# Unsteady laminar compressible boundary layers with vectored mass transfer and an applied magnetic field

TL;DR: In this article, the authors dealt with the laminar unsteady compressible boundary layer flow of an electrically conducting viscous fluid at the stagnation point of an axi-symmetric blunt-nosed body with vectored mass transfer, non-zero dissipation parameter and an applied magnetic field.

Abstract: This paper deals with the laminar unsteady compressible boundary layer flow of an electrically conducting viscous fluid at the stagnation point of an axi-symmetric blunt-nosed body with vectored mass transfer, non-zero dissipation parameter $m(=u^2_s/2H_e)$ and an applied magnetic field. The semi-similar solutions have been obtained numerically using an implicit finite difference scheme in combination with the quasi-linearization technique. The results have been obtained for (i) an accelerating/decelerating stream and (ii) a fluctuating stream. The skin-friction responds to the fluctuation in the free-stream more compared to the heat transfer. In presence of the magnetic field velocity overshoot occurs which gives rise to the point of inflexion in velocity profile and for small wall values of the total enthalpy, the point of inflexion arises in velocity and total enthalpy profiles which may cause instability in the flows but this can be avoided in certain circumstances. Skin friction can be reduced by applying vectored mass transfer more compared to the mass transfer applied normal to the
surface. The results also show that the skin friction coefficient increases but Nusse it number decreases
as the magnetic parameter M increases.

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TL;DR: In this paper, the effects of non-uniform multiple slot injection (suction) and nonuniform wall enthalpy on the steady nonsimilar compressible laminar boundary-layer flow over a two-dimensional and axisymmetric bodies were investigated.

Abstract: An analysis is performed to study the effects of nonuniform multiple slot injection (suction) and nonuniform wall enthalpy on the steady nonsimilar compressible laminar boundary-layer flow over a two-dimensional and axisymmetric bodies. Nonsimilar solution is obtained from the origin of the stream-wise coordinate to the exact point of separation. The difficulties arising out at the starting point of the streamwise coordinate, at the edges of the slots and at the point of separation are overcome by applying the method of quasilinear implicit finite difference scheme with an appropriate selection of finer step sizes along the streamwise direction. It is found that the nonuniform multiple slot injection moves the point of separation downstream but the nonuniform multiple slot suction has the reverse effect. Also, separation can be delayed more effectively by applying nonuniform multiple slot injection as compared to the nonuniform single slot injection. The nonuniform total enthalpy at the wall (i.e., the cooling or heating of the wall in the slots) along the streamwise coordinate has very little effect on the skin friction and thus on the point of separation.

7 citations

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TL;DR: In this paper, non-similar solutions of steady two-dimensional and axi-symmetric compressible laminar boundary layer flows with non-uniform slot injection (or suction) have been obtained, from the starting point of the stream-wise coordinate to the exact point of separation.

Abstract: Non-similar solutions of steady two-dimensional and axi-symmetric compressible laminar boundary layer flows with non-uniform slot injection (or suction) have been obtained, from the starting point of the stream-wise co-ordinate to the exact point of separation. The difficulties arising at the starting point of the stream-wise co-ordinate, and at the edges of the slot and at the point of separation have been overcome by applying the method of quasi-linear implicit finite difference scheme, along with an appropriate selection of finer step size along the stream-wise direction. It is observed that the separation can be delayed by a non-uniform slot suction and also by moving the slot downstream but the effect of non-uniform slot injection is just the opposite. The increase of total enthalpy at the wall causes the separation to occur earlier while cooling delays it. The increase of Mach number shifts the point of separation upstream due to the adverse pressure gradient.

6 citations

### Additional excerpts

...constituting a vectored mass transfer discussed by Venkatachala and Nath [5], Inger and Swean [6], and Roy and Nath [7]....

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TL;DR: In this paper, the authors consider the non-parallel evolution of short waves and show that they can be described in terms of the solution of a system of ordinary differential equations.

Abstract: Buoyant boundary-layer flows, typified by the flow over a heated flat plate, have the curious property that they can exhibit regions of "overshoot" in which the streamwise velocity exceeds its free-stream value. A consequence of this is the streamwise velocity develops a local maximum and is inflectional in nature. It is therefore inviscidly unstable, and the fastest growing wave mode is known to be one whose wavelength is short compared to the boundary-layer thickness. In this work we consider the nonparallel evolution of these short waves and show that they can be described in terms of the solution of a system of ordinary differential equations. Numerical and asymptotic studies enable us to explain the ultimate fate of the wave and show, depending on a key parameter which is a function of the underlying boundary layer, that two possibilities can arise. Nonparallelism may be sufficiently stabilizing so as to extinguish the linearly unstable waves or, in other cases, the mode may intensify but concentrate itself in a very thin zone surrounding the maximum in the streamwise velocity. These findings enable us to give some indication of the part these modes play in the transition to turbulence in buoyant boundary layers.

2 citations

##### References

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01 Jan 1955

TL;DR: The flow laws of the actual flows at high Reynolds numbers differ considerably from those of the laminar flows treated in the preceding part, denoted as turbulence as discussed by the authors, and the actual flow is very different from that of the Poiseuille flow.

Abstract: The flow laws of the actual flows at high Reynolds numbers differ considerably from those of the laminar flows treated in the preceding part. These actual flows show a special characteristic, denoted as turbulence. The character of a turbulent flow is most easily understood the case of the pipe flow. Consider the flow through a straight pipe of circular cross section and with a smooth wall. For laminar flow each fluid particle moves with uniform velocity along a rectilinear path. Because of viscosity, the velocity of the particles near the wall is smaller than that of the particles at the center. i% order to maintain the motion, a pressure decrease is required which, for laminar flow, is proportional to the first power of the mean flow velocity. Actually, however, one ob~erves that, for larger Reynolds numbers, the pressure drop increases almost with the square of the velocity and is very much larger then that given by the Hagen Poiseuille law. One may conclude that the actual flow is very different from that of the Poiseuille flow.

17,321 citations

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30 Nov 1961

TL;DR: In this article, the authors propose Matrix Methods for Parabolic Partial Differential Equations (PPDE) and estimate of Acceleration Parameters, and derive the solution of Elliptic Difference Equations.

Abstract: Matrix Properties and Concepts.- Nonnegative Matrices.- Basic Iterative Methods and Comparison Theorems.- Successive Overrelaxation Iterative Methods.- Semi-Iterative Methods.- Derivation and Solution of Elliptic Difference Equations.- Alternating-Direction Implicit Iterative Methods.- Matrix Methods for Parabolic Partial Differential Equations.- Estimation of Acceleration Parameters.

5,317 citations

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TL;DR: The possibility of reducing stagnation point heat-transfer rates on blunt bodies at hypersonic speeds by means of a magnetic field is considered in this paper, where the modification of the flow within and external to the viscous boundary layer is analyzed.

Abstract: The possibility of reducing stagnation point heat-transfer rates on blunt bodies at hypersonic speeds by means of a magnetic field is considered. The modification of the flow within and external to the viscous boundary layer is analyzed. It is concluded that the primary mechanism which serves to reduce the heat transfer is an alteration of the inviscid flow external to the boundary layer. The existence of a magnetic-fluid boundary layer is indicated for fluids of high electrical conductivit}^ This magnetic boundary layer is seen to parallel viscous boundary-layer theory closely. On the basis of a cursory examination it does not seem practicable to reduce significantly the aerodynamic heating load on flat plates by magnetic techniques unless the electrical conductivity of air is artificially enhanced.

169 citations