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This paper derives approximate ‘sunset’ similarity solutions for receding plane strain and radially symmetric hydraulic fractures in permeable elastic media close to the point of closure. Local analysis is used to show that a receding hydraulic fracture has a linear aperture asymptote $\\hat {w}\\sim \\hat {s}$ in the fracture tip, where $\\hat {s}$ is the distance from the fracture front. Due to the regularity of the linear asymptote, it is possible to determine similarity solutions in the form of power series expansions, which, for integers $N\\ge 2$ and values of the radius decay exponent $\\gamma =1/N$, can be shown to terminate to yield polynomial solutions for the fracture aperture of degree $N$. Of this countable infinity of polynomial solutions, the final aperture profile as the fracture approaches closure is associated with the second-degree polynomial with $\\gamma =1/2$ called the sunset solution. For the reverse time $t^{\\prime }$ measured from closure, the sunset solution is characterized by $w\\sim t^{\\prime }$ and $R\\sim t^{\\prime 1/2}$. Of all the admissible polynomial similarity solutions, the sunset solution is shown to form an attractor, as $t^{\\prime }\\rightarrow 0$, for receding hydraulic fractures associated with a wide variety of points in parametric space. Using the sunset solution, it is possible to estimate the duration of recession, assuming that the fracture aperture and radius at the start of recession are given, and determine how it scales with a dimensionless shut-in parameter. As the fracture approaches closure, the term responsible for coupling the elastic force balance and fluid conservation becomes subdominant to the other terms in the lubrication equation, which reduces to a local kinematic relation between the decaying fracture aperture and the leak-off velocity. This fundamental decoupling of dynamics from kinematics results in the sunset solution being dependent on only a single material parameter – namely the leak-off coefficient. This isolation of the leak-off coefficient by the sunset solution opens the possibility to determine this parameter from laboratory or field measurements.

In the present article, we have obtained the similarity solutions with the help of optimal system of Lie sub-algebras and numerical method for the cylindrical shock wave formed by a moving piston in a non-ideal dusty gas in a rotating medium with the impact of monochromatic radiation. The Lie group theoretic technique is used to obtain the optimal system of Lie sub-algebras for the fundamental equations in the case of 1-D (one-dimensional) flow. By using the Lie group theoretic technique, we are able to drive the similarity and numerical solution for both the power and exponential law shock paths. The similarity solutions with power law shock path in two different cases (i.e., Cases Ia, Ib) are obtained in Case I. Also, in Case II, the similarity solutions exist in two different cases with power law shock path (i.e., Cases IIa, IIc) and in two different cases with exponential law shock path (i.e., Cases IIb, IId) which are discussed in detail. The results presented in Cases Ib, IIa, IIb, IIc, and IId are newly obtained results. The numerical solutions in the power law shock path for Case Ia and the exponential law shock path in Case IIb are performed, and the distribution of the physical flow variables in the flow field region behind the shock front is obtained. The impacts of the several problematic physical parameters on the shock strength and on the physical flow variable are studied. In the present study, it is found that with an increment in the value of the initial ratio of the density of solid particles to the species density of the gas (G1), the shock strength increases, whereas the shock strength decreases with an increment in the gas non-ideality parameter (ω¯). For the lower value of G1 (i.e., for G1=6), the shock strength decreases, whereas for a higher value of G1 (i.e., for G1=20 or 100), the shock strength increases with an increment in the mass concentration of the solid particles in the mixture (μp).

In this present work, the laminar free convection boundary layer flow of a two-dimensional fluid over the vertical flat plate with a uniform surface temperature has been numerically investigated in detail by the similarity solution method. The velocity and temperature profiles were considered similar to all values and their variations are as a function of distance from the leading edge measured along with the plate. By taking into account this thermal boundary condition, the system of governing partial differential equations is reduced to a system of non-linear ordinary differential equations. The latter was solved numerically using the Runge-Kutta method of the fourth-order, the solution of which was obtained by using the FORTRAN code on a computer. The numerical analysis resulting from this simulation allows us to derive some prescribed values of various material parameters involved in the problem to which several important results were discussed in depth such as velocity, temperature, and rate of heat transfer. The definitive comparison between the two numerical models showed us an excellent agreement concerning the order of precision of the simulation. Finally, we compared our numerical results with a certain model already treated, which is in the specialized literature.

In this article, we analyzed the time fractional higher-dimensional nonlinear modified model of wave propagation, namely the (3 + 1)-dimensional Benjamin–Bona–Mahony-type equation. The fractional sense was defined by the classical Riemann–Liouville fractional derivative. We derived firstly the existence of symmetry of the time fractional higher-dimensional equation. Next, we constructed the one-dimensional optimal system to the time fractional higher-dimensional nonlinear modified model of wave propagation. Subsequently, it was reduced into the lower-dimensional fractional differential equation. Meanwhile, on the basis of the reduced equation, we obtained its similarity solution. Through a series of analyses of the time fractional high-dimensional model and the results of the above obtained, we can gain a further understanding of its essence.

