the appropriate characteristic length and velocity scales for B.L. analysis are L and u., respectively. This gives X* = Xx y* = =0-> U∞o du* L Əx* u* = Rearranging and substituting into conservation of mass yields du Əv + ?х ду + น U oo (u*ux) d(v²u.co) Ə(x*L) Əy*L) U∞ dv* L Əy* du* dv* əx* əy* + v* = = 0 + = 0 = 0 V U oo Replicate this process for conservation of momentum and energy to derive the dimensionless B.L. equations.

Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
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Here is the transcription of the text, tailored for an educational website:

---

The appropriate characteristic length and velocity scales for boundary layer (B.L.) analysis are \(L\) and \(u_{\infty}\), respectively. This gives:

\[
x^* = \frac{x}{L}, \quad y^* = \frac{y}{L}, \quad u^* = \frac{u}{u_{\infty}}, \quad v^* = \frac{v}{u_{\infty}}
\]

Rearranging and substituting into conservation of mass yields:

\[
\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0 \rightarrow \frac{\partial (u^* u_{\infty})}{\partial (x^* L)} + \frac{\partial (v^* u_{\infty})}{\partial (y^* L)} = 0
\]

\[
\frac{u_{\infty}}{L} \frac{\partial u^*}{\partial x^*} + \frac{u_{\infty}}{L} \frac{\partial v^*}{\partial y^*} = 0
\]

\[
\frac{\partial u^*}{\partial x^*} + \frac{\partial v^*}{\partial y^*} = 0
\]

Replicate this process for conservation of momentum and energy to derive the dimensionless B.L. equations. 

--- 

(Note: Parts of the text were redacted with black lines and thus not included in the transcription.)
Transcribed Image Text:Here is the transcription of the text, tailored for an educational website: --- The appropriate characteristic length and velocity scales for boundary layer (B.L.) analysis are \(L\) and \(u_{\infty}\), respectively. This gives: \[ x^* = \frac{x}{L}, \quad y^* = \frac{y}{L}, \quad u^* = \frac{u}{u_{\infty}}, \quad v^* = \frac{v}{u_{\infty}} \] Rearranging and substituting into conservation of mass yields: \[ \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0 \rightarrow \frac{\partial (u^* u_{\infty})}{\partial (x^* L)} + \frac{\partial (v^* u_{\infty})}{\partial (y^* L)} = 0 \] \[ \frac{u_{\infty}}{L} \frac{\partial u^*}{\partial x^*} + \frac{u_{\infty}}{L} \frac{\partial v^*}{\partial y^*} = 0 \] \[ \frac{\partial u^*}{\partial x^*} + \frac{\partial v^*}{\partial y^*} = 0 \] Replicate this process for conservation of momentum and energy to derive the dimensionless B.L. equations. --- (Note: Parts of the text were redacted with black lines and thus not included in the transcription.)
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