Shear viscoelasticity of suspensions of biological cells with fluid membrane

In order to consider the effect of membrane fluidity upon the mechanical property of biological cell suspensions, we have calculated the complex intrinsic viscosity [eta*] = [eta'] - i[eta"] of spherical shell structures with material incompressibility in suspension as a function of the dimensionless frequency x = omega eta'a/gamma' together with the parameters of hm = eta m/eta', g = gamma/gamma', h = eta/eta', delta = d/a, where a is the radius of the cell, d is the width of the membrane, eta, eta m, and eta' are the viscosities of the medium, of the membrane and of the internal region of the cell, gamma and gamma' the surface tensions at the outer and the inner side of the membrane respectively, and omega the angular frequency. The result is simply represented by two dispersions as follows: [eta*]/[eta] = A1 + B1/(1 + i omega tau 1) + B2/(1 + i omega tau 2). Here i is the imaginary unit, A1 = 2(1 - h)/(2 + 3h) + O(delta), B1 = 3h/(5 + 5h) + O(delta), B2 = h (19 + 16h)/[5(1 + h) (2 + 3h)] + O(delta), tau 1 = [(5/24) (1 + h) (1 + 1/g) delta-2 + O(delta-1)] a eta'/gamma', tau 2 = [(2 + 3h) (19 + 16h)/[40 (1 + h) (1 + g)] + O(delta)] a eta'/gamma', and [eta] = (5/2) [96hmg + 32g (5 + 5h - 12hm)delta + O(delta 2)]/[96hmg + 32g (5 + 2h - 12hm) delta + O(delta 2)].