An equation to describe the shear stress for the flow of the fluid through the given pipe is to be determined using Navier-Stokes equation. Concept Introduction: Navier-stokes describes the motion of a fluid having constant density and viscosity. In r direction: ρ ( ∂ v r ∂ t + v r ∂ v r ∂ r + v θ r ∂ v r ∂ θ − v θ 2 r + v z ∂ v r ∂ z ) = μ [ ∂ ∂ r ( 1 r ∂ ∂ r ( r v r ) ) + 1 r 2 ∂ 2 v r ∂ θ 2 − 2 r 2 ∂ v θ ∂ θ + ∂ 2 v r ∂ z 2 ] − ∂ p ∂ r + ρ g r ................ (1) In θ direction: ρ ( ∂ v θ ∂ t + v r ∂ v θ ∂ r + v θ r ∂ v θ ∂ θ + v r v θ r + v z ∂ v θ ∂ z ) = μ [ ∂ ∂ r ( 1 r ∂ ∂ r ( r v θ ) ) + 1 r 2 ∂ 2 v θ ∂ θ 2 + 2 r 2 ∂ v r ∂ θ + ∂ 2 v θ ∂ z 2 ] − 1 r ∂ p ∂ θ + ρ g θ ................ (2) In z direction: ρ ( ∂ v z ∂ t + v r ∂ v z ∂ r + v θ r ∂ v z ∂ θ + v z ∂ v z ∂ z ) = μ [ 1 r ∂ ∂ r ( r ∂ v z ∂ r ) + 1 r 2 ∂ 2 v z ∂ θ 2 + ∂ 2 v z ∂ z 2 ] − ∂ p ∂ z + ρ g z ................ (3) Here, ρ = density of the fluid μ = viscosity of the fluid p = pressure of the system v r = velocity of fluid in r direction v θ = velocity of fluid in θ direction v z = velocity of fluid in z direction t = time g r = acceleration due to gravity in r direction g θ = acceleration due to gravity in θ direction g z = acceleration due to gravity in z direction

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Chapter 9, Problem 9.24P

Interpretation Introduction

Interpretation:

An equation to describe the shear stress for the flow of the fluid through the given pipe is to be determined using Navier-Stokes equation.

Concept Introduction:

Navier-stokes describes the motion of a fluid having constant density and viscosity.

In r direction:

ρ(∂vr∂t+vr∂vr∂r+vθr∂vr∂θ−vθ2r+vz∂vr∂z)=μ[∂∂r(1r∂∂r(rvr))+1r2∂2vr∂θ2−2r2∂vθ∂θ+∂2vr∂z2]−∂p∂r+ρgr ................ (1)

In θ direction:

ρ(∂vθ∂t+vr∂vθ∂r+vθr∂vθ∂θ+vrvθr+vz∂vθ∂z)=μ[∂∂r(1r∂∂r(rvθ))+1r2∂2vθ∂θ2+2r2∂vr∂θ+∂2vθ∂z2]−1r∂p∂θ+ρgθ ................ (2)

In z direction:

ρ(∂vz∂t+vr∂vz∂r+vθr∂vz∂θ+vz∂vz∂z)=μ[1r∂∂r(r∂vz∂r)+1r2∂2vz∂θ2+∂2vz∂z2]−∂p∂z+ρgz ................ (3)

Here,

ρ= density of the fluidμ=viscosity of the fluidp=pressure of the systemvr=velocity of fluid in r directionvθ=velocity of fluid in θ directionvz=velocity of fluid in z directiont=timegr=acceleration due to gravity in r directiongθ=acceleration due to gravity in θ directiongz=acceleration due to gravity in z direction

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