Biomedical Physics - Fluid Dynamics and Hemodynamic Notes

Introduction to Fluid Dynamics and Hemodynamic

  • Biomedical Physics course at An-Najah National University, Faculty of Medicine and Health Science, Department of Biophysics and Medical Imaging (7108101).

  • File No. 4, Chapter 3: Fluid Dynamics and Hemodynamic.

Content Layout

  • Sec 3.1: Introduction

  • Sec 3.2: Dynamics of Non-Viscous Fluids

  • Sec 3.3: Dynamics of Viscous Fluids

  • Sec 3.4: Hemodynamic and Circulatory System

Types of Fluids

  • Incompressible Fluid: A fluid whose density is constant; its volume cannot be varied with pressure.

    • Liquids: Particles are very close together, so applying pressure does not significantly change the volume. Mostly incompressible, explaining its use in hydraulic presses.

  • Compressible Fluid:

    • Gases: Experience change in density with changing pressure and temperature.

Viscosity

  • The resistance of a fluid to flow.

  • A viscous fluid is one where we cannot ignore the effects of friction within the fluid (high internal attractive forces between the fluid molecules) and between the fluid and the neighboring interfaces.

Types of Flow

  • Laminar Flow: Fluid moves slowly, and fluid layers slide smoothly and in parallel to each other in a uniform direction.

  • Turbulent Flow: Occurs when fluid moves at high velocities or where there are obstacles in the fluid path that prevent movement. Fluid layers mix with eddies, and its direction is irregular.

Ideal vs. Real Fluids

  • Ideal Fluid:

    1. Incompressible – the density is constant.

    2. Irrotational – the flow is smooth, no turbulence.

    3. Inviscid – fluid has no internal friction (\eta = 0).

    • Ideal fluids do not actually exist in nature but are sometimes used for simplified fluid flow problems.

  • Real Fluid:

    • Fluid has viscosity (\eta > 0) and is compressible. These fluids always offer a certain amount of resistance as they move. All fluids in actual practice are real fluids.

Fluid Conservation Laws

  • Fluids follow conservation laws:

    1. Conservation of Mass: Expressed by the Continuity Equation.

    2. Conservation of Energy: Expressed by Bernoulli's Equation.

Dynamic of Non-Viscous Fluids

  • The Equation of Continuity

  • Bernoulli's Equation

  • Bernoulli's Principal Application

Discharge Rate (Volume Flow Rate, Q)

  • The amount of fluid flowing across some surface, such as a pipe's cross-section, in a given time.

  • Measured in cubic meters per second (m^3/s).

  • For an incompressible fluid, the discharge rate equals the product of the cross-sectional area of the pipe and the fluid velocity. ]

  • Equation: Q = \frac{\Delta Volume}{\Delta time} = \frac{A \Delta x}{\Delta t} = Av Where: Q =

    • Q is the discharge rate.

    • A is the cross-sectional area.

    • v is the fluid velocity.

  • The equation shows that the volume of fluid takes time to pass through the cross-sectional area.

Continuity Equation

  • When a fluid is in motion, it must move in such a way that mass is conserved.

  • The continuity equation for the flow of a fluid with density \rho, velocity v and no source or sink terms may be written as:
    Q1 = Q2
    A1v1 = A2v2

  • This equation implies that when a fluid enters a narrower section, it speeds up; wider hose, slower speed; narrower hose, faster speed.

Continuity Equation in Junction

  • The continuity equation applies even in the case of multiple joined pipes: the sum of the volume flow rates into the junction equals the sum out.
    Q{before} = Q{after}
    Q1 = Q2 + Q3 A1V1 = A2V2 + A3V_3

Example Question 1

  • اذا صار في عنا تفرعv3a =v3b

  • A2=A3a+A3b

  • Conversion:
    1 m^3 = 1000 L

Energy Conservation in Fluid

  • Includes all energy types (Potential and Kinetic).

