MHD oblique stagnation-point flow of a Newtonian fluid

Transkript

1 Z. Angew. Math. Phys. 63 (), 7 9 c Springer Basel AG -75//7- published online November 3, DOI.7/s Zeitschrift für angewandte Mathematik und Physik ZAMP MHD oblique stagnation-point flow of a Newtonian fluid Alessandra Borrelli, Giulia Giantesio and Maria Cristina Patria Abstract. The steady two-dimensional oblique stagnation-point flow of an electrically conducting Newtonian fluid in the presence of a uniform external electromagnetic field (E, H ) is analysed, and some physical situations are examined. In particular, if E vanishes, H lies in the plane of the flow, with a direction not parallel to the boundary, and the induced magnetic field is neglected, it is proved that the oblique stagnation-point flow exists if and only if the external magnetic field is parallel to the dividing streamline. In all cases it is shown that the governing nonlinear partial differential equations admit similarity solutions, and the resulting ordinary differential problems are solved numerically. Finally, the behaviour of the flow near the boundary is analysed; this depends on the Hartmann number if H is parallel to the dividing streamline. Mathematics Subject Classification (). 76W5 76D. Keywords. MHD flow Oblique stagnation-point flow Newtonian fluids.. Introduction Oblique stagnation-point flow appears when a jet of fluid impinges obliquely on a rigid wall at an arbitrary angle of incidence. From a mathematical point of view, such a flow is obtained by combining orthogonal stagnation-point flow with a shear flow parallel to the wall. The steady two-dimensional oblique stagnation-point flow of a Newtonian fluid has been object of many investigations starting from the paper of Stuart in [6]. The oblique solution was later studied by Tamada [7], Dorrepaal [3,5], Wang [9,]; recently, Drazin and Raley [6], and Tooke and Blyth [8] reviewed the problem and included a free parameter associated with the shear flow component, which is related to the pressure gradient. Magnetohydrodynamic stagnation-point flow is an area of investigation discussed by several authors in recent years (see for example [,, 3]). In this class of problems, the fluid is electrically conducting and its motion towards the wall occurs in the presence of an applied electromagnetic field. The aim of this paper is to study how the steady oblique stagnation-point flow of a Newtonian fluid is influenced by a uniform external electromagnetic field (E, H ). The motions we find depend upon how the applied electromagnetic field is oriented relative to the flat boundary. First of all, we consider an inviscid fluid and analyse three cases, which are significant from a physical point of view. In the first two cases, an external constant field, either electric or magnetic, is impressed parallel to the rigid wall. In both cases, we find that an oblique stagnation-point flow exists, and we obtain the exact induced magnetic field. The presence of the electromagnetic field modifies the pressure p, which is smaller than the pressure in the purely hydrodynamical flow. In the third case, we suppose that E vanishes and H lies in the plane of the flow, with a direction not parallel to the boundary. Under the hypothesis that the magnetic Reynolds number is small, we neglect the induced magnetic field, as is customary in the literature. We prove that the oblique stagnation-point flow exists if and only if H is parallel to the dividing streamline. In regard of this result, we point out that the analysis contained in [] appears incorrect, because the Authors carry out their analysis by supposing the external magnetic field orthogonal to the boundary, but we prove in Theorem.6 that in that case the oblique stagnation-point flow does not exist for an inviscid fluid.

2 7 A. Borrelli et al. ZAMP The presence of H parallel to the dividing streamline modifies p, which is smaller than the pressure in the purely hydrodynamical flow. In the second part, we consider the same problems for a Newtonian fluid. As it is customary when one studies the plane stagnation-point flow for a Newtonian fluid, we assume that at infinity the flow approaches the flow of an inviscid fluid for which the stagnation-point is shifted from the origin [6, 5, 8]. The coordinates of this new stagnation-point contain two constants: A and B. A is determined as part of the solution of the orthogonal flow, and B is free. As far as the flow is concerned, in the first two cases we find the same equations of the oblique stagnation-point flow in absence of electromagnetic field, while the induced magnetic field is obtained by direct integration. Hence, the external uniform electromagnetic field does not influence the flow, and modifies only the pressure. Moreover, p has a constant component parallel to the wall proportional to B A. Thisdoesnot appear in the orthogonal stagnation-point flow. This component determines the displacement parallel to the boundary of the uniform shear flow. The flow is obtained for different values of B by numerical integration using a shooting method. We remark that the thickness of the layer affected by the viscosity is larger than that in the orthogonal stagnation-point flow. Finally, in the more general case in which H is parallel to the dividing streamline of the inviscid flow, we find that the flow has to satisfy an ordinary differential problem whose solution depend on H through the Hartmann number M. The numerical integration is provided using a finite-differences method. In this case, A (and so the stagnation-point) depends on M and decreases as M is increased. Further, the influence of the viscosity appears only in a layer near to the wall depending on M whose thickness decreases as M increases from zero. This is standard in magnetohydrodynamics. Some numerical examples and pictures are given in order to illustrate the effects due to the magnetic field. The paper is organized in this way: In Sect., we formulate the problem in the three cases for an inviscid fluid and summarize the results in Theorems.,. and.6. Section 3 is devoted to treat the same physical problems for a Newtonian fluid. Theorems 3., 3.3 and 3. collect our results. Further, we analyse the behaviour of the flow near the wall. This depends on the Hartmann number in the third case. Along the wall, we calculate three important coordinates: the origin which is the stagnation-point, the point of maximum pressure and the point of zero tangential stress (zero skin friction) where the dividing streamline meets the boundary. These points depend on M in the third case. As in electrically inert Newtonian fluid, the ratio of the slope of the dividing streamline at the wall to its slope at the infinity is independent of the angle of incidence; in the last case, it depends on M. In Sect., we numerically integrate the previous problems, and discuss some numerical results.. Inviscid fluids Consider the steady plane MHD flow of an inviscid, homogeneous, incompressible, electrically conducting fluid near a stagnation point occupying the region S, given by S = {x R 3 :(x,x 3 ) R,x > }. (.) The boundary of S, having the equation x =, is a rigid, fixed, non-electrically conducting wall. The equations governing such a flow in the absence of external mechanical body forces are: ρv v = p + μ e ( H) H,

