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The two-dimensional problem of the electrochemical machining of metals with a periodic cathode tool
Journal of Applied Mathematics and Mechanics 76 (2012) 475–481
Contents lists available at SciVerse ScienceDirect
Journal of Applied Mathematics and Mechanics journal homepage: www.elsevier.com/locate/jappmathmech
The two-dimensional problem of the electrochemical machining of metals with a periodic cathode tool夽 N.M. Minazetdinov Naberezhnye Chelny, Russia
a r t i c l e
i n f o
Article history: Received 16 February 2011
a b s t r a c t The problem of the electrochemical machining of metals with a periodic cathode tool in the form of a lattice of plates is formulated and solved. The hydrodynamic analogue of the initial problem is the problem of a plane-parallel potential circulating flow of an ideal incompressible fluid around plate electrodes. Auxiliary schemes are considered in order to specify the initial data and conditions determining the parameters of the problem. The steady-state shapes of the anode boundaries are found. It is shown that different anode boundaries are obtained due to a change in the electric field characteristics provided that the properties of the metal and the electrolyte, the geometry of the electrode tool and its rate of advance are identical. © 2012 Elsevier Ltd. All rights reserved.
Two-dimensional electrochemical machining problems with a symmetrical triangular cathode and with a cathode with a curvilinear part of the boundary have been solved earlier1,2 using a hydrodynamic analogy. The solution of a problem, associated with determining the steady-state shape of the surface of a workpiece in to case of machining with a periodic cathode, is found below within the limits of a model of an ideal process.3 This electrode can be used to form the contoured surfaces of heat exchange equipment.
1. Model of the process A model of an ideal process is used as a first approximation in the theoretical analysis of the electrochemical machining of metals. The electric field in the interelectrode gap is assumed to be potential and the potential u of the field is assumed to be a harmonic function. The boundary of the anode (of the machined surface) and the boundary of the cathode tool (of the machining surface) are equipotential field lines. In the steady state, the shape of the machined surface does not change in a moving system of coordinates associated with the cathode. The distribution of the normal derivative of the potential on the steady-state anode boundary has the form2,3
where is the electrical conductivity of the medium, is the electrochemical equivalent of the metal, is the density of the anode material and is the angle between the velocity vector for the advancement of the cathode Vc and the vector na of the outward normal at a given point of the anode boundary. The constants a0 and a1 characterize the electrolyte properties; values found from experimental data5 obtained in machining 5HXM steel have been given for these constants.4 A two-dimensional model of the process is considered. A system of Cartesian coordinates x1 , y1 is introduced, and this system, which is associated with the cathode, moves in the direction of the ordinate axis. Using the prerequisites for a model of an ideal process,3 we will assume that a complex electric field potential exists in the interelectrode gap
夽 Prikl. Mat. Mekh., Vol. 76, No. 4, pp. 658–666, 2012. E-mail address: [email protected] 0021-8928/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jappmathmech.2012.09.015
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N.M. Minazetdinov / Journal of Applied Mathematics and Mechanics 76 (2012) 475–481
Fig. 1.
where u(z1 ) is the field potential and (z1 ) is the stream function.6 We now introduce the current density i0 , that depends on the velocity at which the cathode advances, the properties of the machined metal and the length H,7 as well as the dimensionless variables
Here, ua and uc are the values of the field potential on the anode and cathode boundaries. We now change to the dimensionless complex potential
In the interelectrode gap region, the function is harmonic and, on the anode and cathode boundaries, it satisfies the conditions1
The electric field in the interelectrode gap is modelled by a fictitious plane-parallel potential flow of an ideal incompressible fluid.8 The velocity of the fictitious flow varies according to the law1 (1.1) where is the argument of the velocity vector. 2. Formulation and solution of the basic problem A cross-section of the interelectrode gap is shown schematically in Fig. 1. The cathode tool is formed by an infinite assembly of identical plates of length 2l lying in a plane, each of which is obtained from the adjacent plate by a parallel translation through the same distance 2 h along the abscissa. The thickness of the plates is negligibly small. We will confine ourselves to considering the region located between the lines of symmetry AF and BF. In this region, CDE is the cathode boundary and the line AB is the required anode boundary. On the line AF there is a branch point P of the equipotential electric field line at which the voltage potential is equal to zero9 and F is an infinitely distant point. The problem of determining the boundary of the flow AB with the given law of variation in the velocity (1.1) is the hydrodynamic analogue. The flow is created by a system of continuously distributed sources along the lines AP and EF and sinks on the lines BC and FP. The velocity of the fictitious fluid flow is equal to zero at the point P (Fig. 1). The flow domain in the physical plane z (Fig. 1) is denoted by Gz and the semicircle of unit radius in the parametric plane t = + i␦ (Fig. 2) is denoted by Gt . We shall seek a conformal mapping z(t) of the domain Gt onto Gz . The correspondence of the points is clear from Figs 1 and 2.
