Summary
Laminar combined convection of non-Newtonian fluids in vertical eccentric annuli, in which the inner and outer walls are held at different constant temperatures is considered and a new economical method of solution for the three-dimensional flow in the annulus is developed. Assuming that the ratio of the radial to the vertical scale, ε, is small, as occurs frequently in many industrial applications, then the governing equations can be simplified by expanding all the variables in terms of ε. This simplification gives rise to the presence of a dominant cross-stream plane in which all the physical quantities change more rapidly than in the vertical direction. The solution trechnique consists of marching in the vertical streamwise direction using a finite-difference scheme and solving the resulting equations at each streamwise step by a novel technique incorporating the Finite Element Method. The process is continued until the velocity, pressure and temperature fields are fully developed, and results are presented for a range of the governing non-dimensional parameters, namely the Grashof, Prandtl, Reynolds and Bingham numbers.
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Abbreviations
- Bn :
-
Bingham number,\(\tau _y *\left( {\frac{{d*}}{{w_m *}}} \right)^n {\rm K}*^{ - 1} \)
- d * :
-
difference between the radii of the outer and inner cylinders,r o *−ri *
- e * :
-
distance between the axes of the inner and outer cylinders
- e :
-
eccentricity,e */d*
- F * :
-
external force acting on the fluid
- g * :
-
acceleration due to gravity
- g * :
-
gravitational vector, (0,0,g *)
- Gr :
-
Grashof number, ɛϱ m *2 g*β*(T 0*−T e*)d*3/η m *2
- K * :
-
consistency of the fluid
- L * :
-
height of the cylinders of the annulus
- n :
-
flow behaviour index
- p * :
-
dimensional pressure
- P :
-
dimensionless pressure gradient
- Pr :
-
Prandtl number, η m */ϱ m *α*
- r i * :
-
radius of the inner cylinder of the annulus
- r o * :
-
radius of the outer cylinder of the annulus
- r T :
-
wall temperature difference ratio,(T i *−Te *)/(To *−Te *)
- Re :
-
Reynolds number, ɛϱ m d*w m */η m *
- T :
-
dimensionless temperature of the fluid,(T *−Te *)/(To *−Te *)
- T *dif :
-
temperature difference between the walls of the annulus
- T e * :
-
temperature at the fluid at the entrance of the annulus
- T i * :
-
temperature at the inner cylinder of the annulus
- T o * :
-
temperature at the outer cylinder of the annulus
- u :
-
dimensionless transverse velocity in thex direction,u */(εwm *)
- U :
-
dimensionless transverse velocity in the annulus,Reu
- u * :
-
fluid velocity vector, (u *, v*, w*)
- v :
-
dimensionless transverse velocity in they direction,v */(εwm *)
- V :
-
dimensionless transverse velocity in the annulus,Rev
- w :
-
dimensionless vertical velocity,w */wm *
- w m :
-
scaling used to non-dimensionalise the vertical velocity
- x :
-
dimensionless transverse coordinate,x */d*
- y :
-
dimensionless transverse coordinate,y */d*
- z :
-
dimensionless vertical coordinate,z */L*
- Z :
-
dimensionless vertical coordinate,z/Re
- Z r :
-
dimensionless distance in the vertical direction where the final wall temperatures are attained,Z r */L*
- α*:
-
dimensional molecular thermal diffusivity
- β*:
-
coefficient of thermal expansion,\( - \frac{1}{{\varrho *}}\frac{{\partial \varrho *}}{{\partial T*}}\)
- \(\dot \gamma *\) :
-
dimensional rate of strain tensor
- \(\overline{\overline \varepsilon } \) :
-
dimensionless ratio of the length scales in the annulus,d */L*
- η*:
-
dimensional apparent non-Newtonian viscosity
- η m *:
-
mean viscosity,\({\rm K}*\left( {\frac{{w_m *}}{{d*}}} \right)^{n - 1} \)
- ϱ*:
-
dimensional fluid density
- ϱ m *:
-
dimensional reference fluid density
- τ*:
-
dimensional stress tensor
- \(\overline{\overline \tau } _y *\) :
-
yield stress
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Ingham, D.B., Patel, N. Developing combined convection of non-Newtonian fluids in an eccentric annulus. Acta Mechanica 121, 35–49 (1997). https://doi.org/10.1007/BF01262522
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DOI: https://doi.org/10.1007/BF01262522