3 Vol. 63 () MHD oblique stagnation-point flow 73 v =, H = σ e (E + μ e v H), E =, E =, H =, in S (.) where v is the velocity field, p is the pressure, E and H are the electric and magnetic fields, respectively, ρ is the mass density (constant > ), μ e is the magnetic permeability and σ e is the electrical conductivity (μ e,σ e = constants > ). We assume that the region S = {x R 3 :(x,x 3 ) R,x < } to be a vacuum (free space), and μ e is equal to the magnetic permeability of free space. To (.) we append the usual boundary condition for v: v = at x =. Further, we suppose that the tangential components of H and E are continuous through the plane x =. We are interested in the oblique plane stagnation-point flow so that v = ax + bx, v = ax, v 3 =, x R, x R +, (.3) with a, b constants (a >). As known, the streamlines of such a flow are hyperbolas whose asymptotes have the equations: x = and x = a b x. These two straight lines are degenerate streamlines too. Our aim is to study how such a flow is influenced by a uniform external electromagnetic field (E, H ). To this end, we consider three cases which, from a physical point of view, are significant... Case I E = E e 3, H =. Let the induced electromagnetic field (E i, H i H) beintheform E i = Ee i + Ee i + E3e i 3, H = h(x )e, where (e, e, e 3 ) is the canonical base of R 3. The boundary conditions require that E i =,E3 i = at x =, h() =. (.) From (.) follows that E E i + E = ψ, where ψ is the electrostatic scalar potential. Moreover, (.) 3 provides ψ = ψ(x 3 )and dψ (x 3 )=aμ e h(x )x + h (x ). dx 3 σ e From this equation we deduce that both members are equal to the same constant. Boundary condition (.) furnishes

4 7 A. Borrelli et al. ZAMP h + aσ e μ e hx = σ e E, x >, ψ = E x 3 + ψ, x 3 R. (.5) The integration of differential problem (.5),(.) 3 gives e h(x )= σ e E e ax with e = σ e μ e = electrical resistivity. As far as the pressure field is concerned, from (.) we get x e at e dt, x R +, (.6) p = ρa (x + x ) μ e h (x )+p, x R, x R +, where h is given by (.6) andp is the pressure in the stagnation point. We observe that the presence of E modifies the pressure field which is smaller than the pressure in the purely hydrodynamical flow. Our results can be summarized in the following: Theorem.. Let a homogeneous, incompressible, electrically conducting inviscid fluid occupy the region S. The steady plane MHD oblique stagnation-point flow of such a fluid has the following form when a uniform external electric field E = E e 3 is impressed: v =(ax + bx )e ax e, H = h(x )e, E = E e 3, p = ρa (x + x ) μ e h (x )+p, x R, x R + where h(x )= σ e E e ax Remark.. We note that the function: x e x x e e at e dt. e t dt =: daw(x), x R + is known as Dawson s integral and has the properties: x<< daw(x) x, x>> daw(x) x. Remark.3. The solution of the problem relative to the electromagnetic field in S is E = E = E e 3 and H = H =. In order to plot the function h, it is suitable to introduce dimensionless parameters and variables. To this end, we denote by V a characteristic velocity and put So (.5) can be written as where is the magnetic Reynolds number. L = V a, = x L, h(l) Ψ() =. σ e LE Ψ ()+R m Ψ() = R m = LV e

5 Vol. 63 () MHD oblique stagnation-point flow 75 x E H x x 3 Fig.. Flow description in Case I 6 Ψ 8 Ψ Ψ.8. Ψ Fig.. Case I: plots showing the graphs of Ψ for R m = 3,,, Therefore, Ψ() = e Rm e Rm t dt. Fig. shows the graphs of the function Ψ for some values of R m.

