scholarly journals A Study of Analytical Solution for the Special Dissolution Rate Model of Rock Salt

2017 ◽  
Vol 2017 ◽  
pp. 1-8
Author(s):  
Xin Yang ◽  
Xinrong Liu ◽  
Wanjun Zang ◽  
Zhiyong Lin ◽  
Qiyun Wang
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Analytical Solution ◽  
Dissolution Rate ◽  
Rock Salt ◽  
Analytical Formula ◽  
Influential Factors ◽  
Dissolution Mechanism ◽  
Rate Model ◽  
Stable Conditions

By calculating the concentration distributions of rock salt solutions at the boundary layer, an ordinary differential equation for describing a special dissolution rate model of rock salt under the assumption of an instantaneous diffusion process was established to investigate the dissolution mechanism of rock salt under transient but stable conditions. The ordinary differential equation was then solved mathematically to give an analytical solution and related expressions for the dissolved radius and solution concentration. Thereafter, the analytical solution was fitted with transient dissolution test data of rock salt to provide the dissolution parameters at different flow rates, and the physical meaning of the analytical formula was also discussed. Finally, the influential factors of the analytical formula were investigated. There was approximately a linear relationship between the dissolution parameters and the flow rate. The effects of the dissolution area and initial volume of the solution on the dissolution rate equation of rock salt were computationally investigated. The results showed that the present analytical solution gives a good description of the dissolution mechanism of rock salt under some special conditions, which may provide a primary theoretical basis and an analytical way to investigate the dissolution characteristics of rock salt.

scholarly journals Approximate Analytical study of unsteady Hybrid nanofluid in the presences of magnetic field with convective boundary condition over a stretching surface

2021 ◽  
Author(s):  
Ali Rehman ◽  
Waris khan
Keyword(s):  
Magnetic Field ◽  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Analytical Solution ◽  
Convective Boundary Condition ◽  
Skin Friction Coefficient ◽  
Nonlinear Ordinary Differential Equation ◽  
Hybrid Nanofluid ◽  
Stretching Surface ◽  
Convective Boundary

Abstract The objective of this researcher paper is to study the analytical solution of unsteady hybrid nanofluid in the presences of magnetic field over a stretching surface. By using similarity transformation the major partial differential equation is converted to a set of nonlinear ordinary differential equation .the analytical method (OHAM) is used to find the approximate analytical solution of the nonlinear ordinary differential equation The BVPh 2.0 package function of MATHEMATICA is used to obtained the numerical results the result of important parameter such as, magnetic parameter, Prandtl number, Eckert number and surface convection parameter for both velocity and temperature profile are plotted and discuss. The BVPh 2.0 package is used to obtained the converges of the problem up to 25 iterations. The skin friction coefficient and Nusselt is explained in table form.

scholarly journals Classification of extinction profiles for a one-dimensional diffusive Hamilton–Jacobi equation with critical absorption

2018 ◽  
Vol 148 (3) ◽  
pp. 559-574
Author(s):  
Razvan Gabriel Iagar ◽  
Philippe Laurençot
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Jacobi Equation ◽  
Threshold Value ◽  
Monotonicity Property ◽  
Fast Diffusion ◽  
Extinction Time ◽  
One Step ◽  
Hamilton Jacobi Equation

A classification of the behaviour of the solutions f(·, a) to the ordinary differential equation (|f′|p-2f′)′ + f - |f′|p-1 = 0 in (0,∞) with initial condition f(0, a) = a and f′(0, a) = 0 is provided, according to the value of the parameter a > 0 when the exponent p takes values in (1, 2). There is a threshold value a* that separates different behaviours of f(·, a): if a > a*, then f(·, a) vanishes at least once in (0,∞) and takes negative values, while f(·, a) is positive in (0,∞) and decays algebraically to zero as r→∞ if a ∊ (0, a*). At the threshold value, f(·, a*) is also positive in (0,∞) but decays exponentially fast to zero as r→∞. The proof of these results relies on a transformation to a first-order ordinary differential equation and a monotonicity property with respect to a > 0. This classification is one step in the description of the dynamics near the extinction time of a diffusive Hamilton–Jacobi equation with critical gradient absorption and fast diffusion.

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