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Mehanika sploshnyh sred

Uravneniya Eilera dlya ideal'noi zhidkosti.

Pri zadannyh vneshnih silah i izvestnyh svoistvah zhidkosti mozhno zapisat', pol'zuyas' 2-m zakonom N'yutona, uravnenie dvizheniya edinicy ob'ema neszhimaemoi nevyazkoi zhidkosti:

$\rho\frac{dv}{dt} = F - {\rm grad }\; {p},$

(3.28)

gde operator grad (gradient) opredelyaetsya kak

${\rm grad} = i\frac{\partial}{\partial x} + j\frac{\partial}{\partial y} + k\frac{\partial}{\partial z}$

(3.29)

Uravnenie (3.28)zapisano v vektornom vide i yavlyaetsya obobsheniem odnomernogo uravneniya (3.3). Raspisyvaya (3.28) dlya treh proekcii skorosti, poluchaem sistemu uravnenii $\rho\left(\frac{\partial v_x}{\partial t} + v_x\frac{\partial v_x}{\partial x} + v_y\frac{\partial v_x}{\partial y} + v_z\frac{\partial v_x}{\partial z}\right) = F_x - \frac{\partial p}{\partial x}$

$\rho\left(\frac{\partial v_y}{\partial t} + v_x\frac{\partial v_y}{\partial x} + v_y\frac{\partial v_y}{\partial y} + v_z\frac{\partial v_y}{\partial z}\right) = F_y - \frac{\partial p}{\partial y}$

(3.30)

$\rho\left(\frac{\partial v_z}{\partial t} + v_x\frac{\partial v_z}{\partial x} + v_y\frac{\partial v_z}{\partial y} + v_z\frac{\partial v_z}{\partial z}\right) = F_z - \frac{\partial p}{\partial z}$ Esli eti uravneniya dopolnit' usloviem nerazryvnosti $\frac{\partial v_x}{\partial x} + \frac{\partial v_y}{\partial y} + \frac{\partial v_z}{\partial z} = 0,$ to my poluchaem polnuyu sistemu uravnenii s chetyr'mya neizvestnymi funkciyami koordinat i vremeni (v_x, v_y, v_z i p). Uravneniya (3.29) nazyvayutsya uravneniyami Eilera i pozvolyayut, v principe, rasschitat' dinamiku zhidkosti. Odnako s matematicheskoi tochki zreniya eta sistema, v otlichie ot mnogih drugih uravnenii v fizike, yavlyaetsya nelineinoi iz-za nalichiya chlenov tipa $v_x\frac{\partial v_x}{\partial x}, \ldots, v_z\frac{\partial v_z}{\partial z}$ . Poetomu integrirovanie etih uravnenii i nahozhdenie iskomyh funkcii predstavlyaet podchas ves'ma slozhnuyu zadachu dazhe pri ispol'zovanii moshnyh EVM. Neslozhno, naprimer, iz (3.30) poluchit' uravnenie Bernulli dlya stacionarnogo techeniya, kogda $\frac{\partial v}{\partial t} = 0$ . Odnako strogii vyvod etogo uravneniya my predostavlyaem chitatelyu prodelat' samostoyatel'no, obrativshis' k rekomendovannoi literature. My zhe budem ispol'zovat' uravneniya (3.30) dlya opisaniya volnovogo dvizheniya zhidkosti i analiza svoistv akusticheskih voln. V zaklyuchenie otmetim, chto chasto sistema (3.30) pishetsya v bolee kompaktnom vide s ispol'zovaniem operatora gradienta. Kazhdoe iz treh uravnenii (3.30) imeet vid $\rho\left(\frac{\partial}{\partial t} + v \cdot {\rm grad }\right) v_{x,y,z} = F_{x,y,z} - ({\rm grad }\; {p})_{x,y,z}.$ Vozvrashayas' k vektornomu predstavleniyu, poluchaem vozmozhnost' zapisat' 4 uravneniya Eilera (3.29) v vide dvuh vektornyh:

$\begin{array}{l} \rho\left[\frac{\partial}{\partial t} + v \cdot {\rm grad }\right] v = F - {\rm grad }\; {p}\\ {\rm div }\; {\bf v} = 0. \end{array}$

(3.31)

Uravnenie nerazryvnosti dlya szhimaemoi zhidkosti

Pri techenii gazov, osobenno pri bol'shih skorostyah, ih plotnost' mozhet zametno, a to i znachitel'no, menyat'sya vo vremeni i v prostranstve. Yasno, chto ob'em vtekayushei zhidkosti mozhet ne byt' ravnym ob'emu vytekayushei zhidkosti cherez poverhnost' kubika, izobrazhennogo na ris. 3.11. Esli takogo ravenstva net, to massa gaza vnutri kubika (a s nei i plotnost') budut so vremenem menyat'sya. Uravnenie (3.24) v etom sluchae stanovitsya nespravedlivym. Odnako i zdes' mozhno zapisat' uravnenie nerazryvnosti, osnovnaya ideya vyvoda kotorogo baziruetsya na balanse massy gaza, sostavlyayushego fizicheskuyu sut' ravenstva (3.1). Potok massy gaza cherez ploshadku dS budet raven $dN_M = \rho v\cdot dS$ . Togda polnyi potok massy gaza cherez bokovuyu poverhnost' elementa ob'ema dxdydz, analogichno (3.27), raven

