Theoretical studies in micropolar and viscoelastic fluids

Abstract

We record below a summary of the results of our investigations. The linear micropolar fluid is a viscous fluid with five additional coefficients of viscosity, apart from the usual Newtonian fluid. In its flow behaviour it resembles greatly the Newtonian fluid in that it does not exhibit the Weissenberg effect, recti? linear flow in pipes of arbitrary cross?section is always possible and it shows no separation in the secondary flow in a simple shearing motion. There are no extra normal stress effects present along the streamlines, and, in fact, these normal stresses are identical with those of a Newtonian fluid with viscosity (2?+K)(2\mu + K)(2?+K). However, these fluids differ from the Newtonian fluids in that they exhibit micro?rotational effects and can support couple stresses and body couples. Besides, the components of shearing stress in these fluids are affected by the vorticity and micro?rotation of the fluid and are no longer symmetric. The most noteworthy feature of these fluids is the micro?rotation. According to Eringen, micro?rotation bears a resemblance to the vorticity inasmuch as only those components of it are non?vanishing which correspond to the non?vanishing components of vorticity and they depend on the same space variable on which the vorticity components depend. We have in Part III assumed that the micro?rotation vector is in itself a measure of rotation and thus its divergence must vanish. In Parts I and II, where there is dependence on only one space coordinate, this is found to happen automatically. Again, the boundary conditions satisfied by these micro?rotations have not been fully established. In Parts I and II, we have taken the micro?rotation to vanish when the fluid is in contact with a rigid body, irrespective of whether the rigid body is in motion or not. This is because we have assumed that the fluid element sticks to the solid boundary and thus is deprived of all internal angular velocity. However, there is much to be said about the boundary condition of “wall vorticity”, as observed in Part III. In the flows studied, we note that when the micro?rotation equals the vorticity near a solid boundary, it equals the vorticity throughout the flow field. The micropolar fluid then behaves like a fluid governed by the indeterminate couple stress theory. The presence of couple stresses is another important char? acteristic of these fluids. We find that these couple stresses assume large values at the boundaries. Probably, they are primarily responsible for boundary effects observed experimentally in these fluids. We aimed at examining if we can explain the behaviour of the highly viscous inelastic and elastic fluids which go by the name of “non?Newtonian fluids” by this theory. We do not observe in any of the flows that we have studied the characteristic non? Newtonian phenomena, like the Weissenberg effect, separation and reversal of secondary flows etc. We attribute this to the fact that the constitutive equations have been linearised in eee and e?\dot{e}e? (the rate?of?strain tensor and micro?deformation tensor) and hence cannot give rise to normal stress effects, so typical of non? Newtonian fluids. If we make less stringent simplifications than what Eringen has done in establishing the theory of simple linear microfluids, and retain terms not only linear in b?eb - eb?e, b??e?\dot{b} - \dot{e}b??e? but also the terms in e2e^2e2 and the products of eee with b?eb - eb?e and b??e?\dot{b} - \dot{e}b??e?, we would obtain a general visco?inelastic fluid exhibiting all the properties of a Reiner–Rivlin fluid as seen from the following constitutive equations: ?=?pI+2?e+?2(b?e)+?4(b?e)2+?(e(b?e))\tau = -pI + 2\mu e + \alpha_2 (b - e) + \alpha_4 (b - e)^2 + \eta(e(b - e))?=?pI+2?e+?2?(b?e)+?4?(b?e)2+?(e(b?e)) and M=?0I+?1(b?e)+?2(b??e?)+?3(be?+e?b?2e)M = \beta_0 I + \beta_1 (b - e) + \beta_2 (\dot{b} - \dot{e}) + \beta_3 (b\dot{e} + \dot{e}b - 2e)M=?0?I+?1?(b?e)+?2?(b??e?)+?3?(be?+e?b?2e) (4.1, 4.2) where ?1,?2,?3,?4,?5 are of the form tr?(e2),tr(b?e)2\alpha_1, \alpha_2, \alpha_3, \alpha_4, \alpha_5 \text{ are of the form } \mathrm{tr}\,(e^2), \mathrm{tr}(b - e)^2?1?,?2?,?3?,?4?,?5? are of the form tr(e2),tr(b?e)2 and ?1,?2,?3,?4\beta_1, \beta_2, \beta_3, \beta_4?1?,?2?,?3?,?4? are constants. For the purpose at hand it is not necessary to change the expression for ppp.