ESTIMATION OF RESISTANCE. 9 1 



Estimation of the Resistance. , When a fluid flows through a tube of uniform calibre, the 

 propelling force, h, diminishes from point to point on account of the uniformly acting resist- 

 ance, hence the sum of the resistance in the whole tube is directly proportional to its length. 

 In a uniformly wide tube, fluid flows through each sectional area with equal velocity, hence v 

 and also F are equal in all parts of the tube. The diminution which h (propelling force) 

 undergoes can only occur from a diminution of pressure D, as F remains the same throughout 

 (and h = F + D). Experiment with the pressure-cylinder shows that the pressure towards 

 the outflow end of the tube gradually diminishes. In a uniformly wide tube, the height of the 

 pressure in the manometers expresses the resistances opposed to the current of fluid, which it has to 

 overcome in its course from the point investigated to the free orifice of efflux. 



Nature of the Resistance. The resistance opposed to the flow of a fluid depends upon the 

 cohesion of the particles of the fluid amongst themselves. During the current, the outer layer 

 of fluid which is next the wall of the tube, and which moistens it, is at rest. All the other 

 layers of fluid, which may be represented as so many cylindrical layers, one inside the other, 

 move more rapidly as we proceed towards the axis of the tube, the axial thread or stream 

 being the most rapidly moving part of the liquid. On account of the movement of the 

 cylindrical layers, one within the other, a part of the propelling energy must be used up. The 

 amount of the resistance greatly depends upon the amount of the cohesive force which the 

 particles of the fluid have for each other ; the more firmly the particles cohere the greater will 

 be the resistance, and vice versa. Hence, the sticky blood-current experiences greater resist- 

 ance than water or ether. 



Heat diminishes the cohesion of the particles, hence it also diminishes the resistance to the 

 onflow. These resistances are first developed by, and result from, the movement of the particles 

 of the fluid, they being, as it were, torn from each other. The more rapid the current, there- 

 fore, i.e., the larger the number of particles of fluid which are pulled asunder in the unit of 

 time, the greater will be the sum of the resistance. As the layer of fluid lying next the tube, 

 and moistening it, is at rest, the material which composes the tube exerts no influence on the 

 resistance. 



Tubes of Unequal Diameter. When the velocity of the current is uniform, the resistance 

 depends upon the diameter of the tube the smaller the diameter the greater the resistance ; 

 the greater the diameter the less the resistance. The resistance in narrow tubes, however, 

 increases more rapidly than the diameter of the tube decreases, as has been proved experi- 

 mentally. In tubes of unequal calibre, at different parts of their course, the velocity of the 

 current varies it is slower in the wide part of the tube and more rapid in the narrow parts. 

 As a general rule, in tubes of unequal diameter the velocity of the current is inversely pro- 

 portional to the diameter of the corresponding section of the tube ; i.e., if the tube be 

 cylindrical, it is inversely proportional to the square of the diameter of the circular transverse 

 section. In tubes of uniform diameter, the propelling force of the moving fluid diminishes 

 uniformly from point to point, but in tubes of unequal calibre it does not diminish uniformly. 

 As the resistance is greater in narrow tubes, of course the propelling force must diminish more 

 rapidly in them than in wide tubes. Hence, within the wide parts of the tube the pressure is 

 greater than the sum of the resistances still to be overcome, while in the narrow portions it is 

 less than these. 



Tortuosities and bending of the vessels add new resistance, and the fluid presses more 

 strongly on the convex side than on the concave side of the bend, and there the resistance to 

 the flow is greater than on the concave side. 



Division of a tube into two or more branches is a source of resistance, and diminishes the 

 propelling power. When a tube divides into two smaller tubes, of course some of the particles 

 of the fluid are retarded, while others are accelerated on account of the unequal velocities of 

 the different layers of the fluid. Many particles which had the greatest velocity in the axial 

 layer come to lie more towards the side of the tube where they move more slowly ; and con- 

 versely many of those lying in the outer layers reach the centre, where they move more 

 rapidly. Hence, some of the propelling force is used up in this process, and the pulling 

 asunder of the particles where the tube divides acts in a similar manner. If two tubes join 

 to form one tube, new resistance is thereby caused, which must diminish the propelling force. 

 The sum of the mean velocities in both branches is independent of the angle at which the 

 division takes place (Jacobson). If a branch be opened from a tube, the principal current is 

 accelerated to a considerable extent, no matter at what angle the branch may be given off. 



63. FLOW IN CAPILLARY TUBES. Poiseuille proved experimentally that the flow in 

 the capillaries is subject to special conditions 



(1) The quantity of fluid which flows out of the same capillary tube is proportioual to the 

 pressure. 



k (2) The time necessary for a given quantity of fluid to flow out (with the like pressure, 

 diameter of tube and temperature), is proportional to the length of the tubes. 



_ (3) The product of the outflow (other things being equal) is as the fourth power of the 

 diameter. 



