Unsteady Free Convection Boundary-layer Flow Research Articles

Lie Symmetry Group for Unsteady Free Convection Boundary-Layer Flow over a Vertical Surface

The Lie symmetry group transformation method was used to investigate the partial differential equations that model the motion of a natural convective unsteady flow past to a non-isothermal vertical flat surface. The one-parameter Lie group transformation was applied twice consecutively to convert the motion governing equations into a system of ordinary differential equations. The obtained system of ordinary differential equations was solved numerically using the Lobatto IIIA formula (implicit Runge–Kutta method). The effect of the Prandtl number on the temperature and velocity profiles is illustrated graphically.

The effects of wall transpiration on the unsteady free convection boundary-layer flow near a stagnation point in a heat generating porous medium

The natural convection boundary-layer flow near a stagnation point on a permeable surface embedded in a porous medium is considered when there is local heat generation within the boundary layer at a rate proportional to (T-T_infty )^p, p ge 1, where T is the fluid temperature and T_infty the ambient temperature. There is mass transfer through the surface characterized by the dimensionless parameter gamma, with gamma >0 for fluid injection and gamma <0 for fluid withdrawal. The steady states are considered where it is found that, for p >1, there is a critical value gamma _c of gamma with solutions existing for gamma ge gamma _c if 1<p<2 and for gamma le gamma _c if p>2. The initial-value problem reveals that, for 1 le p<2, the nontrivial steady states are stable and the solution evolves to this state at large times. However, for p>2 these steady states are unstable and the solution either approaches the trivial state with the local heating dying out or a finite-time singularity develops for sufficiently large initial inputs.

The Unsteady Free Convection Boundary-Layer Flow Near a Stagnation Point in a Heat Generating Porous Medium with Modified Arrhenius Kinetics

The free convection boundary layer on an insulated wall formed by local internal heating through a modified form of Arrhenius kinetics is considered. It is shown to involve two dimensionless parameters, $$\epsilon $$ the activation energy and $$q_0$$ the rate of local heating. Numerical solutions to the initial-value problem are obtained showing that, for relatively weak internal heating (small $$q_0$$ ), a nontrivial flow arises at large times, whereas for larger local heating the solution becomes singular at a finite time. This behaviour is also seen to depend on the size of the initial input. The corresponding steady states, being the possible large time solutions to the initial-value problem, are also treated. These show the existence of a critical value $$q_{0,\mathrm{{crit}}}$$ of $$q_0$$ , dependent on $$\epsilon $$ . These critical values determined numerically showing that there was a finite region of the $$\epsilon {\sim }q_0$$ parameter plane over which steady states cannot be found. Asymptotic forms for both $$\epsilon $$ and $$q_0$$ being small and large are derived.

Unsteady MHD free convection boundary-layer flow of a nanofluid along a stretching sheet with thermal radiation and viscous dissipation effects

In this work, we study the unsteady free convection boundary-layer flow of a nanofluid along a stretching sheet with thermal radiation in the presence of magnetic field. To obtain non-similar equations, continuity, momentum, energy, and concentration equations have been non-dimensionalized by usual transformation. The non-similar solutions are considered here which depend on the magnetic parameter M, radiation parameter R, Prandtl number Pr, Eckert number Ec, Lewis number Le, Brownian motion parameter Nb, thermophoresis parameter Nt, and Grashof number Gr. The obtained equations have been solved by an explicit finite difference method with stability and convergence analysis. The velocity, temperature, and concentration profiles are discussed for different time steps and for the different values of the parameters of physical and engineering interest.

Unsteady free convective boundary-layer flow near a stagnation point in a heat-generating porous medium

The free convection boundary-layer flow near a stagnation point in a porous medium is considered when there is local heat generation at a rate proportional to (T − T ∞) p , (p ≥ 1), where T is the fluid temperature and T ∞ the ambient temperature. Two cases are treated, when the surface is thermally insulated and when heat is supplied at a constant (dimensionless) rate h s from the boundary. If h s = 0 the solution approaches a nontrivial steady state for time t large in which the local heating has a significant effect when p ≤ 2. For p > 2 the effects of the local heating become increasingly less important and the solution dies away, with the surface temperature being of O(t −1) for t large. When h s > 0 and there is heat input from the surface, the solution for p ≤ 2 again approaches a nontrivial steady state for t large and all h s . For p > 2 there is a critical value h s,crit (dependent on the exponent p) of h s such that the solution still approaches a nontrivial steady state if h s h s,crit a singularity develops in the solution at a finite time, the nature of which is analysed.