6 76 A. Borrelli et al. ZAMP x H H x x 3 Fig. 3. Flow description in Case II We note that there is a layer lining the boundary beyond which Ψ vanishes. Its thickness increases if R m decreases; moreover, at = Rm the function Ψ assumes a minimum whose value decreases as R m decreases... Case II E =, H = H e. Let the induced electromagnetic field (E i E, H i )beintheform E = E e + E e + E 3 e 3, H =[h(x )+H ]e. We append the boundary conditions (.). By proceeding as in CASE I, we deduce ψ = ψ(x 3 )and dψ (x 3 )=aμ e [h(x )+H ]x + h (x ). dx 3 σ e From this relation, we get h + a e hx = a e x H, x >, h() = (.7) and ψ = ψ from which E =. The solution of (.7) is h(x )=H (e ax e ), x R +, (.8) so that H = H e ax e e. Moreover, by virtue of (.) we deduce that the pressure field is given by p(x,x )= ρa (x + x ) μ e H e ax e + p, x R, x R +. We observe that the value of the pressure at the stagnation point is p μ e H.

7 Vol. 63 () MHD oblique stagnation-point flow Ψ Fig.. Case II: plot showing Ψ for R m = Finally, also in this case the presence of H modifies the pressure which is smaller than the pressure in the purely hydrodynamical flow. Therefore, we have obtained the following: Theorem.. Let a homogeneous, incompressible, electrically conducting inviscid fluid occupy the region S. The steady plane MHD oblique stagnation-point flow of such a fluid has the following form when a uniform external magnetic field H = H e is impressed: v =(ax + bx )e ax e, H = H e ax e e, E =, p = ρa (x + x ) μ e H e ax e + p, x R, x R +. Remark.5. The solution of the problem relative to the electromagnetic field in S is E = E = and H = H = H e. In dimensionless form, h becomes Ψ() =e Rm, R +, where Ψ() = h(l). H Figure shows that Ψ has an inflection point at =. Rm.3. Case III E =, H = H (cos ϑe +sinϑe ) with ϑ fixed in (,π). Suppose the induced electromagnetic field (E i E, H i )tobeintheform E = E e + E e + E 3 e 3, H i = h (x,x )e + h (x,x )e.

8 78 A. Borrelli et al. ZAMP x H x x 3 Fig. 5. Flow description in Case III Taking into account (.) 3,(.3) we obtain: [ h σ e E = h ] σ e μ e (v h v h + H v sin ϑ H v cos ϑ) e 3. (.9) x x Hence ψ = ψ(x 3 ) and we conclude that E =, aswehaveincaseii. Therefore, from (.) 3 H = σ e μ e v H, from which it follows h (x,x ) h (x,x ) x x = σ e μ e {(ax + bx )[h (x,x )+H sin ϑ]+ax [h (x,x )+H cos ϑ]}. (.) To this equation we must adjoin (.) 6, i.e. : h + h =. (.) x x So (h,h ) satisfies the PDE system (.), (.) with suitable boundary conditions. It is very difficult to find an explicit solution to this differential problem; so we proceed as it is customary in the literature by neglecting the induced magnetic field (h,h ). This approximation is motivated by physical arguments for MHD flow at small magnetic Reynolds number, e.g. in the flow of liquid metals. Then, ( H) H σ e μ e (v H ) H = σ e μ e H [a sin ϑx +(b sin ϑ + a cos ϑ)x ][ sin ϑe +cosϑe ]. On substituting this approximation in (.) we get: p = ρa x B x σ e sin ϑ[a sin ϑx +(bsin ϑ + a cos ϑ)x ], p = ρa x + B x σ e cos ϑ[a sin ϑx +(bsin ϑ + a cos ϑ)x ], p = p = p(x,x ), (.) x 3 with B = μ e H.