  • Pressure exerted by fluid molecules can be related to the transfer of energy into or out of the volume, thus it must be accounted in the conservation law.

  • Higher column of water shows higher pressure; Lower column of water shows lower pressure.

  • Low Velocity, High Pressure; High Velocity, Low Pressure.

Bernoulli's Principal Discovery

  • Daniel Bernoulli, the Swiss mathematician and physicist, published in his famous book Hydrodynamica the results of his experiments about water flow through a tube that dealt with equilibrium, pressure, speeds, and evaluation.

  • He found that as a fluid moves through a region where its speed and/or elevation above the Earth's surface changes, the pressure in the fluid varies with these changes.

  • The general behavior of pressure with speed is true even for gases. As the speed increases, the pressure decreases.

Bernoulli's Principal

  • States: An increase in the speed of a fluid occurs simultaneously with a decrease in static pressure or a decrease in the fluid's potential energy.

  • High velocity, low pressure; Low velocity, high pressure.

  • Thinner region has more flow, so the molecules' speed will result in more flowing than bouncing against each other.

  • Broader regions have lesser flow, so the molecules' speed will result in more "bouncing" / pressure.

Bernoulli's Equation

  • Valid for ideal fluids and most liquids, and for gases when no expansion or compression is happening (no changing in volume).

  • Pressure, potential energy, and kinetic energy are all combined in one equation: P1 + \frac{1}{2} \rho v1^2 + \rho gh1 = P2 + \frac{1}{2} \rho v2^2 + \rho gh2 Where:

    • P = Pressure

    • \rho = Density

    • v = Velocity

    • g = Gravitational Acceleration

    • h = Height

  • With the help of density definition \rho = \frac{m}{V}, we used the energy density expression to refer to energy per unit volume

  • gravitational potential energy per unit volume: \rho gh

  • kinetic energy per unit volume: \frac{1}{2} \rho v^2

  • energy density units is \frac{J}{m^3} = \frac{N}{m^2} = Pa

Examples

  • Here this is a horizontal pipe, h is fixed, so the gravitational potential energy is not changing, thus fluid will move with pressure gradient.

  • When the piston is pulled up, the atmospheric pressure inside the cylinder will decrease. The atmospheric pressure outside pushes the liquid up into the syringe.

  • Liquid moved from high-pressure point to low pressure.

Breathing Techniques

  • The pressure inside the lungs increases and decreases with each breath.

    • a. During inhalation:

      • Diaphragm muscles expand the chest, and the diaphragm moves downward, reducing pressure inside the lungs to less than atmospheric, causing air to flow into the lungs.

    • b. During exhalation:

      • The muscles simply relax, and surface tension in the alveoli creates a pressure inside the lungs larger than atmospheric pressure, forcing air out.

Haemodialysis (غسيل الكلى)

  • Hemodialysis is the process of cleaning the patient's blood outside the body.

Example Question 2

  • A pipe is designed to carry a fluid of density 1500 kg/m3 at a speed of 3 m/s. Any faster than this and the flow could become turbulent, with undesirable results. Any slower than this and the fluid could start to solidify on the sides of the pipe. The fluid is to be carried from a holding tank which is at a pressure of Pt to a manufacturing line at PL = 100 kPa, which is 2.5m below the holding tank. At what pressure must the tank be maintained?

Normal Saline Infusion to Patient (ضخ محلول ملحي للمريض عبر الوريد "IV solution")

  • Normal saline infusion is used for extracellular fluid replacement (e.g., dehydration).

  • This higher position of the IV bag places greater gravitational pressure on the solution, thus the IV infusion rate will speed up.

Blood Donation Process

  • The blood bag is lower than the donor's hand to benefit from the gravitational potential energy difference to speed up the blood flowing.

