Modeling of Inflation of Liquid Spherical Layers. under Zero Gravity

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1 Applied Mathematical Sciences Vol no HIKAI Ltd Modeling of Inflation of Liquid Spherical Layers under Zero Gravity G. I. Kurbatova Saint-Petersburg State University Saint-Petersburg ussia gi_kurb@mail.ru N. N. Ermolaeva Saint-Petersburg State University Saint-Petersburg n.ermolaeva@spbu.ru Copyright 013 G. I. Kurbatova and N. N. Ermolaeva. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. Abstract A mathematical model of the inflation of a spherical layer (shell) made of heat-conducting incompressible Newtonian liquid has been developed and examined. The model presents a set of differential equations the analytical solution of which has been found in the form of the dependences of the velocity of liquid flow and the pressure in the liquid layer on the inner radius of the shell. esults of simulation of the shell inflation process under different conditions of gas pumping are presented and discussed. Keywords: mathematical models; liquid spherical shell; zero gravity; space technology; viscous flow; oligomer

2 6868 G. I. Kurbatova and N. N. Ermolaeva 1. Introduction Solar energy collected by space mirrors appears to be of great importance for energetics of the future [1]. In this study we develop and examine a mathematical model of the inflation of a spherical shell under zero gravity. By the spherical shell we will imply a closed spherical layer made of a special liquid substance. The inflation process underlies one of the methods of producing space mirrors. Schematically the process consists in the following. A portion of the special liquid substance is placed into a camera mounted overboard of a spacecraft with the pressure and temperature in the camera being maintained equal to certain constant values P n and T n. If the liquid substance does not contact any solid surfaces except for a thin tube inserted into the bulk of the liquid surface tension makes it take shape of a ball. The thin tube is used to pump a gas into the very center of the ball that results in the formation of a spherical shell. In practice the special liquid substances used are oligomers chemical compounds demonstrating the following rheological properties. Being in darkness in a wide range of pressures and temperatures they behave like a Newtonian liquid till they are exposed to radiation e.g. sunlight under which oligomers instantly harden. The experiments under terrestrial conditions have demonstrated applicability of this method to produce hollow spherical shells which are used as preforms for spherical space mirrors. Nevertheless in order to develop the real space technology it is necessary to resolve a set of problems one of which pertains to the optimization of the inflation process in particular the conditions of gas pumping into the expanding liquid spherical layer.. Mathematical model The mathematical (phenomenological) model of a heat-conducting incompressible Newtonian liquid is presented by a set of the following equations [3]: div u 0 u u u p u t u divq : D t ct q gradt pi D

3 Modeling of inflation of liquid spherical layers 6869 where u p T are the velocity of a liquid flow the pressure and the temperature is the liquid density is the Laplace operator is the mass density of internal energy and c are the coefficients of heat conductivity dynamic viscosity and heat capacity respectively q is the vector of heat flow I is the unit tensor D is the strain velocity tensor is the stress tensor is the Hamiltonian operator :D is the contraction of and D. Initial conditions In the initial instant of time t 0 we have a motionless spherical liquid layer at temperature T 0 which is just a spherical shell with a thick wall. Thus the initial conditions can be written as G : t r t u ( r t0 ) 0 r G T ( r t ) T r G. 0 0 Let us use the spherical coordinate system (r ) with its origin in the sphere center. (t) and t () are the inner and outer radii of the spherical layer respectively. Given unchangeable (during the inflation process) spherical symmetry of the layer we simplify the foregoing problem as follows 1 ( ru) 0 (1) r r u u p u t r r () T T T u u r The initial conditions become 1 u r t r c r r c r r (3) u 0 r (4) tt T T r (5) tt where u is the radial component of the velocity 0 and 0 are the initial inner and outer radii of the spherical layer. Estimation of the validity of such simplification is beyond this communication. The values and are functions t t. of u p T which are in turn functions of t and r At the initial moment of time t 0 the spherical layer is in equilibrium which is characterized by the equality 0 t k