9 Vol. 63 () MHD oblique stagnation-point flow 79 It is possible to find a function p = p(x,x ) satisfying equations (.) if and only if p = p. (.3) x x x x Taking into account (.), the previous condition furnishes sin ϑ(a cos ϑ + b sin ϑ) =. (.) From (.), we get tan ϑ = a b. (.5) So the MHD oblique stagnation-point flow is possible if and only if H is parallel to the dividing streamline x = a b x. We underline that in particular if H is normal to the plane x = (i.e. ϑ = π/), then there is no pressure that satisfies equations (.) and therefore the oblique stagnation-point flow given by (.3) does not exist. Under the condition (.5), the pressure field has the form p = ρa (x + x ) σ eb a a + b (ax + bx ) + p, x R, x R +. (.6) Our results can be summarized in the following: Theorem.6. Let a homogeneous, incompressible, electrically conducting inviscid fluid occupy the region S. If we impress an external magnetic field H = H (cos ϑe +sinϑe ), <ϑ<π,andwe neglect the induced magnetic field, then the steady MHD oblique plane stagnation-point flow of such a fluid is possible if and only if tan ϑ = a b, i.e. H H = a + b ( be +ae ). Moreover, v =(ax + bx )e ax e, p = ρa (x + x ) σ eb a a + b (ax + bx ) + p, x R, x R +. Remark.7. In order to study oblique stagnation-point flow for Newtonian fluids, it is convenient to consider a more general motion. More precisely, we suppose the fluid obliquely impinging on the flat plane x = A and v = ax + b(x B), v = a(x A), v 3 =, x R, x A, (.7) with A, B = constants. In this way, the stagnation point is not (, ), but the point ( b (B A),A). a In this case, the streamlines are the hyperbolas whose asymptotes are x = a b x +B A and x = A. As it is easy to verify, in the absence of (E, H), the pressure field is given by: p = ρa {[x b a (B A)] +(x A) } + p. We underline that under these new assumptions, Theorems.,. and.6 continue to hold by replacing x,x with x b a (B A), x A, respectively.

10 8 A. Borrelli et al. ZAMP 3. Newtonian fluids Consider now the steady plane MHD flow of a Newtonian, homogeneous, incompressible, electrically conducting fluid near a stagnation point occupying the region S given by (.). In the absence of external mechanical body forces, the MHD equations governing such a flow are the equations (.) where (.) must be replaced with: v v = ρ p + ν v + μ e ( H) H, (3.) ρ where ν is the kinematic viscosity. As far as boundary conditions are concerned, we modify only the condition for v, assuming the no-slip boundary condition v x= =. (3.) Since we are interested to the oblique plane stagnation-point flow, we suppose v = ax f (x )+bg(x ), v = af(x ), v 3 =, x R, x R + (3.3) with f,g unknown functions. The condition (3.) supplies f() =, f () =, g() =. (3.) Moreover, as it is customary when studying stagnation-point flow for a Newtonian fluid, we assume that at infinity, the flow approaches the flow of an inviscid fluid. Therefore, to (3.) we must append also the following boundary conditions lim x + f (x )=, lim x g (x )=. (3.5) + By following the arguments of [6], and [8], we suppose that, at infinity, the Newtonian fluid has the same behaviour of an inviscid fluid which impinges the plane x = A, and has the stagnation-point at ( b (B A),A). In all the following cases, when we will refer to an inviscid fluid, all results obtained in a Sect. have to be modified by replacing x,x with x b a (B A), x A, respectively. In particular, the constants A, B are related to the asymptotic behaviour of f, g at infinity in the following way: lim x + [f(x ) x ]= A, lim [g(x ) x ]= B. (3.6) x + As we will see, A is determined as part of the solution of the orthogonal flow [5], instead B is a free parameter [6]. In order to study the influence of a uniform external electromagnetic field, we consider the three cases analysed in the previous section. 3.. Case I-N By proceeding as well as for an inviscid fluid, from (.) 3, (.) electromagnetic field, we obtain and boundary conditions for the h + a e fh = e E, x >, h() =, ψ(x 3 )= E x 3 + ψ, x 3 R. (3.7)

11 Vol. 63 () MHD oblique stagnation-point flow 8 If we regard f as a known function, the integration of the differential problem (3.7) gives h(x )= σ e E e a x e f(t)dt x e a s e f(t)dt ds, x R +. (3.8) As is easy to verify, ( the induced magnetic fields given by (3.8) and (.6) have the same asymptotic behaviour at infinity ) ee σ e. a(x A) Now we proceed in order to determine p, f, g. Substituting (3.3) into(3.) we obtain: p = p(x,x ), ax (νf + aff af )+b[νg + a(fg f g)] = p, ρ x νaf + a f f + μ e ρ h h = p. (3.9) ρ x Then, by integrating (3.9) 3, we find p(x,x )= ρa f (x ) ρaνf (x ) μ e h (x )+P(x ) where P (x ) is determined supposing that, far from the wall, the pressure p has the same behaviour as for an inviscid electroconducting fluid, whose velocity is given by (.7). Therefore, since the induced magnetic fields given by (3.8), (.6) have the same asymptotic behaviour, we get, by virtue of (3.5), (3.6) P (x )= ρ a [x b a (B A)] + p + ρaν. Finally, the pressure field assumes the form p = ρ a [x b a (B A)x + f (x )] ρaνf (x ) μ e h (x )+p (3.) with p = p + ρaν ρ b (B A). Equation (3.9) together (3.) furnishes ν a f + ff f +=, ν a g + fg f g = B A, (3.) with f() =, f () =, g() =, lim x + f (x )=, lim x g (x )=. (3.) + We remark that (f,g) satisfies the same differential problem that governs the oblique stagnation-point flow in the absence of an electromagnetic field. Hence, the external uniform electromagnetic field does not influence the flow. Moreover, as we can see from (3.), p has a constant component in the x direction proportional to B A which determines the displacement of the uniform shear flow parallel to the wall x =. Therefore, we have obtained the following

