricl coordintes (г, в, z): * ^=к Ц + Ц + z Sz) Эх _ Зу (В.9-1) ь p (dt _,_ dt v o dt Hr{ r f)- г 6>0 ^z j (B.9-) b ПГ д Т 1 71 Э Т 1 V& в Т pl 1 V<i> dt ' v 'dr ' г дв ' r вдф) L It 1 ^ 1 cin fl ^ 1 1 Г Sin в vv OV у 1 67 i I ^ <?</> J дф, (В.9-3) /; This form of the energy eqution is lso vlid under the less stringent ssumptions к = constnt nd {d In piд In T) t,dp/dt = 0. The ssumption p = constnt is given in the tble heding becuse it is the ssumption more often mde. b The function Ф у is given in B.7. The term /и,ф у is usully negligible, except in systems with lrge velocity grdients. B.1O THE EQUATION OF CONTINUITY FOR SPECIES IN TERMS" OF ) [pdtjjdt = -(V Crte coordintes (x, y f z): у ду z dz J (B.10-1) Cylindricl coordintes (r, в, z): (до) а д(х) а 1>0 d> do) Р Vr dt дг 3< r ~de~ + Vz ~dz~) = ~J~dr^r]r ' + ~r~de + ~> (B.10-) Sphericl coordintes (r, в, ф): (до) а да) а да) а р dt Vy дг г дв r в dф ) Г 1 д (r / ) 1 _ r l д dr > r Г Sin в de JO S i n У > ) ~H + r (B.10-3) To obtin the corresponding equtions in terms of J* mke the following replcements: Replce p ш а ] а v r by С Х а Га V* R-X f,rfi 3 1

9 B.ll The Eqution of Continuity for Species A in Terms of OJ A for Constnt ряь лв 851 B.ll THE EQUATION OF CONTINUITY FOR SPECIES A IN TERMS OF <o A FOR CONSTANT" p<3) AB p<d) AB V (D A + r A ] Crte coordintes (x, y, z): (д(о А до) А д(л) А v SWA ), U > A д (х) А + ^ 1 + r Cylindricl coordintes (г, в, z): 1 до) А до) А V e do) A дсол [- *j ^ ^ n 1 1 о (x) A о u> A до) А до) А v 0 ди> А Уф дшл Г1 д ( дс А, 1 д (. (В.11-3) а То obtin the corresponding equtions in terms of x A, mke the following replcements: Replce p co n v r N by с х а v* R Q - x Q Rp 8 1

REMARKS ON ESTIMATING THE LEBESGUE FUNCTIONS OF AN ORTHONORMAL SYSTEM UDC B. S. KASlN

REMARKS ON ESTIMATING THE LEBESGUE FUNCTIONS OF AN ORTHONORMAL SYSTEM UDC B. S. KASlN Мат. Сборник Math. USSR Sbornik Том 106(148) (1978), Вып. 3 Vol. 35(1979), o. 1 REMARKS O ESTIMATIG THE LEBESGUE FUCTIOS OF A ORTHOORMAL SYSTEM UDC 517.5 B. S. KASl ABSTRACT. In this paper we clarify the