Current of a liquid on a pipe of variable section and branched out

S peeds of current of a real liquid in a horizontal pipe of variable section on various sites of a pipe differ also gradients of pressure accordingly differ.

For liquids, with small viscosity, current can differ at low speeds and average levels of pressure differ too (rule Bernoulli). Besides in places of change of section of a pipe the turbulences causing significant losses are formed. Therefore usually static pressure along all pipes of variable section falls and more abruptly on sites of smaller section. The gradient of pressure depends on the general section of the branched out part in the branched out pipe. Under the formula of Newton force F_B of internal friction is proportional to a gradient of speed in a direction of radius of a pipe. The gradient of speed, as shown in figure, has the greatest size near the wall of pipe and the least in its central part. Hence, force of internal friction and losses of energy of a liquid in layers, there are near the wall of pipe, considerably above, than in central. Therefore, if to compare the general losses of energy on internal friction in a continuous pipe and in branched out (at identical general section) it will appear, that in the latter case when the general surface is more than layers, there are near the wall of pipe, these losses considerably above.

It follows as well from formula Gagen-Puazal, according to which hydraulic resistance in inverse proportion to the fourth degree of radius of a pipe. By simple transformations of this formula it is possible to show, that for the branched out pipe consisting from п pipers, resistance to current of a liquid at identical general section and other equal conditions will be in п times more in comparison with not branched out pipe. Accordingly in п time increases also a gradient of pressure.

Definition of speed of blood-groove

There are some methods of definition of speed of a blood-groove. We shall consider physical bases of two of them. The ultrasonic method (ultrasonic flowmeter survey) is based on Doppler’s effect. The signal from the generator 1 of electric fluctuations of ultrasonic frequency acts on a radiator of ultrasound 2 (look figure) and on the device of comparison of frequencies 3. The ultrasonic wave 4 gets into a blood vessel 5 and is reflected from moving erythrocytes 6. The reflected ultrasonic wave 7 gets in the receiver 8 where it will be transformed to electric fluctuation and amplifies. The strengthened electric fluctuation gets in the device 3. Here the fluctuations, the falling and reflected waves are compared, and is allocated Doppler shift of frequency in the form of electric fluctuation: U=U_ocos2πν_Dt

From this formula it is possible to define speed erythrocytes:

In large vessels speed erythrocytes depends on their arrangement concerning an axis: erythrocytes, located close to an axis, move with greater speed, and erythrocytes located close to a wall of a vessel move with smaller speed. The ultrasonic wave can be reflected from different erythrocytes; therefore Doppler shift has an interval of frequencies. Doppler’s effect allows to define average speed of a blood-groove and speed of movement of various layers of blood.

The electromagnetic method (electromagnetic flowmeter survey) measurements of speed of a blood-groove is based on a deviation of moving charges in a magnetic field. Blood is electrically neutral system, it consists from positive and negative ions. Hence, moving blood is a stream of the charged particles which move with a speed v. Force in a magnetic field with induction B acts on a moving electric charge q. F=qvВ

If a charge negative, force is directed to opposite of vector product vВ.

The forces operating from a magnetic field on opposite charges are directed to the opposite parties. The positive charge accumulates about one wall of a blood vessel, about another - negative. Redistribution of charges on section of a vessel will cause occurrence of an electric field. This physical phenomenon refers to effect of the Hall.

Pressure U_H (Hall’s pressure) depends on speed v of movements of ions, i.e. from speed of blood. Thus, measuring this pressure, it is possible to define and speed of a blood-groove. Knowing section S of a vessel, it is easy to calculate volumetric speed of a blood-groove: Practically it is more convenient to use in this method a variable magnetic field (see fig.). It leads to occurrence variable Hall’s pressure U_H which then amplifies and measured.