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Cattaneo-Christov Heat Flux Model for MHD Three-Dimensional Flow of Maxwell Fluid over a Stretching Sheet.

Rubab K, Mustafa M - PLoS ONE (2016)

Bottom Line: The governing partial differential equations even after employing the boundary layer approximations are non linear.It is noticed that velocity decreases and temperature rises when stronger magnetic field strength is accounted.Penetration depth of temperature is a decreasing function of thermal relaxation time.

View Article: PubMed Central - PubMed

Affiliation: School of Natural Sciences (SNS), National University of Sciences and Technology (NUST), Islamabad 44000, Pakistan.

ABSTRACT
This letter investigates the MHD three-dimensional flow of upper-convected Maxwell (UCM) fluid over a bi-directional stretching surface by considering the Cattaneo-Christov heat flux model. This model has tendency to capture the characteristics of thermal relaxation time. The governing partial differential equations even after employing the boundary layer approximations are non linear. Accurate analytic solutions for velocity and temperature distributions are computed through well-known homotopy analysis method (HAM). It is noticed that velocity decreases and temperature rises when stronger magnetic field strength is accounted. Penetration depth of temperature is a decreasing function of thermal relaxation time. The analysis for classical Fourier heat conduction law can be obtained as a special case of the present work. To our knowledge, the Cattaneo-Christov heat flux model law for three-dimensional viscoelastic flow problem is just introduced here.

No MeSH data available.


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pone.0153481.g001: Geometry of the problem.

Mentions: Consider the flow of upper-convected Maxwell fluid induced by an elastic sheet stretching in two lateral directions. The sheet is coincident with the plane z = 0, whereas the fluid occupies the region z ≥ 0. The electric field is absent while induced magnetic field is neglected due to the consideration of small magnetic Reynolds number. The velocities of the stretching sheet along the x− and y− directions are uw(x) = ax and vw(y) = by respectively. The sheet is kept at constant temperature Tw, whereas T∞ is the ambient value of the temperature such that Tw > T∞. Considering the velocity vector V = [u(x, y, z), v(x, y, z), w(x, y, z)] and the temperature T (see Fig 1). The boundary layer equations for three-dimensional flow and heat transfer of Maxwell fluid can be expressed as below:∂u∂x+∂v∂y+∂w∂z=0,(1)u∂u∂x+v∂u∂y+w∂u∂z=ν∂2u∂z2-λ1(u2∂2u∂x2+v2∂2u∂y2+w2∂2u∂z2+2uv∂2u∂x∂y+2vw∂2u∂y∂z+2uw∂2u∂x∂z)-σB02ρ(u+λ1w∂u∂z),(2)u∂v∂x+v∂v∂y+w∂v∂z=ν∂2v∂z2-λ1(u2∂2v∂x2+v2∂2v∂y2+w2∂2v∂z2+2uv∂2v∂x∂y+2vw∂2v∂y∂z+2uw∂2v∂x∂z)-σB02ρ(v+λ1w∂v∂z),(3)ρcp(u∂T∂x+v∂T∂y+w∂T∂z)=-∇.q,(4)where u, v and w are the velocity components along the x−, y− and z− directions respectively, ν is the kinematic viscosity, cp is the specific heat, σ is the electrical conductivity, ρ is the fluid density, T is the fluid temperature, λ1 is the fluid relaxation time and q is the heat flux which satisfies the following relationship [3].q+λ2∂q∂t+V.∇q-q.∇V+(∇.V)q=-k∇T,(5)in which λ2 is the thermal relaxation time and k is the thermal conductivity of the fluid. Following Christov [3], we eliminate q from Eqs (4) and (5) to obtain the following:u∂T∂x+v∂T∂y+w∂T∂z=kρcp∂2T∂z2-λ2[u2∂2T∂x2+v2∂2T∂y2+w2∂2T∂z2+2uv∂2T∂x∂y+2vw∂2T∂y∂z+2uw∂2T∂x∂z+(u∂u∂x+v∂u∂y+w∂u∂z)∂T∂x+(u∂v∂x+v∂v∂y+w∂v∂z)∂T∂y+(u∂w∂x+v∂w∂y+w∂w∂z)∂T∂z](6)Boundary conditions for the present problem are:u=uw(x)=ax,  v=vw(y)=by,  w=0,  T=Tw  at  z=0,u→0,  v→0,  T→T∞ as  z→∞.(7)Considering the following similarity transformationsη=aνz,u=axf′(η),v=ayg′(η),w=-aν[f(η)+g(η)],θ=T-T∞Tw-T∞.(8)Eq (1) is identically satisfied and Eqs (2), (3), (4) and (5) take the following forms:f′′′+(M2β+1)(f+g)f′′-f′2+β2(f+g)f′f′′-(f+g)2f′′′-M2f′=0,(9)g′′′+(M2β+1)(f+g)g′′-g′2+β2(f+g)g′g′′-(f+g)2g′′′-M2g′=0,(10)1Prθ′′+(f+g)θ′-γ(f+g)(f′+g′)θ′+(f+g)2θ′′)=0,(11)f(0)=g(0)=0,f′(0)=1,g′(0)=λ,θ(0)=1,f′(∞)→0,g′(∞)→0,θ(∞)→0,(12)where λ = b/a is the ratio of the stretching rate along the y− direction to the stretching rate along the x− direction, β = λ1a is the non-dimensional fluid relaxation time, γ = λ2a is the non-dimensional relaxation time for heat flux and Pr is the Prandtl number. It can be noticed that when λ = 0, the two-dimensional case is jumped. Further λ = 1 corresponds to the case of axisymmetric flow.


