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Combined convection flow of a power-law fluid in a vertical duct with linearly varying wall temperatures

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Summary

This paper considers steady laminar combined convection flows of power-law fluids between vertical parallel-plates with a uniform temperature gradient applied to the walls. The parabolic equations are written in an implicit finite-difference form and are solved using a marching technique. The governing parameters are Gr, Pr, andn. Various values of Gr andn are considered, including the forced convection solution, Gr=0, and the Newtonian case,n=1. Pr is set at a value of unity in order to present the numerical method and to compare with the existing results for Newtonian fluids. Under certain circumstances reverse flow regions appear, either at the centre or adjacent to the walls of the duct and these are present in the fully-developed flow. These reverse flow problems are dealt with using an iterative technique. In order to assess the effects of recirculation and pseudoplasticity on the flow and heat transfer characteristics, flow profiles, Nusselt numbers and friction factors are presented for various values of the governing parameters.

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Abbreviations

d :

half width of the duct

f :

friction factor

g :

gravitational acceleration

Gr:

Grashof number,gβλd 2n+1|v m|1−2n v m/v m/v 0 2

η:

local heat transfer coefficient of the fluid

H :

finite-difference step size across the duct

k :

thermal conductivity of the fluid

K :

finite-difference step size along the duct

n :

power-law index

N:

number of finite-difference steps across the duct

NN:

number of finite-difference steps along the duct

Nu:

Nusselt number, hd/k

Pr:

Prandtl number,v 0/α|v m/d|n−2 n m/d

Re:

Reynolds number,d n|v m|1−n v m/v 0

T :

temperature

u :

transverse velocity component

U :

dimensionless transverse velocity component,u/v m

v :

streamwise velocity component

V :

dimensionless streamwise velocity component,v/v m

x :

transverse coordinate

X :

dimensionless transverse coordinate,x/d

y :

streamwise coordinate

Y :

dimensionless streamwise coordinate,y/(d Re)

α:

molecular thermal diffusivity of the fluid

β:

coefficient of thermal expansivity of the fluid, (−1/ϱ0) (∂ϱ/∂T)

λ:

increase in temperature along the duct walls over the initial lengthd of the duct

v :

characteristic kinematic viscosity of the fluid

μ:

consistency index of the fluid

ϱ:

density of the fluid

ϑ:

dimensionless temperature, (T−T 0)/(λ Re)+Y

ψ:

dimensionless streamfunction

Ω:

dimensionless vorticity

d :

value at the beginning of the reverse flow region

f :

value at the end of the reverse flow region

i :

transverse finite-difference suffix

j :

streamwise finite-difference suffix

m :

flow average value

0:

reference value

∞:

value asy→∞

d :

value calculated during the downstream iteration sweep

s :

number of the iteration in the reverse flow region

u :

value calculated during the upstream iteration sweep

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Author notes

  1. D. B. Ingham

    Present address: Department of Applied Mathematical Studies, Leeds University, LSZ 9 JT, Leeds, UK

  2. A. T. Jones

    Present address: , RR/56 Well Simulation, Shell, Rijswirk, The Netherlands

Authors and Affiliations

  1. Department of Applied Mathematical Studies, Leeds University, LSZ 9 JT, Leeds, UK

    A. T. Jones

Authors
  1. D. B. Ingham
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Ingham, D.B., Jones, A.T. Combined convection flow of a power-law fluid in a vertical duct with linearly varying wall temperatures. Acta Mechanica 110, 19–32 (1995). https://doi.org/10.1007/BF01215412

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