The method of regularized stokeslets is a powerful numerical method to solve the Stokes flow equations for problems in biological fluid mechanics. A recent variation of this method incorporates a nearestneighbor discretization to improve accuracy and efficiency while maintaining the ease-of-implementation of the original meshless method. This method contains three sources of numerical error, the regularization error associated from using the regularized form of the boundary integral equations (with parameter ε), and two sources of discretization error associated with the force and quadrature discretizations (with lengthscales h f and hq). A key issue to address is the quadrature error: initial work has not fully explained observed numerical convergence phenomena. In the present manuscript we construct sharp quadrature error bounds for the nearest-neighbor discretisation, noting that the error for a single evaluation of the kernel depends on the smallest distance (δ) between these discretization sets. The quadrature error bounds are described for two cases: with disjoint sets (δ > 0) being close to linear in hq and insensitive to ε, and contained sets (δ = 0) being quadratic in hq with inverse dependence on ε. The practical implications of these error bounds are discussed with reference to the condition number of the matrix system for the nearest-neighbor method, with the analysis revealing that the condition number is insensitive to ε for disjoint sets, and grows linearly with ε for contained sets. Error bounds for the general case (δ ≥ 0) are revealed to be proportional to the sum of the errors for each case. arXiv:1806.01560v1 [physics.flu-dyn]
The problem of non-isothermal fluid flow in and around a liquid drop has been studied. The temperature of the fluid is assumed to be non-constant, steady and hence is governed by the Laplace's equation. The thermal and hydrodynamic problems have been solved under nonisothermal boundary conditions assuming Stokes equations for the flow inside and outside the drop. The drag and torque on the droplet in the form of Faxen's laws are presented. The use of the drag formula has been demonstrated by few particular cases. Some important asymptotic limiting cases have been discussed.
In this paper we prove the existence of multiple solutions for a nonlinear nonlocal elliptic PDE involving a singularity which is given aswhere Ω is an open bounded domain in R N with smooth boundary, N > ps, s ∈ (0, 1), λ > 0, 0 < γ < 1, 1 < p < ∞, p − 1 < q ≤ p * s = N p N −ps . We employ variational techniques to show the existence of multiple positive weak solutions of the above problem. We also prove that for some β ∈ (0, 1), the weak solution to the problem is in C 1,β (Ω).
A general method to discuss the problem of an arbitrary Stokes flow (both axisymmetric and non-axisymmetric flows) of a viscous, incompressible fluid past a sphere with a thin coating of a fluid of a different viscosity is considered. We derive the expressions for the drag and torque experienced by the fluid coated sphere and also discuss the conditions for the reduction of the drag on the fluid coated sphere. In fact, we show that the drag reduces compared to the drag on a rigid sphere of the same radius when the unperturbed velocity is either harmonic or purely biharmonic, i.e., of the form r 2 v, where v is a harmonic function. Previously Johnson (J Fluid Mech 110:217-238, 1981), who considered a uniform flow showed that the drag on the fluid coated sphere reduces compared to the drag on the uncoated sphere when the ratio of the surrounding fluid viscosity to the fluid-film viscosity is greater than 4. We show that this result is true when the undisturbed velocity is harmonic or purely biharmonic, uniform flow being a special case of the former. However, we illustrate by an example that the drag may increase in a general Stokes flow even if this ratio is greater than 4. Moreover, when the unperturbed velocity is harmonic or purely biharmonic, and the ratio of the surrounding fluid viscosity to the fluid-film viscosity is greater than 4 for a fixed value of the viscosity of the ambient fluid, we determine the thickness of the coating for which the drag is minimum. (2000). 76D07.
Mathematics Subject Classification
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