Convection of a micropolar fluid with stretch

SUMMARY. — As a model for the Bénard convection in the asthenosphere the problem of the hydrodynamic stability of an infinite horizontal layer is calculated. The layer consists of a micropolar fluid with streich. The field equations for the velocity vector, microrotation vector, microstretch, microinertia, density, temperature, and pressure form a system of eleven partial differential equations for the determination of eleven unknown scalar functions. We succeed in decoupling the system and reducing the problem to an ordinary differential equation. The analytical solution can be given for the special case of a micropolar Boussinesq fluid.


INTRODUCTION
When investigating flow phenomena in the Earth's mantle, a Newtonian fluid is mostly used as constitutive equation. This assumption is fully justified as an approximation and has often been used successfully ( 3 ' 5 > 9 > 14 ). From shear tests of possible mantle  ( 17 ). In another approach ( 15 ), used to gain information on the convection current pattern in the Earth's mantle, the question as to the constitutive equation is evaded. Some simple assumptions are made regarding the kinematics of the stream lines which appear to he plausible from the point of view of fluid mechanics. These assumptions, together with the geometry of the mantle, lead to certain possible modes of flow which would create a topography on the surface of the Earth which is similar to the ob- served one. Recently, the low-velocity layer of the upper mantle has been assumed to be partly molten f 1 ). If the geometrical connection between the two phases exists in the form (see Fig. 1) suggested by Stocker and Ashby ( 12 ), a micropolar fluid may be assumed as a constitutive equation. In such a medium, in addition to the three translatory degrees of freedom of conventional continuum mechanics, three rotational degrees of freedom are assigned to each spatial point with the help of which the rotation of the solid grains of Fig. 1 may be described. While some authors have already tried to work with micropolar elastic media (e. g., Teisseyre ( 13 ), Bosclii ( 2 )), Cosserat fluids have been introduced only recently ( 16 ) into geophysics. The author ( 10 ) calculated the Benard convection in the asthenosphere, assuming, apart from the term with the buoyancy forces, that the micropolar fluid is incompressible.
In the present paper the problem is to be tackled in a more general way. Grains and intermediate fluid are no longer assumed to be incompressible. Consequently, the Boussinesq approximation is dropped, stretch and microstretch are introduced, which results in an increase of the number of degrees of freedom.

-GOVERNING EQUATIONS
The new model of the asthenosphere consists of a horizontal layer of a micropolar fluid with stretch of the thickness h. The lower surface is kept constant at a temperature T" and the upper surface at a temperature Ti, where Tn > Tt. We employ a rectangular cartesian coordinate system xi, x2, X3, the origin being positioned in the lower boundary plane and x3 being directed upwards. Because the general theory of simple microfluids has too many degrees of freedom to solve a special problem of motion with justifiable calculation efforts, we use the following simplifications (Eringen ( 7 )). Let the microinertia tensor iki have the following form: iki = --j Ski [1] where 11 for k = 1 Ski = 1 0 for k ^ 1 and j is a scalar quantity, i. e., the fluid is microisotropic. Let the gyration tensor nm have -I independent scalar functions instead of 9: llkl = n Ski + Cklr llr [3] where for (k]r) cycl -= ( 1 2 3) for ( Vectors are indicated by bold-face letters or by arrows over the letters. v or vr = velocity, ei = unit vector in the 1-direction. Equation [5] shows that the total derivative of the vector £ with respect to time can be subdivided into an isotropic microstretch and a microrotation. The microrotation generally is not identical with the classical rotation vector.
(Or = -erkl Vl,k. [0] An index followed by a comma means a partial differentiation with respect to space variable .\k, e. g., vi,k = -.
The relationships [8] to [14] are eleven scalar equations for the determination of eleven scalar unknowns: Vk, m, n, j, p, T, p. Compared with the fundamental equations of a micropolar fluid without stretch ( 1(i ), n and j are additional sought functions, f, d, and d* are to be given quantities. As Eringen ( 7 ) and Erdogan (°) have found, the material constants are subject to certain restrictions which are necessary and sufficient to ensure the validity of the principle of entropy: [17] Let us now make some highly justified simplifications. The quantities v, n and n are so small in the asthenospliere that quadratic and mixed quadratic terms thereof may be neglected. [24] 2.

