Lie-group method of solution for steady two-dimensional boundary-layer stagnation-point flow towards a heated stretching sheet placed in a porous medium

Boutros, Y. Z.; Abd‐el‐Malek, Mina B.; Badran, Nagwa A.; Hassan, Hossam S.

doi:10.1007/s11012-006-9014-x

Cited by 44 publications

(24 citation statements)

References 11 publications

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“…This phase depends greatly on the bottom bathymetry. As a model of this initial tsunami displacement, we consider the wave presented by George [7] which is given by (27).…”

Section: Tsunami Propagation Wavementioning

confidence: 99%

“…In fact, since the water at  x 250 km seems likely to acquire instantaneously, a velocity different from zero, Figure 6. (26) for an initial flow representing the tsunami propagation wave given by (27), in case of neglecting and including the Coriolis force at t = 400, 1200 and 3000 sec respectively. As mentioned before, in case of neglecting the Coriolis force, the velocity vector is in the direction of the wave propagation and minor oscillations will appear in the results when the wave propagates over the topography, Figure 8.…”

Section: The Velocity Component In the X-directionmentioning

confidence: 99%

“…Figures 10 and 11 illustrate the behavior of the velocity v over the topography ) (x  representing a hump given by (25) for an initial flow representing the dam break wave give n by (28) and a rising up function given by (26) for an initial flow representing the tsunami propagation wave given by (27) with the effect of Coriolis force. the dam break wave given by (28) in case of neglecting and including the Coriolis force at t = 400, 1200 and 1800 sec respectively at y = 0.…”

Section: The Velocity Component In the Y-directionmentioning

confidence: 99%

“…the dam break wave given by (28) in case of neglecting and including the Coriolis force at t = 400, 1200 and 1800 sec respectively at y = 0. (26) for an initial flow representing by the tsunami propagation wave given by (27) in case of neglecting and including the Coriolis force at t = 400, 1200 and 3000 sec respectively at y = 0. Since the water at 250  x km seems likely to acquire instantaneously, it is plausible that a shock would be created instantly on the upstream side and a relatively small propagated wave in the negative X-direction due to bottom topography as shown in the circle in Figure 12.…”

Section: The Velocity Component In the Y-directionmentioning

confidence: 99%

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Numerical Solution of the Rotating Shallow Water Flows with Topography Using the Fractional Steps Method

Hassan¹,

Ramadan²,

Hanna³

2010

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The two-dimensional nonlinear shallow water equations in the presence of Coriolis force and bottom topography are solved numerically using the fractional steps method. The fractional steps method consists of splitting the multi-dimensional matrix inversion problem into an equivalent one dimensional problem which is successively integrated in every direction along the characteristics using the Riemann invariant associated with the cubic spline interpolation. The height and the velocity field of the shallow water equations over irregular bottom are discretized on a fixed Eulerian grid and time-stepped using the fractional steps method. Effects of the Coriolis force and the bottom topography for particular initial flows on the velocity components and the free surface elevation have been studied and the results are plotted.

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“…This phase depends greatly on the bottom bathymetry. As a model of this initial tsunami displacement, we consider the wave presented by George [7] which is given by (27).…”

Section: Tsunami Propagation Wavementioning

confidence: 99%

Section: The Velocity Component In the X-directionmentioning

confidence: 99%

Section: The Velocity Component In the Y-directionmentioning

confidence: 99%

Section: The Velocity Component In the Y-directionmentioning

confidence: 99%

See 2 more Smart Citations

Numerical Solution of the Rotating Shallow Water Flows with Topography Using the Fractional Steps Method

Hassan¹,

Ramadan²,

Hanna³

2010

Self Cite

View full text Add to dashboard Cite

show abstract

“…Lie-group method is applied to the equations of motion, energy and diffusion for determining symmetry reductions of partial differential equations, [12][13][14][15][16][17][18][19][20][21][22][23][24]. Exact solutions for the temperature have been obtained and we used them to get the velocity and the concentration using MATLAP package.…”

Section: Introductionmentioning

confidence: 99%

Lie Group Method for Studying the Heat Generation Effect on Freeconvection Laminar Boundary-layer Flow Over a Vertical Flat Plate

Na²,

Hs³

et al. 2015

J Generalized Lie Theory Appl

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The nonlinear equations of heat and mass transfer in two-dimensional free-convection, laminar, boundary layer flow of a viscous incompressible fluid over a vertical plate with thermophoresis and heat generation effect have been considered. We apply Lie-group method for determining symmetry reductions of partial differential equations. Liegroup method starts out with a general infinitesimal group of transformations under which the given partial differential equations are invariant. The determining equations are a set of linear differential equations, the solution of which gives the transformation function or the infinitesimals of the dependent and independent variables. After the group has been determined, a solution to the given partial differential equations may be found from the invariant surface condition such that its solution leads to similarity variables that reduce the number of independent variables of the system. The effect of the heat generation parameter He, the Prandtl number Pr, the Schimted number Sc, the thermophoretic parameter τ, the solutal Grashof number Gc and the thermal Grashof number Gr on velocity, concentration and temperature have been studied and the results are plotted.

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Application of differential transform method to unsteady free convective heat transfer of a couple stress fluid over a stretching sheet

Kumar

Reddy

Kumar

et al. 2018

Heat Trans. Asian Res.

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In the present article, the transient rheological boundary layer flow over a stretching sheet with heat transfer is investigated, a topic of relevance to non‐Newtonian thermal materials processing. Stokes couple stress model is deployed to simulate non‐Newtonian characteristics. Similarity transformations are utilized to convert the governing partial differential equations into nonlinear ordinary differential equations with appropriate wall and free stream boundary conditions. The nondimensional boundary value problem emerging is shown to be controlled by a number of key thermophysical and rheological parameters, namely the rheological couple stress parameter (β), unsteadiness parameter (A), Prandtl number (Pr), buoyancy parameter (λ). The semi‐analytical differential transform method (DTM) is used to solve the reduced nonlinear coupled ordinary differential boundary value problem. A numerical solution is also obtained via the MATLAB built‐in solver “bvp4c” to validate the results. Further validation with published results from the literature is included. Fluid velocity is enhanced with increasing couple stress parameter, whereas it is decreased with unsteadiness parameter. Temperature is elevated with couple stress parameter, whereas it is initially reduced with unsteadiness parameter. The flow is accelerated with increasing positive buoyancy parameter (for heating of the fluid), whereas it is decelerated with increasing negative buoyancy parameter (cooling of the fluid). Temperature and thermal boundary layer thickness are boosted with increasing positive values of buoyancy parameter. Increasing Prandtl number decelerates the flow, reduces temperatures, increases momentum boundary layer thickness, and decreases thermal boundary layer thickness. Excellent accuracy is achieved with the DTM approach.

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Lie-group method of solution for steady two-dimensional boundary-layer stagnation-point flow towards a heated stretching sheet placed in a porous medium

Cited by 44 publications

References 11 publications

Numerical Solution of the Rotating Shallow Water Flows with Topography Using the Fractional Steps Method

Numerical Solution of the Rotating Shallow Water Flows with Topography Using the Fractional Steps Method

Lie Group Method for Studying the Heat Generation Effect on Freeconvection Laminar Boundary-layer Flow Over a Vertical Flat Plate

Application of differential transform method to unsteady free convective heat transfer of a couple stress fluid over a stretching sheet

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