The problem of motion of a uniformly heated spheroidal hydrosol particle is solved. An expression which allows estimation of the resistance force of the spheroidally shaped solid hydrosol particle at arbitrary differences between the temperatures of the particle surface and the region at a distance from it is obtained with account for the dependence of the viscosity on the temperatureyesBS

Malai, N. V.

DSpace at Belgorod State University

MOTION OF A HEATED SPHEROIDAL PARTICLE AT LOW REYNOLDS NUMBERSN. V. Malai UDC 533.72The problem o f  motion o f  a uniformly heated spheroidal hydrosol particle is solved. An expression which al­lows estimation o f the resistance force o f  the spheroidally shaped solid hydrosol particle at arbitrary differ­ences between the temperatures o f  the particle surface and the region at a distance from it is obtained with account for the dependence o f  the viscosity on the temperature.We consider the motion of a uniformly heated spheroidal solid particle (an oblate spheroid) in a viscous fluid in the gravitational field. By a heated particle is meant a particle whose mean surface temperature greatly exceeds in magnitude the temperature of the surrounding medium. Heating of the particle surface can be caused, for example, by the occurrence of a volume chemical reaction, absorption of electromagnetic radiation, etc.Of all the parameters of liquid transfer, it is only the coefficient of dynamic viscosity that is strongly related to the temperature [1]. To allow for the dependence of the viscosity on the temperature we use expression (1), which makes it possible to describe a viscosity change over a wide range of temperatures with any required accuracy (at /•„ = 0 this formula can be reduced to the known Reynolds relation [1])M-Uq =  M-o. 1+ I X/7=1TjiqT -  1 exp <-AT liqT -  1 ( 1 )Let us change to the reference system related to the particle. The problem in essence is reduced to analysis of the plane-parallel flow of a liquid (at a velocity II,„(11,,, || Oz)) past a motionless spheroid. The flow past the sphe­roid is described in the spheroidal coordinate system (e, r|, (p) with origin at the center of the hydrosol particle.At low Reynolds numbers the distributions of the velocity Uiiq, described by the following system of equations [2, 3]:the pressure /J|R|. and the temperature 7:|K| areiqV A i q  =  h i q  A U liq +  2  № q V ) U liq +  tV ^ l iq  X r 0 t U liq] divlliiq = 0 ; (2)dlV fluqVTfoj) = 0 .In solving this system of equations, the boundary conditions(3)Uiiq= 0 ,  Tliq = Ts for e = e0Uiiq ~ cos r| ee -  £/„ sin r| , T]ia^ T „ ,  P lja-> P „  for e ^ ° o .cr| > 1 liq liq(4)(5)are taken into account.The force affecting the particle from the side of the flow is determined from the formulaBelgorod State University, Belgorod, Russia; email: malay@bsu.ru. Translated from Inzhenerno-Fizicheskii Zhumal, Vol. 75, No. 3, pp. 44-46. May-June, 2002. Original article submitted November 14, 2000; revision submit­ted November 26, 2001.F z = .sinh e-  Py.n COS Tl +  Ope cos ri   o CT1 sin ri4 coshe 1dS, (6)2 2where dS= c  cosh e sin r\cfc\d§ is the differential surface element and o ee and o eT| are the components of the stress tensor in the spheroidal coordinate system [2].In order to find the force affecting the heated spheroidally shaped solid particle we must know the tempera­ture field in its vicinity. Integration of Eq. (3) with the corresponding boundary conditions yieldsJaQ%q = 1 + arcctan^, (7)where /|R| = T\n /T„. "k= sinhe, and y=-ts 1Vi + ^2 arcctan Xo the particle surface, ts = 7S/ 7 M and Xq = bQ/c .With account for (7) expression (1) takes the formis the dimensionless parameter characterizing the heating ofoo1 + X  Fn To (arcctan X f exp j -  Y0 arcctan Aj ( AJ } To-  c ao77=1 V /(8)Since the viscosity depends only on the radial coordinate 'k (formula (8)), we find the solution of the system of equations (2) by the method of separation of variables, expanding the fields of velocity and pressure in Legendre and Gegenbauer polynomials [2]. In particular, for the components of the mass velocity Uiiq we obtained the following expressions, which satisfy the boundary conditions (5):£ / „ 2  U~ 2 t/e(e,Tl) = ^ c0STi[C + A 1G1+A2G2\ ,  t/T|(e,ri) = - ^ s i n r i  [c + A 1G3 +A2G4] (9)where.