class: center, middle, inverse, title-slide .title[ # Lecture 14 ] .subtitle[ ## Fluids Around <br> Bernoulli ] .author[ ### Dr. Christopher Kenaley ] .institute[ ### Boston College ] .date[ ### 2024/02/29 ] --- class: top # Fluids <!-- Add icon library --> <link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.14.0/css/all.min.css"> .pull-left[ Today we'll .... - Consider Mass - Consider Energy - MP3 questions ] .pull-right[ <!-- --> ] --- class: top # Two Important Conditions .pull-left[ `$$\tau=\mu \frac{du}{dy}=v\frac{d(\rho u)}{dy}$$` <img src="img/shearvelocity.png" width="400" /> ] .pull-right[ No-slip condition (fluids adhere to surfaces) <img src="img/noslip.png" width="400" /> ] --- class: top # Two More Important Conditions . . . Laws Really .pull-left[ - Conservation of mass, a.k.a., the Principle of Continuity - Conservation of energy, a.k.a., Bernoulli's Principle ] .pull-right[ <img src="img/Bernoullipic.jpeg" width="400" /> ] --- class: top # Conservation of Mass: Principle of Continuity .pull-left[ Rate of mass flow in must equal rate of mass flow out (water is incompressible) `\(\frac{v_{out}}{t}=\frac{v_{in}}{t}\)` `\(u_{out}A_{out}=u_{in}A_{in}\)` <img src="img/facet.jpeg" width="300" /> ] .pull-right[ <img src="img/FluidContinuity.png" width="500" /> ] --- class: top # Conservation of Mass: Principle of Continuity .pull-left[ Rate of mass flow in must equal rate of mass flow out (water is incompressible) `\(\frac{v_{out}}{t}=\frac{v_{in}}{t}\)` `\(u_{out}A_{out}=u_{in}A_{in}\)` `\(\Sigma u_{in}A_{in}=\Sigma u_{out}A_{out}\)` ] .center[ <img src="img/continflow.png" width="650" /> ] --- class: top # Conservation of Mass: Principle of Continuity .pull-left[ Rate of mass flow in must equal rate of mass flow out (water is incompressible) `\(\frac{v_{out}}{t}=\frac{v_{in}}{t}\)` `\(u_{out}A_{out}=u_{in}A_{in}\)` `\(\Sigma u_{in}A_{in}=\Sigma u_{out}A_{out}\)` ] .pull-right[ <img src="img/bloodvel.jpg" width="450" /> ] --- class: top # Conservation of Energy: Bernoulli .pull-left[ What is the relationship between fluid motion and pressure? - Flow is steady - Fluid is incompressible - Fluid is inviscid . . . `\(\mu=0\)` ] .center[ <img src="img/contenergy.png" width="650" /> ] --- class: top # Conservation of Energy: Bernoulli .pull-left[ What is the relationship between fluid motion and pressure? - Potential energy (PE=mgh) - Kinetic energy .... `\(KE =mu^2/2\)` - Mechanical work (W=Fd=PAd) - Along a streamline PE + KE + W = constant ] .pull-right[ `\((P_2-P_1)/\rho+(u_2^2-u_1^2)/2=0\)` <img src="img/contenergy.png" width="650" /> ] --- class: top # Conservation of Energy: Bernoulli .pull-left[ What is the relationship between fluid motion and pressure? - Potential energy (PE=mgh) - Kinetic energy .... `\(KE =mu^2/2\)` - Mechanical work (W=Fd=PAd) - Along a streamline PE + KE + W = constant ] .pull-right[ Written another way <img src="http://hyperphysics.phy-astr.gsu.edu/hbase/imgmec/bernoul.gif" width="650" /> ] --- class: top # Conservation of Mass and Energy Putting it all together Predict the pressure of blood along the length of the vascular system --- class: top # Conservation of Mass and Energy Putting it all together .center[ <img src="https://www.neuromon.eu/images/vasculardiameter.jpg" width="500" /> ] --- class: center, middle # Thanks! Slides created via the R package [**xaringan**](https://github.com/yihui/xaringan).