Fig. 2.
N.M. Minazetdinov / Journal of Applied Mathematics and Mechanics 76 (2012) 475–481
477
Fig. 3.
The complex potential W(t) = (t) + i(t) satisfies the conditions
(2.1) where is the polar angle in the parametric plane t. The function (t) takes constant values on the lines EF, BC, AP and PF. Without loss of generality, we shall assume that
The domain of the change in the complex potential is shown in Fig. 3. Using the method of conformal mapping,6 we find the derivative of the complex potential
(2.2) Integrating equality (2.2) along a semicircle of infinitesimal radius with centre at the point t = -f in the plane of the variable t, using the theory of residues6 we find
(2.3) Integrating equality (2.2) in the intervals [− , 0] and [− 1, − p] respectively, we obtain
(2.4) where
In the hydrodynamic interpretation of the problem, the parameter 0 is related to the circulation along a closed contour around the boundary of the cathode by the equality 0 = − /2.7 The parameter 0 determines the amount of fictitious fluid flowing between the anode (the workpiece) and the point P where the flow branches. The value of the parameter = 1 − 0 characterizes the electric current I flowing across the part of the anode boundary AB (s is a curved abscissa).
We now introduce the Zhukovskii function10
(2.5)
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N.M. Minazetdinov / Journal of Applied Mathematics and Mechanics 76 (2012) 475–481
where V0 = a + b is the value of the fictitious flow velocity at the point B(t = 1) and we represent it in the form of a sum10 (2.6) where * (t) = r* − i* , r* = ln(V* /V0 ) is the Zhukovskii function for the flow according to the given scheme with the condition V* /V0 on the ¯ t . On the boundary of anode boundary AB, while (t) is a function that is analytic in the domain Gt and continuous in the closed domain G the domain Gt , the functions (t) and * (t) satisfy the conditions
(2.7) Using Chaplygin’s method of singular
points,10
we find
(2.8) Taking account of equality (2.6) and boundary conditions (2.7), we find the non-linear boundary-value problem (2.9) (2.10) for the function (t). Here,
By virtue of condition (2.10), the function (t) is expanded in a power series with real coefficients (summation is carried out from k = 1 to =∞ everywhere) (2.11) We now introduce the function
From formulae (2.2), (2.5).(2.6), (2.8) and (2.11) we will have the equality
(2.12) Integrating equality (2.12) along a semicircle of infinitesimal radius with centre at the point t = -f using the theory of residues, we find the distance between the lines AF and BF
(2.13) Integrating equality (2.12) in the intervals [− d, 0] and [− , − d], we respectively find the lengths L1 and L2 of the parts CD and DE
(2.14) The coefficients of expansion (2.11) are determined by the collocation method in such a way that condition (2.9) is satisfied at a finite number of points on the required anode boundary. Since the position of the point P is unknown, there are only three geometric conditions, defined by Eqs (2.13) and (3.14), for determining the parameters d, , f and p for known values of the half-length l of the plates and the distance h between the lines AF and EF. We now introduce an additional condition, giving the value of the parameter 0 and using the second formula from relations (2.4). In the limiting case when the point P coincides with the infinitely distant point F, the amount of fictitious fluid flowing between the boundaries of the cathode tool and a stream line passing through the point P will be minimal, and the value of the function = 2 at the point P will correspond to the smallest value of the parameter 0 . Taking account of the fact that the value of the parameter 0 is less than unity, we obtain the conditions (2.15)
N.M. Minazetdinov / Journal of Applied Mathematics and Mechanics 76 (2012) 475–481
479
Fig. 4.
