Stokes flow with slip and Kuwabara boundary conditions

2002 ◽  
Vol 112 (3) ◽  
pp. 463-475 ◽  
Author(s):  
Sunil Datta ◽  
Satya Deo
Keyword(s):  
Boundary Conditions ◽  
Stokes Flow

scholarly journals From diffusive mass transfer in Stokes flow to low Reynolds number Marangoni boats

2021 ◽  
Vol 44 (1) ◽  
Author(s):  
Hendrik Ender ◽  
Jan Kierfeld
Keyword(s):  
Symmetry Breaking ◽  
Boundary Conditions ◽  
Spontaneous Symmetry Breaking ◽  
Stokes Flow ◽  
Peclet Number ◽  
Swimming Velocity ◽  
Marangoni Flow ◽  
Péclet Number ◽  
Constant Flux ◽  
Flux Boundary Conditions

AbstractWe present a theory for the self-propulsion of symmetric, half-spherical Marangoni boats (soap or camphor boats) at low Reynolds numbers. Propulsion is generated by release (diffusive emission or dissolution) of water-soluble surfactant molecules, which modulate the air–water interfacial tension. Propulsion either requires asymmetric release or spontaneous symmetry breaking by coupling to advection for a perfectly symmetrical swimmer. We study the diffusion–advection problem for a sphere in Stokes flow analytically and numerically both for constant concentration and constant flux boundary conditions. We derive novel results for concentration profiles under constant flux boundary conditions and for the Nusselt number (the dimensionless ratio of total emitted flux and diffusive flux). Based on these results, we analyze the Marangoni boat for small Marangoni propulsion (low Peclet number) and show that two swimming regimes exist, a diffusive regime at low velocities and an advection-dominated regime at high swimmer velocities. We describe both the limit of large Marangoni propulsion (high Peclet number) and the effects from evaporation by approximative analytical theories. The swimming velocity is determined by force balance, and we obtain a general expression for the Marangoni forces, which comprises both direct Marangoni forces from the surface tension gradient along the air–water–swimmer contact line and Marangoni flow forces. We unravel whether the Marangoni flow contribution is exerting a forward or backward force during propulsion. Our main result is the relation between Peclet number and swimming velocity. Spontaneous symmetry breaking and, thus, swimming occur for a perfectly symmetrical swimmer above a critical Peclet number, which becomes small for large system sizes. We find a supercritical swimming bifurcation for a symmetric swimmer and an avoided bifurcation in the presence of an asymmetry.

NON-NEWTONIAN STOKES FLOW WITH FRICTIONAL BOUNDARY CONDITIONS

2007 ◽  
Vol 12 (4) ◽  
pp. 483-495 ◽  
Author(s):  
Fouad Saidi
Keyword(s):  
Fluid Flow ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Variational Inequalities ◽  
Newtonian Fluid ◽  
Stokes Flow ◽  
Weak Solutions ◽  
Existence And Uniqueness ◽  
Boundary Value

In this work we deal with the boundary value problem for the non‐Newtonian fluid flow with boundary conditions of friction type, mostly by means of variational inequalities. Among others, theorems concerning existence and uniqueness or non‐uniqueness of weak solutions are presented.

Three-dimensional eddy structure in a cylindrical container

1997 ◽  
Vol 342 ◽  
pp. 97-118 ◽  
Author(s):  
P. N. SHANKAR
Keyword(s):  
Flow Field ◽  
Boundary Conditions ◽  
Stokes Flow ◽  
Three Dimensional ◽  
Transcendental Equation ◽  
Cylindrical Container ◽  
Circular Section ◽  
Eddy Structure ◽  
Translatory Motion

We consider Stokes flow in a cylindrical container of circular section induced by the uniform translatory motion of one of the endwalls. This flow field is of interest because it is possible to get reliable analytical descriptions of important three-dimensional structures such as the primary and corner eddies. It is shown, using a result of Tran-Cong & Blake, that separable solutions exist which can be combined to yield vector eigenfunctions that satisfy the sidewall boundary conditions provided the eigenvalues satisfy the transcendental equationformula here

Method of regularized sources for axisymmetric Stokes flow problems

2016 ◽  
Vol 26 (3/4) ◽  
pp. 1226-1239 ◽  
Author(s):  
Kai Wang ◽  
Shiting Wen ◽  
Rizwan Zahoor ◽  
Ming Li ◽  
Božidar Šarler
Keyword(s):  
Analytical Solution ◽  
Fundamental Solution ◽  
Boundary Conditions ◽  
Stokes Flow ◽  
Regularization Parameter ◽  
Fine Grid ◽  
Content Type ◽  
Flow Problems ◽  
Singular Source ◽  
Analytical Expressions

Purpose – The purpose of this paper is to find solution of Stokes flow problems with Dirichlet and Neumann boundary conditions in axisymmetry using an efficient non-singular method of fundamental solutions that does not require an artificial boundary, i.e. source points of the fundamental solution coincide with the collocation points on the boundary. The fundamental solution of the Stokes pressure and velocity represents analytical solution of the flow due to a singular Dirac delta source in infinite space. Design/methodology/approach – Instead of the singular source, a non-singular source with a regularization parameter is employed. Regularized axisymmetric sources were derived from the regularized three-dimensional sources by integrating over the symmetry coordinate. The analytical expressions for related Stokes flow pressure and velocity around such regularized axisymmetric sources have been derived. The solution to the problem is sought as a linear combination of the fields due to the regularized sources that coincide with the boundary. The intensities of the sources are chosen in such a way that the solution complies with the boundary conditions. Findings – An axisymmetric driven cavity numerical example and the flow in a hollow tube and flow between two concentric tubes are chosen to assess the performance of the method. The results of the newly developed method of regularized sources in axisymmetry are compared with the results obtained by the fine-grid second-order classical finite difference method and analytical solution. The results converge with a finer discretization, however, as expected, they depend on the value of the regularization parameter. The method gives accurate results if the value of this parameter scales with the typical nodal distance on the boundary. Originality/value – Analytical expressions for the axisymmetric blobs are derived. The method of regularized sources is for the first time applied to axisymmetric Stokes flow problems.

Sign in / Sign up

Export Citation Format

Share Document