A spin-vorticity relation for unidirectional plane flows of micropolar fluids

A SPIN-VORTICITY RELATION FOR UNIDIRECTIONAL PLANE FLOWS OF MICROPOLAR FLUIDS KENNETH A. KLINE Wayne State University, Detroit, MI 48202,U.S.A. Abstract-The field equations of micropolar fluid theory are applied to consider transient unidirectional plane flows. The relative angular velocity is introduced and maximum and minimum principles of parabolic partial differential equations are utilized to establish a spin-vorticity relation which is valid for very general boundary and initial conditions on spin. I. INTRODUCTION THE EQUATIONS governing

the flow of micropolar fluids involve a velocity vector field and also a micro-rotation or spin vector field. The purpose of the present note is to establish a kinematical result for unidirectional plane flows of micropolar fluids. This work follows in spirit our recent study[l] in that we attempt to circumvent the difficulty involved in establishing the validity of any specific spin boundary condition by extracting from the theory conclusions which hold for a broad class of spin boundary conditions. In the next section the field equations of micropolar fluid theory are recorded. The nature of previous kinematical results pertaining to velocity, vorticity and spin, and which hold for general spin boundary conditions are discussed. In the final section the equations governing time dependent, unidirectional plane flows of micropolar fluids are recorded. A spin-vorticity relation, valid for very general boundary and initial conditions on spin, is established by making use of maximum and minimum principles of parabolic equations, For completeness we note here that the theory of micropolar ffuids is a special case of the theory of structured fluids. The governing equations of structured fluid theory can be obtained from an energy postulate by apptication of invariance requirements[2,3]. Differences of notation and terminology aside, these equations can also be obtained from direct postulates, and were given originally by Eringen and Suhubi[4] and Eringen [5]. 2. MICROPOLAR

FLUIDS

The momenta and continuity equations for incompressible micropolar fluids are recorded below using the notation from Eringen’s[6] basic work on the subject -P,i + (K, + &)ui.jj + Kveijkvk,j+ /$ = p&,

(a, + pv)vi.ik + yvvk.ii + K8kijUj.i- kirk vi,: = 0,

+ plk = pjtik,

(2.1) (2.2) (2.3)

in which P denotes the pressure, v the velocity vector, v the micro-rotation or spin vector, f the body force, I the body couple, p the mass density and t/(j) the constant volume averaged radius of gyration of the substructure. Further, the vorticity vector w is defined by 0;

=

tei,kvk.j

(2.4)

where e denotes the alternating tensor. The viscosities used in [ 11are related to the viscosities in Eringen’s [6] formulation as given above by 2p=2p,+~,; &L,=K,; 2(nt+It2+n3)=y,; -2n2=a,;-2n3=&. Constitutive equations for the stress and couple stress tensors are given in Section 4 of [6]. 131

K. A. KLINE

132

Following Eringen[6] we define parameter k, of dimension (length))’ by

2Ku

k2-

cm

(1+7))Y” in which, for convenience, we introduce the dimensionless parameter n q=Li.Kv

(2.6)

+2/-L’

Equations (2.1)-(2.3) suffice to determine the velocity v, spin v and pressure P when suitable boundary and initial conditions are given. Unfortunately the proper form of boundary condition to be used involving spin v is still undetermined. Although the no spin boundary condition, used for example by Eringen[6] in the solution of the Poiseuille flow problem, seems reasonable for many flow situations, a wide variety of other boundary conditions has been investigated. Kline[l] recently observed that significant information can be extracted from the theory without the need to specify a spin boundary condition. In particular, using a cylindrical (r, 8, z) coordinate system with base vectors e,, ee and e, to characterize steady flow in a circular tube of the form v = u(r)e,, v = fl(r)e, with vorticity 1 dv a=oee, o=-----Cl, 2dr Kline[l] proved, for any spin boundary condition on n(r), if spin exceeds vorticity (0 > w) at any point in the flow field, then (i) R 2 0 throughout the flow, and (ii) the dimensionless velocity profile (velocity divided by velocity u(0) on the tube centerline) is blunter than parabolic. Also R < o at any one point implies 0 s w throughout the flow and that the dimensionless velocity profile is less blunt than parabolic. Predictions regarding apparent viscosity and discussion concerning the applicability of micropolar fluid theory to suspension rheology may be found in Ref.[l]. Here we wish only to observe that one can study the theory, while avoiding the uncertainty of a spin boundary condition, by exploiting the fundamental nature of the governing field equations. Cowin[7] has recently obtained some general spin-vorticity results for a class of steady flows by making use of properties of elliptic partial differential equations. In the next section we consider an initial value problem and use a maximum principle for parabolic equations to obtain certain spin-vorticity results for plane flows. 3. UNIDIRECTIONAL

PLANE

FLOWS

Consider flow in the (x, y) plane with velocity and spin vectors in the form v = u(y, t)e,,

v = Wy, t)el.

