FUNDAMENTALS
Fundamentals of Fluid Mechanics
Course Title: BIOE 412
Prepared by: Engr. Kim Dowell D. PanganibanInstitution: Batangas State University
Objectives of Learning
Fluid Kinematics
Viscosity
Newtonian and Non-Newtonian Fluids
Dimensionless Numbers of Biofluid Mechanics
Steady vs. Unsteady Flow and Laminar vs. Turbulent Flow
Boundary Conditions and No Slip Boundary
Compressible and Incompressible Flows
Stress Tensor
Viscoelasticity and Viscoplasticity
Basic Equations of Fluid Mechanics
Conservation of Mass, Momentum, and Energy
Navier-Stokes, Bernoulli, Hagen-Poiseuille Equations
Steady Flow Along Tube
Pulsatile Flow in Rigid and Elastic Tubes
Resistance, Compliance, and Inertance
Two-Phase Flows
I. Fluid Kinematics
Definition
Fluid kinematics deals with describing the motion of fluids without considering the forces causing them to move, focusing solely on the path and behavior of the fluid elements themselves.
Key Concept
Fluid characteristics such as velocity and acceleration are functions of position and time, and these parameters are essential for understanding fluid behavior in various applications.
1.1 Lagrangian vs. Eulerian Descriptions
Lagrangian:
Tracks properties (e.g., velocity, pressure) of individual fluid particles as they move through space and time.
More suited for solid mechanics where tracking discrete particles is feasible.
Eulerian:
Focuses on flow properties at specific points in space rather than the motion of individual particles.
More realistic for analyzing and predicting fluid behavior in a control volume, especially in fluid dynamics.
1.2 Flow Visualization
Types of Flow Lines:
Streamlines: Curves that represent the instantaneous movement of fluid particles; they are tangent to the velocity vector at every point.
Pathlines: The actual trajectory that a fluid particle follows through space over time.
Streaklines: The locus of fluid particles that have passed through a particular point in the flow over time.
Timelines: Represents the position of adjacent particles at a specific time that move together, useful for visualizing changes in flow structure.
II. Viscosity
Definition
Viscosity is a measure of a fluid's internal friction and its resistance to deformation, critical for understanding how fluids flow under various conditions.
Ideal Fluid
An ideal fluid is a theoretical fluid that exhibits no resistance to shear and has zero viscosity (e.g., superfluidity).
2.1 Shear Stress and Viscosity Coefficient
Relationship of shear stress (τ) to shear rate (γ):
For linear fluids, the constant of proportionality is defined as the viscosity coefficient (μ).
This relationship describes how much force is needed to move one layer of fluid over another.
III. Newtonian & Non-Newtonian Fluids
Ideal Fluid
An incompressible fluid that has a constant viscosity equal to zero.
Newtonian Fluids
These fluids maintain a constant viscosity, regardless of the applied shear force, leading to predictable and linear behavior (e.g., water, air).
Non-Newtonian Fluids
Viscosity depends on shear stress and may change under different flow conditions. These fluids can be categorized as:
Time-independent: Viscosity remains constant or changes in a predictable way regardless of time (e.g., Bingham fluids).
Time-dependent: Viscosity changes based on how long the shear is applied (e.g., yogurt).
3.1 Examples of Non-Newtonian Fluids
Bingham Plastics: Requires a yield stress to begin flowing (e.g., ketchup).
Pseudoplastic Fluids: Viscosity decreases as shear rate increases (e.g., paint).
Dilatant Fluids: Viscosity increases with an increase in shear rate (e.g., cornstarch suspension).
IV. Dimensionless Numbers of Biofluid Mechanics
Definition
Dimensionless numbers are ratios that characterize fluid dynamics without physical units, facilitating comparison among different fluid behaviors.
Examples
Reynolds number (Re)
Womersley number (α)
Strouhal number (St)
Dean number (De)
Stokes number (Stk)
4.1 Reynolds Number (Re)
Formula:
Re = (ρ * V * Dh) / μWhere:
ρ = density
V = fluid velocity
Dh = hydraulic diameter
μ = dynamic viscosity
V. Steady vs. Unsteady Flow & Laminar vs. Turbulent Flow
Steady Flow
Fluid properties at any given point do not change with time.
Unsteady Flow
Fluid properties change over time, impacting the analysis of flow behavior.
Laminar Flow
Characterized by smooth, orderly motion with parallel streamlines, typically occurring at low Reynolds numbers.
Turbulent Flow
Chaotic and unpredictable with random fluctuations in velocity, commonly observed at high Reynolds numbers.
VI. Boundary Conditions & No Slip Boundary
No Slip Condition
This condition states that the fluid velocity at a solid boundary (e.g., the wall of a tube) equals that of the boundary itself, meaning that the fluid 'sticks' to the solid surface.
VII. Compressible & Incompressible Flows
Compressible Flow
Occurs when density changes significantly, such as gases flowing at speeds near the speed of sound.
Incompressible Flow
Density remains nearly constant, typically observed in liquids like water, where slight changes in pressure do not significantly affect density.
VIII. Stress Tensor
Definition
Stress in fluid mechanics is defined as force per unit area represented in tensor form, which aids in understanding the fluid's behavior under applied forces.
Types:
Normal stress,
Shear stress,
Isotropic stress.
IX. Viscoelasticity & Viscoplasticity
Viscoelasticity
Materials that exhibit both viscous and elastic properties (e.g., blood) that responds to stress in a non-linear manner and can recover shape after deformation.
Viscoplasticity
Fluids that require a yield stress to flow and often display non-linear responses (e.g., emulsions) that demonstrate time-dependent behavior.
X. Basic Equations of Fluid Mechanics
The fundamental equations include the conservation laws of mass, momentum, and energy, forming the groundwork for modeling and solving fluid behavior problems in engineering contexts.
XI. Conservation of Mass, Momentum, and Energy
Principles
Mass: Matter cannot be created or destroyed within a closed system.
Momentum: The total momentum of a system remains constant unless an external force acts upon it.
Energy: Energy is conserved in a system; it can change forms but cannot be created or destroyed.
XII. Navier-Stokes, Bernoulli, Hagen-Poiseuille Equations
Navier-Stokes Equation
Models fluid motion by accounting for viscosity and external forces acting on the fluid, crucial for predicting flow behavior in various scenarios.
Bernoulli’s Equation
Describes the relationship between pressure, velocity, and elevation in fluid theory, often applied in ideal fluid flow analysis.
Hagen-Poiseuille Equation
Characterizes laminar flow in cylindrical pipes, allowing engineers to calculate flow rates based on viscosity and geometric factors.
XIII. Steady Flow Along Tubes
Characteristics
Fluid properties in steady flow do not change with time, governed by Poiseuille's law which is essential for understanding flow through narrow passages or tubes.
XIV. Pulsatile Flow in Rigid and Elastic Tubes
Studying the behavior of blood flow under oscillating pressure conditions, highlighting differences in dynamics between rigid and elastic walls, which is crucial in cardiovascular biomechanics.
XV. Resistance, Compliance, & Inertance
Resistance
Impedance to flow due to friction present in vessels, affecting how easily fluids can move through them.
Compliance
The ability of a vessel to change volume in response to pressure changes, influencing the dynamics of fluid movement.
Inertance
Refers to the effects of momentum in fluid flow, particularly relevance in systems such as the cardiovascular system where acceleration plays a critical role.
XVI. Two-Phase Flows
Definition
Fluids composed of two dissimilar phases, often encountered in biological contexts, such as the flow of blood which contains both liquid and cellular components.