4 6870 G. I. Kurbatova and N. N. Ermolaeva Pg Pn where P g is the pressure inside the spherical layer is the surface tension. In this expression the Laplace law is taken into consideration and P g is a function of time. One can see that the motion equation Eq. () coincides with the Euler equation so that in view of the adopted model the liquid flow proves to be potential and the Laplacian of the velocity vector is equal to zero. However owing to the liquid adhesion taking place on solid surfaces in most boundary problems the flow of viscous liquids cannot be considered irrotational that is a potential flow. The problem under study constitutes an exception due to the specific boundary conditions described below. Boundary conditions Boundary conditions specify kinematics and dynamics as well as temperature behavior at the outer and inner surfaces of the layer. The kinematic boundary conditions are given by 0 k : d d t t t u r dt u r dt. (6) The dynamic boundary conditions are based on the generalization of the Laplace law which determines a value of the step in the normal component of the stress vector at a curved surface. For the outer and inner surfaces of the layer they are written as u ( p ) Pg (7) r r u ( p ) Pn. (8) r r Temperature at the outer and inner surfaces should obey the following conditions T T T r g (9) r r T r r r 4 4 T T r n T T n. (10) Here and are the heat exchange coefficients at the outer and the inner surfaces of the spherical layer respectively T g and T n are the temperatures inside and outside the liquid spherical shell respectively is the Stefan-Boltzmann constant and is the emissivity of the liquid. Conservation of the liquid volume V yields an additional condition:

5 Modeling of inflation of liquid spherical layers const kk; kk V (11) 3 where kk is the constant. 3. Splitting of the model It is well known that if density of a liquid is considered constant (it is our assumption) the general problem expressed by Eqs. (1)-(11) can be split into the hydrodynamic and heat parts which can be solved separately. In such a case the velocity field in the liquid layer found from the hydrodynamic part of the problem is used in solving the heat part of the problem. 4. Calculation of the velocity and pressure fields From the continuity equation Eq. (1) it follows u c() t r. The function c(t) can be found from the kinematic boundary condition Eq.(6) and it has the form field as c() t which allows writing an expression for the velocity u (1) r where a dot added over the variable denotes the time derivative. For a potential flow one can use the Cauchy-Lagrange integral that is u p Ft () (13) t which allows finding the pressure field p using the velocity field u that is Eq. (1). Here is the velocity potential given by / r. Taking into account Eq. (1) we obtain 1 1 ( r t) ( t) r. (14) Let us express the pressure p(rt) via the function (t) and its derivatives. To do this we need to determine F(t) at the inner surface of the spherical liquid layer. Taking r= and using the boundary conditions Eq. (6) and Eq. (7) as well as Eq. (14) we can rewrite the Cauchy-Lagrange integral Eq. (13) as follows

6 687 G. I. Kurbatova and N. N. Ermolaeva P g 4 d( r t) F() t. (15) d Omitting algebraic manipulations we write an expression for p(rt) in dimensionless form: Am A s p( r t) ( t) Pn 1 Ap Ap A p r (16) 4 1 A p r r t t t. 0 k In Eq. (16) all designations have the same meanings but they are dimensionless. The dimensionless groups A m A s and A p are expressed in terms of the already introduced model parameters and characteristic quantities t x p x and r x. These groups are 4tx tx pxtx m s 3 p rx rx rx A A A. The derived expressions for the velocity Eq. (1) and for the pressure Eq. (16) give us the full solution of the hydrodynamic part of the problem in parametric form that is u u t t r p p t t t Pg t r. Now basing on the set of equations Eq. (1) Eq. () and Eqs. (6)-(8) let us derive a differential equation controlling time dependence of that is function (t). To do this we use Eq. (1) and rewrite the motion equation Eq. () in terms of as follows 4 p. 5 r r r r Let us integrate this equation over r from to using the boundary conditions Eq. (7) and Eq. (8) and introducing the following designations ht () H H t 1 and h() t K. t () (1 K 3 ) 1 3 The value of H is the ratio of the layer thickness to its outer radius. After a series of algebraic manipulations we arrive at the desired equation in dimensionless form:

7 Modeling of inflation of liquid spherical layers H ( H 4H 6 H ) A ( 3 3 ) m H H H ( H) A A s p Pg Pn kk K. r 3 x For Eq. (17) we can formulate the Cauchy problem ( t ) ( t ) 0 (18) where 0 is the dimensionless initial inner radius of the spherical layer. The so-called control function of our problem is function Q(t) which is the timedependent flow rate of the gas injected into the hollow of the expanding spherical liquid layer. We assume that there is no pressure gradient in the shell hollow and diffusion of the gas through the liquid layer is negligible. Then for any time t[t 0 t k ] the mass m(t) of the gas injected inside the liquid layer is calculated by t 0 0. m( t) m t Q( ) d For an adiabatic process and T=const we can write the Poisson equation: P g g P 0 0 where (19) (17) g is the gas density 0 and P 0 are the gas density and pressure at t=t 0 and γ is the adiabatic exponent which for a diatomic gas is equal to 4/3. The gas density can be expressed via Q(t) and (t) as follows g t 0 0 m t Q() d g () t. 4 3 t () 3 Then keeping designations of the dimensionless values the same we can write an expression for P g (t) as P () t P g 1 A Q( ) d t q (0) qt 0 x As 1 1 where Aq is the dimensionless group P0 Pn and mt ( 0 ) A p 0 0 q 0 is the characteristic value of the flow rate of the injected gas.