-LINEARIZED GOVERNING EQUATIONS
Let us now assume that the principle of exchange of stabilities ( 4 ) is valid, i. e., the current is considered to be stationary at the beginning. In the marginal state at the beginning of convection the mean state is assumed to be equal to the equilibrium state, and all variables should be representable as sums of equilibrium quantities which are functions of X3 only, and of small perturbations which are functions of XI, x2, x3, and t: In order to obtain the linearized fundamental equations, we now substitute [26] into the equations [18j to [27], the mean, state vari-ables being known now from the statics. We renounce here the introduction of the Boussinesq approximation usual at this point, since it is exactly here where the theory is to be extended by stretch and microstretch. We obtain: + v . Vj -2nj = 0, (a -f-¡3) VV . n + y V 2 n + x V X v -2x n = p j , ot I Oil V 2 n = V 2 n + -p j -, 'n-Ot 3_ _ . 3n 2a0 k V 2 T' = v. T + c p V . v, ax3 p" 8 g T' e3 -V p' + X" V n + (X + 2p + x) VV . [55] The curl of equation [  T" -Ti -(hxa -x3)-. -X3 [63] [64] [65] [66] -X3 - V-va =[V;+ ^ ) va = (D* -a 2 ) v:  [86] [87] [88] Equations [83] to [88] show that, even if we neglect internal heat sources, the solution w(0 cannot be analytically given for a micropolar stretch fluid, since the f4, • • •, fa explicitely contain the quantity X, also in that case. Therefore, we may just as well solve the full problem [74] to [82] in the numerical calculations, with q ^ 0. The main problem, namely to reduce the system ([8] to [14]) from eleven partial differential equations with eleven unknown functions and four independent variables to an ordinary differential equation with one sought function and one independent variable, however, has been solved. Finally, we will derive the boundary conditions for w.

-BOUNDARY CONDITIONS
It is obvious that for convection in the asthenosphere the case of fixed, rigid boundaries is most important, so that we will exclusively deal with this case here. D'w + Ar D 3 w + Cr w = 0 for Ç = 0 and Ç = 1. [105] Thus, the boundary conditions the solution w of [74] must satisfy at the boundaries L -0 and Ç = 1 can be summarized as follows: w = Dw = (D 2 +Aj-D+Bc)w = (D 4 +A-D 3 +Cc)w = 0. [106] In the special case of an incompressible micropolar fluid without internal heat sources the six constants A>-, and CV disappear.

CONCLUSIONS
Whatever the principal driving mechanism for the motion of the lithospheric plates may be, it is obvious that lattice and radiative thermal diffusivities of mantle rock are not sufficient to explain the heat flux observed at the surface of the Earth. Therefore, there must be convection in the mantle and particularly in the asthenosphere. If it is assumed ( 4 ) that the low-velocity layer is partly molten and that there is a geometrical connectivity between the solid and liquid phases in the form (see Fig. 1) suggested by Stocker and Ashby ( 12 ), a layer of a micropolar stretch fluid with internal heat sources may be introduced as a model for the asthenosphere. Hence, the Bonard problem of this model has to be solved, the case of fixed, rigid boundaries being of interest. The dynamic fundamental equations form a system of eleven partial differential equations. The essential object of the present paper is to show how the equations can be decoupled and the problem be reduced to an ordinary differential equation, which is a suitable starting point for numerical calculations. In all the calculations the Boussinesq approximation has not been used and stretch and microstretch have been considered. In the special case of a micropolar fluid without stretch the solution can even be given analytically.