Hereunder is presented a resume of the author's recent work on hemorheology.
The problem concerns in the first place the influence of the plasmatic zone upon the relationship of the quantity of blood and its pressure flowing through minute vessels. When blood is regarded as a Bingham body with plastic viscosity ηB and yield value fB, the pressure-flow relationship is given in eq. (6) as P>pB/γ and in eq. (8) as P<pB/γ, where ξ=pB/P, pB=2LfB/R and γ=1-(δ/R). Here δ is the thickness of the plasmatic zone, R is the radius of the vessel, and P is the difference in pressure between two cross-sections at distance L. Similar pressure-flow relationships (11) and (13) were also obtained of the blood obeying Casson's equation with Casson viscosity ηc and Casson yield value fc.
The problem concerns in the second place the flow behavior of blood in capillaries with permeable walls. Blood is regarded, for simplicity, as an incompressible Newtonian fluid with viscosity η, and an approximate solution of Navier-Stokes equation was obtained under the boundary conditions u=0 and v=k(p-α) at the wall. Here u and v are respectively the axial and radial component of velocity, p is the hydrostatic pressure, k is the permeability coefficient and α is a constant. Starling's law was assumed with regard to filtration and reabsorption of water. The streamlines are shown in Fig. 9. It is shown that the flow Q becomes minimum at a distance LΔα/Δp from the arterial end of the capillary, where L is the length of the capillary, Δα=pa-α and Δp=pa-pv. Here pa and pv are respectively the pressure at the arterial and venous end of the capillary. In case where filtration and reabsorption of water balance, Δα/Δp must be equal to 1/2, that is, α=(pa+pv)/2.
The problem concerns lastly the circumferential tension T in a thick-walled blood vessel. The tension is given in T=p1r1'-p2r2', where r1' and r2' are respectively the inner and the outer radius of the vessel under the internal pressure p1 and the external pressure p2. The formula holds quite generally, irrespective of whether the wall is homogeneous or inhomogeneous, whether the wall is isotropic or anisotropic, whether the elasticity of the wall is Hookean or nonlinear. The distribution of the circumferential stress τ in the wall with Hookean elasticity was discussed in detail on the basis of classical theory of elasticity. It is shown that τ is not always positive throughout the wall even if p1 is greater than p2. Three cases actually occur: (a) τ is always positive throughout the wall, (b) τ is positive in the inner region, while it is negative in the outer region, and (c) τ is always negative throughout the wall. Introducing non-dimensional parameters difined by k=p1/p2 and s=r2/r1, sk-plane (s>1, k>1) can be divided by two curves k=(1+s2)/2 and k=2s2/(1+s2) into three regions A, B and C which correspond to the cases a, b and c, respectively. It is clear that the circumferential tension T is not always positive in more general cases.
