Bioengineering Fundamentals Module 12

Fluid Statics

Attachment Forces

• Components of attachment forces: F1,x, F1,y and F2,x, F2,y

• Position vectors: r1,x, r1,y and r2,x, r2,y

• Use of right-hand rule to determine directions of cross products

• Moment arm crossing in +x direction with force component in +y results in +z direction vector

• Interpretation of -(r1,yF1,x)z: magnitude is r1,yF1,x pointing in -z direction

• System is called statically indeterminate if unknowns exceed equations

Static Fluid Behavior

• Steady-state system with no mass transfer reduces to equation 6.6-1: a Fu = 0

• Viscosity does not affect behavior of static fluids

Cube in a Fluid

• Cube with sides dx, dy, dz within a still fluid of density r

• Body force due to gravity on mass of fluid within

• Pressure forces act on all faces of the cube

• Pressure force is balanced by gravitational force

Pressure Calculation

• Equation 6.6-2 describes forces in z-direction: a Fu z = Fu p + Fu g = 0

• Differential pressure, dP, captures pressure difference between opposing faces

• dP/dz = -rg indicates pressure varies with height, independent of lateral position

• For x-direction: dP/dx = 0 indicates pressure does not change with x-position

• For y-direction: analysis similar to x-direction shows pressure independent of y-position

• Pressure only acts in z-direction on static fluids

Hydrostatic Pressure Difference

• Relationship given as ∆P = -rg ∆z

• Pressure difference between two points in a static fluid depends on height, density, gravitational constant

• Area over which pressure acts does not affect pressure gradient

Effects of Height on Pressure

• Example of hydrostatic pressure difference from shoulder to ankle

• Pressure in ankle is greater due to greater hydrostatic head

Force Due to Hydrostatic Pressure

• Containers with same base area but different heights exert same pressure force on base due to hydrostatic pressure

• Fup = APbase = Arg(x + y)

• Total force on a container at base equals its weight plus pressure force from the annulus at height y

Isolated, Steady-State Systems

Overview

• Isolated systems have no mass transfer across boundaries, resulting in momentum balance equations

• Conservation of momentum in steady-state conditions corresponds to no changes in momentum

• Equations also represent internal forces acting in equilibrium

Application to Platelet Adhesion

• Problem analyzes adhesion of platelets while ignoring external forces

• Platelets injected into a solution in specific velocities

Perfectly Elastic Collisions

• Perfectly elastic collisions maintain kinetic energy before and after impact

• Conservation of momentum used in perfectly elastic collisions to calculate individual velocities necessary

Impulse Forces

• Useful in evaluating impact situations or forces that act over short times

• Total impulse is calculated using change in momentum over time interval

Steady-State Systems with Mass Movement

Mass Flow Dynamics

• Introduction of mass across system boundaries means momentum change occurs due to acceleration of fluid movement

• Differential form of conservation of momentum must account for flowing fluids

Resultant Forces

• Resultant force calculations essential for analyzing fluid movement stability

• Forces considered for mass flow into system and pressure maintained during movement

Reynolds Number

Understanding Fluid Flow

• The Reynolds number (Re) characterizes flow regimes (laminar or turbulent)

• Re is calculated using formula: Re = rvD/m

• Transition from laminar to turbulent flow at established Re values (2100 < Re < 4000)

Example of Tracheal Airflow

• Analyzing airflow in the trachea using volumetric flow rate and trachea diameter to estimate Reynolds number

Mechanical Energy Accounting

Mechanical Energy Definition

• Accounting for mechanical energy is essential in fluid systems

• Mechanical energy sums kinetic energy, potential energy, and work done on the fluid

• Importance of mechanical energy accounting follows principles of conservation but does not assume mechanical energy is conserved directly.