A spectral method which employs trigonometric functions and Chebyshev polynomials is used to compute the steady, incompressible laminar flow past a circular cylinder. Linear stability methods are used to formulate a pair of decoupled generalized eigenvalue problems for the growth of symmetric and asymmetric (about the dividing streamline) perturbations. We show that, while the symmetric disturbances are stable, the asymmetric perturbations become unstable at a Reynolds number about 40 with a Strouhal number about 0.12. The critical conditions are found to depend on the size of the computational domain in a manner similar to that observed in the laboratory.

Numerical calculations are presented for the steady three-dimensional structure of thermal convection of a fluid with strongly temperature-dependent viscosity in a bottom-heated rectangular box. Viscosity is assumed to depend on temperature T as exp (− ET), where E is a constant; viscosity variations across the box r (= exp (E)) as large as 105 are considered. A stagnant layer or lid of highly viscous fluid develops in the uppermost coldest part of the top cold thermal boundary layer when r > rc1, where r = rc1 ≡ 1.18 × 103Rt0.309 and Rt is the Rayleigh number based on the viscosity at the top boundary. Three-dimensional convection occurs in a rectangular pattern beneath this stagnant lid. The planform consists of hot upwelling plumes at or near the centre of a rectangle, sheets of cold sinking fluid on the four sides, and cold sinking plume concentrations immersed in the sheets. A stagnant lid does not develop, i.e. convection involves all of the fluid in the box when r < rc1. The whole-layer mode of convection occurs in a three-dimensional bimodal pattern when r > rc2 = 3.84 × 106Rt−1.35. The planform of the convection is rectangular with the coldest parts of the sinking fluid and the hottest part of the upwelling fluid occurring as plumes at the four corners and at the centre of the rectangle, respectively. Both hot uprising plumes and cold sinking plumes have sheet-like extensions, which become more well-developed as r increases. The whole-layer mode of convection occurs as two-dimensional rolls when r < min (rc1, rc2). The Nusselt number Nu depends on the viscosity at the top surface more strongly in the regime of whole-layer convection than in the regime of stagnant-lid convection. In the whole-layer convective regime, Nu depends more strongly on the viscosity at the top surface than on the viscosity at the bottom boundary.

Steady thermal convection of an infinite Prandtl number, Boussinesq fluid with temperature‐dependent viscosity is systematically examined in a three‐dimensional, basally heated spherical shell with isothermal and stress‐free boundaries. Convective flows exhibiting cubic (𝓁 2, 𝓂 = {0, 4}) and tetrahedral (𝓁 = 3, 𝓂 = 2) symmetry are generated with a finite‐volume numerical model for various combinations of Rayleigh number Ra (defined with viscosity based on the average of the boundary temperatures) and viscosity contrast 𝓇μ (ratio of maximum to minimum viscosities). The range of Ra for which these symmetric flows in spherical geometry can be maintained in steady state is sharply reduced by even mild viscosity variations (𝓇μ ≤ 30), in contrast with analogous calculations in Cartesian geometry in which relatively simple, three‐dimensional convective planforms remain steady for 𝓇μ ≈ 104. The mild viscosity contrasts employed place some solutions marginally in the sluggish‐lid transition regime in Ra‐𝓇μ parameter space. Global heat transfer, given by the Nusselt number N𝓊, is found to obey a single power law relation with Ra when Ra is scaled by its critical value. A power law of the form N𝓊 ∼ (Ra/Racrit)1/4 (Racrit is the minimum critical value of Ra for the onset of convection) is obtained, in agreement with previous results for isoviscous spherical shell convection with cubic and tetrahedral symmetry. The calculations of this paper demonstrate that temperature‐dependent viscosity exerts a strong control on the nature of three‐dimensional convection in spherical geometry, an effect that is likely to be even more important at Rayleigh numbers and viscosity contrasts more representative of the mantles of terrestrial planets. The robustness of the N𝓊‐Ra relation, when scaled by Racrit, is important for studies of planetary thermal history that rely on parameterizations of convective heat transport and account for temperature dependence of mantle viscosity.

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Independent pseudo-spectral and Galerkin numerical codes are used to investigate three-dimensional infinite Prandtl number thermal convection of a Boussinesq fluid in a spherical shell with constant gravity and an inner to outer radius ratio equal to 0.55. The shell is heated entirely from below and has isothermal, stress-free boundaries. Nonlinear solutions are validated by comparing results from the two codes for an axisymmetric solution at Rayleigh number Ra = 14250 and three fully three-dimensional solutions at Ra = 2000, 3500 and 7000 (the onset of convection occurs at Ra = 712). In addition, the solutions are compared with the predictions of a slightly nonlinear analytic theory. The axisymmetric solution is equatorially symmetric and has two convection cells with upwelling at the poles. Two dominant planforms of convection exist for the three-dimensional solutions: a cubic pattern with six upwelling cylindrical plumes, and a tetrahedral pattern with four upwelling plumes. The cubic and tetrahedral patterns persist for Ra at least up to 70000. Time dependence does not occur for these solutions for Ra [les ] 70000, although for Ra > 35000 the solutions have a slow asymptotic approach to steady state. The horizontal and vertical structure of the velocity and temperature fields, and the global and three-dimensional heat flow characteristics of the various solutions are investigated for the two patterns up to Ra = 70000. For both patterns at all Ra, the maximum velocity and temperature anomalies are greater in the upwelling regions than in the downwelling ones and heat flow through the upwelling regions is almost an order of magnitude greater than the mean heat flow. The preferred mode of upwelling is cylindrical plumes which change their basic shape with depth. Downwelling occurs in the form of connected two-dimensional sheets that break up into a network of broad plumes in the lower part of the spherical shell. Finally, the stability of the two patterns to reversal of flow direction is tested and it is found that reversed solutions exist only for the tetrahedral pattern at low Ra.