$dN_M = dxdydz\cdot {\rm div }(\rho {\bf v}),$

(3.32)

gde - $\rho v$ novoe vektornoe pole. Esli etot potok polozhitel'nyi, to massa vnutri elementa $m = \rho dxdydz$ budet ubyvat' za schet umen'sheniya vo vremeni plotnosti . Poetomu, zapisyvaya uslovie balansa massy v vide

$dxdydz\cdot {\rm div }(\rho {\bf v}) = -dxdydz\frac{\partial\rho}{\partial t}.$

(3.33)

my poluchaem (posle sokrasheniya na dxdydz) odno iz fundamental'nyh uravnenii gidrodinamiki - uravnenie nerazryvnosti szhimaemoi zhidkosti:

$\frac{\partial\rho}{\partial t} + {\rm div }(\rho {\bf v}) = 0$

(3.34)

Sleduet otmetit', chto pri $\rho$ =const eto uravnenie perehodit v (3.24).

Ris. 3.11.

V elektrodinamike eto uravnenie yavlyaetsya takzhe fundamental'nym. V samom dele, esli rech' idet o dvizhushihsya zaryadah, ob'emnaya plotnost' kotoryh ravna $\rho$ , to uravnenie (3.34) yavlyaetsya matematicheskim vyrazheniem universal'nogo zakona sohraneniya zaryada.

Uravneniya Eilera i uravnenie Bernulli dlya szhimaemoi zhidkosti.

Dinamika szhimaemoi zhidkosti baziruetsya takzhe na 2-m zakone N'yutona, zapisannom dlya edinicy massy zhidkosti. Ravnodeistvuyushaya sil davleniya i vneshnih sil sozdaet uskorenie edinicy massy, poetomu

$\left(\frac{\partial}{\partial t} + v \cdot {\rm grad }\right) v = - \frac{1}{\rho} {\rm grad } p + F,$

(3.35)

gde F - vneshnyaya sila, deistvuyushaya na edinicu massy. Dlya opredeleniya 5-ti neizvestnyh velichin (v_x, v_y, v_z, p i $\rho$ ) neobhodimo dopolnit' (3.35) material'nym uravneniem, svyazyvayushim plotnost' i davlenie:

$p = p(\rho).$

(3.36)

Sistema (3.34) - (3.36) nosit nazvanie uravnenii Eilera dlya szhimaemoi zhidkosti. Ogromnoe kolichestvo zadach gazodinamiki reshaetsya na osnove analiza etih uravnenii. Vospol'zuemsya uravneniem (3.35) i poluchim uravnenie Bernulli. Dlya etogo vidoizmenim pravuyu chast' (3.35), vvedya vspomogatel'nuyu funkciyu ${\cal P}$ (2.27), i uchtem (2.29). Togda (3.35) primet vid

$\left(\frac{\partial}{\partial t} + v{\rm grad }\right) v = - {\rm grad }({\cal P} + U_1).$

(3.37)

Pri stacionarnom techenii $\frac{\partial v}{\partial t} = 0$ . V napravlenii osi trubki toka (vdol' krivolineinoi koordinaty $\ell$ ) mozhno zapisat'

$v\frac{d}{d\ell}v = - \frac{d}{d\ell}({\cal P} + U_1).$

(3.38)

Poskol'ku potencial'naya energiya edinicy massy $U_1 (\ell) = U_1 (h) = gh + {\rm const}$ , a ${\cal P}(\ell) = \int\limits_{p(\ell)}^{p_1 (\ell_1)} \frac{dp}{\rho}$ , to, po analogii s (3.13), perepishem (3.38) v vide:

$\frac{d}{d\ell}\left(\frac{v^2}{2} + \int\limits_{p(\ell)}^{p_1 (\ell_1)} \frac{dp}{\rho} + gh\right) = 0$

(3.39)

Integriruya (3.39) vdol' trubki toka, poluchaem uravnenie Bernulli dlya szhimaemoi zhidkosti

$\frac{v^2}{2} + \int\limits_{p(h)}^{p_1 (h_1)} \frac{dp}{\rho} + gh = {\rm const}$

(3.40)

Zdes' h - polozhenie po vertikali secheniya trubki toka s koordinatoi $\ell$ . Ochevidno, chto $p(\ell) = p(h), p_1(\ell_1) =p_1(h_1)$ . Postoyannaya v (3.40) opredelyaetsya zadaniem skorosti v₁ i vysoty h₁ v fiksirovannom sechenii s koordinatoi $\ell_1$ . S uchetom etogo, uravnenie (3.40) obretaet vid

