Biofluid Mechanics Final Review

Drag Forces

*Parallel with fluid flow (against the flow) This force develops to resist the motion of the object. 𝕯= ∫dFx= ∫pcosθ dA + ∫ 𝜏 sinθ dA

Lift forces

*Perpendicular with fluid flow. A lift force is generated when a surface experience a difference in pressure between its two surfaces, where the region of higher pressure pushes harder than the region of lower pressure generating a lift! ℒ= ∫dFy= -∫p*sinθ dA + ∫ 𝜏*cosθ dA

What assumptions can we make for flow in pipes to simplify analysis?

- Steady flow -Incompressible flow -Inviscid flow

Orifice meter

- also known as pipe orifice and orifice plate - Because the pressure loss is non negligible, we must account for it in our solution and use the modified version of the Bernoulli equation. -The coefficient of Discharge (Cd) is the ratio between the Therefore flowrate is : Q=Cd A0√(2(𝓅1-𝓅2)/ρ(1-β⁴) ) where β=d2/d1

Methods for measuring flowrate

-Orifice meter -Nozzle meter -Venturi meter All of these work on the same principle: By taking something that we can easily measure like pressure we can find more difficult quantities like average velocity, volumetric flow rate, and mass flow rate. By creating a pressure drop we can mathematically relate these quantities using the continuity equation and the Bernoulli equations.

Flow visualization techniques

1. Optical method - Optical property of the flow media: Refractive Index (RI) 2. Energy Addition Method -The spark lines are generated by applying a high voltage with a high frequency between two electrodes perpendicular to gas flow regions of interest and producing the air ionization via the electrical discharge. 3. Flow Seeding Method -smoke -dye -particles (hydrogen bubbles, PIV, PTV, etc.)

venturi meter

A flow measuring device placed in a pipe. The device consists of a tube whose diameter gradually decreases to a throat and then gradually expands to the diameter of the pipe. The flow is determined on the basis of the difference in pressure (caused by different velocity heads) between the entrance and throat of the Venturi meter. Use Original Bernoulli equation and the continuity equation to find flow rate or just find the velocity of the fluid. Q=A2√(2(𝓅1-𝓅2)/ρ(1-(A2/A1)²) )

Doppler ultrasonic meter

A flowmeter that transmits an ultrasonic pulse diagonally across a flow stream, which reflects off turbulence, bubbles, or suspended particles and is detected by a receiving crystal.

Darcy-Weisbach equation

An equation used in fluid mechanics to find the pressure change cause by friction within a pipe or conduit

Bernoulli Equation (simply)

Assumptions -Steady State System -Incompressible Fluid -No frictional loss -Velocities are constants -No change in internal energy The equation states that the sum of the kinetic energy, potential energy, and pressure energy of a fluid particle is constant along a streamline. This means that if the velocity of the fluid increases at one point, the pressure must decrease at that point to maintain the constant total energy. The Bernoulli equation is widely used in fluid mechanics to analyze the behavior of fluids flowing through pipes, channels, and other systems. *Due to assumptions above all equations equal a total head H=constant Three forms: Pressure definition: H=p+ 1/2ρu²+γz Distance definition: H=p/γ +u²/2g+z Energy Density definition: H=p/ρ+1/2u²+gz

How pitot tubes work

At its most basic a pitot tube is measuring two different pressures: the static pressure and the stagnation pressure (or total pressure). The stagnation pressure is a combination of the static pressure and the dynamic pressure. Therefore by having one channel that collects static pressure by having a channel perpendicular to the flow and another channel that collects the dynamic and static pressure by combining a channel parallel and perpendicular with the flow, we can find the velocity: Bernoulli tells us: static pressure + dynamic pressure= total pressure or 𝓅ₛ+ 1/2 ρ v² =𝓅ₜ ∴ v=√(2(𝓅ₜ-𝓅ₛ)/ρ)

Prandtl Boundary Layer Theory

Boundary layer theory is a mathematical theory that is used to predict the behavior of fluid flows near solid surfaces. It is based on the assumption that the fluid flow near a solid boundary is affected by the presence of the boundary, and that the effects of the boundary on the flow can be represented by a thin layer of fluid called the boundary layer. This layer is characterized by a gradient in the velocity of the fluid, with the velocity being zero at the solid boundary and increasing to the free-stream velocity of the flow far from the boundary. Boundary layer theory is important in fluid mechanics because it allows engineers to accurately predict the behavior of fluid flows in situations where the presence of solid boundaries is important. For example, it can be used to predict the drag on an object moving through a fluid, the heat transfer between a solid surface and a fluid, and the behavior of fluid flows in complex geometries such as the flow of air over an airplane wing.