An exact solution to the unsteady free-convection boundary-layer flow past an impulsively started vertical surface with newtonian heating

An exact solution of the unsteady free-convection boundary-layer flow of an incompressible fluid past an infinite vertical plate with the flow generated by Newtonian heating and impulsive motion of the plate is presented. The resulting governing equations are nondimensionalized and their solutions are obtained in a closed form with the help of the Laplace transform technique. A parametric study of the roles of all involved parameters is conducted and a representative set of numerical results for the velocity, temperature, and skin friction is illustrated graphically. The physical aspects of the problem are discussed.

Analytical solutions for unsteady free convection in porous media

Analytic solutions for two of the similarity cases identified by Johnson and Cheng (1978) for the unsteady free-convection boundary-layer flow over an impermeable vertical flat plate adjacent to a fluid saturated porous medium are given in the present paper. These are the solutions corresponding to an exponential (e sup a 2 t sup) and a power-law (t m) variation of the surface temperature, respectively. They represent exact solutions for doubly infinite plates and approximate solutions for semi-infinite plates. In the latter cases their validity is restricted to the so-called `conduction regime' of the flow. It is shown that in the power law case, physical solutions only exist in the range m>−1 of the temperature exponent and they can be expressed in terms of Kummer's confluent hypergeometric functions. For m ≥ 0 exponentially decaying unique solutions were found, while in the range −1<m<0 both exponentially and algebraically decaying multiple solutions occur. The origin of the multiple solutions as well as the feasibility conditions of all the above mentioned solutions is discussed in detail.

Free convection near a stagnation point in a porous medium resulting from an oscillatory wall temperature

A detailed theoretical study of unsteady free convection boundary-layer flow near the stagnation point of a two-dimensional cylindrical surface embedded in a fluid-saturated porous medium is presented when the surface temperature oscillates about a mean value above ambient. Both numerical and asymptotic solutions are employed to solve the governing equations for general values of the frequency ω and amplitude A of the surface temperature oscillations. Results for the heat transfer rate, boundary-layer thickness and temperature profiles are obtained for both small and large values of A and also for slow and fast oscillations. It is found that outside the thin boundary layer on the surface, a steady flow is induced for large times and fast oscillations. This steady boundary-layer flow is studied in detail for large values of the amplitude A.

Unsteady free convection flow in the stagnation-point region of a rotating sphere

The unsteady free convection boundary-layer flow in the forward stagnation-point region of a sphere, which is rotating with time-dependent angular velocity in an ambient fluid, has been studied. Both constant wall temperature and constant heat flux conditions have been considered. The non-linear coupled parabolic partial differential equations governing the flow have been solved numerically using an implicit finite-difference scheme. The skin friction and the heat transfer are enhanced by the buoyancy force. The effect of the buoyancy force is found to be more pronounced for smaller Prandtl numbers than for larger Prandtl numbers. For a given buoyancy force, the heat transfer increases with an increase in Prandtl number, but the skin friction decreases.

Application of the Group-Theoretical Method to Physical Problems

The concept of the theory of continuous groups of transformations has attracted the attention of applied mathematicians and engineers to solve many physical problems in the engineering sciences. Three applications are presented in this paper. The first one is the problem of time-dependent vertical temperature distribution in a stagnant lake. Two cases have been considered for the forms of the water parameters, namely water density and thermal conductivity. The second application is the unsteady freeconvective boundary-layer flow on a non-isothermal vertical flat plate. The third application is the study of the dispersion of gaseous pollutants in the presence of a temperature inversion. The results are found in closed form and the effect of parameters are discussed.

Three-dimensional darcian free convection near a stagnation point on an isothermal surface

The unsteady free convection boundary-layer flow in a fluid-saturated porous medium near a general three-dimensional stagnation point is considered. It is shown that the problem can be reduced to an equivalent two-dimensional problem by a simple transformation of variables. The form of solution then depends only on the sign of the quantity a + b, where a and b are the principal curvatures of the body at the stagnation point.

Semisimilar solutions for unsteady free-convective boundary-layer flow on a vertical flat plate

Semisimilar solutions for unsteady free-convective boundary-layer flow on a vertical flat plate