Fluid Mechanics: Fundamentals and Applications
Fluid Mechanics: Fundamentals and Applications
4th Edition
ISBN: 9781259696534
Author: Yunus A. Cengel Dr., John M. Cimbala
Publisher: McGraw-Hill Education
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Chapter 10, Problem 110P

Calculate the nine components of the viscous stress tensor in cylindrical coordinates (see Chap. 9) for the velocity field of Prob. 10-109. Discuss.

Expert Solution & Answer
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To determine

The nine components of viscous stress tensor in cylindrical coordinate system.

Answer to Problem 110P

The nine components of viscous stress tensor in cylindrical coordinate system are 0 for τrr, μΓ2πr2 for τrθ, 0 for τrz, μΓ2πr2 for τθr, 0 for τθθ, 0 for τθz, 0 for τzr, 0 for τzr, 0 for τzθ.

Explanation of Solution

Given information:

Velocity component for radial direction.

  ur=0

Velocity component for angular direction.

  uθ=Γ2πr

Here, radius for angular velocity is r, constant is Γ.

Velocity component in vertical direction.

  uz=0

Write the expression for nine component viscous stress tensor in cylindrical co-ordinates.

  τ=[ τ rr τ rθ τ rz τ θr τ θθ τ θz τ zr τ zθ τ zz]  ...... (I)

Here, viscous stress tensor in purely radial direction is τrr, viscous stress tensor in angular direction is τrθ, viscous stress tensor in vertical direction is τrz, viscous stress tensor in angular direction is τθr, viscous stress tensor in purely angular direction is τθθ,

viscous stress tensor in radial direction is τθz, viscous stress tensor in radial direction τzr, viscous stress tensor in vertical direction is τzθ, viscous stress tensor in purely vertical direction

Write the expression for viscous stress tensor in radial direction.

  τrr=2μ(rurr)  ...... (II)

Here, viscosity is μ.

Write the expression for viscous stress tensor in angular direction.

  τrθ=μ(rr( u θ r)+1rurθ)  ...... (III)

Write the expression for viscous stress tensor in vertical direction.

  τrz=μ(urz+uzr)  ...... (IV)

Write the expression for viscous stress tensor in angular direction.

  τrθ=μ(1ruθθ+urr)  ...... (V)

Write the expression for viscous stress tensor in vertical direction.

  τθz=μ(uθz+1ruzθ)  ...... (VI)

Write the expression for viscous stress tensor in purely vertical direction.

  τzz=2μ(ruzz)  ...... (VII)

Calculation:

Substitute 0 for ur in Equation (II).

  τrr=2μ(r ( 0 ) r)=2μ(0)=0

Substitute 0 for ur, Γ2πr for uθ in Equation (III).

  τrθ=μ(r r( Γ 2πr r )+1r ( 0 ) θ)=μ( rΓ 2π r( r 2 )+0)=μΓ2πr2

Substitute 0 for ur, 0 for uz.in Equation (IV).

  τrz=μ( ( 0 ) z+ ( 0 ) r)=0

Substitute 0 for ur, Γ2πr for uθ in Equation (V).

  τrθ=μ(1r ( Γ 2πr ) θ+0r)=μ(1rΓ 2πr ( 1 ) θ+0)=0

Substitute Γ2πr for uθ, 0 for uz in Equation (VI).

  τθz=μ( ( Γ 2πr ) z+1r ( 0 ) θ)=μ(Γ 2πr ( 1 ) z+0)=0

Substitute 0 for uz in Equation (VII).

  τzz=2μ(r ( 0 ) z)=2μ(0)=0

Substitute 0 for τrr, μΓ2πr2 for τrθ, 0 for τrz, μΓ2πr2 for τθr, 0 for τθθ, 0 for τθz, 0 for τzr, 0 for τzθ, 0 for τzz in Equation (I) to find nine components of the viscous stress tensor in cylindrical co-ordinates.

  τ=[0 μΓ 2π r 2 0 μΓ 2π r 2 00000]

Thus, the nine components of viscous stress tensor in cylindrical coordinate system are 0 for τrr, μΓ2πr2 for τrθ, 0 for τrz, μΓ2πr2 for τθr, 0 for τθθ, 0 for τθz, 0 for τzr, 0 for τzr, 0 for τzθ.

Conclusion:

The nine components of viscous stress tensor in cylindrical coordinate system are 0 for τrr, μΓ2πr2 for τrθ, 0 for τrz, μΓ2πr2 for τθr, 0 for τθθ, 0 for τθz, 0 for τzr, 0 for τzr, 0 for τzθ.

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