12 8 A. Borrelli et al. ZAMP Theorem 3.. Let a homogeneous, incompressible, electrically conducting Newtonian fluid occupy the region S. The steady MHD oblique plane stagnation-point flow of such a fluid has the following form when a uniform external electric field E = E e 3 is impressed: v =[ax f (x )+bg(x )]e af(x )e, H = h(x )e, E = E e 3, p = ρ a [x b a (B A)x + f (x )] ρaνf (x ) μ e h (x )+p, x R, x R +, where (f,g) satisfies the problem (3.),(3.) and h(x ) is given by (3.8). If we put ( a ν ν f a = ν x, φ() = ( ) a ν γ() = ν g a, Ψ() = ) a, a h ( ν a ), ν σ e E then we can write problem (3.7), (3.), and (3.) in dimensionless form φ + φφ φ +=, γ + φγ φ γ = β α, Ψ + R m φψ=, φ() =, φ () =, γ() =, Ψ() =, lim + φ () =, lim + γ () =, (3.3) where a a β = ν B, α = ν A, R m = ν = magnetic Reynolds number. e Notice that the function φ influences the functions γ and Ψ but not viceversa. The function φ satisfies the well-known Hiemenz equation [6, 5]. We recall that Hiemenz stagnation flow cannot be analytically solved, but only by a numerical integration. Existence and uniqueness of the solution to Hiemenz stagnation flow were shown by Hartmann [8], Tam [], and Craven and Peletier []. As far as problem (3.3),(3.3) 6,(3.3) 9 is concerned, the solution is formally obtained as [6] γ() =(α β)φ ()+Cφ ()Φ() (3.) with C = φ ()[γ () φ ()(α β)], φ() = {[φ (s)] e s φ(t)dt} ds. (3.5) We have numerically computed the functions γ,γ (as well as φ, φ,φ ) for various values of the parameter β, as we will see in Sect.. Precisely, we have taken β α = 5 α, α,, α,5 α (as in [8]). Other Authors (e.g. Stuart [6], and Tamada [7]) take β = α, while Dorrepaal takes β =[3,5]. We note that the constant C, given by (3.5), contains α, φ (), γ () which are not assigned. Their values are determined by numerical integration of problem (3.3). Finally, the induced magnetic field in dimensionless form is given by: Ψ() = e Rm φ(s)ds e Rm t φ(s)ds dt, R +.

13 Vol. 63 () MHD oblique stagnation-point flow 83 Remark 3.. There are along the wall x =three important coordinates: the origin x =which is the stagnation-point for the flow, the point x = x p of maximum pressure and the point x = x s of zero tangential stress (zero skin friction) where the dividing streamline of equation ξφ()+ b ν γ(s)ds =, ξ = a a x (3.6) meets the boundary [3, 5]. In consideration of (3.) and(3.3), we see that ν ν x p = b (β α), a3 x γ () s = b a 3 φ (). (3.7) We note that x p does not depend on h and the ratio ( as we will see in Sect. ) x p =(α β) φ () x s γ () is the same for all angles of incidence. Finally, we recall that studying the small- behaviour of γ(s)ds φ() [3], the slope of the dividing streamline at the wall is given by [6]: 3a[φ ()] m s = b[(β α)φ () + γ ()] and does not depend on the kinematic viscosity. Thus, the ratio of this slope to that of the dividing streamline at infinity (m i = a ) is the same for all oblique stagnation-point flows and is given by b m s = 3 [φ ()] m i [(β α)φ () + γ ()]. (3.8) This ratio is independent of a and b, depending only upon the constant pressure gradient parallel to the boundary through B A. [5] 3.. Case II-N By proceeding as one would with an inviscid fluid, from (.) 3,(.) and boundary conditions for the electromagnetic field, we get h + a e fh = a e fh, x >, h() =, ψ(x 3 )=ψ E =. (3.9) The integration of (3.9) leads to h(x )=H (e a x e f(t)dt ), x R +, (3.) so that H = H e a x e f(s)ds e. The pressure field, as is easy to verify, becomes p = ρ a [x b a (B A)x + f (x )] ρaνf (x ) μ e [h(x )+H ] + p, (3.) p = p + ρaν ρ b (B A) and (f,g) satisfies problem (3.). Therefore, in this case as well, the external uniform electromagnetic field does not influence the flow.

