Laminar boundary layers [Fluid Mechanics #13]

Summary

This video delves into the concept of laminar boundary layers in fluid mechanics, focusing on external flows where fluid interacts with a solid surface. It explains how boundary layers form due to viscosity and the no-slip condition at the wall, leading to a velocity gradient near the surface. The video outlines the theoretical approach to analyzing these layers, including assumptions and boundary conditions. It also discusses the significance of boundary layer thickness and how it influences forces like drag and lift. Prandtl and Blasius' solutions for the velocity profile within these layers are highlighted, along with the calculation of wall shear stress and drag force, emphasizing the importance of understanding boundary layers in various applications such as aerodynamics and climate modeling.

Takeaways

• 🌊 The video discusses external flows, which are open on one or more sides and can extend infinitely in at least one direction, unlike enclosed flows.
• 🔍 The focus is on boundary layers, which are regions of velocity gradient near a surface due to viscosity and the no-slip condition at the wall.
• 🎯 The video aims to solve for the velocity and pressure fields in laminar boundary layers, with turbulent boundary layers to be covered in a subsequent video.
• 📚 External flows include boundary layers, jets, wakes, and shear layers, all of which require specific analysis in fluid mechanics.
• 🛠 Assumptions made for analysis include incompressible, steady, two-dimensional, and two-component flow, with no body forces.
• 📉 Observational assumptions are used to simplify the conservation equations, focusing on the gradual growth region of the boundary layer.
• 🧮 The conservation of mass and momentum equations are simplified using these assumptions, leading to a manageable set of equations for boundary layer analysis.
• 📉 Prandtl and Blasius solved the simplified equations, revealing the self-similar shape of the laminar boundary layer's velocity profile.
• 📏 Three methods to measure boundary layer thickness are discussed: disturbance thickness, displacement thickness, and momentum thickness.
• 💨 The video concludes with the calculation of wall shear stress and drag force due to the fluid on the surface, emphasizing the importance of understanding boundary layers in various applications.

Q & A

What is the primary focus of the video?

The primary focus of the video is on laminar boundary layers, which are regions of lower velocity fluid that grow on an external surface in a flow field due to viscosity.

What are the key differences between enclosed flows and external flows?

Enclosed flows have boundaries on all sides, while external flows can be open and extend effectively into infinity in at least one direction. External flows often exhibit growth or development in the streamwise direction, leading to more complicated flows.

What is a boundary layer in the context of fluid mechanics?

A boundary layer is a thin layer of fluid in the immediate vicinity of a bounding surface where the effects of viscosity are significant, resulting in a velocity gradient from the surface to the free-stream velocity.

Why is the no-slip condition important in boundary layer analysis?

The no-slip condition is important because it states that the fluid velocity at the wall is equal to the wall's velocity, which is crucial for understanding the velocity profile near the surface and the associated shear stress.

What assumptions are typically made when analyzing laminar boundary layers?

When analyzing laminar boundary layers, it is assumed that the fluid is incompressible, steady, two-dimensional, and two-component. Additionally, it is assumed that there are no body forces, and the flow develops in the x-direction.

How do Prandtl and Blasius contribute to the understanding of boundary layers?

Prandtl and Blasius are credited with pioneering boundary layer research. They derived and solved the conservation equations for a laminar boundary layer, leading to the understanding of the self-similar shape of the velocity profile.

What are the three main strategies to measure the boundary layer height?

The three main strategies to measure the boundary layer height are disturbance thickness (δ), displacement thickness (δ*), and momentum thickness (θ), each providing a different perspective on the boundary layer's impact on the flow.

How is the Reynolds number used in boundary layer analysis?

The Reynolds number is used to characterize the flow and determine when a flow transitions to turbulence. For a smooth flat plate, the transition is estimated to occur around a Reynolds number of five hundred thousand.

What is the significance of the skin friction coefficient in boundary layer analysis?

The skin friction coefficient (c_f) is a non-dimensional parameter that represents the wall shear stress relative to the flow inertia. It is important for understanding the drag force on a surface due to the fluid flow.

How does the drag force on a surface due to a boundary layer get calculated?

The drag force on a surface due to a boundary layer is calculated by integrating the shear stress over the surface area. For laminar boundary layers, this is a function of the velocity, plate length, and the Reynolds number based on the plate length.

