Flow of MHD Casson carbon nanotubes with mass and heat transfer under Marangoni thermosolutal convection in a porous medium: analytical solution

The flow of a Casson fluid with thermal radiation and Marangoni thermosolutal convection transpiration is analyzed in this study. The carbon nanotube particles are immersed inside the fluid to achieve better thermal efficiency. Temperature gradients and solute concentrations define surface tension. Figure 1 shows a schematic representation of fluid flow.

Suppose the surface of the fluid moves towards X axis. The governing equations can be defined as follows taking into account the above-mentioned premises (See34.35).

$$frac{partial u}{{partial x}} + frac{partial v}{{partial y}} = 0.$$

(1)

$$ufrac{partial u}{{partial x}} + vfrac{partial v}{{partial y}} = nu_{nf} left( {1 + frac{1} {Lambda }} right)frac{{partial^{2} u}}{{partial y^{2} }} – left( {frac{{mu_{nf} }}{{ rho_{nf} K}} + frac{{sigma_{nf} B_{0}^{2} }}{{rho_{nf} }}} right),$$

(2)

begin{aligned} ufrac{partial T}{{partial x}} + vfrac{partial T}{{partial y}} = & ,frac{{kappa_{nf } }}{{left( {rho C_{P} } right)_{nf} }}frac{{partial^{2} T}}{{partial y^{2} }} + frac{{mu_{nf} }}{{left( {rho C_{P} } right)_{nf} }}left( {frac{partial u}{{partial y} }} right)^{2} + left( {frac{{mu_{nf} }}{{left( {rho C_{P} } right)_{nf} K}} + frac{{sigma_{nf} B_{0}^{2} }}{{left( {rho C_{P} } right)_{nf} }}} right)u^{2} & – ,frac{1}{{left( {rho C_{P} } right)_{nf} }}frac{{partial q_{r} }}{partial y} + frac{{Q_{0} }}{{left( {rho C_{P} } right)_{nf} }}left( {T – T_{infty } } right), end{aligned}

(3)

$$ufrac{partial C}{{partial x}} + vfrac{partial C}{{partial y}} = Dfrac{{partial^{2} C}}{{ partial y^{2} }} – Gleft( {C – C_{infty } } right),$$

(4)

Here, the term Casson’s fluid is used to characterize the non-Newtonian fluid. The magnetic term and the porous medium term are used for many scientific and technological phenomena. The effect of porous medium and thermal radiation play a major role in fluid flow as these effects in fluid flow are used in many industrial and real life applications. The heat source/sink in the fluid flow influences the heat transfer characteristics because there is a substantial difference in temperature between the surface and the fluid. Moreover, the combination of mass transfer and heat source/sink helps to overcome the problem of boundary layer separation.

The surface tension as well as the heat and mass limits are given by (see36,37,38)

$$sigma = sigma_{0} left[ {1 – gamma_{T} left( {T – T_{infty } } right) – gamma_{C} left( {C – C_{infty } } right)} right],$$

(5)

The surface tension coefficients for heat and mass respectively are given by

$$gamma_{T} = – frac{1}{{sigma_{0} }}left( {frac{partial sigma }{{partial T}}} right)_{T} ,,,,,{text{and}},,,gamma_{C} = – frac{1}{{sigma_{0} }}left( {frac{ partial sigma }{{partial C}}} right)_{T} .$$

(6)

The terms of Eqs. (1–6) are specified in this section under Parts List.

Associate B. Cs

$$mu left( {frac{partial u}{{partial y}}} right)_{y = 0} = – left( {frac{partial sigma }{{partial x}}} right)_{y = 0} = sigma_{0} left( {gamma_{T} left( {frac{partial T}{{partial x}}} right) _{y = 0} + gamma_{C} left( {frac{partial C}{{partial x}}} right)_{y = 0} } right),$$

(seven)

$$left. begin{gathered} Vleft( {x,0} right) = V_{0} ,,,,,,,,,,,,,, ,,,,,,,,,,uleft( {x,infty } right) = 0 hfill Tleft( {x,0} right) = T_{infty } + T_{0} X^{2} ,,,,,,,,Tleft( {x,infty } right) = T_{infty } hfill Cleft( {x,0} right) = C_{infty } + C_{0} X^{2} ,,,,,,Cleft( {x, infty } right) = C_{infty } hfill end{gathered} right}.$$

(8)

here (X = frac{x}{L})and (L = – frac{mu nu }{{sigma_{0} T_{0} gamma_{T} }}) is the characteristic length, (T_{0} ,,{text{and}},,C_{0}) are constants. In addition, the following transformations are defined.

$$left. {begin{array}{*{20}l} {psi left( {x,y} right) = nu Xfleft( eta right),,,,eta = frac{y}{L}} hfill {Tleft( {x,y} right) = T_{infty } + T_{0} X^{2} theta left( eta right)} hfill {C = C_{infty } + C_{0} X^{2} phi left( eta right)} hfill end{array} } right} ,$$

(9)

Using the dimensional form the velocity components are given by

$$u = frac{nu }{L}f_{eta } left( eta right),,,,,v = – frac{nu }{L}fleft ( eta right).$$

(ten)

The value (q_{r}) can be defined based on the Rosseland approximation as follows (See39,40,41,42.

$$q_{r} = – frac{{4sigma^{*} }}{{3alpha_{r} }}frac{{partial T^{4} }}{partial y},$$