14 8 A. Borrelli et al. ZAMP Thus, we obtain the following: Theorem 3.3. Let a homogeneous, incompressible, electrically conducting Newtonian fluid occupy the region S. The steady MHD oblique plane stagnation-point flow of such a fluid has the following form when a uniform external magnetic field H = H e is impressed: v =[ax f (x )+bg(x )]e af(x )e, H =[H + h(x )]e, E =, p = ρ a [x b a (B A)x + f (x )] ρaνf (x ) μ e [h(x )+H ] + p, x R, x R + where (f,g) satisfies the problem (3.),(3.), andh(x ) is given by (3.). In dimensionless form, h(x ) becomes Ψ() =e Rm φ(t)dt, R +, (3.) where Ψ() = h( ν a ). H Of course, Remark 3. continues to hold in this case Case III-N Taking into account the results obtained for an inviscid fluid, we assume H H = a + b ( be +ae ), E =. By means of the same arguments of CASE III, we deduce E = H = σ e μ e (v H) and we neglect the induced magnetic field, replacing (3.) with the equation: v v = ρ p + ν v + μ e ρ (v H ) H. (3.3) We substitute (3.3) into(3.3) to determine p, f, g. This yields p = p(x,x ), ax (νf + aff af a σ e ρ B a + b f ) +b[νg + a(fg f g) a σ e B ρ a + b (g f)] = p, ρ x νaf + a f f + σ e B ρ a + b [a bx f + ab (g f)] = ρ The integration of (3.) 3 gives p = ρa f (x ) ρaνf (x ) σ e B a + b [a bx f(x )+ab where P (x ) has to be found in CASE I-N, II-N. x (g(s) f(s))ds]+p (x ) p. (3.) x

15 Vol. 63 () MHD oblique stagnation-point flow 85 After some calculations, we obtain with p constant. So the pressure field is: P (x )= ρ a ( + a ρ σ e B ) a + b [x b a (B A)] + p p = ρ a [x b a (B A)x + f (x )] ρaνf (x ) σ eb { a + b a bx f(x )+ab x (g(s) f(s))ds +a 3 [x b } a (B A)x ] + p, x R, x R +. (3.5) The constant p represents the pressure at the stagnation point. Then, (3.) supplies ν a f + ff f ++M ( f )=, ν a g + fg gf + M (f g) =(+M )(B A), (3.6) where aσe M =B ρ(a + b = Hartmann number. ) We append boundary conditions (3.), and (3.5) to the system (3.6). We remark that, unlike the previous cases, the external electromagnetic field modifies the flow, and if M = the system (3.6) reduces to the system (3.). Theorem 3.. Let a homogeneous, incompressible, electrically conducting Newtonian fluid occupy the region S. If we impress the external magnetic field H H = a + b ( be +ae ) and if we neglect the induced magnetic field, then the steady MHD oblique plane stagnation-point flow of such a fluid has the form v =[ax f (x )+bg(x )]e af(x )e, E =, p = ρ a [x b a (B A)x + f (x )] ρaνf (x ) σ eb { a + b a bx f(x )+ab x (g(s) f(s))ds +a 3 [x b } a (B A)x ] + p, x R, x R +, where (f,g) satisfies problem (3.6), (3.) and(3.5). System (3.6) in dimensionless form becomes φ + φφ φ ++M ( φ )=, γ + φγ φ γ + M (φ γ) =(+M )(β α) (3.7) and we adjoin boundary conditions (3.3) 9. Notice the one-way coupling that the function φ influences the function γ but not viceversa.

16 86 A. Borrelli et al. ZAMP φ φ φ Fig. 6. Case I-N: plot showing the behaviour of φ (Hiemenz function), φ,φ Remark 3.5. We recall that the solution of the differential problem (3.7), (3.3), (3.3) 5 and (3.3) 8 exists and it is unique as proved by Hoernel in [9]. As far as γ is concerned, it satisfies a linear second-order non-homogeneous differential equation, if we regard φ as a known function. After some calculations, we obtain that γ is formally expressed by γ() =(α β)φ ()+Cφ ()Φ() +M φ () Φ(s)φ(s)φ (s)e s φ(t)dt ds Φ() φ(s)φ (s)e s φ(t)dt ds with C, Φ() are given by (3.5) again. We note that if M =, then (3.8) reduces to (3.). (3.8) Remark 3.6. The points x = x p of maximum pressure and x = x s of zero tangential stress on x = are formally the same as in CASE I-N and II-N. However, these points depend on M. Finally, the slope of the dividing streamline at the wall is given by: 3a[φ ()] ( + M )b[(β α)φ () + γ ()].. Numerical results and discussion In this section, we discuss the numerical solutions of the problems studied in CASES I, II, III-N... Case I-N We have solved problem (3.3) numerically by using a shooting algorithm. Figure 6 shows the graphics of Hiemenz function and its derivatives. As one can see, lim + φ ()=, lim + φ () =.At =., one has φ =.99 and, if >., then φ = α with α =.679. From the numerical integration, we get φ () =.36. Our results are consistent with the previous studies.