Outlines

00:00

🌊 Introduction to External Flows and Boundary Layers

This paragraph introduces the concept of external flows, contrasting them with enclosed flows by highlighting their open nature and potential to extend infinitely in at least one direction. The focus is on boundary layers, which are regions of velocity gradient near a surface due to viscosity and the no-slip condition at the wall. The aim is to understand laminar boundary layers, which are critical for analyzing fluid forces on external vehicles, aerodynamics, and atmospheric flows. The paragraph sets up the theoretical framework for studying boundary layers, including assumptions such as incompressibility, steady flow, two-dimensionality, and the absence of body forces. It also discusses the importance of the boundary layer's growth, velocity profile, and its impact on force exerted on a surface.

05:02

🔍 Assumptions and Conservation Equations in Boundary Layer Analysis

The second paragraph delves into the assumptions and boundary conditions necessary for solving the equations governing boundary layer flow. It acknowledges the need to move beyond the steady and 2D assumptions due to the development of the boundary layer in the streamwise direction. Key boundary conditions include the no-slip and no-penetration conditions at the wall and the behavior of the flow as it moves away from the wall towards infinity. The paragraph outlines the conservation of mass and momentum equations, simplifying them using the assumptions made. It introduces observational assumptions to further simplify the equations, focusing on the dominance of certain terms based on the growth characteristics of the boundary layer. The result is a set of equations that, while complex, can be solved numerically to understand the behavior of boundary layers.

10:04

📏 Quantifying Boundary Layer Thickness and Growth

This paragraph discusses the methods for quantifying the thickness and growth of boundary layers. It introduces three main strategies for measuring boundary layer height: disturbance thickness, displacement thickness, and momentum thickness. Each method is defined and explained in the context of its application and significance. The paragraph also touches on the historical context of boundary layer research, crediting Prandtl and Blasius for their pioneering work in solving the boundary layer equations and defining the self-similar shape of the laminar boundary layer. The importance of the Reynolds number in boundary layer analysis is highlighted, particularly its role in determining when a flow transitions to turbulence.

15:07

💨 Shear Stress, Drag Force, and Practical Implications of Boundary Layers

The fourth paragraph focuses on the practical implications of boundary layer analysis, particularly in terms of shear stress and drag force. It explains how the velocity field within the boundary layer contributes to wall shear and the resulting drag force. The concept of the skin friction coefficient is introduced as a non-dimensional measure of wall shear stress relative to flow inertia. The paragraph also discusses the general drag coefficient and how it relates to the force of drag, the surface area, and the flow inertia. The importance of understanding boundary layers in aerodynamics, climate modeling, and other industries is emphasized, highlighting the relevance of the equations and concepts discussed in the video.

20:08

🏁 Conclusion and Summary of Laminar Boundary Layers

In the final paragraph, the video concludes with a summary of the key points covered regarding laminar boundary layers. It reiterates the importance of understanding these layers in fluid mechanics and their impact on various fields such as aerodynamics and climate modeling. The paragraph also emphasizes the significance of the theoretical approach and the assumptions made in the analysis. It acknowledges the complexity of boundary layers and their role in phenomena like lift and drag on aircraft wings, as well as the onset of separation. The video ends with an encouragement for viewers to appreciate the intricacies of boundary layer analysis and its broader applications.

Keywords

💡Enclosed Flows

Enclosed flows refer to fluid movements that are confined by boundaries on all sides, such as channels or pipes. These flows are analyzed by solving for the velocity fields (u, v, w) and pressure fields in both laminar and turbulent states. In the video, enclosed flows are contrasted with external flows, which are open in at least one direction.

💡External Flows

External flows are fluid flows that are open and extend into infinity in at least one direction. Examples include boundary layers, jets, wakes, and shear layers. In the video, external flows are described as being more complex than enclosed flows, often exhibiting development in the streamwise direction.

💡Boundary Layer

A boundary layer is the region near a solid surface where the fluid velocity changes from zero (due to the no-slip condition) to the free-stream velocity. This layer is important in understanding how fluid forces act on surfaces. The video focuses on the growth of laminar boundary layers and how they influence drag and surface forces.

💡No-Slip Condition

The no-slip condition refers to the physical phenomenon where the fluid velocity at the wall of a solid surface is equal to the velocity of the wall itself, usually zero. This is due to friction between the fluid and the wall. In the video, the no-slip condition is essential for defining the behavior of the boundary layer near the surface.