Cattaneo-Christov Heat Flux Model for MHD Three-Dimensional Flow of Maxwell Fluid over a Stretching Sheet.

Rubab K, Mustafa M - PLoS ONE (2016)

Geometry of the problem.
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4836741&req=5

pone.0153481.g001: Geometry of the problem.
Mentions: Consider the flow of upper-convected Maxwell fluid induced by an elastic sheet stretching in two lateral directions. The sheet is coincident with the plane z = 0, whereas the fluid occupies the region z ≥ 0. The electric field is absent while induced magnetic field is neglected due to the consideration of small magnetic Reynolds number. The velocities of the stretching sheet along the x− and y− directions are uw(x) = ax and vw(y) = by respectively. The sheet is kept at constant temperature Tw, whereas T∞ is the ambient value of the temperature such that Tw > T∞. Considering the velocity vector V = [u(x, y, z), v(x, y, z), w(x, y, z)] and the temperature T (see Fig 1). The boundary layer equations for three-dimensional flow and heat transfer of Maxwell fluid can be expressed as below:∂u∂x+∂v∂y+∂w∂z=0,(1)u∂u∂x+v∂u∂y+w∂u∂z=ν∂2u∂z2-λ1(u2∂2u∂x2+v2∂2u∂y2+w2∂2u∂z2+2uv∂2u∂x∂y+2vw∂2u∂y∂z+2uw∂2u∂x∂z)-σB02ρ(u+λ1w∂u∂z),(2)u∂v∂x+v∂v∂y+w∂v∂z=ν∂2v∂z2-λ1(u2∂2v∂x2+v2∂2v∂y2+w2∂2v∂z2+2uv∂2v∂x∂y+2vw∂2v∂y∂z+2uw∂2v∂x∂z)-σB02ρ(v+λ1w∂v∂z),(3)ρcp(u∂T∂x+v∂T∂y+w∂T∂z)=-∇.q,(4)where u, v and w are the velocity components along the x−, y− and z− directions respectively, ν is the kinematic viscosity, cp is the specific heat, σ is the electrical conductivity, ρ is the fluid density, T is the fluid temperature, λ1 is the fluid relaxation time and q is the heat flux which satisfies the following relationship [3].q+λ2∂q∂t+V.∇q-q.∇V+(∇.V)q=-k∇T,(5)in which λ2 is the thermal relaxation time and k is the thermal conductivity of the fluid. Following Christov [3], we eliminate q from Eqs (4) and (5) to obtain the following:u∂T∂x+v∂T∂y+w∂T∂z=kρcp∂2T∂z2-λ2[u2∂2T∂x2+v2∂2T∂y2+w2∂2T∂z2+2uv∂2T∂x∂y+2vw∂2T∂y∂z+2uw∂2T∂x∂z+(u∂u∂x+v∂u∂y+w∂u∂z)∂T∂x+(u∂v∂x+v∂v∂y+w∂v∂z)∂T∂y+(u∂w∂x+v∂w∂y+w∂w∂z)∂T∂z](6)Boundary conditions for the present problem are:u=uw(x)=ax,  v=vw(y)=by,  w=0,  T=Tw  at  z=0,u→0,  v→0,  T→T∞ as  z→∞.(7)Considering the following similarity transformationsη=aνz,u=axf′(η),v=ayg′(η),w=-aν[f(η)+g(η)],θ=T-T∞Tw-T∞.(8)Eq (1) is identically satisfied and Eqs (2), (3), (4) and (5) take the following forms:f′′′+(M2β+1)(f+g)f′′-f′2+β2(f+g)f′f′′-(f+g)2f′′′-M2f′=0,(9)g′′′+(M2β+1)(f+g)g′′-g′2+β2(f+g)g′g′′-(f+g)2g′′′-M2g′=0,(10)1Prθ′′+(f+g)θ′-γ(f+g)(f′+g′)θ′+(f+g)2θ′′)=0,(11)f(0)=g(0)=0,f′(0)=1,g′(0)=λ,θ(0)=1,f′(∞)→0,g′(∞)→0,θ(∞)→0,(12)where λ = b/a is the ratio of the stretching rate along the y− direction to the stretching rate along the x− direction, β = λ1a is the non-dimensional fluid relaxation time, γ = λ2a is the non-dimensional relaxation time for heat flux and Pr is the Prandtl number. It can be noticed that when λ = 0, the two-dimensional case is jumped. Further λ = 1 corresponds to the case of axisymmetric flow.

Bottom Line: The governing partial differential equations even after employing the boundary layer approximations are non linear.It is noticed that velocity decreases and temperature rises when stronger magnetic field strength is accounted.Penetration depth of temperature is a decreasing function of thermal relaxation time.

View Article: PubMed Central - PubMed

Affiliation: School of Natural Sciences (SNS), National University of Sciences and Technology (NUST), Islamabad 44000, Pakistan.

ABSTRACT
This letter investigates the MHD three-dimensional flow of upper-convected Maxwell (UCM) fluid over a bi-directional stretching surface by considering the Cattaneo-Christov heat flux model. This model has tendency to capture the characteristics of thermal relaxation time. The governing partial differential equations even after employing the boundary layer approximations are non linear. Accurate analytic solutions for velocity and temperature distributions are computed through well-known homotopy analysis method (HAM). It is noticed that velocity decreases and temperature rises when stronger magnetic field strength is accounted. Penetration depth of temperature is a decreasing function of thermal relaxation time. The analysis for classical Fourier heat conduction law can be obtained as a special case of the present work. To our knowledge, the Cattaneo-Christov heat flux model law for three-dimensional viscoelastic flow problem is just introduced here.

No MeSH data available.


Related in: MedlinePlus