In fluid dynamics, the Ostroumov flow, also known as the Ostroumov–Birikh–Hansen–Rattray flow describes fluid motion driven by horizontal density gradients within horizontal channels, pipes, or open water bodies such as rivers and estuaries. The flow is named after Georgy Andreyevich Ostroumov (1952), R. V. Birikh (1966), Donald V. Hansen and Maurice Rattray Jr (1965). Unlike the Poiseuille flow or the Couette flow, the velocity profile in the Ostroumov flow is a cubic function of the coordinate normal to gravity.
Planar channel Consider a two-dimensional planar channel of width 2 h {\displaystyle 2h} and their walls located at z = − h {\displaystyle z=-h} and z = + h {\displaystyle z=+h} . The gravity vector is given by g = − g e z {\displaystyle \mathbf {g} =-g\mathbf {e} _{z}} , where g {\displaystyle g} is the gravitational acceleration. Suppose that there exists a horizontal density gradient in the fluid, i.e., ρ = ρ ( x , y , t ) {\displaystyle \rho =\rho (x,y,t)} with a characteristic length scale l {\displaystyle l} . Such gradients can be induced by some scalar field such as temperature or solute concentration, present within the fluid. Whenever horizontal density gradients exists within a fluid, mechanical equilibrium is impossible and thus, fluid motion occurs. Furthermore, we work in the usual lubrication theory or Hele-Shaw flow limit
ϵ ≡ h l ≪ 1 , ρ r U b h μ r h l ≪ 1 {\displaystyle \epsilon \equiv {\frac {h}{l}}\ll 1,\quad {\frac {\rho _{r}U_{b}h}{\mu _{r}}}{\frac {h}{l}}\ll 1}
where U b {\displaystyle U_{b}} is the characteristic velocity scale associated with the flow induced by the buoyancy forces and ( ρ r , μ r ) {\displaystyle (\rho _{r},\mu _{r})} are the reference values of fluid density and viscosity. If Δ ρ {\displaystyle \Delta \rho } represents a characteristic density difference then U b {\displaystyle U_{b}} is given by
U b = Δ ρ g h 3 μ r l , G r = ρ r U b h μ r = ρ r Δ ρ g h 4 μ r 2 l {\displaystyle U_{b}={\frac {\Delta \rho gh^{3}}{\mu _{r}l}},\quad Gr={\frac {\rho _{r}U_{b}h}{\mu _{r}}}={\frac {\rho _{r}\Delta \rho gh^{4}}{\mu _{r}^{2}l}}}
where G r {\displaystyle Gr} is a Grashof number. In the lubrication limit ( ϵ → 0 {\displaystyle \epsilon \to 0} and ϵ G r → 0 {\displaystyle \epsilon Gr\to 0} ) under consideration, the variable-density ( ρ = ρ ( x , y , t ) {\displaystyle \rho =\rho (x,y,t)} ), variable viscosity ( μ = μ ( x , y , t ) {\displaystyle \mu =\mu (x,y,t)} ) Navier–Stokes equations reduces to
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