Biofluid Mechanics

The Human Circulation, Second Edition

Krishnan B. Chandran, Stanley E. Rittgers, Ajit P. Yoganathan

• Publisher: CRC Press
• Year: 2012
• ISBN:
  9781439898161
• Edition: 2
• Topic: Medicine
• Subtopic: Biotechnology in Medicine

Designed for senior undergraduate or first-year graduate students in biomedical engineering, Biofluid Mechanics: The Human Circulation, Second Edition teaches students how fluid mechanics is applied to the study of the human circulatory system. Reflecting changes in the field since the publication of its predecessor, this second edition has been ex

Part I

Fluid and Solid Mechanics and Cardiovascular Physiology

1 Fundamentals of Fluid Mechanics

1.1 INTRODUCTION

Before considering the mechanics of biological fluids in the circulation, it is necessary to first consider some key definitions and specific properties. Once established, we will use these “pieces” to construct important laws and principles which are the foundation of fluid mechanics. To begin, we will define what we mean by a fluid. In general, a material can be characterized as a fluid if it deforms continuously under the action of a shear stress produced by a force that acts parallel to the line of motion. In other words, a fluid is a material that cannot resist the action of a shear stress. Conversely, a fluid at rest cannot sustain a shearing stress. For our applications, we will treat a fluid as a continuum (i.e., it is a homogeneous material) even though both liquids and gases are made up of individual molecules. On a macroscopic scale, however, fluid properties such as density, viscosity, and so on are reasonably considered to be continuous.

1.2 INTRINSIC FLUID PROPERTIES

The intrinsic properties of a fluid are considered to be its density, viscosity, compressibility, and surface tension. We will discuss and define each of these individually.

1.2.1 DENSITY

Density, commonly denoted by the symbol, ρ, is defined as the mass of a fluid per unit volume and has units of [M/L³]. In the meter/kilogram/second (MKS) system, this would be represented by (kg/m³) or by (g/cm³) in the centimeter/gram/second (CGS) system where 1 g/cm³ = 10³ kg/m³. Values of density for several common biofluids are

ρ_water = 999 kg/m³ at 15°C (≈1 g/cm³)

ρ_air = 1.22 kg/m³ at standard atmospheric temperature and pressure

ρ_{whole blood} = 1060 kg/m³ at 20°C (6% higher than water)

A related property of a fluid is its specific gravity, denoted as SG, which is defined as its density divided by the density of water at 4°C (a reference value that is quite repeatable). Thus, for whole blood at 20°C, SG = 1.06. The specific weight of a fluid, denoted as γ (not to be confused with

, the symbol for rate of shear), is the weight of a fluid per unit volume, or ρg. For blood at 20°C, γ = 1.04 × 10⁴ N/m³.

1.2.2 VISCOSITY

As we said earlier, a fluid is defined as a material that deforms under the action of a shear force. The viscosity of a fluid (or its “stickiness”), denoted by the symbol, μ, is related to the rate of deformation that a fluid experiences when a shear stress is applied to it. Just as with a solid, fluid shear stress is defined as shear force per unit area applied tangentially to a surface and is denoted by τ. To illustrate this, consider two parallel plates each of cross-sectional area A (cm²) with fluid of viscosity μ between them as shown in Figure 1.1.

If a tangential force F_t is applied to the top plate as shown, it will result in the plate moving with a velocity U (cm/s) relative to the lower plate. The fluid adjacent to the top plate will move with the same velocity as that of the plate since the fluid is assumed to stick to the plate (known as the “no-slip” condition). Similarly, the fluid adjacent to the bottom plate will be at rest since it sticks to a stationary surface. Thus, a velocity gradient, or change in velocity per unit change in height, is produced within the fluid as shown. The shearing force F_t divided by the area, A, over which it acts is defined as the shearing stress, τ, having the units [ML⁻¹T⁻²]. The velocity gradient, also referred to as the rate of shear,

, is the ratio U/h where h is the distance between the two parallel plates. Thus, the rate of shear has the dimension of s⁻¹. In general, the rate of shear is defined as ∂u/∂y where y is the distance perpendicular to the direction of shear as shown in the figure. The viscous properties of all fluids are defined by the relationship between the shear stress and the rate of shear over a range of shear rates.

The relationship between viscous shear stress, τ, viscosity, μ, and shear rate, ∂u/∂y, for flow in the x-direction is given in Equation 1.1 with the derivative taken in the y-direction, perpendicular to the direction of flow:

The coefficient in this relationship is known as the dynamic viscosity [ML⁻¹T⁻¹] and is usually expressed as either Pa · s (kg/m · s) in the MKS system or as Poise (g/cm · s) in the CGS system. In many biological applications, it is convenient to define a centiPoise (cP) where 1 cP = 0.01 P due to the relatively low values of this property. The constitutive relationship expressed in Equation 1.1 can be plotted in a shear stress versus rate of shear plot as shown in Figure 1.2. (Note: Kinematic viscosity, υ = μ/ρ, is commonly used to condense the density and viscosity into a single variable, especially for liquids where the density is relatively constant.)

FIGURE 1.1 Fluid subjected to simple shearing stress.

FIGURE 1.2 Shear stress versus rate of shear plots for Newtonian and non-Newtonian fluids.

A fluid in which the viscosity is constant is known as a Newtonian fluid and the relationship between viscous shear stress and shear rate is represented in the figure by a straight line passing through the origin with a slope equal to μ. In reality, many fluids do not follow this ideal linear relationship. Those fluids in which the shear stress is not directly proportional to the rate of shear are generally classified as non-Newtonian fluids. In this case, the ratio of shear stress to the rate of shear at any point of measurement is referred to as the apparent viscosity, μ_app. The apparent viscosity is not a constant but depends on the rate of shear at which it is measured. There are several classes of non-Newtonian fluids whose constitutive relationships are shown in Figure 1.2. For example, many fluids that exhibit a nonlinear relationship between shear stress and rate of shear and pass through the origin are expressed by the relationship

where n ≠ 1. Such fluids are classified as power law fluids. Another class of fluids is known as Bingham plastics because they will initially resist deformation to an applied shear stress until the shear stress exceeds a yield stress, τ_y. Beyond that point, there will be a linear relationship between shear stress and rate of shear. The constit...