Inspection of thermal jump conditions on nanofluids with nanoparticles and multiple slip effects

0

Flow Description

A stable two-dimensional flow of nanofluid with nanoparticles (iron oxide (left( {Fe_{3} O_{4} } right))Zirconium dioxide (left( {ZrO_{2} } right))and Titanium (left( {Ti} right))) through a disk in the existence of boundary conditions of slip and thermal radiation is considered. The cylindrical coordinate system is (left( {r,varphi ,z} right)). the (left({u,w} right)) are components of the velocity in the directions of (left( {r,varphi ,z} right)) as shown in Fig. 1. In the axial direction of the disk, a magnetic force (left( {B_{0} } right)) of constant intensity is provided. Moreover, we can ignore the induced magnetic field by assuming a low magnetic Reynolds number. In the existence of a nanofluid phase model, heat transport is also embedded at the disc surface.

Figure 1

Flow geometry of the considered problem.

Nonlinear dimensional equations

The main governing PDEs are (Iqbal et al.42):

$$u_{r} + frac{u}{r} + w_{z} = 0.$$

(1)

$$ rho_{nf} left( {uu_{r} + frac{{v^{2} }}{r} + wu_{z} } right) = – p_{r} + mu_{nf } left( {u_{rr} + frac{1}{r}u_{r} – frac{u}{{r^{2} }} + u_{zz} } right) – sigma_{ nf} B_{0}^{2} u, $$

(2)

$$rho_{nf} left( {uv_{r} + frac{uv}{r} + wv_{z} } right) = mu_{nf} left( {v_{rr} + frac {1}{r}v_{r} – frac{v}{{r^{2} }} + v_{zz} } right) – sigma_{nf} B_{0}^{2} v, $$

(3)

$$ rho_{nf} left( {uw_{r} + ww_{z} } right) = – p_{r} + mu_{nf} left( {w_{rr} + frac{1} {r}w_{r} + w_{zz} } right), $$

(4)

$$left( {rho C_{p} } right)_{{nf}} left( {uT_{r} + wT_{z} } right) = k_{{nf}} left( { T_{{rr}} + frac{1}{r}T_{r} + T_{{zz}} } right) – q_{{r_{z} }} ,$$

(5)

With

$$ q_{r} = – frac{{4sigma^{*} }}{{3k^{*} }}T_{z}^{4} = – frac{{16sigma^{* } }}{{3k^{*} }}T^{3} T_{z} . $$

Conditions to the limits

With boundary constraints

$$left. begin{gathered} u = L_{1} u_{z} ,v = L_{1} v_{z} + rOmega ,w = 0,T = T_{w} + L_{2} T_{z} ,at,z = 0, hfill u to 0,,,,,,,,,v to 0,,,,,, ,T to T_{infty } ,,,,,p to p_{infty } ,,,,as,,,,,,,z to infty hfill end{gathered} right}.$$

(6)

Transform Variables

The following transformation variables are

$$left. begin{gathered} zeta = zsqrt {frac{{U_{0} }}{{rv_{f} }}} u = rOmega f^{prime}left( zeta right) ,v = rOmega gleft( zeta right),w = – 2sqrt {Omega v_{f} } gleft( zeta right), hfill p = p_{infty } – Omega mu_{f} Pleft( zeta right),T = T_{infty } + left( {T_{w} – T_{infty } } right)theta left( zeta right) hfill end{gathered} right}. $$

(7)

Right here, (left( {u,v& w} right)) speed components (left( {rho_{nf} } right)) are the density, (left( {mu_{nf} } right)) is the dynamic viscosity, and (left( {sigma_{nf} } right)) its nanofluid electrical conductivity, (left( {L_{1} } right)) wall slip coefficient, (left( {L_{2} } right)) temperature jump coefficient, (left( P right)) is the pressure, and (left( {U_{0} = Omega r} right)) free current velocity, respectively.

Dimensionless equations

The dimensionless results of the main governing equations are

$$2frac{{v_{nf} }}{{v_{f} }}f^{primeprimeprime} – f^{{prime}{2}} + g^{2} + 4ff ^{primeprime} – frac{{rho_{f} }}{{rho_{nf} }}M^{2} f^{prime} = 0.$$

(8)

$$2frac{{v_{nf} }}{{v_{f} }}g^{primeprime} + 2fg^{prime} – 2f^{prime}g – frac{{rho_ {f} }}{{rho_{nf} }}M^{2} g = 0.$$

(9)

$$ frac{{v_{nf} }}{{v_{f} }}f^{primeprime} + ff^{primeprime} – frac{{rho_{f} }}{ {rho_{nf} }}frac{dP}{{dzeta }} = 0, $$

(ten)

$$frac{{left( {rho C_{p} } right)_{f} }}{{left( {rho C_{p} } right)_{nf} }}left ( {frac{{k_{nf} }}{{k_{f} }} + Rd} right)theta^{primeprime} + Pr ftheta^{prime} = 0,$ $

(11)

With

$$left. begin{gathered} fleft( 0 right) = 0,f^{prime}left( 0 right) = alpha f^{primeprime}left( 0 right),g left( 0 right) = 1 + alpha g^{prime}left( 0 right),theta left( 0 right) = 1 + beta theta^{prime}left( 0 right), hfill f^{prime} to 0,,,,,,,,,,,,g to 0,,, ,,,,,,,,,P to 0,,,,,,,,,,,theta to 0, ,,,,,,,,,,,,When,,,,zeta to infty hfill end{gathered} right }.$$