💡Free-Stream Velocity

Free-stream velocity, denoted as u-infinity, is the velocity of the fluid far from the influence of any solid surface, where the flow is unaffected by boundary layers. In the video, free-stream velocity is used to explain the velocity profile in external flows and serves as a reference point for measuring the boundary layer's velocity changes.

💡Reynolds Number

The Reynolds number is a dimensionless quantity used to predict flow patterns in different fluid flow situations. It is calculated as the product of density, velocity, and length scale, divided by viscosity. In the video, the Reynolds number is used to distinguish between laminar and turbulent boundary layers and to estimate the transition to turbulence.

💡Laminar Flow

Laminar flow is a smooth, orderly fluid motion characterized by parallel layers of fluid. In the video, laminar flow is studied in boundary layers, with a focus on the velocity profile and shear stress in this region. The video contrasts laminar flow with turbulent flow, which is more chaotic.

💡Momentum Thickness

Momentum thickness, denoted as θ, is a measure of the momentum deficit in a boundary layer. It is the height at which the flow would have to extend to conserve the momentum lost due to the boundary layer. The video explains how this concept is used to quantify drag forces on surfaces.

💡Skin Friction Coefficient

The skin friction coefficient (Cf) is a dimensionless number representing the ratio of wall shear stress to the fluid's inertia. It helps quantify the effect of shear stress on a surface due to viscous forces. In the video, this coefficient is used to describe the forces acting on surfaces within a laminar boundary layer.

💡Prandtl and Blasius

Ludwig Prandtl and his student Blasius were pioneering researchers in the field of boundary layer theory. In 1908, they solved the laminar boundary layer equations, providing a self-similar velocity profile for laminar flows. Their work, mentioned in the video, forms the foundation for much of modern boundary layer analysis.

Highlights

Introduction to external flows, which are open on one or more sides and can extend to infinity.

Focus on boundary layers, the layer of velocity gradient near a surface due to viscosity and the no slip condition at the wall.

Exploration of the importance of boundary layers in understanding fluid forces on external vehicles, aerodynamics, and atmospheric flows.

Assumptions made for analysis, including incompressible, steady, two-dimensional flow without body forces.

Explanation of the no slip boundary condition at the wall, where the fluid velocity must match the wall's velocity.

Discussion on the gradual growth of boundary layers and the importance of viscosity in these layers.

Introduction of the conservation of mass and momentum equations for boundary layer analysis.

Simplification of the momentum equations using observational assumptions about the growth characteristics of boundary layers.

Derivation of the final form of the conservation equations for a laminar boundary layer by Prandtl and Blasius.

Presentation of the self-similar shape of the laminar boundary layer and its velocity profile.

Quantification of boundary layer height using disturbance thickness, displacement thickness, and momentum thickness.

Calculation of the wall shear stress resulting from the boundary layer and its impact on drag force.

Introduction of the skin friction coefficient and its role in non-dimensionalizing the wall shear stress.

Explanation of how to calculate the total drag force on a surface due to the fluid's boundary layer.

Discussion on the significance of laminar boundary layers in aerodynamics, including their role in lift and drag on aircraft wings.

Highlight of the role of boundary layer analysis in climate modeling and predicting global flow patterns.

Conclusion summarizing the importance of understanding laminar boundary layers in fluid mechanics.