(3.1)

If body force and couple are negligible field equations (2.1) and (2.2) yield &,I, v”+l+T) 0”

-(dP/dx) au 1+7j =at*’ l+q,jaR =--217 *at*

where t* is the dimensionless time r* = f(K, + CL,) )‘,a() 9( ay ’ P

(3.2)

(3.3)

A spin-vorticity

relation

for unidirectional

plane flows of micropolar

fluids

133

and pressure gradient -dP/dx is regarded as a known function of time. Continuity eqn (2.3) is identically satisfied by (3.1). Following Cowin[8] we introduce the difference H between spin and vorticity

H=R-(-iv’).

(3.4)

Then, assuming sufficiently smooth velocity and spin fields, differentiating yields

which, combined

(3.2) with respect to y

with (3.4) in (3.3) produces

H”_k2HJl+dkzaH+~ 22772

l_i(‘+vw 2[

If j_ 2-m

271

22qat*’

1au

(3.6)

(3.7)

then eqn (3.6) reduces to

(3.8) Now, observe from (3.6) that relation (3.7) is a necessary condition to allow spin equal vorticity (H = 0) to be a possible solution of the field equations for a general flow. In this case (3.2) reduces to the classical linear momentum equation for viscous fluids from which one can determine the velocity field with pressure gradient specified. As noted earlier [9], H = 0 implies a microstructure which is behaving materially. We thus assume that relation (3.7) holds. The field equations to allow determination of velocity u and relative angular velocity H are then (3.5) and (3.8). For example, these equations govern transient (and steady) flow through a straight channel and Couette flow, as well as Stokes’ first and second problems for micropolar fluids satisfying relation (3.7). A spin -vorticity relation Consider flow to take place (and eqns (3.5) and (3.8) to hold) in a domain E of the (y, t *) plane E:{O
(3.9)

where Y and T may be as large as we please. The boundary of E consists of four sides, on the following three of which boundary and initial conditions are to be specified S,:{y=O,Oct*sT}, Sz:{y= S,:{Ocy<

Y,Ost*sT},

(3.10)

Y,t*=0}.

Consider boundary and initial conditions for the spin R on sides S,, Sz, S, to be of any form such that on these three sides either H 2 0 or H s 0. One can then conclude that If H > 0 at any single point of flow domain E, then H 3 0 throughout E Similarly if there is a point in E such that H < 0, then H s 0 throughout E. The above result is simple to establish but nonetheless quite remarkable. It says that if the spin boundary and initial conditions are such that relative angular velocity H is either

134

K.

A.KLINE

nonnegative or nonpositive on sides S,, St, and S, then, in flow development for example, if at any time t say there is a point j, 0 < j < Y, at which If > 0 (H < 0), then it must be that H will remain nonnegative (nonpositive) for all later times and throughout the entire y flow field. This follows since time T in the result may be as large as we choose. This result does not hold, of course, if nonconstant external body couple fields are applied. It does hold, however, for a body force field of the form f = f(t)e*. The proof follows easily by contradiction. Assume at a point (y,, tT) of E that H > 0 while at another point (y2, ti) of E H < 0. On sides S,, Sz and S, of E it is specified that either H 2 0 or H < 0. Thus in either case the assumed values of H at points (y,, t’C)and (y2, t T) imply that H has either a positive maximum or a negative minimum at an interior point of E or on side S,: (0 c y < Y, t* = T} of the boundary. Since H satisfies eqn (3.8) in E, either possibility contradicts known maximum and minimum principles of parabolic equations[lO]. Thus it must be that if H > 0 at (y,, tt), then H 2 0 throughout E. Acknowledgemenl-Support

of this work by the National Science Foundation

is gratefully acknowledged

REFERENCES [1] K. A. KLINE, Trans. Sac. Rheol. 19, 139 (1975). [2] S. J. ALLEN, C. N. DeSILVA and K. A. KLINE, Phys. Fluids 10, 2551 (1967); II, 1590 (1968). [3] K. A. KLINEandC. N. DeSILVA,In InelasticBehaviorofSolids (Editedby M.F. Kanninen,W. F. Adler,A. R. Rosenfield and R. I. Idee), p. 327. McGraw-Hill, New York (1970). [4] A. C. ERINGEN and E. S. SUHUBI, Inr. 1. Engng Sci. 2. 189 (1%4). [5] A. C. ERINGEN, Int. J. Engng Sci. 2, 205 (1964). (61 A. C. ERINGEN, 3. Math. Mech. 16, 1 (1966). [7] S. C. COWIN, Trans. Sot. Rheol. 20, 195 (1976). [8] S. C. COWIN, Phys. Fluids 11, 1919 (1968). [9] S. J. ALLEN and K. A. KLINE, Trans. Sot. Rheol. 14, 39 (1970). [IO] M. H. PROTTER and H. F. WEINBERGER, Maximum Principles in Diflerential Equations, Chap. 3. Prentice-Hall, Englewood Cliffs, New Jersey (1%7). (Received 10 July 1975)