In this paper we study a stationary generalized bioconvection problem given by a Navier–Stokes type system coupled to a cell conservation equation for describing the hydrodynamic and micro-organisms concentration, respectively, of a culture fluid, assumed to be viscous and incompressible, and in which the viscosity depends on the concentration. The model is rewritten in terms of a first-order system based on the introduction of the shear-stress, the vorticity, and the pseudo-stress tensors in the fluid equations along with an auxiliary vector in the concentration equation. After a variational approach, the resulting weak model is then augmented using appropriate redundant parameterized terms and rewritten as fixed-point problem. Existence and uniqueness results for both the continuous and the discrete scheme as well as the respective convergence result are obtained under certain regularity assumptions combined with the Lax–Milgram theorem, and the Banach and Brouwer fixed-point theorems. Optimal a priori error estimates are derived and confirmed through some numerical examples that illustrate the performance of the proposed technique.

在本文中，我们研究了由Navier-Stokes类型系统给出的平稳广义生物对流问题，该系统与细胞守恒方程相结合，分别描述了假定为粘性和不可压缩的培养液的流体动力学和微生物浓度。粘度取决于浓度。根据流体方程中切应力，涡度和拟应力张量以及浓度方程中的辅助矢量的引入，根据一阶系统对模型进行了重写。在变分方法之后，然后使用适当的冗余参数化项来扩充所得的弱模型，并将其重写为定点问题。连续和离散方案的存在性和唯一性结果以及各自的收敛结果是在一定的规则性假设下结合Lax-Milgram定理以及Banach和Brouwer不动点定理获得的。最佳的先验误差误差估计值是通过一些数字示例得出的，并通过一些数值示例来说明所提出技术的性能。