A Ricci-type flow on globally null manifolds and its gradient estimates

AbstractSweet whey is a waste product from the dairy industry that is difficult to manage. High hopes are fostered regarding its neutralization in the methane fermentation. An economically viable alternative to a typical mesophilic fermentation seems to be the process involving psychrophilic bacteria isolated from the natural environment. This study aimed to determine the feasibility of exploiting psychrophilic microorganisms in methane fermentation of sweet whey. The experiments were carried out under dynamic conditions using Bio Flo 310 type flow-through anaerobic bioreactors. The temperature inside the reactors was 10 ± 1 °C. The HRT was 20 days and the OLR was 0.2 g COD/dm3/day. The study yielded 132.7 ± 13.8 mL biogas/gCODremoved. The CH4 concentration in the biogas was 32.7 ± 1.6%, that of H2 was 8.7 ± 4.7%, whereas that of CO2 reached 58.42 ± 2.47%. Other gases were also determined, though in lower concentrations. The COD and BOD5 removal efficiency reached 21.4 ± 0.6% and 17.6 ± 1.0%, respectively.

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Gradient estimates for divergence form parabolic systems from composite materials

Calculus of Variations and Partial Differential Equations ◽

10.1007/s00526-021-01927-5 ◽

2021 ◽

Vol 60 (3) ◽

Author(s):

Hongjie Dong ◽

Longjuan Xu

Keyword(s):

Composite Materials ◽

Parabolic Systems ◽

Divergence Form ◽

Gradient Estimates

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A note on the solution of the Navier-Stokes equations for a spherically symmetric expansion into a very low pressure

Journal of Fluid Mechanics ◽

10.1017/s0022112073001606 ◽

1973 ◽

Vol 59 (2) ◽

pp. 391-396 ◽

Cited By ~ 2

Author(s):

N. C. Freeman ◽

S. Kumar

Keyword(s):

Shock Wave ◽

Reynolds Number ◽

Stokes Equation ◽

Stokes Equations ◽

Low Pressure ◽

Navier Stokes ◽

Spherically Symmetric ◽

Navier Stokes Equation ◽

Navier Stokes Equations ◽

Type Flow

It is shown that, for a spherically symmetric expansion of a gas into a low pressure, the shock wave with area change region discussed earlier (Freeman & Kumar 1972) can be further divided into two parts. For the Navier–Stokes equation, these are a region in which the asymptotic zero-pressure behaviour predicted by Ladyzhenskii is achieved followed further downstream by a transition to subsonic-type flow. The distance of this final region downstream is of order (pressure)−2/3 × (Reynolds number)−1/3.

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