What is Hele-Shaw model?
Hele-Shaw flow is defined as Stokes flow between two parallel flat plates separated by an infinitesimally small gap, named after Henry Selby Hele-Shaw, who studied the problem in 1898. Various problems in fluid mechanics can be approximated to Hele-Shaw flows and thus the research of these flows is of importance.
What is potential flow equation?
Potential flow is an idealized model of fluid flow that occurs in the case of incompressible, inviscid, and irrotational flow. The velocity potential of a potential flow satisfies Laplace’s equation: ∇2→ϕ=0.
What is the difference between Couette and Poiseuille flow?
In Couette flow, one plate is moving with respect to the other plate, and that relative motion drives the shearing action in the fluid between the plates. In Poiseuille flow, the plates are both stationary and the flow is driven by an external pressure gradient.
What is stream and potential function?
Velocity potential function and stream function are two scalar functions that help study whether the given fluid flow is rotational or irrotational. Both the functions provide a specific Laplace equation. The fluid flow can be rotational or irrotational flow based on whether it satisfies the Laplace equation or not.
What is the governing equation of Hele-Shaw flow?
The governing equation of Hele-Shaw flows is identical to that of the inviscid potential flow and to the flow of fluid through a porous medium ( Darcy’s law ). It thus permits visualization of this kind of flow in two dimensions. A schematic description of a Hele-Shaw configuration. ). When the gap between plates is asymptotically small
Why is approximation important to Hele-Shaw flow?
Approximation to Hele-Shaw flow is specifically important to micro-flows. This is due to manufacturing techniques, which creates shallow planar configurations, and the typically low Reynolds numbers of micro-flows.
Do Hele-Shaw solutions exist?
For zero-surface-tension Hele-Shaw, local-in-time existence of classical solutions is known, and global existence for weak so- lutions in the injection case. Only local results are known with positive surface tension. 25 Solution behaviour 26 Blow-up and regularisation
How do you find the Hele-Shaw law using variational arguments?
A variational argument (varyNwithVflxed) then shows that the boundary moves according to the Hele-Shaw law. There is also a connection with moments. The point charges analogy gives Zρ(z0)d2z0 z0− z ∂V ∂z = 0 (note the Cauchy transform). The ‘usual’ form forVis V(z) =−1 2|z| 2(Gaussian) + X tkz k wheretk=Mk/kare identifled with themoments. Thus ∂V ∂z