Hydrogen Bubble method

By adding an electrically charged cathode wire to our flow and insulating different regions of the wire, we can generate a column of hydrogen bubbles that can be emitted for a certain "pulse duration" and different numbers of pulses Since the duration of each pulse will be the same, we can analyze the flow by looking at the length of the hydrogen bubbles. The longer the length the faster that region of the flow. We know this to be true from a simple duration of velocity as v=length/time.

Euler equations for sinus of Valsalva

By simplifying the Euler equations to one dimension we can get ρ(∂u/∂t+u*∂u/∂x)=-∂p/∂x in the x-direction. Then using the continuity equation, we can further simplify 0=∂u/∂x+∂v/∂y+∂w/∂z→ ∂u/∂t=-1/ρ*∂p/∂x As we can clearly see this changing velocity function is dependent upon a pressure gradient. Originally when flow is fast and passing through the aorta we flow from high to low pressure, but if, as we did in the analog diagram, pinch the flow, the sinus will generate an inverted region of high and low pressure preventing the valve from opening and stopping the possibility of back flow. In simple terms because the velocity is able to decelerate the pressure gradient was able to inverse helping the valve stay closed. This is unique to the aorta because of the shape of the valve. The mitral valve for examples just uses flaps and opens to a much larger chamber.

Flow development around a sphere

Characteristics with increasing flow: (flow increases as you move down) - No separation -Steady separation bubble -Oscillating Karman vortex -Laminar boundary layer, wide turbulent wake -Turbulent boundary layer, narrow turbulent wake

Major head loss (Moderate values of Re)

Colebrook equation: (implicit solution) 1/√f=-2.0log((ε∕D)/3.7+2.51/(Re√f)) *Requires use of a moody chart to solve. It would take you too long to try to figure out what combination works. Haaland equation: (explicit solution) 1/√f=-1.8log[((ε∕D)/3.7)^1.11+6.9/Re]

What even are streamlines, streaklines, and pathlines?

Definitions: One simple way to describe the difference between these three types of lines is to think of them as representing different moments in time. Pathlines show the entire history of a fluid particle, from its starting point to its current location. Streaklines show the current position of a fluid particle at a specific point in time. And streamlines show the future path of a fluid particle if it were to continue moving along with the flow. In experiment: In the lab you will have a flow channel with a particle seeder that will eject particles into the flowing fluid. Along with one or two cameras for taking different exposures of the material. Pathlines: For path lines we seed a SINGLE particle and take MUTIPLE exposures. This is because we want to observe the path of an individual particle. The line is simple a trace of the given particle at given time intervals. Streaklines: We seed MULTIPLE particles at designated time periods with a SINGLE exposure. This is because we are interested in the path of a particle at any particular point in time. The line is created by passing through particles that have passed through the common point. Streamlines: We seed MULTIPLE particles and take a DOUBLE exposure image. This means by overlaying the images when can draw a tangent line between two particles pairs and determine the velocity of the particles. By definition a streamline is a curve that is tangent to the velocity field. IN STEADY FLOWS THE pathline, streakline, and streamline will all follow the same path!!!!

How entrance flow conditions affect the loss coefficient

Depending on the shape of the entrance we can generate different losses. The entrance to a pipe or duct can have a significant effect on the loss coefficient in fluid flows. This is because the flow at the entrance is typically not fully developed, and may be affected by the shape of the entrance and the boundary conditions at the entrance. For example, if the entrance to a pipe is rounded or flared, it can cause the flow to separate from the walls of the pipe, resulting in an increase in the loss coefficient. On the other hand, if the entrance is sharp or abrupt, it can cause the flow to become turbulent, which can also increase the loss coefficient. In general, the loss coefficient at the entrance to a pipe or duct is typically larger than the loss coefficient in the fully developed region of the flow.