Application of Bernoulli's Equation in the Circulatory System

  1. Stenosis: Is the case of narrowing a blood vessels by plaque deposits (كتل ترسبية).

    • The blood velocity increased with the area decreasing, also the pressure decreases which may result in further narrowing, leading the artery to close entirely. Also, the flow will become more turbulent, possibly damaging the arterial wall.

  2. Aneurysms: Is a localized expanding in a blood vessel (balloon-like bulge) (توسع الاوعية الدموية).

    • As the radius increases, the blood velocity decreases and its pressure increases. As the wall is already likely to be weakened, the chances of a rupture in the vessel wall increase.

Homework 1

  • An aneurysm forms in a small blood vessel through which blood travels at 3 m/s. The radius of the blood vessel increases by 20%. What is the increase in pressure inside this aneurysm? (blood density= 1060 kg/m^3.)

  • Answer: \Delta P = 2477 Pa

Dynamic of Viscous Fluids

  • Outline

    • Viscosity

    • Poiseuille's Law

    • Reynold's Number

Viscosity

  • Viscosity is the resistance of the fluid to flow.

  • It have a unit of Pa.sec = \frac{N.S}{m^2} = 10 Poise

  • Affected by the temperature beside many other factors

  • Its symbol is \eta (eta)

  • Flowing along the axis of the tube moves more rapidly than the liquid near the wall. Also, there is a layer that stuck to the wall and remains stationary.

Viscous Fluid Dynamic

  • Flowing viscous fluid along a pipe requires a pushing force to overcome the fluid's friction with the pipe wall or within its layer (fluid resistance). This force is maintained by a pressure difference.

  • Anyone who drinks a fruit cocktail knows that it is much harder to pull through a straw than a glass of cola, and they often come served with a short straw that has a larger diameter to get a larger amount of cocktail.

  • The narrower the pipe, the smaller amount of fluid moved.

  • The longer the pipe, the smaller amount of fluid moved.

  • The higher the viscosity of the fluid, the smaller amount of fluid moved.

  • Blood Viscosity (\eta) Resistance

    • Vessel Length (l) Resistance

  • Vessel Radius (r) Resistance

  • And all these cases a larger pressure is needed

Poiseuille's Law

  • All of the previous factors are related by Poiseuille's equation Q = \frac{(P1 – P2) \pi r^2}{8 \eta l} Where:

    • Q is the volume flow rate

    • \eta fluid viscosity

    • For a cylindrical pipe, length is l and its radius is r

    • the pressure difference between the two ends

Example Question 3

  • A drug is being delivered into a patient's arm at a rate of 10 mL/min. The drug is moved from a syringe through a 5 cm long needle with an internal diameter of 1 mm to the patient's arm. If the patient's blood pressure is 15 kPa, what must the pressure in the needle be? Note that the drug viscosity is 8.90 \times 10^{-4} Pa.s

Reynold's Number

  • The speed at which the flow becomes turbulent depends on the fluid characteristic like viscosity and density, in addition to the dimensions and shape of the pipe it is flowing through.

  • Its important to know when the fluid flowing becomes turbulence; many applications depend on this information, especially in Hemodynamic, also in measuring blood pressure or during the construction of manufacturing lines.

  • Reynolds number, R_e, is a dimensionless number which takes account of all of the related fluid properties and determines whether the fluid flow is laminar or turbulent.

Reynold's Number Formula

R_e = \frac{\rho vd}{\eta}

  • R_e is the Reynold's number, \rho is the fluid density, \eta is the fluid viscosity, d is the pipe diameter, v is fluid velocity

    • a. if R_e \leq 2000 → the flow is laminar

    • b. If 2000 < R_e < 3000 → transitional flow

    • c. if R_e > 3000 → the flow is turbulent.

Example Question 4

  • A drug is being delivered into a patient's arm at a rate of 319 mL/min. This drug has a density of 1050 kg/m^3 and viscosity of 8.90×10^-4 Pa.s. Determine its flow type when its move from a syringe through a needle with an internal diameter of 1 mm?

