Hydrodynamic and hydromagnetic stability

This paper experimentally investigates the heat transport and structure of convection in a high Prandtl number fluid layer whose viscosity varies by up to a factor of 300 between the boundary temperatures. An appropriate definition of the Rayleigh number R uses the viscosity at the average of the top and bottom boundary temperatures. With rigid boundaries and heating from below, the Nusselt number N normalized with the Nusselt number N0 of a constant-viscosity fluid decreases slightly as the viscosity ratio increases. The drop is 12% at a variation of 300. A slight dependence of N/N0 on R is consistent with a decrease in the exponent in the relation N ∝ Rβ from its constant-viscosity value of 0·281 to 0·25 for R [lsim ] 5 × 104. This may be correlated with a transition from three- to two-dimensional flow. At R ∼ 105 and viscosity variation of 150, the cell structure is still dominated by the horizontal wavelength of the marginally stable state. This is true with both free and rigid upper boundaries. The flow is strongly three-dimensional with a free upper boundary, while it is nearly two-dimensional with a rigid upper boundary.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112076000876

Thermal convection with strongly temperature-dependent viscosity

It is pointed out that the understanding of convection in large-Prandtl-number Boussinesq fluids with uniform properties and contained in simple geometries is virtually complete. Present efforts are typically directed towards relaxing some of the original assumptions by going to lower Prandtl number, more complicated geometries, variable material properties, or introducing new dynamical processes such as the Lorentz forces. A description is given of experiments which are concerned with the effect on convection of relaxing the assumption of a uniform viscosity. The reported experiments were designed to measure both the horizontally averaged temperature as a function of depth and the heat transfer of convection over a range of viscosity variations up to 100,000.

Heat transfer and horizontally averaged temperature of convection with large viscosity variations

Studies of non-Newtonian thermal convection have been made to determine the importance of non-Newtonian viscosity on flow in the earth's mantle. Finite difference solutions have been obtained with a viscosity law representing the sum of deformation rates due to diffusion and dislocation creep. Non-Newtonian flow structures differ only slightly from corresponding Newtonian flows. The principal effect of the non-Newtonian viscosity is to increase the effective Rayleigh number. An effective Rayleigh number based on a strain rate squared averaged viscosity provides a good correlation between Newtonian and non-Newtonian flows over a wide range of Rayleigh numbers.

Studies of finite amplitude non‐Newtonian thermal convection with application to convection in the Earth's mantle

Publisher Summary This chapter presents the field of thermal convection in non-Newtonian fluids. The thermal convection in external flows of inelastic fluids is relatively well understood in the chapter. Thermal convection in viscoelastic fluids has been studied only in a restricted way. An approximate correlating equation for mixed convection in external flows on non-Newtonian fluids is presented. The chapter explores the theoretical indications to show that there are likely to be tremendous advantages in adding drag-reducing polymers to impart viscoelastic properties to materials undergoing the process of thermal convection under turbulent conditions. The problem of buoyancy-induced secondary flow in heated horizontal tubes appears to be most challenging from a theoretical viewpoint. There are certain equations that correlate heat transfer data empirically, but the situation is not wholly satisfactory. The possibilities of oscillatory convection in viscoelastic fluids appear to exist under certain circumstances. The chapter concludes that thermal convection experiments are difficult to conduct, therefore there appears to be a massive generation of data correlating overall heat transfer coefficients with process variables but practically very little information on the velocity and temperature fields.

Thermal Convection in Non-Newtonian Fluids*

Buoyancy-driven convection in horizontal layers of dilute water solutions of ionic and nonionic polyacrylamides confined between rigid, parallel, conducting surfaces was studied experimentally. By determining the temperature difference at the point where convection first sets in, values of the zero shear rate viscosity were calculated in good agreement with those obtained from rheogoniometer data. Beyond this point, the heat transfer characteristics of the solutions and the associated convective flow patterns were found to be rather similar to those of a viscous Newtonian fluid forRayleigh numbers up to 5 × 104, the range covered by the present experiments. However, theNusselt numbers at any givenRayleigh number were slightly but consistently higher than those of a Newtonian fluid of comparable viscosity. Also, the regularity of the flow patterns was more pronounced.

A. Acrivos

Papers

Experiments on buoyancy driven convection in non-Newtonian fluid.