Laminar flow

Flows in parallel lines in a smooth progression.

Major head loss (Laminar)

For laminar flow: f=64/Re

Minor Losses

Head loss associated with bends, fittings, valves, etc.

Tubular Pinch Effect

In 1961 experiments by Segre and Silberberg proved that spherical particles in pipe flow experienced inertial migration or the tubular pinch effect. The best explanation we have for this effect is that two opposing dominating lateral lift forces are developing: (technically there are others) -Wall lift force Fw -Shear gradient lift force Fₛ The adjacent wall repels the particles away from the wall and the curvature of the velocity profile's interaction with the particle direct the particle away from the channel Here is my best explanation for why the particles are arranging in an annulus centered position: In a Poiseuille flow the parabolic nature generated by the lag of the particles means that the fluid is flowing much faster on side of the particle and much slower on the other. This generates a region of low pressure on the wall side of the ball pushing it towards the wall until the sphere begins to experience the changing flow field in the presence of the wall by decelerating the particle and driving it away from the wall

Why does a fully developed laminar flow does not experience acceleration?

In a fully developed laminar flow, the fluid flows in a series of parallel layers, or laminae, with no mixing between the layers. The velocity of each layer is constant, and there is no acceleration of the fluid particles within a given layer. This is because the viscous forces within the fluid are sufficient to counteract any acceleration that might otherwise occur due to the applied pressure gradient. In other words, the viscous forces within the fluid act to dampen any acceleration, resulting in a steady and uniform flow.

Gorlin equations

In an attempt to characterize valve stenosis, Richard Gorlin used his knowledge of fluid mechanics to figure out the effective orifice area of the valve. EOA=A_0=Q_m/C_d √(ρ/2∆p)

Special case: flow past a sphere

In class the main application of BLT (boundary layer theory) was to a sphere. Because of the symmetrical nature of a sphere, we only need to observe a cross section of the sphere. When a fluid moves past a sphere the pressure is the highest at the front then it move around the surface where it speeds up. As we move across the surface there is a point of separation where the flow begins to separate and slow down. Known as an adverse pressure gradient. This generates a reverse velocity and recirculation flow. To extend this area of attached boundary layer we can create a turbulent boundary layer that carry much more momentum and can stay attached to the surface much longer. This is why a golf ball can fly so much further because we extend the boundary layer that is moving across the surface before we have separation (I.E. a smaller wake area).

Turbulent flow

Irregular flow with random variations in pressure.

Bernoulli equation for viscous fluid

One way to account for viscous effects is to use the concept of major and minor head loss. Major head loss refers to the energy loss due to viscous effects that occurs in a fluid as it flows through a pipe or other duct. This loss can be significant in some cases and must be taken into account in order to accurately predict the behavior of the fluid flow. Minor head loss, on the other hand, refers to the smaller energy losses that occur due to changes in the velocity or direction of the fluid flow. Therefore we write: p₁/γ +u₁²/2g+z₁=p₂/γ +u₂²/2g+z₂+hL *where hL=hL major + hL minor We will analyze the given situation to determine if hL minor or major dominants. We will not typically observe both at the same time.

How PIV works

PIV takes a statistical Eulerian-type approach, and PTV takes a more intensive Lagrangian approach. Let us look into how each is conducted. In PIV, as I stated, we are not necessarily interested in each individual particle; instead, we use a camera to snap a photo of a specific area with two light pulses. Then using the statistical principle of correlation, each pixel of the image is interrogated until there is a correlation between the two pixels and thus gives us the change in time, which will provide us with the local velocity.

PIV vs PTV

PIV: Characteristics- for small particles, with a high seeding density due to use of mathematical principle of cross-correlation and divides image into small regions. This results in a velocity vector map that is evenly distributed PTV: Characteristics: for relatively larger particles (like RBCs) at a low seeding density to accurately track each individual particle. Velocity vectors are distributed unevenly due to being a physical representation of the random distribution of particles.