Transcripts

hi

and welcome to another video in fluid

mechanics

up until this point we have dealt

primarily with enclosed flows

these are flows that have boundaries on

all sides and include both

channels and pipes in analysis of these

flows we have come

to solve the velocity field u v and w

and also learned a bit about the

pressure field as well

we studied these flows in both laminar

and turbulent states

today we move on to the world of

external flows

unlike enclosed flows these flows can be

open and extend effectively into

infinity in at least one direction

specifically we'll be interested in

boundary layers the layer velocity

gradient near the surface due to

viscosity acting on the fluid

and the no slip condition at the wall

like with enclosed flows we will try and

solve for the velocity field and the

pressure field

today our focus will be on laminar

boundary layers and we'll save turbulent

boundary layers for the next video

let's jump right in external flows are

our next focus

these flows are open on one or more

sides and often exhibit growth or

development in the streamwise direction

leading to more complicated flows than

we saw with enclosed flows

these types of flows include boundary

layers a velocity gradient near a solid

surface

also jets a concentration of moving air

wakes which is a velocity deficit region

that's often

unsteady and downstream of a body moving

in a fluid

and shear layers where two flows with

different velocity come together

all these types of flows exist in the

world of fluid mechanics and warrant

specific analysis

today our aim is to better understand

the laminar boundary layer

consider a plate moving in a fluid or a

fluid passing by a solid plate

there is some free stream velocity u

infinity

near the wall we have what's called the

no slip boundary condition

physically this means velocity super

close to the wall

has to have the velocity of the wall

itself due to friction

so u at y equals zero is zero

now we know far from the wall the

velocity field does not feel the

pressure of the wall so the velocity

field remains u infinity which is the

free stream velocity

in our fluid there must be a continuous

region where the fluid changes from the

zero velocity

at the wall to the free stream velocity

far from the wall

this region is called the boundary layer

and it grows as flow

moves downstream boundary layers are

essentially molecular diffusion which is

viscosity

creeping out slowly further and further

from the wall

in these layers viscosity is deemed to

be important

due to the meaningful vertical velocity

gradients

these layers are generally quite small

relative to the surface moving through

the fluid

but despite their small size they are

absolutely critical in understanding how

fluid forcing works

on external vehicles and aerodynamics

heat transfer

and even atmospheric flows

our goal is to be able to describe how a

boundary layer grows downstream

what the thickness is what the velocity

profile looks like

and how that imparts a force on a

surface

as we did with all of our flows let's

see how far we can get with theory using

just the conservation equations

let's set up our flow with a schematic

of a boundary layer

we have a free stream velocity coming in

u infinity

the dashed white line is the line that

represents the upper bound of the

boundary layer

we can mark our u equals zero wall

condition

and that's it notice that here the flow

goes past the surface

remember this still applies to moving

vehicles we have just attached our

coordinate

system to the vehicle itself

to start analysis we're going to need

some assumptions

here we will use the some familiar

assumptions and some newer ones midway

through our analysis

first our fluid is incompressible so

density is a constant

second the fluid is steady which means

nothing changes in time in our flow

we also assume it's two dimensional and

two-component

so nothing substantial happens in the

out-of-plane direction

remember the steady and the 2d2c

assumptions are very common for laminar

flows

at this point we almost instinctively

write fully developed

but for boundary layers and most

external flows this is not the case

these flows develop in the x direction

and we can no longer

ignore x gradients in the velocity field

so regrettably we have to leave this

assumption behind because we can't use

it

last let's assume there are no body

forces for now

after assumptions which help us simplify

our equations

we will need boundary conditions which

we need to solve the equations

at the wall we have the no slip and no

penetration conditions

meaning the streamwise and vertical

velocities are zero

where they meet the surface these are

specifically wall conditions

however now that our fluid is unbound in

the positive y direction

we have some new conditions we can

utilize as it goes off into infinity

we take the limits on the flow behavior

as y approaches infinity or gets so far

from the wall

it no longer feels the wall as y goes to

infinity the streamline's velocity

becomes the free stream velocity u

infinity

as a result the free stream is an

idealized location where there is no

vertical velocity

so we set v to be zero

lastly we also just say there is no

pressure gradient in the free stream

direction as

y goes to infinity as the free stream is

just a constant

x velocity this condition on the

pressure gradient will be critical in

being able to deal with the pressure

terms later on

as always we make our start with the

conservation of mass for an