17 Vol. 63 () MHD oblique stagnation-point flow γ γ Fig. 7. Case I-N: Figures 7 and 7 show γ and γ with, from above, β α = 5 α, α,,α, 5 α, respectively β α γ () C Table. CASE I-N: γ (),C, xp x s, ms m i, γ for some values of β α x p x s m s m i γ Figures 7,and7 show the profiles of γ(),γ (), for some values of β α, i.e. β α = 5 α, α,, α,5 α. As far as γ () is concerned, we show in Table its values for different β (β = 5,, α,α, 5). We note that the constant C, given by (3.5), contains α, φ (), γ () which are not assigned, but their values have been determined by the numerical integration of problem (3.3). Table points out that C has always the same value,.79, according to the value determined by Stuart [6] and Glauert [7] by means of asymptotic estimate of the integral Φ() at infinity. In Table, we list also the numerically computed values of (denoted by γ )atwhichγ =.99 (if β α ), or γ =. (if β α<). So when > γ, then γ = β. We have that γ is greater than.. Hence the influence of the viscosity appears only in a layer lining the boundary whose thickness is ν γ. We remark that the layer affected by the viscosity is proportional to and its thickness is larger a ν than in the orthogonal stagnation-point flow, which is. (see Figs. 6, 7). a Observing again Table, we notice that x s (given by (3.7) ) has the sign of b if β α> and the sign of b if β α. Moreover, if b is positive (negative) x s increases (decreases) as β α increases. As far as x s is concerned, if β α increases from a negative value to zero, x s decreases and so x s approaches the origin, otherwise, as β α increases from zero to a positive value, x s increases and so x s departs from the origin. Figures 8, 8, 8 3 show the streamlines and the points ν ξ p = a x p, ξ s = for b =andβ α = α,, α, respectively. a ν a x s (.)

18 88 A. Borrelli et al. ZAMP ξ s ξ ξ p p ξ ξ ξ s ξ ξ s ξ p Fig. 8. Case I-N: plots showing the streamlines and the points ξ p,ξ s for b =andβ α = α,, α, respectively a Ψ Ψ Fig. 9. Case I-N: plots showing Ψ with R m =, 6 Figure 9 shows the behaviour of the induced magnetic field Ψ with R m =, that is similar to the behaviour of Ψ in CASE I (inviscid fluid). Figure 9 shows the behaviour of Ψ with R m = 6 ;for [,.6] the graph is approximately linear, because in this interval, for very small values of R m,the equation (3.3) 3 reduces to Ψ =.

19 Vol. 63 () MHD oblique stagnation-point flow Ψ Fig.. Case II-N: plot showing Ψ for R m = Table. CASE III-N: dependence of α and φ () on M M α φ () Case II-N In this case the ordinary differential problem governing (φ, γ) is the same as in CASE I-N; we have only to compute Ψ(), given by (3.). Figure shows that Ψ has a similar behaviour as in CASE II..3. Case III-N We have solved the problem (3.7) with the boundary conditions (3.3) 9 by using a finite-differences method, because the usual shooting does not produce the desired accuracy []. We remark that the values of α and φ () depend on M, as we can see from Table. More precisely, α decreases and φ () increases as M is increased from, as we would expect physically. Table 3 shows numerical results of some parameters significant from a physical point of view for M =,,, 5, and β α = 5 α, α,, α,5 α. As far as the dependence of γ () on M is concerned, from Table 3 we can see that its value increases when M increases if β α<, otherwise it decreases. In Fig., we can see the profiles φ, φ,φ for M =, while Fig. 3 shows the behaviour of φ for different M. Figures, show the profiles of γ(),γ (), for M = and for some values of β α, i.e. β α = 5 α, α,, α,5 α. In Figs., 5, 6, we provide the behaviour of γ for different M when β α is fixed.

20 9 A. Borrelli et al. ZAMP M β α γ () C Table 3. CASE III-N: descriptive quantities for some value of M γ () φ () x p x s m s m i φ γ φ φ φ Fig.. Case III-N: plots showing φ, φ,φ for M = We have plotted the profiles of φ, φ,φ,γ,γ for M = because they have an analogous behaviour for M. Moreover, we observe (see Table 3) that the constant C in (3.8) has approximately always the same value if we fix M and it decreases as M decreases.

21 Vol. 63 () MHD oblique stagnation-point flow γ γ Fig.. Case III-N: plots showing γ,γ with M = and, from above, β α = 5 α, α,, α, 5 α, respectively M= M=5 M= M= M= φ Fig. 3. Case III-N: plots showing φ for different M In Table 3, we also list, for some M and β α = 5 α, α,,α, 5 α, the values of φ, γ beyond which φ = α and γ = β, respectively. We note that γ is greater than the corresponding value of ν φ ; so the influence of the viscosity appears only in the region < γ, i.e. x < a γ. Further we underline that the thickness of this layer depends on M and decreases when M increases (as easily seen in Figs. 3,, 5, 6). This effect is normal in magnetohydrodynamics. Finally we notice that the points x p,x s, given by (3.7), lie on the same side of the origin. Their location depends on M and β α, as seen in Table 3. Figures 7 show the streamlines and the points ξ p,ξ s for b a =,β α = α,, α,and M =, 5. We can observe that x p tends to as M x s increases.

