A standard method to measure the mechanical properties of red cells is to suspend the cells in solutions much more viscous than blood plasma and to measure their elongation when the suspension is subjected to simple shear flow (1, 2). In addition to being elongated, the membrane moves around the elongated shape thus inducing an eddy flow within the cytoplasm. The membrane motion has been termed tank-tread motion (1). The elongation is usually quantified by an elongation index , where *L* and *B* denote the length and the width of the elongated cell.

Measurements of EI of different blood samples are usually performed using a constant viscosity of the suspending medium (η_{0}) under variation of the mean shear rate . Such data are often plotted versus the shear stress of the undisturbed shear flow suggesting that EI scales with τ. If, on the other hand, the same blood sample was tested varying both η_{0} and , it was shown that EI does not scale with τ. Using four viscosities between 12 and 51 mPas, EI was found to scale with (3). An exponent greater unity can also be inferred from other experimental results (4, 5).

To explain this finding, the following hypothesis is put forward. (i) Besides elastic stresses, viscous stresses resist an elongation of tank-treading red cells. (ii) A variable contribution of the viscous stresses being three dimensional in the cytoplasm and two dimensional in the membrane is responsible for an exponent greater unity.

To rationalize the hypothesis, we consider the following experiment. We first measure EI with a certain set of and η_{0}. Then we decrease by a certain amount and increase η_{0} to such an extent that EI remains constant. The elastic stresses and bending moments in the membrane remain essentially constant as EI did not change. The viscous stresses, on the other hand, decrease as the tank-tread frequency decreases with decreasing (6). Therefore, the required increase of η_{0} is less than the reciprocal of the decrease in . This in turn, gives η_{0} a greater weight than and as a consequence an exponent greater than unity in the experimentally determined scaling law.

Of course, endowing η_{0} with an exponent is one of two choices. One could have set as well the exponent to in which case its value would have been less than unity.

Based on the hypothesis the following prediction is made. At sufficiently large η_{0}, the viscous contribution can be neglected against the elastic contribution, i.e., the exponent approaches unity with increasing η_{0}. It follows that above a threshold value of η_{0}, the elongation of tank-treading red cells is essentially determined by the membrane elasticity alone.

To test the hypothesis, this prediction was checked by repeating the previous experiment (3) with the following modifications. First, the viscosity range was extended to about 100 mPas. Second, the exponent was determined as a function of η_{0} instead of a single value for the whole viscosity range as previously. Third, the initial rise of the elongation curve was used instead of its middle part.

We will show that the prediction is confirmed by our experiments and give an estimate for the threshold value for η_{0}, above which a moderate elongation of tank-treading red cells solely depends on the membrane elasticity. As a side observation, we report that the variation of the elongation is large both intraindividually and interindividually.