毛細血管中での血液流動

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Blood flowing in capillary cannot be treated as a homogeneous fluid, since the diameter of capillary is of the same dimension as that of red cells. The hematocrit is about 45%. Therefore, when blood flows in capillary, the width of the "compartment" between the two adjacent red cells will be nearly equal to the thickness of the red cell. For the sake of simplicity, we shall make assumptions as follows: (1) the surfaces of red cells are perpendicular to the axis of capillary; (2) since the "compartments" are very narrow, there can be no relative motion of plasma to be contained between the red cells with respect to them; (3) the capillary is an infinite small cylindrical tube; (4) the flow of blood in capillary is steady; (5) blood is incompressible; (6) the effect of gravity is negligible; (7) there is no slip on the wall of the tube. From the above assumptions, we may treat the flow of blood in capillary as if an infinite rod flows steadily through an infinite cylinder filled with plasma. Let us take a cylindrical coordinate system (r,θ,z), the z-axis being the axis of the capillary. Then the velocity component v_z will be a function of r alone. Solving the equations of continuity and of momentum, we get the discharge Q=πΔp(R^4-r_0^4)/8η. If blood is regarded as a homogeneous fluid of apparent viscosity η_b, the discharge will be given by Q_0=πΔpR^4/8η_b according to the Poiseuille equation. Consequently, the ratio of Q to Q_0 is given by Q/Q_0=η_0/η・{1-(r_0/R^4}. The apparent viscosity η_a of blood in our model is related with the discharge by the relation Q=πΔpR^4/8η_a. By equating this expression with the obtained formula, we get η_a/η={1-(r_0/R)^4}^<-1>. This relationship has been derived by Whitmore from a different stand point of view under certain conditions.
社団法人日本材料学会の論文
1967-07-15

毛細血管中での血液流動

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