(11)

where the ambient temperature J4 expands in terms of the Taylor series as

$$T^{4} = T_{infty }^{4} + 4T_{infty }^{3} left( {T – T_{infty } } right) + 6T_{infty }^{ 2} left( {T – T_{infty } } right)^{2} + cdots$$

(12)

when the higher order elements of this equation are ignored, this results in

$$T^{4} cong 3T_{infty }^{3} + 4T_{infty }^{3} T.$$

(13)

By applying Eq. (8) in eq. (6), then the first order derivative of the heat flux can be given by

$$frac{{partial q_{r} }}{partial y} = – frac{{16sigma^{*} T_{infty }^{3} }}{{3alpha_{r } }}frac{{partial^{2} T}}{{partial y^{2} }}.$$

(14)

Therefore, eq. (3) can be rewritten as

begin{aligned} ufrac{partial T}{{partial x}} + vfrac{partial T}{{partial y}} = & ,frac{{kappa_{nf } }}{{left( {rho C_{P} } right)_{nf} }}frac{{partial^{2} T}}{{partial y^{2} }} + frac{{mu_{nf} }}{{left( {rho C_{P} } right)_{nf} }}left( {frac{partial u}{{partial y} }} right)^{2} + left( {frac{{mu_{nf} }}{{left( {rho C_{P} } right)_{nf} K}} + frac{{sigma_{nf} B_{0}^{2} }}{{left( {rho C_{P} } right)_{nf} }}} right)u^{2} & , + frac{1}{{left( {rho C_{P} } right)_{nf} }}frac{{16sigma^{*} T_{infty }^{ 3} }}{{3k^{*} }}frac{{partial^{2} T}}{{partial y^{2} }} + frac{{Q_{0} }}{{ left( {rho C_{P} } right)_{nf} }}left( {T – T_{infty } } right). end{aligned}

(15)

Using Eqs. (9) and (10) in the equations. (2) and (3) to get

$$varepsilon_{1} left( {1 + frac{1}{Lambda }} right)f_{eta eta eta } + varepsilon_{2} left( {ff_{eta eta } – f_{eta }^{2} } right) – left( {varepsilon_{1} Da^{ – 1} + varepsilon_{3} Q} right)f_{eta } = 0 ,$$

(16)

$$left( {varepsilon_{5} + R} right)theta_{eta eta } + Pr varepsilon_{4} left( {ftheta_{eta } + left( {I – 2f_{eta } } right)theta } right) + Ecleft( {varepsilon_{1} f_{eta eta }^{2} + left( {varepsilon_{1} Da^ { – 1} + varepsilon_{3} Q} right)f_{eta } } right) = 0.$$

(17)

$$phi_{eta eta } + Scleft( {fphi_{eta } – left( {delta + 2f_{eta } } right)phi } right) = 0,  (18) the B. C is reduced to$$begin{aligned} fleft( 0 right) = & V_{C} ,,,,f_{eta } left( infty right) = 0,,,, ,,,,,f_{eta eta } left( 0 right) = – 2left( {1 + M_{a} } right) theta left( 0 right) = & 1,,,,theta left( infty right) = 0,,,,phi left( 0 right) = 1,,,, phi left( infty right) = 0, end{aligned}$$(19) here (V_{C} = – frac{gamma }{L}{text{v}}_{0}) is the mass transpiration, here (V_{C} = 0), (V_{C} > 0) and (V_{C} indicates respectively the cases of aspiration, injection and non-permeability. (Pr = frac{kappa }{{mu C_{P} }},,,Sc = frac{nu }{D},,{text{and}}, delta = frac{{GL^{2} }}{nu }) indicates the Prandtl number, Schmidt number, chemical reaction coefficient respectively. (R = frac{{16sigma^{*} T_{infty }^{3} }}{{3alpha_{r} kappa }}) is the radiation number, (I = frac{{Q_{0} L^{2} }}{{varepsilon_{4} rho C_{P} nu }}) is the heat source or sink parameter, (Da^{ – 1} = frac{{L^{2} }}{K}) is the inverse of Darcy’s number, (Q = frac{{sigma B_{0}^{2} L^{2} }}{rho nu }) is the Chandrasekhar number, (Ec = frac{{gamma^{2} }}{{L^{2} T_{0} C_{P} }}) is the Eckert number, and finally (Ma = frac{{Ma_{C} }}{{Ma_{T} }}) is the Marangoni number (thermosolutal surface tension ratio), (Ma_{C} = frac{{sigma_{0} gamma_{C} C_{0} LC_{P} }}{kappa }) and (Ma_{T} = frac{{sigma_{0} gamma_{T} T_{0} LC_{P} }}{kappa }) are the solutal and thermal Marangoni numbers. Quantities of carbon nanofluid used in the equations. (16) and (17) can be defined as (See43.44)$$varepsilon_{1} = frac{{mu_{nf} }}{{mu_{f} }},,,varepsilon_{2} = frac{{rho_{nf} }} {{rho_{f} }},,,varepsilon_{3} = frac{{sigma_{nf} }}{{sigma_{f} }},,,varepsilon_{4} = frac{{left( {rho C_{P} } right)_{nf} }}{{left( {rho C_{P} } right)_{f} }},, ,varepsilon_{4} = frac{{kappa_{nf} }}{{kappa_{f} }}