Similarity solution

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

Multiple exact solutions for micropolar slip flow and heat transfer of a bidirectional moving plate

Abstract: The analysis of micropolar fluid flow over a deforming permeable surface is carried out taking into account velocity slip with constant and linearly growing temperature field conditions. Upon using the boundary layer approximation and the method of similarity transformation, the constitutive equations are remodeled into coupled, nonlinear differential equations which are then solved for the exact solutions of fluid flow and heat transport under different physically acceptable parametric values. In particular, the physical domains of mass transfer and micropolar parameters in determining the existence, singleness and multipleness of exact solutions play a leading role. The examination of critical values for the mass transfer parameter exhibits the borderline for the existence and nonexistence of solutions. Unique solution is detected for the stretching sheet, whereas the shrinking sheet demonstrates twofold solutions. Exact fluid flow and heat transfer solutions under special parametric effects are also considered. The need to examine the prominent physical features of the flow system, closed form formulas for velocity, angular velocity, heat transfer, skin friction and reduced Nusselt number are derived, which are for analysis purpose presented graphically.

Steady rise of a small spherical gas bubble along the axis of a cylindrical pipe at high Reynolds number

Abstract: Steady irrotational flow of inviscid liquid of density ρl around a spherical gas bubble which lies on the axis of a cylindrical pipe is investigated using the analysis of Smythe (Phys. Fluids 4 (1961) 756). The bubble radius b=qa is assumed small compared to the pipe radius a, and the interfacial tension between gas and liquid is γ. Far from the bubble, in the frame in which the bubble is at rest, the liquid velocity along the pipe is v0, whereas the liquid velocity at points on the wall closest to the bubble is Uzw=v0(1+1.776q3+⋯). The decrease in wall pressure as the bubble passes is therefore Δp=1.776ρlv02q3. When the Weber number W=2bv02ρl/γ is small, the bubble deforms into an oblate spheroid with aspect ratio χ=1+9W(1+1.59q3)/64. If the fluid viscosity μ is non-zero, and the Reynolds number Re=2v0ρlb/μ is large, a viscous boundary layer develops on the walls of the pipe. This decays algebraically with distance downstream of the bubble, and an exponentially decaying similarity solution is found upstream. The drag D on the bubble is D=12πμv0b(1−2.21Re−1/2)(1+1.59q3)+7.66μv0bRe1/2q9/2, larger than that given by Moore (J. Fluid Mech. 16 (1963) 161) for motion in unbounded fluid. At high Reynolds numbers the dissipation within the viscous boundary layers might dominate dissipation in the potential flow away from the pipe walls, but such high Reynolds numbers would not be achieved by a spherical air bubble rising in clean water under terrestrial gravity.

Particle paths and phase plane for time-dependent similarity solutions of the one-dimensional Vlasov-Maxwell equations

Abstract: The phase trajectories of particles in a plasma described by the one-dimensional Vlasov-Maxwell equations are determined qualitatively, analyzing exact general similarity solutions for the cases of temporally damped and growing (sinusoidal or localized) electric fields. The results of numerical integration in both untransformed and Lie-group point-transformed coordinates are presented in extensive graphs and characterized in detail. The implications of the present analysis for the stability of BGK equilibria are explored, and the existence of nonlinear solutions arbitrarily close to and significantly different from the BGK solutions is demonstrated.