Homework 2

  • Determine the flow type for the case mentioned in Q.3. If you know that Note that the drug density is 1070 kg/m3

  • Answer: Re = 254 → Laminar Flow

Circulatory System and Hemodynamic

  • Outline

    • Circulatory System Dynamic

    • Blood viscosity

    • Blood Pressure Measurements

    • Cardiac Output

Circulatory System Dynamic

  • The vascular system includes five types of blood vessels

Capillary Exchange

  • Blood pressure is higher than osmotic pressure. Liquid part of blood is forced out to form tissue fluid.

  • Osmotic pressure is higher than blood pressure. Tissue fluid is reabsorbed.

Blood Flow, Blood Pressure, and Resistance

  • The relationships among blood vessels that can be compared include:

    • (a) vessel diameter

    • (b) total cross-sectional area

    • (c) average blood pressure

    • (d) velocity of blood flow.

Blood Viscosity

  • Blood is a heterogeneous mixture, consisting of various substances such as RBCs and plasma. Its not like a simple liquid, as the blood cells are physically separate from the plasma.

  • As a result of this, its behavior is more complicated. Its viscosity is not constant but depends on a number of factors such as hematocrit.

  • Hematocrit is the volume fraction of the blood composed of red blood cells.

  • Higher hematocrit ratio means denser blood, which leads to higher viscosity.
    *Normal Blood:
    * 37%-47% hematocrit
    * 42%-52% hematocrit

  • As the viscosity of blood increases, pushing blood through the circulatory system becomes harder. And so, its velocity decreases, and its pressure increases.

  • Higher blood sugar causes an increase in blood viscosity.

Haematocrit Levels

  • Males generally have a higher hematocrit (47% versus 42% for women) → This is a possible factor in their higher rates of hypertension (high blood pressure) and consequently greater risk of heart disease and strokes.

  • At high altitude, the number of RBCs is increased as a response to hypoxia (an inadequate supply of oxygen). This leads to a higher hematocrit ratio, higher blood viscosity, and higher blood pressure.

  • Blood viscosity is also affected by the carrier-container dimensions. As in anaphylactic shock, when the release of histamine into the blood vessels causes expanding in it → Which causes a reduction in blood flow speed, stacking of RBC, causing the blood to behave like solid particles suspended in a liquid, and thus an increase in blood viscosity.

Blood Pressure (BP)

  • Blood pressure is the pressure produced by the circulating blood on the walls of blood vessels.

  • It is usually expressed in terms of the systolic pressure (maximum BP during one heartbeat) over diastolic pressure (minimum BP in between two heartbeats).

Blood Pressure Measurements

  • Normal blood pressure in a resting adult is ~ 120/80 mmHg.

  • A sphygmomanometer is a device used to measure arterial blood pressure.

  • It consists of an inflatable cuff, a measuring unit, and a mechanism for inflation, which may be a manually operated bulb and valve or a pump operated electrically and used in conjunction with a stethoscope.

Sphygmomanometer Working Principle

  • When the pressure in a sphygmomanometer cuff is released, a clinician can hear the Korotkoff sounds.

Cardiac Output (CO)

  • Amount of blood pumped by each ventricle contraction per minute

  • cardiac output = stroke volume X heart rate CO(ml/min) = SV(ml/beat) \times HR(beats/min)

    • Stroke Volume (SV):

      • Volume of blood ejected by each ventricle/beat

Factors Affecting Heart Rate (HR)

  • Autonomic innervation

  • Hormones

  • Fitness levels

  • Age

Factors Affecting Stroke Volume (SV)

  • N Heart size

  • Fitness levels

  • Gender

  • Contractility

  • Duration of contraction

  • Preload (EDV)

  • Afterload (resistance)

  • Stroke Volume (SV) = EDV - ESV

  • Cardiac Output (CO) = HR × SV