Question
The laminar boundary layer for a fluid is assumed to be cubic, such that $u / U=C_{1}+C_{2}(y / \delta)+C_{3}(y / \delta)^{3} .$ If the free-stream velocity $U$ starts at $y=\delta,$ determine the constants $C_{1}, C_{2},$ and $C_{3}$.
Step 1
First, let's consider the boundary conditions for the laminar boundary layer. There are three conditions we need to satisfy: a) At the surface (y = 0), the fluid velocity u must be zero due to the no-slip condition. This means: $u(0) = 0$ b) At the edge of the Show more…
Show all steps
Your feedback will help us improve your experience
James Kiss and 76 other Physics 101 Mechanics educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
The laminar boundary layer for a fluid is assumed to be parabolic, such that $u / U=C_{1}+C_{2}(y / \delta)+C_{3}(y / \delta)^{2}$. If the free-stream velocity $U$ starts at $y=\delta,$ determine the constants $C_{1}, C_{2},$ and $C_{3}$.
A boundary layer for laminar flow of a fluid over the plate is to be approximated by the equation $u / U=C_{1}(y / \delta)+C_{2}(y / \delta)^{2}+C_{3}(y / \delta)^{3} . \quad$ Determine the constants $C_{1}, C_{2},$ and $C_{3}$ using the boundary conditions when $y=\delta, u=U ;$ when $y=\delta, d u / d y=0 ;$ and when $y=0$ $d^{2} u / d y^{2}=0 .$ Find the thickness of the boundary layer as a function of $x$ and $\operatorname{Re}_{x}$ using the momentum integral equation.
The viscous boundary layer velocity profile shown in Fig. 2.15 can be approximated by a cubic equation, \[u(y)=a+b\left(\frac{y}{\delta}\right)+c\left(\frac{y}{\delta}\right)^{3}\] The boundary condition is $u=U$ (the free stream velocity) at the boundary edge $\delta$ (where the viscous friction becomes zero). Find the values of $a, b,$ and $c .$
Transcript
Watch the video solution with this free unlock.
EMAIL
PASSWORD