Similarity solution

About: Similarity solution is a research topic. Over the lifetime, 2074 publications have been published within this topic receiving 59790 citations.

Group invariant solution for a pre-existing fluid-driven fracture in impermeable rock

Abstract: The propagation of a two-dimensional fluid-driven fracture in impermeable rock is considered. The fluid flow in the fracture is laminar. By applying lubrication theory a partial differential equation relating the half-width of the fracture to the fluid pressure is derived. To close the model the PKN formulation is adopted in which the fluid pressure is proportional to the half-width of the fracture. By considering a linear combination of the Lie point symmetries of the resulting non-linear diffusion equation the boundary value problem is expressed in a form appropriate for a similarity solution. The boundary value problem is reformulated as two initial value problems which are readily solved numerically. The similarity solution describes a preexisting fracture since both the total volume and length of the fracture are initially finite and non-zero. Applications in which the rate of fluid injection into the fracture and the pressure at the fracture entry are independent of time are considered.

Similarity solutions for the stagnation-point flow and heat transfer over a nonlinearly stretching/shrinking sheet

Abstract: This paper presents a numerical analysis of a stagnation-point flow towards a nonlinearly stretching/shrinking sheet immersed in a viscous fluid. The stretching/shrinking velocity and the external flow velocity impinges normal to the stretching/shrinking sheet are assumed to be in the form U ~ xm, where m is a constant and x is the distance from the stagnation point. The governing partial differential equations are converted into ordinary ones by a similarity transformation, before being solved numerically. The variations of the skin friction coefficient and the heat transfer rate at the surface with the governing parameters are graphed and tabulated. Different from a stretching sheet, it is found that the solutions for a shrinking sheet are non-unique for m > 1/3.

A similarity solution for the axisymmetric viscous‐gravity jet

Abstract: A local similarity solution to the viscous‐gravity jet valid for large Reynolds number flows is given. The jet is divided into an inner core where axial gradients are small relative to the outer annular region. A similarity transformation is found for the outer region, reducing the resulting differential equations to a two‐point boundary value problem. This solution is matched to the inner solution through the boundary conditions. This approach effectively eliminates the mathematical difficulties associated with the unknown free surface and the stress singularity at the exit.

Infinite series symmetry reduction solutions to the modified KdV–Burgers equation

Abstract: From the point of view of approximate symmetry, the modified Korteweg-de Vries–Burgers (mKdV–Burgers) equation with weak dissipation is investigated. The symmetry of a system of the corresponding partial differential equations which approximate the perturbed mKdV–Burgers equation is constructed and the corresponding general approximate symmetry reduction is derived; thereby infinite series solutions and general formulae can be obtained. The obtained result shows that the zero-order similarity solution to the mKdV–Burgers equation satisfies the Painleve II equation. Also, at the level of travelling wave reduction, the general solution formulae are given for any travelling wave solution of an unperturbed mKdV equation. As an illustrative example, when the zero-order tanh profile solution is chosen as an initial approximate solution, physically approximate similarity solutions are obtained recursively under the appropriate choice of parameters occurring during computation.

Generalized Beltrami field modeling disk‐jet system

Abstract: A generalized Beltrami field, including the effect of inhomogeneous density, can model the singular structure of thin‐disk and collimated‐jet combination. On an accretion disk, the singularity at the origin (r = 0) of the Keplerian rotation (Vθ∝r−1/2) is the determinant of the geometry and mechanics of accompanying jet: the collimation of the jet is the consequence of the alignment —so‐called Beltrami condition— of the flow velocity and the “generalized vorticity” that appears as an axle penetrating the disk (the vorticity is generalized to combine with magnetic field as well as to subtract the friction force causing the accretion). We construct an analytical solution of the simplified version of the model by the method of similarity solution.