incompressible fluid

we can only remove the term dwdz due to

the assumption that the flow is two

dimensional

and that's as far as we can get we can't

say anything about the v

velocity as we would for the enclosed

flows because we have two gradients here

to keep track of

so we move right on to our conservation

of momentum in the x direction to see

what that tells us

let's write them out in their entirety

the material derivative representing

flow acceleration is on the left

then that is balanced by the forces of

pressure viscosity

and a body force on the right to

simplify

let's apply our assumptions we're going

to be able to remove four terms here the

du dt term because the flow is steady

both the derivatives in the z direction

because the flow is two dimensional

and the body force term and what we're

left with is a rather intimidating

balance of momentum terms

pressure and viscosity terms let's call

this equation one

if we did the same process on the

conservation momentum in the y direction

and applied our assumptions we would

find a similar result here

these two equations are really hard to

work with in the current state

in order to move forward we need to be

able to do something else to make them

more simple

so we're going to make some

observational assumptions that are based

on

observations we've made for this type of

flow in the past

let's consider the growth

characteristics of the boundary layer

and maybe we can say something about the

sizes of some terms compared to others

in the beginning of the boundary layer

development growth is rapid

however not long into development the

growth gets dramatically more gradual

the bulk of the flow is gradual growth

so let's claim that

in our analysis we're considering only

this gradual growth region

this leads us to some more possible

assumptions

because growth is slow in the x

direction we claim that the velocity

field

changes in the wall normal direction y

are much larger than changes in the

streamwise

direction x remember vertically the

boundary layer is small

so things have to change rapidly to go

from zero velocity to the free stream

velocity in the y direction

this means that if we ever see these two

terms together in an equation

for example d u d y versus d u d x

we assume one dominates the other the d

u d y

term and we remove the smallest term d u

d

x keep in mind here you can only compare

like terms meaning only u terms compared

to u terms

and v terms compare to v terms for

example

we can't compare d u d x and d v d

y next we're going to assume that the

streamlines velocity

is much larger than other velocity

components which is typically a fine

claim for these flows since the majority

of the flow is in the streamwise

direction

soon we will be comparing terms and

deciding which to get rid of based on

size

so let's sort the terms into whether

they are considered to be big

small or tiny u and y derivatives of the

u velocity are the biggest

next up we have v and y derivatives on v

as well as the streamwise derivatives on

u these are small

but not too small last we have the tiny

terms

streamwise gradients of v

we can organize these like this because

of generally how we've observed boundary

layers to behave

now we can write both conservation of

momentum equations we have

equations 1 and 2 and label each term

with their size

in equation one we have some terms that

are big times as small

and some that are just small and some

that are just

big let's keep the big and the big time

small

terms we remove only the smallest term

in the equation

in equation two we have some tiny terms

now

we keep everything that is small or a

big

times a tiny term

we can remove the smallest term which is

just tiny

notice this is the same term we removed

above

now if we compare equation 1 to equation

2 term by term

we notice that all terms in equation one

are much bigger than their partner term

in equation two

and we don't know anything about the

pressure terms

however if we know all the terms in

equation one

are much greater than equation two we

also know that the pressure gradient in

1 is bigger than the pressure gradient

in 2.

this is important if we say dp dx

is much much bigger than dpdy then we

know that p

is not a function of y and that it is

approximately only a possible function

of x

so p is a constant in y

and if we pair this with our boundary

condition on the pressure gradient which

says as

y goes to infinity the pressure gradient

goes to zero

this means the pressure gradient is zero

everywhere because it's a constant in

y essentially

through this deduction we can ignore the

streamwise pressure gradient

this brings us to our final form of the

conservation equations for a laminar

boundary layer

they're still rather complicated but

they're totally solvable numerically

back in 1908 these were originally

derived and solved by prandtl and his

student blasius

a famous pair of researchers who

pioneered boundary layer research

their solutions led to the following

discretized form of the boundary layer

shape

here's a table showing their results the

left column is effectively the

y-coordinate

the profile is self-similar in x meaning

it doesn't change

shape in x it just grows this is why we

see the x variable come into the y

column

because the x variable is acting to

stretch the vertical coordinate

next to this is the velocity normalized

by the free stream

so it goes to one far from the wall

if you plotted this data it would

represent the shape of a laminar

boundary layer

[Music]

