### Educational Content on Dimensionless Boundary Layer EquationsBelow are the steps to deriving the dimensionless forms of the Boundary Layer (B.L.) equations by substituting dimensionless variables/parameters. The primary focus is on performing algebraic manipulations and considering the conservation of the mass equation.#### Equations:1. **Continuity Equation:** \[ \frac{\partial u^*}{\partial x^*} + \frac{\partial v^*}{\partial y^*} = 0 \]2. **Momentum Equation:** \[ u^* \frac{\partial u^*}{\partial x^*} + v^* \frac{\partial u^*}{\partial y^*} = -\frac{\partial p^*}{\partial x^*} + \frac{1}{Re} \frac{\partial^2 u^*}{\partial y^{*2}} \]3. **Energy Equation:** \[ u^* \frac{\partial T^*}{\partial x^*} + v^* \frac{\partial T^*}{\partial y^*} = \frac{1}{Re \, Pr} \frac{\partial^2 T^*}{\partial y^{*2}} \]#### Dimensionless Variables:- \( p^* = \frac{p}{\rho_0 u_0^2} \)- \( Re = \frac{u_0 L}{\nu} \)- \( Pr = \frac{\nu}{\alpha} \)#### Procedure:1. **Substitution:** - Introduce dimensionless variables/parameters into the dimensional B.L. equations.2. **Algebraic Manipulation:** - Carry out requisite algebraic transformations to simplify the equations.3. **Conservation of Mass:** - The conservation of mass equation is given by: \[ \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0 \]This framework aids in the understanding of fluid dynamics and heat transfer in boundary layers, emphasizing the transformation from dimensional to dimensionless forms for analytical convenience.

Consider all dimensionless parameters.Consider expression for incompressible, 2D continuity…

du* əv* + ах ду du' du u". ·+v². əx* dy* ·+vº ƏT* əx* where p* = P, Re₁ = ², and Pr = Derive the dimensionless forms of the B.L. equations by substituting the various dimensionless variables/parameters into the dimensional B.L. equations and performing the requisite algebraic manipulations consider the conservation of mass equation: ƏT* 1 8²T* dy Re, Pray² dp' 1 0²u* dx* Re, Əy*² du əv əx əy = 0

du* əv* + ах ду du' du u". ·+v². əx* dy* ·+vº ƏT* əx* where p* = P, Re₁ = ², and Pr = Derive the dimensionless forms of the B.L. equations by substituting the various dimensionless variables/parameters into the dimensional B.L. equations and performing the requisite algebraic manipulations consider the conservation of mass equation: ƏT* 1 8²T* dy Re, Pray² dp' 1 0²u* dx* Re, Əy*² du əv əx əy = 0

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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Question

### Educational Content on Dimensionless Boundary Layer Equations

Below are the steps to deriving the dimensionless forms of the Boundary Layer (B.L.) equations by substituting dimensionless variables/parameters. The primary focus is on performing algebraic manipulations and considering the conservation of the mass equation.

#### Equations:

1. **Continuity Equation:**
\[
\frac{\partial u^*}{\partial x^*} + \frac{\partial v^*}{\partial y^*} = 0
\]

2. **Momentum Equation:**
\[
u^* \frac{\partial u^*}{\partial x^*} + v^* \frac{\partial u^*}{\partial y^*} = -\frac{\partial p^*}{\partial x^*} + \frac{1}{Re} \frac{\partial^2 u^*}{\partial y^{*2}}
\]

3. **Energy Equation:**
\[
u^* \frac{\partial T^*}{\partial x^*} + v^* \frac{\partial T^*}{\partial y^*} = \frac{1}{Re \, Pr} \frac{\partial^2 T^*}{\partial y^{*2}}
\]

#### Dimensionless Variables:
- \( p^* = \frac{p}{\rho_0 u_0^2} \)
- \( Re = \frac{u_0 L}{\nu} \)
- \( Pr = \frac{\nu}{\alpha} \)

#### Procedure:

1. **Substitution:**
- Introduce dimensionless variables/parameters into the dimensional B.L. equations.

2. **Algebraic Manipulation:**
- Carry out requisite algebraic transformations to simplify the equations.

3. **Conservation of Mass:**
- The conservation of mass equation is given by:
\[
\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0
\]

This framework aids in the understanding of fluid dynamics and heat transfer in boundary layers, emphasizing the transformation from dimensional to dimensionless forms for analytical convenience.

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