Schrödinger’s operator relations combined with Einstein’s special relativistic energy-momentum equation produce the linear Klein–Gordon partial differential equation. Here, we extend both the operator relations and the energy-momentum relation to determine new families of nonlinear partial differential relations. The Planck–de Broglie duality principle arises from Planck’s energy expression e=hν, de Broglie’s equation for momentum p=h/λ, and Einstein’s special relativity energy, where h is the Planck constant, ν and λ are the frequency and wavelength, respectively, of an associated wave having a wave speed w=νλ. The author has extended these relations to a family that is characterised by a second fundamental constant h′ and underpinned by Lorentz invariant power-law particle energy-momentum expressions. In this note, we apply generalized Schrödinger operator relations and the power-law relations to generate a new family of nonlinear partial differential equations that are characterised by the constant κ=h′/h such that κ=0 corresponds to the Klein–Gordon equation. The resulting partial differential equation is unusual in the sense that it admits a stretching symmetry giving rise to both similarity solutions and simple harmonic travelling waves. Three simple solutions of the partial differential equation are examined including a separable solution, a travelling wave solution, and a similarity solution. A special case of the similarity solution admits zeroth-order Bessel functions as solutions while generally, it reduces to solving a nonlinear first-order ordinary differential equation.

PurposeThe purpose of the present article is to obtain the similarity solution for the shock wave generated by a piston propagating in a self-gravitating nonideal gas under the impact of azimuthal magnetic field for adiabatic and isothermal flows.Design/methodology/approachThe Lie group theoretic method given by Sophus Lie is used to obtain the similarity solution in the present article.FindingsSimilarity solution with exponential law shock path is obtained for both ideal and nonideal gas cases. The effects on the flow variables, density ratio at the shock front and shock strength by the variation of the shock Cowling number, adiabatic index of the gas, gravitational parameter and nonidealness parameter are investigated. The shock strength decreases with an increase in the shock Cowling number, nonidealness parameter and adiabatic index, whereas the strength of the shock wave increases with an increase in gravitational parameter.Originality/valuePropagation of shock wave with spherical geometry in a self-gravitating nonideal gas under the impact of azimuthal magnetic field for adiabatic and isothermal flows has not been studied by any author using the Lie group theoretic method.

In this paper, the Lie symmetry analysis and the dynamical system method are performed on an integrable evolution equation for surface waves in deep water 2 k g u xxt = k 2 u x - 3 2 k ( u u x ) xx . All of the geometric vector fields of the equation are presented, as well as some exact similarity solutions with an arbitrary function of t are obtained by using a special symmetry reduction and the dynamical system method. Different kinds of traveling wave solutions also be found by selecting the function appropriately.

This paper studies the Lie point symmetries, conservation laws and invariant solutions to the space-fractional Prandtl equation with the Riemann–Liouville derivative. Exploiting the classical Lie symmetry analysis method extended to fractional equations, three vector fields are calculated, which were used to reduce the fractional partial differential equation into an ordinary differential equation through similarity transformation. Further, it is found that the fractional Prandtl equation is nonlinearly self-adjoint. Therefore, the nonlinear self-adjointness method not requiring the existence of a Lagrangian is utilized to explore the conservation laws for this equation. Three conservation laws are constructed, and one of them is a trivial conservation law. Finally, due to the difficulties of obtaining analytical solution for the ordinary equation, similarity solutions are presented numerically.

This work presents a similarity solution for boundary layer flow through a porous medium over a stretching porous wall. Two considered wall boundary conditions are power-law distribution of either wall temperature or heat flux which are general enough to cover the isothermal and isoflux cases. In addition to momentum, both first and second laws of thermodynamics analyses of the problem are investigated. Independent numerical simulations are also performed for verification of the proposed analytical solution. The results, from the two independent approaches, are found to be in complete agreement. A comprehensive parametric study is presented and it is shown that heat transfer and entropy generation rates increase with Reynolds number, Prandtl number, and suction to the surface.

In the paper, we discuss some similarity solutions of the time-fractional Burgers system (TFBS). Firstly, with the help of the Lie-point symmetry and the corresponding invariant variables, we transform the TFBS to a fractional ordinary differential system (FODS) under the case where the time-fractional derivative is the Riemann–Liouville type. The FODS can be approximated by some integer-order ordinary differential equations; here, we present three such integer-order ordinary differential equations (called IODE-1, IODE-2, and IODE-3, respectively). For IODE-1, we obtain its similarity solutions and numerical solutions, which approximate the similarity solutions and the numerical solutions of the TFBS. Secondly, we apply the numerical analysis method to obtain the numerical solutions of IODE-2 and IODE-3.