**Problem 1.** The thickness \( \delta \) of the boundary layer for a fluid passing over a flat plate depends upon the distance \( x \) from the plate’s leading edge, the free-stream velocity \( U \) of the flow, and the density \( \rho \) and viscosity \( \mu \) of the fluid. Determine the relation between \( \delta \) and these parameters. Hint: take \( q = \delta \) and \( \mu \).**Diagram Explanation:**The diagram illustrates a flat plate over which fluid is flowing. A boundary layer forms along the surface of the plate, represented by a curved line. The diagram includes:- The horizontal axis labeled as \( x \) indicating the direction along the plate.- The vertical axis labeled as \( y \) indicating the distance perpendicular to the plate. - The free-stream velocity \( U \) is shown above the boundary layer as parallel arrows indicating the direction and speed of the fluid outside the boundary layer.- Inside the boundary layer, the velocity is denoted by \( u \), with smaller arrows indicating slower velocities near the plate.- The thickness of the boundary layer \( \delta \) is marked as the distance from the plate's surface to where the fluid velocity nearly equals the free-stream velocity \( U \).The task is to find the relationship between \( \delta \), \( x \), \( U \), \( \rho \), and \( \mu \).

There are 5 variables present δ, x,v,ρ,μ and the unknown function is fδ, x,v,ρ,μ=0. Using the M-L-T system, δ=Lx=LU=LT-1ρ=ML-3μ=ML-1T-1 Here wee can see that all the dimensions are used so the value of m is 3 therefore the number of π terms used will be n-m = 5-3=2. Dimensional Analysis: Here the separating variables are x,v,ρ, thus the q variables are δ for π1 and μ for π2. Now calculating the values of π1, π1=xavbρcδ=LaLaT-bMcL-3cπ1=McLa+b-3cT-b on comparing M:0=c L: a+b-3c+1T:0=-b on solving we get a=-1, b=0 and c=0 thus π1=x-1v0ρ0δπ1=δx now solving the value of π2 π2=xdveρfμ=LdLeT-eMfL-3fML-1T-1π1=Mf+1Ld+e-3f-1T-e-1 on comparing M:0=f+1L:0= d+e-3f-1T:0=-e-1 on solving we get d=-1,e=-1 and f=-1 thus π2=x-1v-1ρ-1μπ2=μxvρ or xvρμ=Re therefore the function can be written as fδx, Re=0 solving for the value of δ δx=fReδ=xfRe

Engineering

Mechanical Engineering

Problem 1. The thickness 8 of the boundary layer for a fluid passing over a flat plate depends upon the distance x from the plate's leading edge, the free-stream velocity U of the flow, and the density p and viscosity u of the fluid. Determine the relation between 8 and these parameters. Hint: take q = S and µ. y U u X

Problem 1. The thickness 8 of the boundary layer for a fluid passing over a flat plate depends upon the distance x from the plate's leading edge, the free-stream velocity U of the flow, and the density p and viscosity u of the fluid. Determine the relation between 8 and these parameters. Hint: take q = S and µ. y U u X

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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Concept explainers

Properties of Fluids

The term fluid represents a type of substance that can easily change its shape under any external pressure/force and can flow. Fluids can acquire any shape in which it is stored and can resist normal stresses but cannot resist shear stress in rest condition. Whenever external shear stress is applied to fluids, it starts to flow. The fluids may be liquid or gas. Example- water, air, etc.

Question

**Problem 1.** The thickness \( \delta \) of the boundary layer for a fluid passing over a flat plate depends upon the distance \( x \) from the plate’s leading edge, the free-stream velocity \( U \) of the flow, and the density \( \rho \) and viscosity \( \mu \) of the fluid. Determine the relation between \( \delta \) and these parameters. Hint: take \( q = \delta \) and \( \mu \).

**Diagram Explanation:**

The diagram illustrates a flat plate over which fluid is flowing. A boundary layer forms along the surface of the plate, represented by a curved line. The diagram includes:

- The horizontal axis labeled as \( x \) indicating the direction along the plate.
- The vertical axis labeled as \( y \) indicating the distance perpendicular to the plate.
- The free-stream velocity \( U \) is shown above the boundary layer as parallel arrows indicating the direction and speed of the fluid outside the boundary layer.
- Inside the boundary layer, the velocity is denoted by \( u \), with smaller arrows indicating slower velocities near the plate.
- The thickness of the boundary layer \( \delta \) is marked as the distance from the plate's surface to where the fluid velocity nearly equals the free-stream velocity \( U \).

The task is to find the relationship between \( \delta \), \( x \), \( U \), \( \rho \), and \( \mu \).

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