Biofluidics
INTRODUCTION (CHAPTER 1)
Multiscale modeling: a system is described and analyzed across
multiple spatial and or temporal scales at the same time phenomena at
smaller scales influence behavior at larger scales (& omgekeerd), so a
single model at only one scale is often insufficient
Thermal conductivity k: measures how well a material conducts heat
(bv metals: k = high)
Convective heat transfer coefficient h: describes how efficiently heat
is removed/supplied by a fluid (heat transfer between solid surface & fluid)
Thermal diffusivity alfa: how fast heat spreads through a material
compared to how much heat the material can store high = heat spreads
quickly measures speed of T equalization inside material
NUMBERS:
Fourrier number: how far heat has diffused through a material over time
(>>1 = SS)
Biot number: compares internal conduction resistance to external
convection resistance is heat transfer limited by conduction on inside or
convection on outside?
Low Bi = easy conduction (k high)
Reynolds number: characterizes nature of fluid flow laminar/turbulent,
compares inertial forces & viscous forces
High Re = inertia dominates turbulent flow
Low Re = viscosity dominates laminar flow
Prandtl number: compares momentum diffusion to thermal diffusion in
fluid does momentum diffuse faster than heat?
High Pr = momentum move faster
Low Pr = heat diffuses faster
Nusselt number: measures how strong convection is compared to pure
conduction at surface how much does fluid motion enhance heat
transfer compared to conduction only?
High Nu = thin thermal boundary layer, strong convection
= 1 = pure conduction through fluid
>1 = convection enhances a bit
1
,Grashoff number: measures strength of buoyancy driven flow (for free
convection) compares buoyancy forces (due to density differences) vs
viscous forces
<<1: free convection negligible
Laminar/turbulent free convection
Rayleigh number: how strong free convection is when buoyancy & heat
diffusion are taken into account
High Ra = strong free convection
Low Ra = heat transfer mainly by conduction
Peclet number: compares convective transport to diffusive transport
<<1: diffusion
>>1: convection
Womersley number: dimensionless; describes the importance of
pulsatile (time-dependent) flow effects compared to viscous effects in
oscillatory flows, especially blood flow in arteries displays parabolic
tendencies (chapter 6)
<1 = low frequency, parabolic velocity profile (capillaries)
>10 = high frequency, flat profile (aorta)
o Proportional with vessel radius R
Dean number: quantifies the importance of secondary flow in curved
pipes or blood vessels (secondary flow = velocity of fluid in
radial/tangential direction)
Low = flow like in a straight pipe
High = very asymmetric profile
Sherwood number: characterizes convective mass transfer at a surface
(mass transfer analog of Nu-number)
= 1 = transport dominated by diffusion
>1 = convection enhances mass transfer
Schmidt number: compates momentum diffusion to mass diffusion
(mass transfer analog of Pr-number)
High Sc = momentum diffuses faster than mass (thin concentration
boundary layer)
Low Sc = mass diffuses quickly
Knudsen number: tells you whether a fluid can be treated as a
continuum or whether molecular effects become important
< 10-2 = continuum
>10 = free molecular flow
2
, CHAPTER 2
Heat transfer: transfer of energy due to T difference
Conduction, convection, radiation (dia 33-34)
Temperature: measure of movement of molecules
Fourrier’s law: heat flux is proportional to (negative) T gradient
Newton’s law of cooling: flux at boundary is proportional to difference
of T at surface & T in fluid
Boundary conditions: what happens at the boundary of something
(zonder deze heb je oneindig veel opl)
Dirichlet (fixed T): T is fixed & known at boundary: T = Ts
Neumann (prescribed flux): flux at boundary is prescribed, you
know how much heat flows through boundary
o When insulated/adiabatic boundary: flux = 0
Convection: heat transfer at boundary is governed by convection to
a surrounding fluid heat loss depends on T difference between
surface & fluid
o Warmte die door conduction uit materiaal komt & aan
oppervlakte komt (dus eerst conduction in solid) = warmte die
door convectie via fluid wordt afgevoerd
Radiation: heat transfer by thermal radiation between surface &
surroundings (no medium needed) Stefan-Boltzmann law gives
heat flux at surface
o Warmte die door conduction uit materiaal komt & aan
oppervlakte komt (dus eerst conduction in solid = warmte die
door radiation wordt afgevoerd
Steady state: when T no longer changes in time, cte in time (it may vary
in space tho) vanaf dia 43
Free vs forced convection:
- Free: no external forces cause flow of fluid, flow because of density
differences
o Gr, Ra
- Forced: with external forces higher h
o Re
- Both: Nu (to get h for convection)
CHAPTER 3
3
INTRODUCTION (CHAPTER 1)
Multiscale modeling: a system is described and analyzed across
multiple spatial and or temporal scales at the same time phenomena at
smaller scales influence behavior at larger scales (& omgekeerd), so a
single model at only one scale is often insufficient
Thermal conductivity k: measures how well a material conducts heat
(bv metals: k = high)
Convective heat transfer coefficient h: describes how efficiently heat
is removed/supplied by a fluid (heat transfer between solid surface & fluid)
Thermal diffusivity alfa: how fast heat spreads through a material
compared to how much heat the material can store high = heat spreads
quickly measures speed of T equalization inside material
NUMBERS:
Fourrier number: how far heat has diffused through a material over time
(>>1 = SS)
Biot number: compares internal conduction resistance to external
convection resistance is heat transfer limited by conduction on inside or
convection on outside?
