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AbstractIn the circulation system the blood vessels have many generations of branching or subdivision. Similarly in the lung the airway bifurcates as many as 23 generations. The flow of blood or air into each new branch is subjected to a change in the entry region. It is important to know how the transition to Poiseuille flow takes place. In this article the axisymmetric low-Reynolds-number entry flow in the inlet region of a circular cylindrical tube is studied on the basis of the Stokes' approximation. The purpose of this study is to find a theoretical solution of the entry flow at very low Reynolds numbers, for which no previous solution is known to the authors. The present solution shows that the entry length at low Reynolds number is simply of the order of the tube radius. For microcirculation (in blood vessels with diameters smaller than 100 μ) the Reynolds number is of the order of 1 or smaller, and the low-Reynolds number approximation is valid. Similarly, for air flow in the respiratory bronchioles and alveolar ducts and sacs of the lung the Mach number is essentially zero and the Reynolds number lies in the range 1–10−2 and the present analysis is also applicable.

AbstractA quantitative evaluation of lung injury due to impact loading is of general interest. Hemorrhage and edema are the usual sequelae to traumatic pulmonary impact. To gain some quantitative understanding of the phenomena, we perfused excised rabbit lung with Macrodex at isogravimetric condition and monitored lung weight continuously after impact. It is shown that a factor of importance is the rigidity of the surface on which the lung rests. The rate of lung weight increase is smaller if the lung was ‘freely’ supported on a soft cloth, more if it was supported on a rigid plate. This suggests the influence of stress wave reflection. The critical condition correlates with the initial velocity of impact at the surface of the lung, or with the maximum deflection. For a freely supported lung, the rate of lung weight increase was 22% of the initial total lung weight per h after impact when the impact velocity was 11.5 m s−1, 30% when the velocity was 13.2 m s−1, several 100% at 13.5 m s−1, signaling massive lung injury. Since the velocity of sound in rabbit lung is 33.3 m s−1 when the inflation (transpulmonary) pressure is 10 cm H2O, the critical velocity of 13.5 m s−1 corresponds to a Mach number of 0.4. The maximum surface displacement of the lung is almost linearly proportional to the initial velocity of impact. The exact cause of edema and hemorrhage is unknown; we hypothesize that it is due to tensile stress in the alveolar wall caused by the impact.

AbstractEquations describing the movement of fluid between vascular and tissue spaces in a pulmonary alveolar sheet are formulated. Solutions are given for the two-dimensional case in which arterioles and venules are parallel. The theory predicts that at a steady state the volume of interstitial water and the pulmonary capillary blood volume are both simply correlated with the arteriolar pressure and, thus, simply correlated to each other. This conclusion is reached for a wide range of variation (over three orders of magnitude) of the permeability constant and for a variety of morphological geometry of the tissue space. Theoretical effects on interstitial water volume due to variations in membrane permeability, arterial, venous, and alveolar pressures, sheet geometry, the elastic compliance coefficient of the sheet thickness versus blood pressure, the lymph flow, alveolar transudation, and the compliance constant for the tissue pressure are presented. It is shown that if the arteriolar pressure and the tissue pressure are properly accounted for, other parameters such as the left atrium pressure, membrane permeability, or lymph flow play only secondary roles as far as interstitial water and capillary blood volume are concerned.

AbstractA detailed measurement of histological specimens of the lungs of the cat shows that each terminal precapillary vessel (arteriole) supplies, on the average, 24.5 pulmonary alveoli; each terminal postcapillary vessel (venule) drains, on the average, 17.8 alveoli. These numbers link pulmonary alveolar blood flow in capillary sheets with the flow in pulmonary arteries and veins which are cylindrical tubes. They are key numbers needed for hemodynamic analysis. In the literature, these numbers are variously speculated to be 1 or smaller; thus our results correct, even though only for the cat, an important concept.

Publisher SummaryThis chapter discusses the pseudo-elasticity of living tissues; soft tissues, such as arteries, muscles, skin, lung, and ureter have been considered. The mechanical properties of these tissues are qualitatively similar. It focuses on arteries. As a material, arteries are inelastic. They do not meet the definition of an elastic body, which requires that there should be a single-valued relationship between stress and strain. Arteries show hysteresis when they are subjected to cyclic loading and unloading. When held at a constant strain, they show stress relaxation. When held at a constant stress, they show creep. They are anisotropic. Their stress-strain-history relationships are non-linear. Their properties vary with the sites along the arterial tree, ageing, short- or long-term effects of drugs, hypertension, and inervation or denervation. An approach to non-linear elasticity uses the incremental law: a linearized relationship between the incremental stresses and strains obtained by subjecting a material to a small perturbation about a condition of equilibrium.

As an idealized problem of the motion of blood in small capillary blood vessels, the low Reynolds number flow of plasma (a newtonian fluid) in a circular cylindrical tube involving a series of circular disks is studied. It is assumed in this study that the suspended disks are equally spaced along the axis of the tube, and that their centers remain on the axis of the tube and that their faces are perpendicular to the tube axis. The inertial force of the fluid due to the convective acceleration is neglected on the basis of the smallness of the Reynolds number. The solution of the problem is derived for a quasi-steady flow involving infinitesimally thin disks. The numerical calculation is carried out for a set of different combinations of the interdisk distance and the ratio of the disk radius to the tube radius. The ratio of the velocity of the disk to the average velocity of the fluid is calculated. The different rates of transport of red blood cells and of plasma in capillary blood vessels are discussed. The average pressure gradient along the axis of the tube is computed, and the dependence of the effective viscosity of the blood on the hematocrit and the diameter of the capillary vessel is discussed.

Publisher SummaryThis chapter discusses the problems of blood flow and the constitutive equations of various tissues involved, the geometrical configurations and dimensions of the system, and the possible boundary conditions. For the blood flow problem, the blood, the blood vessels, the tissues surrounding the blood vessels, the geometry of the vascular system, and the driving forces are considered. Blood circulation is the most popular subject in biomechanics. The value of mechanics has been shown in the development of heart-assist devices. The development of artificial heart valves demonstrated the importance of fluid mechanics. In microcirculation, research mechanics offers a powerful tool. It offers details about the blood flow that are difficult to obtain by conventional means at the disposal of physiologists. The chapter presents the scope of the existing literature on physiologically important problems. The mathematical framework of the problems of wave propagation in arteries and peristalsis in small vessels are also discussed.

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