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Marangoni forced convective Casson type nanofluid flow in the presence of Lorentz force generated by Riga plate

  • * Corresponding author: Anum Shafiq

    * Corresponding author: Anum Shafiq 
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  • The present communication aims to investigate Marangoni based convective Casson modeled nanofluid flow influenced by the presence of Lorentz forces instigated into the model by an aligned array of magnets in the form of Riga pattern. The exponentially decaying Lorentz force is considered using the Grinberg term. On the liquid - gas or liquid - liquid interface, a realistic temperature and concentration distribution is considered with the assumption that temperature and concentration distributions are variable functions of $ x $. The set of so-formulated governing problems under the umbrella of Navier Stokes equations is transformed into nonlinear ODEs using suitable transformations. Homotopy approach is implemented to achieve convergent series solutions for the said problem. Influence of active fluid parameters such as Casson parameter, Brownian diffusion, Prandtl number, Thermophoresis and others on flow profiles is analyzed graphically. The fluctuation in local physical quantities such as heat and mass flux rates, is noticed to check the significance of current fluid model in many industrial as well as engineering procedures using nanofluids. The outcomes indicate that the effective Lorentz force assists the fluid motion that results in an augmented velocity profile with incremental values of modified Hartman number. Furthermore, incremental data of Casson parameter motivates significant reduction in velocity profile.

    Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.


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  • Figure 1.  Geometry of the problem

    Figure 2.  HAM Curves

    Figure 3.  Variations noted in $ f'(\eta) $ for incremental $ \beta $

    Figure 4.  Variations noted in $ f'(\eta) $ for incremental $ Q_1 $

    Figure 5.  Variations noted in $ f'(\eta) $ for incremental $ r $

    Figure 6.  Variations noted in $ \theta(\eta) $ for incremental $ Q_1 $

    Figure 7.  Variations noted in $ \theta(\eta) $ for incremental $ N_b $

    Figure 8.  Variations noted in $ \theta(\eta) $ for incremental $ Nt $

    Figure 9.  Variations noted in $ \theta(\eta) $ for incremental $ Ec $

    Figure 10.  Variations noted in $ \phi(\eta) $ for incremental $ Sc $

    Figure 11.  Variations noted in $ \phi(\eta) $ for incremental $ Nt $

    Figure 12.  Variations noted in Nusselt number for incremental $ Nt $

    Figure 13.  Variations noted in Sherwood number for incremental $ Sc $

    Table Ⅰ.  Convergence

    Orders $-f^{\prime \prime }$ $-\theta ^{\prime }$ $-\phi ^{\prime }$
    $1$ $1.141912$ $0.8995612$ $1.5211265$
    $5$ $1.256923$ $0.9566101$ $1.3211122$
    $10$ $1.300122$ $1.1223114$ $1.1886111$
    $15$ $1.300222$ $1.2623532$ $0.9956332$
    $20$ $1.300222$ $1.3112112$ $0.8915622$
    $25$ $1.300222$ $1.3112211$ $0.8915512$
    $30$ $1.300222$ $1.3112211$ $0.8915512$
    $35$ $1.300222$ $1.3112211$ $0.8915512$
    $40$ $1.300222$ $1.3112211$ $0.8915512$
    $50$ $1.300222$ $1.3112211$ $0.8915512$
     | Show Table
    DownLoad: CSV

    Table Ⅱ.  Comparison of Nusselt and Sherwood numbers with Hayat et al. [11]

    $\beta$ $Q_1 $ $Sc$ $Nb$ $Nt$ $Pr$ $Ec$ ${Re}_{x}^{-1/2}Nu_{x}$ ${Re}_{x}^{-1/2}Nu_{x} $ ${Re}_{x}^{-1/2}Sh_{x}$
    Current Hayat et al. [11]
    $0.35$ $0.1$ $0.1$ $0.5$ $0.5$ $0.3$ $0.2$ $0.7801$ $0.7799$ $0.6232$
    $0.4$ $0.7810$ $0.7816$ $0.6200$
    $0.6$ $0.7820$ $0.7833$ $0.6155$
    $0.6$ $0.0$ $0.1$ $0.5$ $0.5$ $0.3$ $0.2$ $0.7790$ $0.7855$ $0.5662$
    $0.2$ $0.7770$ $0.7829$ $0.5524$
    $0.4$ $0.7720$ $0.7778$ $0.5412$
    $0.6$ $0.1$ $0.0$ $0.5$ $0.5$ $0.3$ $0.2$ $0.7901$ $0.8100$ $0.7100$
    $0.2$ $0.7799$ $0.7836$ $0.7100$
    $0.4$ $0.7400$ $0.7566$ $0.7101$
    $0.6$ $0.1$ $0.1$ $0.1$ $0.5$ $0.3$ $0.2$ $0.8101$ $--$ $0.6525$
    $0.3$ $0.8002$ $--$ $0.6580$
    $1.5$ $0.7800$ $--$ $0.6612$
    $0.6$ $0.1$ $0.1$ $0.5$ $0.1$ $0.3$ $0.2$ $0.8536$ $--$ $0.5412$
    $0.3$ $0.8825$ $--$ $0.5300$
    $0.5$ $0.9122$ $--$ $0.5121$
    $0.6$ $0.1$ $0.1$ $0.5$ $0.5$ $1.2$ $0.2$ $1.1121$ $1.1532$ $1.0021$
    $2.2$ $1.2112$ $1.4241$ $1.1121$
    $3.2$ $1.3001$ $1.6521$ $1.2231$
    $0.6$ $0.1$ $0.1$ $0.5$ $0.5$ $0.3$ $0.2$ $0.7921$ $0.8081$ $0.7101$
    $0.3$ $0.7600$ $0.7833$ $0.7200$
    $0.4$ $0.7401$ $0.7586$ $0.7405$
     | Show Table
    DownLoad: CSV
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