(l)1 V- 0G2 = - ~  X,(2)— —  E-3, ( i )n=0(n + 3) In —  -  1 X, ^2  ^ „2G3 = G1 + — - — , G4 = G2 + — -— Cj2 , h  = s'rih Eo. = c Vcosh2 e -  sin2 r| , 0® = 1, A0 = 12a  2an±—  £  [(» + 4 -  k) {(» + 1 -  k) 4 1} + a ? }  + a ? ]  e £ t  (« > i ) , e;n (n + 5)£=11(« -  2) (« + 3)n^  {(« + 2 -  &) [(« + 1 -  k) a[ + a[ + a[  ^j Qn_k +k=in■ P £  [(2» -  2k + 3) a f  + a f ]  e ^ 2 -  6a;k=0(n > 3), ef = -  J  [2 ( a f  + a f ’) + a f 1 + taj41] ,p = - i [ { 3  (2a® + a f )  + a f ) e f  - 2  ( o f  + o f ) -  o f  - 6 a f ].Table 1. Dependence of the Coefficient K  on the Mean Temperature of the Surface 7S and the Ratio of the Semiaxes ao/boa0/ b 0 T* K273 283 303 333 343 353 3630.73 0.947 0.705 0.393 0.163 0.121 0.089 0.0650.9 0.980 0.727 0.397 0.158 0.116 0.086 0.062[(»-2)/2](1) ^  i V  /  n -2 k -2  (2) . „  n= C „+12 > (-1 )   , a„ = (n -  2) C„ -  ynC„ , +^  7 (2k + 1) (2k + 3) (2k + 5) " V 7 ” ,0 n~Xk=0[(" ^ )/2] * (4 ^+ 5 )C n_2jt_2 [(" l )/2] * 1+ 12 X  (-1 )  Ç2& + 1) (2k + 3) (2& + 5) _  3 ^  ^  (2£ + 3) (2k + 5) ^  Cn~2k~2 ~k=0 k=0„ (3)-  7oC^-2k-3 + (« -  2k -  4) c„_2,_4] (n > 1), a„ = -  2 (« + 2) Cn + 2loCn_x - 2 ( n - 2 )  C ^ 2 +K^ )/2] k Cn-2k-2 {{nT ] k i k + 2)(Ak+ 5 ) v+ 12 X  ( : ) ( 2 k +5 ) +6  ^  ( ^  (2/fe+3)(2/fe+5)[(” Cn-2k-2 \ Cn-2k-l +k=0 k=0+ ( n - 2 k - 4 ) C n_2k_ ^  (n> 1), oi4) = ^  [YoVi-(«-!)(«-2) V2]k 2s  / j l / j l / j ! ... is\ b i h h h  - j s > 5 2/,+34+5/*+...+s7 =k1 3  5 s/ = / 1+ / 3 + / 5 + . . . + / i , / ^ _ i = ( - i )""1 c ( X ! 1)The integral part /c/2 is denoted by |/c/2|.The force affecting the spheroid due to viscous stresses is determined by integration of expression (6) over the surface of the spheroid, and with account for (9) it isFz = ÔÎIOo (10)where K=  — ,  —--------- — exp J -  —  a0 arccot h) \ and G\ and G2 are the first derivatives of the correspond-3 i l  + ^ [ G 2G \ - G lG{] [ c \ing functions with respect to 'k.The effect of the temperature of the particle surface on the resistance force is determined by the coefficient K. As an example, Table 1 presents the results of numerical calculations of the dependence of the coefficient K  on the mean temperature of the spheroid surface and the ratio of the semiaxes for solid particles suspended in water at Too = 273 K, A = 5.119, and /•„ = 0 for n>  1. The numerical analysis showed that heating of the spheroid surface sub­stantially affects the resistance force.In the lim it when y —>0 (small temperature differences in the vicinity of the spheroid), Gj = 1/(3À ), g \  = -1 / k 4. G2 = 1 /k .  G2 = —1/X2, and ao = bo = R; the coefficient is K = 1 and formula (10) becomes the Stokes for­mula for a spherical solid particle with radius R [2].In order to obtain the expression for the resistance coefficient of an extended spheroid, in (10) we must re­place A. by ik and c by -  ic (i is the imaginary unit).Let us consider the motion of a spheroidal particle in the gravitational field. Equating the gravity force, with account for the buoyancy force, to the resistance force (10), we obtain the velocity of its gravitational fall:, T a0b0 Gl 4  -  G2^l UY iu„  = (Pp -  Pliq) g  V l + X0 -------------- -J------- exp —  a0 arcctan X0 (11)3 |^  Gi [ c jThus, formulas (10) and (11) allow estimation of the hydrodynamic force affecting a uniformly heated spheroidal particle and the velocity of its gravitational force with account for the dependence of the viscosity on the temperature, which is represented in the form of an exponential-power series at arbitrary temperature differences in its vicinity.NOTATION1-1°° = WiqCT’oo); A, /•„ -  const; U0o = | Us, I; 7S. mean temperature of the spheroid surface; ee and cn. unit vec­tors of the spheroidal coordinate system; oq and h(). semiaxes of the spheroid; A,i;q, thermal conductivity of the liquid; Poo and Tec, undisturbed pressure and temperature in the liquid; nz, unit vector in the direction of the z axis; p, density. Subscripts: p, particle; values of physical quantities taken at a distance from the particle (at infinity); s, spheroid.REFERENCES1. S. Bretsznajder, Wlasnosci Gazow i Cieczy [Russian translation], Moscow (1966).2. J. Happel and H. Brenner, Low-Reynolds Number Hydrodynamics [Russian translation], Moscow (1976).3. L. D. Landau and E. M. Lifshits, Mechanics o f  Continuous Media [in Russian], Moscow (1958).

Motion of a heated spheroidal particle at low Reynolds numbers

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Motion of a heated spheroidal particle at low Reynolds numbers

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