Diffusion of liquid hydrogen in time-dependent MHD mixed convective flow

  • P. M. PatilEmail author
  • A. Shashikant


An innovative study on liquid hydrogen diffusion in time-dependent mixed convection flow is carried out in the presence of magnetic field effects. In fact, this is the first approach to analyze such flow problems, and also this is the first research paper to study the non-uniform heat sink/source and nonlinear chemical reaction in the presence of liquid hydrogen diffusion. Initially, the governing equations are reduced to dimensionless form by using non-similar transformations and are linearized by applying quasilinearization technique. Then, the finite difference approximation is utilized to discretize the resulting equations. The mixed convection is analyzed along with exponentially stretching surface through various graphs on profiles as well as gradients. The results display that the non-uniform heat source parameter increases the fluid velocity as well as temperature, and the magnetic parameter reduces the friction at the wall. Specifically, the skin friction coefficient decreases about 40% in the presence of magnetic field. The mass transfer rate increases for high-order chemical reaction and for destructive chemical reaction rate. The mass transfer rate is found to be high for the diffusion of liquid nitrogen than that for the diffusion of liquid hydrogen. In fact, the mass transfer rate increases about 22% for the diffusion of liquid nitrogen. This study can assist the design engineers who are working in pertain to the diffusion of liquid gases in mixed convection regimes. Also, the obtained data in the present study can be more useful for future investigations about time-dependent mixed convection nanofluid flow problems.


Exponentially stretching surface Liquid hydrogen Magnetohydrodynamic (MHD) flow Nonlinear chemical reaction Non-uniform heat source/sink Unsteady effect 

List of symbols

\( f \)

Dimensionless stream function

\( N = \frac{{\lambda_{1} }}{\lambda } \)

Buoyancy ratio parameter

\( B^{*} \)

Temperature-dependent heat sink/source

\( U_{\infty } \)

Free stream velocity constant

\( T_{{{\text{w}}_{0} }} \)

Reference temperature

C, D

Species concentration and mass diffusivity, respectively

\( A^{*} \)

Space-dependent heat source/sink

\( C_{{{\text{w}}_{0} }} \)

Reference concentration

\( B_{0} \)

Magnetic field strength

\( R \)

Dimensionless species concentration

\( Q \)

Dimensionless temperature

\( T_{\text{w}} \)

Temperature at the wall

\( C_{\infty } \)

Ambient species concentration

\( C_{\text{f}} \)

Skin friction coefficient

\( k \)

Thermal conductivity

\( C_{\text{p}} \)

Specific heat due to constant pressure

\( g \)

Acceleration due to gravity

\( {\text{Re}} = \frac{{U_{0} L}}{\nu } \)

Reynolds number

\( C_{\text{w}} \)

Concentration at the wall

\( {\text{Nu}} \)

Nusselt number

\( U_{0} \)

Reference velocity

\( P \)

Dimensionless velocity

\( L \)

Characteristic length

\( k_{1} \)

Chemical reaction rate

\( x,y \)

Cartesian coordinates

\( { \Pr } = \frac{\nu }{{\alpha_{\text{m}} }} \)

Prandtl number

\( T_{\infty } \)

Ambient temperature

\( t \)


\( M = \,\frac{{B_{0} }}{{U_{0} }}\sqrt {\frac{\sigma \nu }{\rho }} \)

Magnetic parameter

\( {\text{Sh}} \)

Sherwood number

\( v \)

Velocity component in the \( y \)-direction

\( n \)

Order of chemical reaction

\( {\text{Sc}} = \frac{\nu }{D} \)

Schmidt number

\( U_{\text{w}} \)

Stretching wall velocity

\( T \)


\( u \)

Velocity component in the \( x \)-direction

\( {\text{Ec}} = \frac{{U_{0}^{2} }}{{C_{\text{p}} (T_{{{\text{w}}_{0} }} - T_{\infty } )}} \)

Eckert Number

\( q^{\prime\prime\prime} = \left( {\frac{{kU_{\text{w}} \left( {x,t} \right)}}{x\nu }} \right)\left[ {A^{*} \left( {T_{\text{w}} - T_{\infty } } \right)P + B^{*} (T - T_{\infty } )} \right] \)

Non-uniform heat sink/source

\( {\text{Gr}} = \frac{{g\beta (T_{{{\text{w}}_{0} }} - T_{\infty } )L^{3} }}{{\nu^{2} }} \), \( {\text{Gr}}^{*} = \frac{{g\beta^{*} (C_{{{\text{w}}_{0} }} - C_{\infty } )L^{3} }}{{\nu^{2} }} \)

Grashof numbers due to temperature and concentration, respectively

Greek symbols

\( \lambda = \frac{\text{Gr}}{{{\text{Re}}^{2} }} \), \( \lambda_{1} = \frac{{{\text{Gr}}^{*} }}{{{\text{Re}}^{2} }} \)

Buoyancy parameters due to temperature and concentration gradients, respectively

\( \varepsilon = \,\frac{{U_{\infty } }}{{U_{0} }} \)

Ratio of mainstream velocity to the reference velocity

\( \alpha_{\text{m}} \)

Thermal diffusivity

\( \alpha \)

Unsteady parameter

\( \xi ,\eta ,\tau \)

Transformed variables

\( \Delta \)

Chemical reaction parameter

\( \beta ,\beta^{*} \)

Volumetric coefficients of thermal and concentration expansions, respectively

\( \phi \left( \tau \right) \)

Unsteady function of \( \tau \)

\( \nu \)

Kinematic viscosity

\( \rho \)


\( \psi \)

Stream function

\( \sigma \)

Electrical conductivity



This work is supported by the University Grants Commission (UGC), New Delhi under the Grant No. F. 510/3/DRS-III/2016 dated 29-02-2016 by UGC-SAP-DRS-III.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2019

Authors and Affiliations

  1. 1.Department of MathematicsKarnatak UniversityDharwadIndia

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