The American Society for Testing and Materials (ASTM) is an independent organization devoted to the development of standards.

American Society for Testing and Materials

The boundary layer equations for plane, incompressible, and steady flow are 

$$\matrix{ {u{{\partial u} \over {\partial x}} + v{{\partial u} \over {\partial y}} = - {1 \over \varrho }{{\partial p} \over {\partial x}} + v{{{\partial ^2}u} \over {\partial {y^2}}},} \cr {0 = {{\partial p} \over {\partial y}},} \cr {{{\partial u} \over {\partial x}} + {{\partial v} \over {\partial y}} = 0.} \cr }$$

Boundary Layer Theory

Boundary-layer flow of a nanofluid past a stretching sheet

The flow of homogeneous fluids through porous media: analogies with other physical problems

Natural convective boundary-layer flow of a nanofluid past a vertical plate

Settling of bidispersed small solid particles in a dilute suspension containing gyrotactic micro-organisms

Natural convection about a vertical plate embedded in a bidisperse porous medium

Forced convection with laminar pulsating flow in a channel or tube

Fluid mechanics and heat transfer in the interface region between a porous medium and a fluid layer: a boundary layer solution.

The aim of this paper is to develop a theory describing the onset of convection instability (called here nanofluid bioconvecion) that is induced by simultaneous effects produced by oxytactic microorganisms, nanoparticles, and vertical temperature variation. The theory is developed for the situation when the nanofluid occupies a shallow horizontal layer of finite depth. The layer is defined as shallow as long as oxygen concentration at the bottom of the layer is above the minimum concentration required for the bacteria to be active (to actively swim up the oxygen gradient). The lower boundary of the layer is assumed rigid, while at the upper boundary both situations when the boundary is rigid or stress free are considered. Physical mechanisms responsible for the slip velocity between the nanoparticles and the base fluid, such as Brownian motion and thermophoresis, are accounted for in the model. A linear instability analysis is performed, and the resulting eigenvalue problem is solved analytically using the Galerkin method.

Andrey V. Kuznetsov

Papers

Settling of bidispersed small solid particles in a dilute suspension containing gyrotactic micro-organisms

Natural convection about a vertical plate embedded in a bidisperse porous medium

Forced convection with laminar pulsating flow in a channel or tube

Fluid mechanics and heat transfer in the interface region between a porous medium and a fluid layer: a boundary layer solution.

Nanofluid bioconvection: interaction of microorganisms oxytactic upswimming, nanoparticle distribution, and heating/cooling from below