**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

See similar textbooks

Similar questions

Consider the steady, laminar flow of two liquids, A and B, with viscosities MA = μ and MB 2μ, respectively, between infinite parallel plates located at z = 1 and z = −ɛ, as shown in the diagram below. The plates are stationary and the liquids do not mix. There is an applied pressure gradient acting on liquid A, given by Vp = -Gi (where G> 0 is constant), and the effects of gravitation can be assumed to be negligible. Z 0 MA = μ U₁(z) = μB = 2µ UB(Z) = = (a) Write down the mathematical consequences of the following assumptions: (i) The flow is two-dimensional. (ii) There is no variation in the direction into the page. (iii) The flow is steady. (iv) There is no variation of velocity parallel to the plates. Hence write down the continuity equation for either of the two liquids and show that the fluid velocities in liquid A and in liquid B are given by uд = µ₁ (²) i and uß = uß(z) i, respectively. (b) State the boundary conditions at the upper and lower plates, and terface z = at (c) Solve…
Q6: Using Raleigh’s method, Analyze an equation to determine the resistance, P of a sphere of diameter, D moving with a velocity, v in an incompressible and unlimited medium of specific mass, ρ and coefficient of viscosity μ
I need right solution with clear calculations.
The fluid particles do not rotate about their own axis in an irrotational fluid flow. Select one: True False
A flow is non-uniform when the magnitude of the parameters varies from point to point along the flow path. TRUE FALSE
An incompressible viscous flow is contained between two parallel plates separated from each other by distance b. as shown in Figure 1. The flow is caused by the movement of the upper plate which has a velocity U, while the bottom plate is fixed. If U =7 m/s and b= 1 cm, and there is no pressure gradient in the flow direction. A.) Start with Navier-Stokes equations and determine the velocity at the point x = 3 cm and y= 0.41 cm. The value of the velocity is.B.) Calculate the magnitude of the vorticity at the same point. The magnitude value of vorticity. C.) Calculate the rate of angular deformation at the same point. The angular deformation value
The block of weight W = 50 N is pulled by a weight W0 = 60 N along the surface of a table by means of an inextensible cable and a pulley. Between the block and the table there is a uniform layer of oil of viscosity m = 0.1 Pa-s and clearance h = 2 mm. The contact area is 0.01 m2. Find the terminal velocity U if the table is very long.
Oil Coating: A long, continuous belt is pulled upwards through a chemical oil bath at velocity V0. The belt has rectangular cross-section and has length (L), width into the paper (W). The belt picks up a film of oil of thickness h, density ρ, and dynamic viscosity μ. Gravity g tends to make the oil drain down, but the movement of the belt keeps the fluid from running off completely. Assume fully developed, steady, laminar, incompressible and two-dimensional flow of oil to answer the following questions. Assume that no pressure gradient is needed in the vertical direction to drive the film flow. Also assume that the shear stress at the air-oil interface is zero (free shear condition). Assume no-slip condition for the fluid in contact with the moving belt. Justify any other assumptions you may make. Show all steps. (a) Derive an expression for the two-dimensional velocity field inside the oil film in terms of the known parameters. Clearly indicate your co-ordinates and origin. You must…
between two infinite surfaces with lateral cross-section It is coated with glycerin with an absolute viscosity of μ = 1.025 Ns / m2. This a cube parallel to two surfaces and with sides of c x c x c cm slides downwards. If the density of the cube is ρ = 254 kg / m3 then U Find the velocity (m / s). (g = 9.81 m / s2)
Problem 1 – A laminar flow fluid of known density (ρ) and viscosity (μ) flows between twoparallel plates with different velocities in the same direction. The top plate has a velocity Utop inthe positive x direction. The bottom plate has a velocity Ubot in the positive x direction. The twoplates are a distance of “a” apart. There is a pressure gradient in the x direction ( ). Derivean expression of the velocity and shear stress profiles between the two plates
fluid flows with velocity profile of u = 5 y + 1, calculate the shear force due to viscosity if the viscosity of the fluid is 0.2 Pa.s