Similarity solution

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

Similarity solution of the boundary-layer equations for laser heated flows

Abstract: The laminar boundary-layer equations with local similarity approximation are solved in order to estimate the heat transfer rate to the wall in a laser-heated rocket thruster using pure hydrogen, where the average temperature of the hot plasma core is about 14,000 K and the core Reynolds number based on nozzle throat diameter is about 2000. Under these conditions, the density-viscosity product at the wall can be 10 times the freestream value, and the hydrogen is completely dissociated. Hence the boundary layer equations with variable transport properties are solved by a quasi-linearization technique, and the equilibrium properties of hydrogen are used in the calculations. Velocity profiles are obtained, and the wall shear and wall stagnation enthalpy gradient are plotted against the pressure gradient parameter.

Nonclassical Symmetry Analysis of Boundary Layer Equations

Abstract: The nonclassical symmetries of boundary layer equations for two-dimensional and radial flows are considered. A number of exact solutions for problems under consideration were found in the literature, and here we find new similarity solution by implementing the SADE package for finding nonclassical symmetries.

Non-Newtonian pseudoplastic fluids: Analytical results and exact solutions

Abstract: A one layer model of laminar non-Newtonian fluids (Ostwald–de Waele model) past a semi-infinite flat plate is revisited. The stretching and the suction/injection velocities are assumed to be proportional to x 1 / ( 1 − 2 n ) and x −1 , respectively, where n is the power-law index which is taken in the interval ( 0 , 1 2 ) . It is shown that the boundary-layer equations display both similarity and pseudosimilarity reductions according to a parameter γ , which can be identified as suction/injection velocity. Interestingly, it is found that there is a unique similarity solution, which is given in a closed form, if and only if γ = 0 (impermeable surface). For γ ≠ 0 (permeable surface) we obtain a unique pseudosimilarity solution for any 0 ≠ γ ≥ − ( ( n + 1 ) / 3 n ( 1 − 2 n ) ) n / ( n + 1 ) . Moreover, we explicitly show that any pseudosimilarity solution exhibits similarity behavior and it is, in fact, similarity solution to a modified boundary-layer problem for an impermeable surface. In addition, the exact similarity solution of the original boundary-layer problem is used, via suitable transverse translations, to construct new explicit solutions describing boundary-layer flows induced by permeable surfaces.

Simple waves in Hertzian chains.

Abstract: The discrete system of equations for a chain consisting of a large number of spheres interacting via the Hertz force of index 3/2 in strain is examined in the very long wavelength limit, yielding an effective medium description. The resulting continuum second-order equation of motion possesses a subset of simple waves obeying a first-order equation of reduced index 5/4. These simple waves appear not to have examined before. For a given initial strain, the simple wave solution prescribes initial sphere centroid velocities. Together the initial strain and velocities are used in the second-order discrete system. Results for shock wave development compare very well between the second-order discrete system (minus physically valid oscillations) and the reduced first-order equation. A second-order simulation of colliding waves examines the ability of waves to pass through each other, with a phase advance accruing during the collision process. An arbitrary initial condition is shown to evolve toward a universal similarity solution proportional to (x/t)(4). A closed-form solution is given including the complete history of the waveform, shock location, and amplitude.