Gradient hypothesis

Crane, R.K. (1975). The gradient hypothesis and other models of carrier-mediated active transport. Rev. Physiol. Biochem. Pharmacol. 78, 101-149. 

The third term of Eq. (3) contains eji, which is generally modeled in terms of turbulent dispersion in a manner analogous to the well-known gradient hypothesis of Boussinesq, as proportional to the gradient of holdup in the z direction, the constant of proportionality being referred to as the turbulent dispersion coefficient ... 

According to the ideas of Boussinesq [19] [20], the first turbulent closures was based on the gradient hypothesis and the coefficient of eddy viscosity. 

Nevertheless, the turbulent dispersion force acting on bubbles in turbulent liquid flow is commonly modeled using the gradient hypothesis (e.g., [20]) ... 

Summarizing these results, one has good evidence that at least in Ehrlich cells the accumulative amino acid transport is coupled to an Na flux in the same direction (cotransport) as postulated by the gradient hypothesis, while no hard evidence can be presented for primary active transport of amino acids. 

The earliest observation on a role of Na CP on resorption of sugars was reported by Reid 80 years ago. This information, however, was forgotten when 20 years ago the role of Na" " in sugar resorption was rediscovered. These investigations led to the formulation of Crane s gradient hypothesis [62] for sugar resorption by... 

B. Distribution of Inducing Factors in Embryos of the Gastrula Stage and the Gradient Hypothesis... 

According to the ideas of Boussinesq [19, 20], the first turbulent closures was based on the gradient hypothesis and the coeflflcient of eddy viscosity. The eddy concept was thus introduced. An eddy still eludes precise definition, but in one interpretation it is conceived to be a turbulent motion, localized within a region of a certain size, that is at least moderately coherent over this region. The region of a large eddy can also contain smaller eddies. 

The second term on the RHS of (3.375) represents the thermal transport due to the fluctuations in velocity and internal energy. This term can be modeled in accordance with the gradient hypothesis and the Boussinesq transport coefficient concept. That is, we approximate the covariance by ... 

The covariance is modeled by use of the gradient hypothesis and the Boussinesq turbulent transport coefficient concept ... 

Randolph and Larson [106], p. 74, claimed that evidence suggests that such random fluctuations in growth rate and the flow have been proven to exist in certain crystallization problems [40]. In problems in which such effects are significant, it might be proposed that these turbulent covariance terms can be approximated by use of the gradient hypothesis and a proportionality coefficient may be introduced in accordance with the Boussinesq hypothesis. In the work of Randolph and Larson [106], the associated proportionality coefficients were denoted the growth and flow difiusivities. 

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