Low Bi = easy conduction (k high)
Reynolds number: characterizes nature of fluid flow laminar/turbulent,
compares inertial forces & viscous forces
High Re = inertia dominates turbulent flow
Low Re = viscosity dominates laminar flow
Prandtl number: compares momentum diffusion to thermal diffusion in
fluid does momentum diffuse faster than heat?
High Pr = momentum move faster
Low Pr = heat diffuses faster
Nusselt number: measures how strong convection is compared to pure
conduction at surface how much does fluid motion enhance heat
transfer compared to conduction only?
High Nu = thin thermal boundary layer, strong convection
= 1 = pure conduction through fluid
>1 = convection enhances a bit
1
,Grashoff number: measures strength of buoyancy driven flow (for free
convection) compares buoyancy forces (due to density differences) vs
viscous forces
<<1: free convection negligible
Laminar/turbulent free convection
Rayleigh number: how strong free convection is when buoyancy & heat
diffusion are taken into account
High Ra = strong free convection
Low Ra = heat transfer mainly by conduction
Peclet number: compares convective transport to diffusive transport
<<1: diffusion
>>1: convection
Womersley number: dimensionless; describes the importance of
pulsatile (time-dependent) flow effects compared to viscous effects in
oscillatory flows, especially blood flow in arteries displays parabolic
tendencies (chapter 6)
<1 = low frequency, parabolic velocity profile (capillaries)
>10 = high frequency, flat profile (aorta)
o Proportional with vessel radius R
Dean number: quantifies the importance of secondary flow in curved
pipes or blood vessels (secondary flow = velocity of fluid in
radial/tangential direction)
Low = flow like in a straight pipe
High = very asymmetric profile
Sherwood number: characterizes convective mass transfer at a surface
(mass transfer analog of Nu-number)
= 1 = transport dominated by diffusion
>1 = convection enhances mass transfer
Schmidt number: compates momentum diffusion to mass diffusion
(mass transfer analog of Pr-number)
High Sc = momentum diffuses faster than mass (thin concentration
boundary layer)
Low Sc = mass diffuses quickly
Knudsen number: tells you whether a fluid can be treated as a
continuum or whether molecular effects become important
< 10-2 = continuum
>10 = free molecular flow
2
, CHAPTER 2
Heat transfer: transfer of energy due to T difference
Conduction, convection, radiation (dia 33-34)
Temperature: measure of movement of molecules
Fourrier’s law: heat flux is proportional to (negative) T gradient
Newton’s law of cooling: flux at boundary is proportional to difference
of T at surface & T in fluid
Boundary conditions: what happens at the boundary of something
(zonder deze heb je oneindig veel opl)
Dirichlet (fixed T): T is fixed & known at boundary: T = Ts
Neumann (prescribed flux): flux at boundary is prescribed, you
know how much heat flows through boundary
o When insulated/adiabatic boundary: flux = 0
Convection: heat transfer at boundary is governed by convection to
a surrounding fluid heat loss depends on T difference between
surface & fluid
o Warmte die door conduction uit materiaal komt & aan
oppervlakte komt (dus eerst conduction in solid) = warmte die
door convectie via fluid wordt afgevoerd
Radiation: heat transfer by thermal radiation between surface &
surroundings (no medium needed) Stefan-Boltzmann law gives
heat flux at surface
o Warmte die door conduction uit materiaal komt & aan
oppervlakte komt (dus eerst conduction in solid = warmte die
door radiation wordt afgevoerd
Steady state: when T no longer changes in time, cte in time (it may vary
in space tho) vanaf dia 43
Free vs forced convection:
- Free: no external forces cause flow of fluid, flow because of density
differences
o Gr, Ra
- Forced: with external forces higher h
o Re
- Both: Nu (to get h for convection)
CHAPTER 3
3