(12)

Reduced settings

$$left. {left( {Pr = frac{{mu_{f} left( {C_{p} } right)_{f} }}{{k_{f} }}} right),left ( {M^{2} = frac{{sigma_{nf} B_{0}^{2} }}{{Omega rho_{f} }}} right),left( {alpha = L_{1} sqrt {frac{Omega }{{v_{f} }}} } right),left( {beta = L_{2} sqrt {frac{Omega }{{v_ {f} }}} } right),left( {Rd = frac{{4sigma T_{infty }^{3} }}{{k^{*} k_{f} }}} right)} right}.$$

(13)

Right here (left( M right)) is the magnetic parameter, (left({Rd} right)) is the thermal radiation parameter, (left( beta right)) is the thermal slip parameter,(left( alpha right)) is the velocity slip parameter, and (left( {Pr } right)) is the Prandtl number.

The engineering parameters are:

$$left. begin{gathered} C_{f} left( { = frac{{sqrt {tau_{r}^{2} + tau_{theta }^{2} } }}{{rho_{nf } left( {rOmega } right)^{2} }}} right), hfill Nu_{r} left( { = frac{{k_{nf} }}{{k_{ f} }}frac{{rq_{w} }}{{left( {T_{w} – T_{infty } } right)}}} right) hfill end{gathered} right}.$$

(14)

The dimensionless results of the engineering parameters

$$left. {left. {tau_{w} = mu_{nf} left( {u_{z} + w_{r} } right)} right|_{z = 0} ,left. {tau_{theta } = mu_{nf} left( {v_{z} + w_{r} } right)} right|_{z = 0} ,left. {q_{w} = – k_{nf} left( {T_{z} } right)} right|_{z = 0} } right}.$$

(15)

$${text{Re}}_{r}^{frac{1}{2}} C_{f} = frac{{mu_{nf} }}{{mu_{f} }} left( {f^{primeprime}left( 0 right)^{2} + g^{prime}left( 0 right)^{2} } right)^{frac{1 }{2}} ,$$

(16)

$$ {text{Re}}_{r}^{{ – frac{1}{2}}} Nu_{r} = – frac{{k_{nf} }}{{k_{f} } }Rdtheta^{prime}left( 0 right), $$

(17)

Right here ({text{Re}}_{r} left( { = frac{{2Omega r^{2} }}{{v_{f} }}} right)) is the local Reynolds number.

Digital scheme

The system of ODE (08–11) of the flow model with boundary conditions (12) is investigated numerically using the efficiency and strength of numerical computation in terms of Lobatto IIIA technique (bvp4c) and the tool MATLAB computing. The graphical results generated demonstrate the difference in momentum, pressure and temperature profiles with respect to different physical factors. System of ordinary differential equations (08–11) transformed into first-order ordinary differential equations for a solution using Lobatto IIIA.

To leave

$$left. begin{collected} f = q_{1} ,f^{prime} = q_{2} ,f^{primeprime} = q_{3} ,f^{primeprimeprime} = q ^{prime}_{3} ,g = q_{4} ,g^{prime} = q_{5} ,g^{primeprime} = q^{prime}_{5} , hfill P = q_{6} ,P^{prime} = q^{prime}_{6} ,theta = q_{7} ,theta^{prime} = q_{8} , theta^{primeprime} = q^{prime}_{8} hfill end{gathered} right},$$

(18)

$$q^{prime}_{3} = left( {q_{2}^{2} – q_{4}^{2} – 4q_{1} q_{3} + frac{{rho_ {f} }}{{rho_{nf} }}M^{2} q_{2} } right)frac{{v_{f} }}{{2v_{nf} }},$$

(19)

$$q^{prime}_{5} = left( { – 2q_{1} q_{5} + 2q_{2} q_{4} + frac{{rho_{f} }}{{ rho_{nf} }}M^{2} q_{2} } right)frac{{v_{f} }}{{2v_{nf} }},$$

(20)

$$q^{prime}_{6} = left( { – frac{{v_{nf} }}{{v_{f} }}q_{3} – q_{1} q_{3} } right)frac{{rho_{nf} }}{{rho_{f} }},$$

(21)

$$q^{prime}_{8} = frac{{ – Pr q_{1} q_{8} frac{{left( {rho C_{p} } right)_{nf} }}{{left( {rho C_{p} } right)_{f} }}}}{{left( {frac{{k_{nf} }}{{k_{f} }} + Rd} right)}},$$

(22)

With

$$left. begin{gathered} q_{1} left( 0 right) = 0,q_{2} left( 0 right) = alpha q_{3} left( 0 right),q_{4} left( 0 right) = 1 + alpha q_{5} left( 0 right),q_{7} left( 0 right) = 1 + beta q_{8} left( 0 right) , hfill q_{2} to 0,,,,,,,,,,,q_{4} to 0,,,,, ,,,,,,,q_{6} to 0,,,,,,,,,,,,q_{7} to 0 ,,,,,,,,,,,,,,,,,zeta to infty hfill end{gathered} right}.$$

(23)

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