these equations and a ton of analysis

and boundary layers

require the reynolds number for boundary

layers the reynolds number is the

product of density

the free stream velocity which acts as

our velocity scale

and the x location of interest which is

sometimes our length scale

all divided by the viscosity

the reynolds number tell us when flows

transition to turbulence

for a smooth flat plate flows this is

estimated to be around five hundred

thousand

now that we know the boundary layer

shape let's determine how we can discuss

its size we measure the boundary layer

height with three main strategies

depending on our needs

first the most simple strategy is to

define a physical distance above the

surface and mark its height

this is called the disturbance thickness

delta and is the y

location where the velocity in the

profile is equal to 99

of the free stream velocity sometimes

people use 95 percent instead of 99 but

regardless it's the same procedure

people use this because it's simple and

easy to find and measure

the second strategy is called the

displacement thickness

consider the boundary layer the velocity

deficit is the region in purple

where the flow velocity is less than the

free stream velocity

this space has an area which we're going

to call area a

we can equate this area a to a rectangle

that has

one length as the free stream velocity

which is known and

some height which is the displacement

thickness which we're trying to find

physically interpret this as the height

from the ground

that the surface would have to extend in

order to push the fluid back into being

uniform

it's a bit harder to calculate and

requires the integral of the velocity

profile

people use this in aircraft design where

they consider this delta star as part of

the surface and ignore viscosity above

this distance

lastly we have the most complex way to

define the height the momentum thickness

similar to the displacement thickness we

consider the area beneath

velocity profile but instead we're

considering the momentum deficit so u

squared

and not the velocity deficit

this means if we found a rectangle with

an area a

equal to the momentum deficit the length

of the rectangle is u infinity squared

which is known

and the height of the rectangle is our

theta the momentum thickness

the calculation is similar and also

needs an integral

this is generally used when considering

the surface drag

because the momentum deficit is more

heavily connected to drag force than

just

velocity

for laminar flows we already have the

solution numerically and we can exactly

define the behavior of these three

thicknesses as they vary in the

streamwise direction

and they're all functions of the local

reynolds number

as you can see the boundary layer grows

downstream non-linearly

because x appears in the square root of

the reynolds number

so now we have a good handle on the

boundary layer size shape

and growth in aerodynamics and in

industry

we're generally interested in the force

the fluid applies on the moving surface

boundary layers have a wall shear and

the wall shear results in a shear stress

which makes a drag force

recall newton's law of viscosity if we

calculate it

at the wall we have the shear stress at

the wall

numerically we can do this derivative to

our flow field and come up with another

estimated expression for laminar

boundary layers

interestingly you might notice that the

further downstream you go

the less the fluid pulls on the surface

this is because as the boundary layer

grows the velocity gradient near the

wall

gets smaller

this is popularly represented in

non-dimensional form

where a form of the flow inertia one

half rho u squared

is used to normalize this stress this is

the skin friction coefficient

c sub f and it defines the behavior for

the wall skin frictions

relative to the flow inertia much like

reynolds number this is similar to

viscosity versus inertia

it's important to keep in mind the skin

friction coefficient is different from a

drag coefficient the skin friction

coefficient is just used as to

non-dimensionalize

the wall shear you cannot get drag from

it directly

however we can turn this shear stress

due to friction

into a drag force

it's important to point out that because

our shear stress is now a function of x

we can no longer use the shortcut that

total wall forcing is just the shear

stress times the wall area

we have to move to the general

definition where the force is the

integral of tau

over an area let's assume flow is

uniform in the z

direction and give the plate a width w

by assuming it's uniform w can come out

of the integral

we integrate from 0 to l which is the

plate length

over dx the final expression tells us

the wall force is a function of the

velocity

plate length and the reynolds number

based on the length

it's important to note that for boundary

layers when there is a subscript on the

reynolds number it's typically referring

to the length scale you use in the

reynolds number calculation

sometimes you'll see re sub l or re sub

x

or even r e sub theta which is a

reynolds number based on the momentum

thickness

this drag force can also be

non-dimensionalized using the flow

inertia

the general drag coefficient is the

force of drag divided by the area of the

surface times one half density times

velocity squared specifically for

laminar boundary layers this transforms

into the following function of reynolds

number based on plate length

and that's it for the equations these

equations are valuable to keep in your

tool belt when

analyzing boundary layers because even

when flow is turbulent we need to use

these to calculate the laminar portion

of the plate as we'll see in the next

video

in practice you will can certainly come

across boundary layers in some way shape

or form

in aerodynamics they are the direct

cause of lift and drag on aircraft wings

the growth of the boundary layer leads

to a phenomena called separation

which can dramatically change the forces

a body feels due to a fluid

last boundary layer analysis is the root

of climate modeling

trying to predict flow patterns on the

planet

so understanding what a boundary layer

is and how to work with it

is critical and that's it

let's review today we introduce laminar

boundary layers

a specific external flow that is a

region of lower velocity fluid that

grows on an

external surface in a flow field due to

viscosity

we used our typical theoretical approach

to understanding a flow with some unique

assumptions and boundary

conditions in the end of our analysis

using some creative assumptions and

comparing the size of different terms in

our equations

we were able to arrive at a solvable set

of equations that define the

velocity prandtl and blasius solve these

equations to get the self-similar shape

of the laminar boundary layer known as

the velocity profile

the boundary layer height was quantified

using three separate methods the

disturbance thickness the displacement

thickness

and the momentum thickness we then use

the velocity field to calculate the

stress that acts on the surface

and the resulting drag force on the

plate due to the fluid

while laminar boundary layers are

typically overshadowed by turbulent

boundary layers

they are still a critical aspect of

fluid mechanics and analysis

i hope you enjoyed the video and thanks

for watching

you