How PTV works

PTV, as stated, takes a more laborious approach in using a lagrangian-type method; it physically tracks the motion of individual particles in the fluid. First, we locate all the particles in a specific reference frame; then, in the next frame of our video, the particles are matched by looking at size, shape, placement, and many other parameters. From this, we can calculate the local velocities of each particle and then generate the total velocity field using spline integration.

Analysis of typical shear flow devices

Parallel-plate flow chamber: homogeneity of the force stimulus, simplicity of the equipment, ease of medium sampling/exchange, and ease of access to the culture (both physically and for microscope visualization). However, the cells experience an uneven shear that starting from a predetermined maximum value at the entrance and falls to zero at the exit Cone-and-plate viscometer: Since both the local relative velocity and the separation between the cone and plate surfaces vary linearly with the radial position, this configuration achieves spatially homogeneous shear stress on both surfaces, as well as the fluid in the gap. Depending on the cone taper and the imposed angular velocity, a wide range of shear stresses can be achieved, extending even into the turbulent flow regime Parallel disk viscometer: do not result in a uniform shear stress across the entire monolayer Orbital shaker: do not result in a uniform shear stress across the entire monolayer Tubular capillary tube: capillary tubes do not yield sufficient amounts of cells for some bioassay analysis Cylindrical Couette Shear: Can only be used for non-adherent cells like RBCs.

What is an inviscid flow?

Remembering back to the Navier-Stokes equation: ρDV/Dt=-∇p+ρg+μ∇²V *For inviscid flows the μ∇²V terms are negligible. This means there is no resistance to shearing forces, and it can flow without any friction or drag. Inviscid flow is a theoretical concept that is used to simplify fluid

What does incompressible flow even mean?

Remembering back to the Navier-Stokes equation: ρDV/Dt=-∇p+ρg+μ∇²V *In incompressible flows the fluid density remains constant the Navier-Stokes equation simplifies to a form that does not include the fluid density as a variable. This is because, in an incompressible flow, the fluid density remains constant and therefore does not change over time. As a result, the equation is easier to solve in this case.

What does steady flow even mean?

Remembering back to the Navier-Stokes equation: ρDV/Dt=-∇p+ρg+μ∇²V *The material derivative or substantial derivative, D/Dt, is used to describe the physical quantity of momentum of a material element with respect to not just time, but also space. This generates a space-time dependent macroscopic velocity field and serves a link between the Eulerian and Lagrangian descriptions. We can define(≡) the material derivative as: Dφ/Dt≡∂φ/∂t+ ẋ•∇φ where ẋ≡dx/dt, i.e a function of position φ≡ a function both position and time *When we have a steady flow the ∂φ/∂t term go to zero because the particles in space are not changing their velocity with time only position! In simple terms if we look at a point in space it will always be the same velocity, the velocity only changes if we move position or reference frame.

Major vs Minor loss

The Bernoulli equation that we originally discussed is an oversimplified version of fluid systems. We must account for some sort of loss that is helping to drive the fluid flow (i.e. there must be a drop in pressure that is causing flow to push forwards) At its most simple: Major loss: Considerable loss due to frictional losses in the pipe with time. Minor loss: Losses due to local valves, bends, tees, filters, changes in cross sectional area, etc.

How to read a Moody Chart

The Moody chart is a graphical tool used to determine the friction factor for a fluid flow in a pipe or other duct. The friction factor is a dimensionless quantity that is used in the Darcy-Weisbach equation, which is used to calculate the major head loss that occurs due to viscous effects in a fluid flow. To read a Moody chart, you will need to know the Reynolds number and the relative roughness of the pipe or duct. The Reynolds number is a dimensionless quantity that is used to predict the type of fluid flow that will occur, while the relative roughness is a measure of the roughness of the pipe or duct wall. Once you have these values, you can use the Moody chart to determine the friction factor. To do this, first locate the value of the Reynolds number on the horizontal axis of the chart. Next, locate the value of the relative roughness on the vertical axis. The point where these values intersect will give you the value of the friction factor. This value can then be used in the Darcy-Weisbach equation to calculate the major head loss for the given fluid flow.