22 9 A. Borrelli et al. ZAMP γ M= γ.. M= M= M= M=5 5 M=5 M= M= M= M= Fig.. Case III-N: plots showing γ for different M. In the first picture β α = 5 α, in the second β α = α.. γ..9.8 M= M=5 M= M= γ M= M=5 M= M= M=.7.6 M= Fig. 5. Case III-N: plots showing γ for different M. In the first picture β α =, in the second β α = α 5 5 M=5 M= M= M= γ M= Fig. 6. Case III-N: plots showing γ for different M with β α =5 α

23 Vol. 63 () MHD oblique stagnation-point flow ξ ξ s p 3 3 ξ ξ ξ s p 3 3 ξ ξ s ξ p 3 3 ξ.5 ξ ξ s p 3 3 ξ ξ ξ s p 3 3 ξ ξ ξ s p 3 3 ξ Fig. 7. Case III-N: Figs, (7),3,5 show the streamlines and the points ξ p,ξ s for b =andβ α = α,, α, respectively, a and M =.Figures(7),,6 for M =5 References. Ariel, P.D.: Hiemenz flow in hydromagnetics. Acta Mech. 3, 3 3 (99). Craven, A.H., Peletier, L.A.: On the uniqueness of solutions of the Falkner-Skan equation. Mathematika 9, 9 33 (97)

24 9 A. Borrelli et al. ZAMP 3. Dorrepaal, J.M.: An exact solution of the Navier-Stokes equation which describes non-orthogonal stagnation-point flow in two dimension. J. Fluid Mech. 63, 7 (986). Dorrepaal, J.M., Mosavizadeh, S.: Steady compressible magnethohydrodynamic flow near a point of reattachment. Phys. Fluids, 5 58 (998) 5. Dorrepaal, J.M.: Is two-dimensional oblique stagnation-point flow unique?. Can. Appl. Math. Q. 8, 6 66 () 6. Drazin, P., Riley, N.: The Navier-Stokes Equations. A Classification of Flows and Exact Solutions. London Mathematical Society, Lecture notes series 33, Cambridge University Press, (7) 7. Glauert, M.B.: The Laminar Boundary Layer on Oscillating Plates and Cylinders. J. Fluid Mech., 97 (956) 8. Hartmann, P.: Ordinary Differential Equations. Siam, Madison () 9. Hoernel, J.D.: On the similarity solutions for a steady MHD equation. Commun. Nonlinear Sci. Numer. Simulat. 3, (8). Grosan, T., Pop, I., Revnic, C., Ingham, D.B.: Magnetohydrodynamic oblique stagnation-point flow. Acta Mech., (9). Mahapatra, T.R., Gupta, A.S.: Magnetohydrodynamic stagnation-point flow towards a stretching sheet. Acta Mech. 5, 9 96 (). Mahapatra, T.R., Nandy, S.K., Gupta, A.S.: Analytical solution of magnetohydrodynamic stagnation-point flow of a power-low fluid towards a stretching surface. Appl. Math. Comput. 5, (9) 3. Na, T.Y.: Computational Methods in Engineering Boundary Value Problems. Academic Press, Massachusetts (979). Tam, K. Kuen: A note on the existence of a solution of the Falkner-Skan Equation. Canad. Math. Bull. 3, 5 7 (97) 5. Schlichting, H., Gersten, K.: Boundary Layer Theory 8th revised and Enlarged. Springer, Berlin (3) 6. Stuart, J.T.: The viscous flow near a stagnation point when the external flow has uniform vorticity. J. Aerospace Sci. 6, 5 (959) 7. Tamada, K.J.: Two-dimensional stagnation-point flow impinging obliquely on an oscillating flat plate. J. Phys. Soc. Jpn. 7, 3 3 (979) 8. Tooke, R.M., Blyth, M.G.: A note on oblique stagnation-point flow. Phys. Fluids, 3 (8) 9. Wang, C.Y.: Stagnation flows with slip: exact solutions of Navier-Stokes equations. ZAMP 5, 8 89 (3). Wang, C.Y.: Similarity stagnation point solutions of the Navier-Stokes equations-review and extension. Eur. J. Mech. B-Fluids 7, (8) Alessandra Borrelli, Giulia Giantesio and Maria Cristina Patria Dipartimento di Matematica Università di Ferrara via Machiavelli 35 Ferrara Italy brs@unife.it G. Giantesio gntgli@unife.it M. C. Patria pat@unife.it (Received: July, )