Real yields force

Plastic fluids are Newtonian or pseudoplastic liquids that exhibit a yield value (Fig. 3a and b, curves C). At rest they behave like a solid due to their interparticle association. The external force has to overcome these attractive forces between the particles and disrupt the structure. Beyond this point, the material changes its behavior from that of a solid to that of a liquid. The viscosity can then either be a constant (ideal Bingham liquid) or a function of the shear rate. In the latter case, the viscosity can initially decrease and then become a constant (real Bingham liquid) or continuously decrease, as in the case of a pseudoplastic liquid (Casson liquid). Plastic flow is often observed in flocculated suspensions. 

The net area of this intimate contact is called the real area of contact Areai. It is assumed that plastic flow occurs at most microscopic points of contact, so that the normal, local pressures correspond to the hardness aj, of the softer of the two materials that are in contact. The (maximum) shear pressure is given by the yield strength cry of the same material. The net load L and the net shear force Fs follow by integrating aj, and cry over the real area of contact Areai. That is, L = cs, Arca and Fs = ayAreai. Hence, the plastic deformation scenario results in the following (static) friction coefficient ... 

As regards tannin, the parts of the cortex, or true bark, in which it is mostly contained, are the exterior layers of the portion known as the liber, and the interior of the cortical tissue—the inside portions of the former, and the most exterior of the latter, yielding very little of this principlo. The same observation is true of other matters, such as quinine and the like. The various dyCB are seated frequently in the exterior portion of the cortical tissue. The sap always ascends through the cellulose of the real hark and as this fluid is the source from which tannin is socreted, it is evident that thcrO will be more of it in the bark, when the flow is greater than at other periods. Experiments have proved this to bo the caso as regards oak, and the same observation applies to the barks of other trees, such as the willow, elm, pine, birch, bcecli, et cetera, with equal force.. ... 

An obvious question that may occur to the reader is why the very simple method of integrating the viscous dissipation function has not been used earlier for calculation of the force on a solid body. The answer is that the method provides no real advantage except for the motion of a shear-stress-free bubble because the easily attained inviscid or potential-flow solution does not generally yield a correct first approximation to the dissipation. For the bubble, Vu T=0(l) everywhere to leading order, including the viscous boundary layer where the deviation from the inviscid solution yields only a correction of 0(Re x 2). For bodies with no-slip boundaries, on the other hand, Vu T is still 0(1) outside the boundary layer, but inside the boundary layer Vu T = O(Re). When integrated over the boundary layer, which is G(Re k2) in radial thickness, this produces an ()( / Re) contribution to the total dissipation,... 

The force of cohesion, i.e. the maximum value of attractive force between the particles, may be determined by a direct measurement of force, F required to separate macroscopic (sufficiently large) particles of radius r, brought into a contact with each other. Such a measurement yields the free energy of interaction (cohesion) in a direct contact, A (h0) = Ff n r,. Due to linear dependence of F on r, one can then use F, to evaluate the cohesive force F2 = (r2/r )Fx, acting between particles in real dispersions consisting of particles with the same physico-chemical properties but of much smaller size, e.g. with r2 10 8 m (i.e. in the cases when direct force measurements can not be carried out). At the same time, in agreement with the Derjaguin equation... 

Clearly, Eq. 8.8-9 gives the same equilibrium requirement as before (see Eq. 8.8-4). whereas Eq. 8.8-10 ensures that the stoichiometric constraints are satisfied in solving the problem. Thus the Lagrange multiplier method yields the same results as the direct substitution or brute-force approach. Although the Lagrange multiplier method appears awkward when applied to the very simple problem here, its real utility is for complicated problems in which the number of constraints is large or the constraints are nonlinear in the independent variables, so that direct substitution is very difficult or impossible. 

---