Reynolds number

The Reynolds number is defined as the ratio of the inertial forces to the viscous forces in a fluid. Inertial forces refer to the tendency of a body to resist changes in its state of motion, while viscous forces are the frictional forces that act on a moving body when it is in a fluid. The relative importance of these forces can be quantified using the Reynolds number. For example, if the Reynolds number is low, the viscous forces will dominate, and the flow will be laminar. On the other hand, if the Reynolds number is high, the inertial forces will dominate, and the flow will be turbulent. By calculating the Reynolds number, engineers can predict the type of flow that will occur in a given situation, which is useful for designing systems that involve fluid flow.

Average velocity

The average velocity of a fluid is often used to calculate the flow rate, which is the volume of fluid that flows through a given area per unit time. The flow rate, Q, is equal to the average velocity of the fluid multiplied by the cross-sectional area of the flow. For example, if a fluid is flowing through a pipe with a cross-sectional area of 1 square meter and an average velocity of 2 meters per second, the flow rate would be 2 cubic meters per second.

Continuity equation

The equation states that the mass flow rate of a fluid, or the amount of fluid passing through a given area per unit time, must remain constant at all points in the flow. This means that if the fluid is flowing into a certain region, it must also be flowing out of that region at the same rate. The continuity equation is often used in conjunction with the Navier-Stokes equation to analyze the motion of a fluid. It helps to ensure that the mass of the fluid is conserved as it flows through a system. At its most fundamental: ΣQin=ΣQout However, when the velocity is not normal to the differential area surface, we can use a differential analysis: dQ=V⋅dA integral definition: Q=∫V⋅dA

What is the purpose of the sinus of Valsalva?

The sinus of Valsalva is at the root of the aorta and together makes up three sinuses.1The sinus of Valsalva regulates the dynamic part of the aorta helping to regulate pressure and flow. Specifically, it helps to prevent back flow.

Turbulent to laminar boundary layer transition around a sphere

The transition from a turbulent to a laminar boundary layer can have a significant impact on the size of the wake region and the drag force on a sphere moving through a fluid. Turbulent boundary layers have more momentum due to the mixing that occurs, which causes them to stick to the surface for longer. In contrast, laminar boundary layers tend to separate from the surface sooner. When the flow separates from the surface of the sphere, a recirculation flow known as the wake region is generated behind the sphere. The wake region is a region of low pressure, and the smaller this region the better the reduction in drag. This is beneficial because if there is a region of high pressure in front of the sphere, a small wake region will help to reduce the drag force on the sphere. This is because pressure is fundamentally a force over an area, so reducing the size of the wake region will reduce the region of low pressure (weaker force) and overall reduce drag force on the sphere.

Karman Vortex

The vortex forms when the fluid flow separates from the surface of the obstacle, creating a low-pressure region behind the obstacle. This low-pressure region causes the fluid to spiral around the obstacle, forming the characteristic vortex pattern.

Ultrasonic Flow meter

Two types you have to know: 1. Transit time 2. Doppler meters

How exit flow conditions affect the loss coefficient

When we move from a pipe to a sufficiently large tank, we assume the velocity at the exit is 0 thus all of the kinetic energy is dissipated through the viscous effects as the stream of the fluid mixes making K=1

transit time ultrasonic meter

a flowmeter consisting of two sets of transmitting and receiving crystals, one set aimed diagonally upstream and the other aimed diagonally downstream

Loss coefficient, Kₗ

h_(L minor)/((u^2∕2g) ) a measure of the resistance to flow in a pipe or duct. It is defined as the ratio of the pressure drop across a section of the pipe or duct to the square of the flow rate through that section. The loss coefficient is used to predict the pressure drop that will occur in a pipe or duct when a fluid is flowing through it, and is an important parameter in the design of fluid flow systems. It is typically represented by the symbol "K," and can vary depending on the type of fluid being flowed, the geometry of the pipe or duct, and the flow rate.

Force balance between pressure and viscous forces (fully developed laminar flow)

p_1∙πr^2-(p_1-∆p)∙πr^2-τ∙2πrl=0