$\frac{v^2}{2} + \int\limits_{p(h)}^{p_1 (h_1)} \frac{dp}{\rho} + gh = \frac{v_1^2}{2} + gh_1.$

(3.41)

Dlya prakticheskogo ispol'zovaniya uravneniya Bernulli neobhodimo znanie material'noi svyazi mezhdu p i $\rho$ . Dlya sluchaya neszhimaemoi zhidkosti ( $\rho$ = const) uravnenie (3.41) perehodit v (3.15). Esli rech' idet o potoke gaza, to pri ego bystrom szhatii (uvelichenie plotnosti) gaz budet nagrevat'sya. Iz-za plohoi teploprovodnosti gaza teplo ne budet uspevat' uhodit' iz nagretyh oblastei. Poetomu dlya ustanovleniya material'noi svyazi $p=p(\rho)$ vospol'zuemsya adiabaticheskim priblizheniem:

$\frac{p}{p_1} = \left(\frac{\rho}{\rho_1}\right)^\gamma,$

(3.42)

gde pokazatel' adiabaty $\gamma >1$ . Takaya svyaz' poluchaetsya iz pervogo nachala termodinamiki i uravneniya sostoyaniya ideal'nogo gaza (2.32) pri uslovii otsutstviya teploobmena mezhdu nagretoi oblast'yu i okruzhayushei sredoi. Davlenie v (3.42) vozrastaet s plotnost'yu bystree, chem pri izotermicheskom processe, tak kak $\gamma >1$ . V kurse molekulyarnoi fiziki budet pokazano. chto $\gamma = C_p/C_V$ (C_p i C_V - teploemkosti pri postoyannyh davlenii i ob'eme sootvetstvenno). Dlya vozduha, sostoyashego glavnym obrazom iz dvuhatomnyh gazov, $\gamma$ = 1,4. Esli podstavit' (3.42) v (3.41) i vypolnit' prosteishee integrirovanie, to mozhno zapisat' vyrazhenie dlya raspredeleniya davleniya vdol' trubki toka:

$p = p_1\left\{ 1 - \frac{\gamma - 1}{\gamma}\frac{\rho_1}{p_1}\left[\frac{1}{2}\left( v^2 - v_1^2\right) + g(h - h_1)\right]\right\}^{\frac{\gamma}{\gamma - 1}}.$

(3.43)

Ne umalyaya obshnosti, budem schitat' trubku toka gorizontal'noi (h=h₁). Polozhim dalee skorosti techeniya takimi, chto

$\frac{1}{2}\left| v^2 - v_1^2\right| < \frac{1}{\gamma - 1}\frac{\gamma p_1}{\rho_1} = \frac{c_1^2}{\gamma - 1},$

(3.44)

gde $c_1 = \sqrt{\gamma\frac{p_1}{\rho_1}}$ - parametr, imeyushii razmernost' skorosti. Kak my uvidim neskol'ko pozdnee, skorost' zvuka v gaze

$c = \sqrt{\gamma\frac{p}{\rho}}.$

(3.45)

Pri normal'nyh usloviyah dlya atmosfery s=336 m/s. V etom sluchae (3.43) mozhno razlozhit' v ryad:

$p = p_1\left\{ 1 - \frac{\rho_1}{p_1}\left(\frac{v^2 - v_1^2}{2}\right) + \frac{1}{2\gamma}\frac{\rho_1^2}{p_1^2}\left(\frac{v^2 - v_1^2}{2}\right)^2 + \ldots\right\}.$

(3.46)

Esli prenebrech' kvadratichnym chlenom v (3.46), to raspredelenie davlenii sootvetstvuet techeniyu neszhimaemoi zhidkosti s plotnost'yu $\rho_1$ =const. Kvadratichnyi chlen nachinaet davat' vklad v raspredelenie davlenii pri skorostyah potoka, soizmerimyh so skorost'yu zvuka s₁. Podstaviv (3.42) v (3.43), poluchaem raspredelenie plotnosti vdol' trubki toka:

$\rho = \rho_1\left\{ 1 - \frac{\gamma - 1}{\gamma}\frac{\rho_1}{p_1}\left[\frac{1}{2}\left( v^2 - v_1^2\right) + g(h - h_1)\right]\right\}^{\frac{1}{\gamma - 2}}$

(3.47)

Dlya gorizontal'noi trubki toka i pri uslovii (3.44) raspredelenie plotnosti (3.47) imeet vid:

$\rho = \rho_1\left( 1 - \frac{v^2 - v_1^2}{2c_1^2} + \ldots\right)$

(3.48)

Takim obrazom, izmenenie plotnosti gaza neobhodimo prinimat' v uchet tol'ko pri skorostyah techeniya, sopostavimyh po poryadku velichiny so skorost'yu zvuka, opredelyaemoi, kak sleduet iz (3.45), davleniem i plotnost'yu v etom potoke. Esli zhe skorost' techeniya $v\ll c$ , to szhimaemost'yu gaza mozhno prenebrech' i operirovat' s nim, kak i s zhidkost'yu.

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