8 6874 G. I. Kurbatova and N. N. Ermolaeva The equation Eq. (17) controls the time dependence of and in term of function Q(t) it takes the form ( H) As H ( H 4H 6 H ) A ( 3 3 ) m H H H 4 t 3 A p 1 Aq Q( ) d (1) 0 P 0 P. 3 n Properties of the differential equation Eq. (1) depend on the values of the dimensionless groups and on the selected flow rate Q(t). Note that for the majority of the problem parameters this nonautonomous nonlinear differential equation is stiff. In addition in the initial and final stages of the inflation process the equation Eq. (1) remaining still stiff demonstrates different properties. When t t k the thickness of the layer h(t) 0 while the outer radius of the layer increases. This leads to the appearance of a small functional parameter H(t) at the highest derivative of so that Eq. (1) becomes similar to a singularly perturbed one. An analysis of Eq. (1) and algorithm of its solution are presented in [45]. Fig.1. Probe dimensionless functions Q 1 (t) and Q (t). The resultant algorithm allows getting the function (t) for any given flow rate Q(t) and different model parameters. Thus for given Q(t) one can firstly find the function (t) and then using the dependences Eq. (16) and Eq. (1) calculate the pressure and velocity fields. Two probe dimensionless functions Q 1 (t) and Q (t) (Fig.1) have been used to model the evolution of shell expansion that is the functions 1 (t) and (t) (Fig.) using Eq. (1) at the following values of the dimensionless groups: A m = A s =16.66 A p = A q = P n =0.9 and K= Time dependences of the pressure p 1 and p in the liquid layer at

9 Modeling of inflation of liquid spherical layers 6875 r=(t)+h(t)/ which are calculated from the functions Q 1 (t) 1 (t) and Q (t) and (t) correspondingly are presented in Fig.3. Fig.. Dependences 1 (t) and (t) for t[01] (a) and t[010] (b) For all the tried dependences Q(t) with t<0 min is typically the appearance of a small minimum in the dependences p(rt) at the beginning of the process. Our calculations have shown that in the absence of outside pressure that is at P n =0 the pressure in the liquid layer at the beginning of the process is close to zero. It appears that this case is out of the model validity. If P n >0 or t>10 min this case does not arise. From the view point of the real space technology time of the shell inflation should not exceed 0 min thus in accordance with our simplified model we conclude that the process of shell inflation must be performed under condition P n 0. Fig. 3. Time dependences of p 1 and p in the liquid layer

10 6876 G. I. Kurbatova and N. N. Ermolaeva 5. Conclusions A mathematical model of the inflation of a spherical liquid layer (shell) under zero gravity is developed. Solutions of the hydrodynamic part of the problem namely the fields of pressure and liquid flow velocity have been found as functions of the inner radius of the shell. Modeling of the shell inflation process was performed for two different conditions of gas pumping. Analysis of the adopted one-dimensional model allowed clarifying the role of the pressure outside the spherical liquid layer and concluding that controllable shell inflation is possible if this pressure is not zero. Acknowledgements The authors are grateful to Professors N. V. Egorov and V. N. Emeljanov for constant attention to this work and to Director of Corning Scientific Center the honorary member of Optical Society of America A.V. Dotsenko for stimulating discussions. eferences [1] L.V. Leskov V. A. Vanke A. V. Luk yanov Space power systems Ed. Mechanical Engineering Moscow [] M. Nagatomo S. Sasaki Y. Naruo V. A. Vanke Solar power systems (SPS) investigations at the Institute of Space and Astronautical Science of Japan Phys. Usp. 37 (4) (1994) [3] C. Truesdell A first course in rational continuum mechanics Maryland The Johns Hopkins University Baltimore 197. [4] N.N. Ermolaeva G. I. Kurbatova Mathematical model of expanding liquid shell Gazette of St.-Petersburg State University series 10 (3) (009) [5] G.I. Kurbatova About solution methods of one stiff ordinary differential equations system Gazette of St.-Petersburg State University series 10 (4) (008) eceived: November 8 013

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