that are necessary for selecting the values of the parameter 0 . To determine the value of 2 , we will consider the problem for the above-mentioned limiting case. 3. Limiting case When account is taken of the equality f = p, the domain Gt of variation in the parametric variable t = + i␦ is identical to the domain shown in Fig. 2. On the boundary of the domain Gt , the imaginary part of the function W(t) satisfies conditions analogous to (2.1) and its real part takes the constant values:
on the lines AP, BC and EP. The domain of variation in the complex potential W is shown in Fig. 4, its derivative has the form (2.2) when f = p and, as a result of integration in the intervals [− p, − ] and [− , 0] respectively, we obtain
The formulae defining the Zhukovskii function (2.5) remain unchanged and are identical to (2.7) - (2.11). As in the general case, we find the geometric characteristics, including the distance between the lines AF and EF and the lengths of the segments CD and DE using formulae (2.12) - (2.14), where it is necessary to put f = p. In this case, the parameters d, and p are completely determined from the geometric conditions (2.13) and (2.14) for known values of l and h. 4. An auxiliary problem The quantity h, characterizing the distance between the cathode plates, occurs in the list of quantities that are specified for the numerical solution of the problem. When h increases, the mutual effect of the periodically arranged plates decreases, and then, starting from a certain critical value h = h* , each plate of the cathode carries out electrochemical machining independently of the adjacent electrodes. The condition (4.1) therefore has to be taken into account when specifying the value of h. In order to determine h* , we consider the auxiliary problem of electrochemical machining with a single plate electrode of length 2l. The left-hand symmetric part of the interelectrode gap and the scheme of the stream lines of the fictitious flow are shown in Fig. 5. The distance between the lines AF and EF is equal to h* . We separate the required anode boundary into two parts. In the part AB, the fictitious flow velocity varies according to the law (1.1). Along the part AF, the velocity decreases from a constant value V = a at point A to zero at the infinitely distant point F. A smooth separation condition11 is satisfied at point A. The method of solution in the case of electrochemical machining with a symmetric triangular shaped cathode has been presented earlier.1 The domain Gt of variation in the parametric variable t = + i␦, taking account of the fact that the branch point P is missing in this scheme, is identical to the domain shown in Fig. 2. The imaginary part of the function W(t) takes constant values on the anode and cathode boundaries and its real part is constant on the lines of symmetry BC and EF:
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N.M. Minazetdinov / Journal of Applied Mathematics and Mechanics 76 (2012) 475–481
Fig. 5.
The domain of variation of the complex potential is a rectangle with sides equal to unity and 3 . The derivative of the complex potential has the form (2.2), where now
The parameter 3 and the function (t) are determined using the formulae
The conditions and formulae determining the function (t) remain unchanged and are identical to relations (2.9) - (2.11). The condition for smooth separation at the point A has the form1 (4.2) The geometric characteristics of the flow are determined from the parametric relation
whence we find the distance h* and the lengths L1 and L2 of the segments CD and DE respectively,
(4.3)
(4.4) In this case, the parameters d, and p are determined from the smooth separation condition (4.1) and the geometric conditions (4.3) for a known value of the half-length of the plate. 5. Results of calculations In order to solve the auxiliary problem, the dimensionless geometric quantity l (the half-length of the plate), the coefficients a0 and a1 , characterizing the electrolyte properties, and the characteristic current density i0 are given. The auxiliary problem was solved for (5.1) As a result, we have
N.M. Minazetdinov / Journal of Applied Mathematics and Mechanics 76 (2012) 475–481
481
Fig. 6.
The anode boundary is the upper dashed curve in Fig. 6. Calculations for the limiting case were then carried out for the same values of the parameters (5.1) and a distance h = 0.9 that satisfies condition (4.1). The results are as follows:
The plot of the anode boundary is the lower dashed curve in Fig. 6. The basic problem is solved taking account of condition (2.15). Calculations were carried out for three values of the parameter 0 = 0.70, 0.80, 0.85 and with the same values of the specified parameters as in the preceding case. The results of the calculation of the parameters 0 , 1 and , the ordinates yA and yP of the points A and P, and of the distance L3 = |yA − yP | are given on the right-hand side of Fig. 6. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
Minazetdinov NM. A hydrodynamic interpretation of a problem in the theory of the dimensional electrochemical machining of metals. J Appl Math Mech 2009;73(1):41–7. Minazetdinov NM. A scheme for the electrochemical machining of metals by a cathode tool with a curvilinear part of the boundary. J Appl Math Mech 2009;73(5):592–8. Davydov AD, Kozak Ye. High-Speed Electrochemical Shaping. Moscow: Nauka; 1990. Kotlyar LM, Minazetdinov NM. Modelling of the Process of the Electrochemical Machining of a Metal for the Technological Preparation of the Process on Machine Tools with Computer Numerical Control. Moscow: Akademia; 2005. Sedykin FV, Orlov BP, Matasov VF. Investigation of the anode current efficiency in electrochemical machining using a constant and pulsed voltage potential. Tekhnologiya Mashinostroyeniya Tula 1975;39:3–10. Lavrent’ev MA, Shabat BV. Methods of the Theory of Functions of a Complex Variable. Moscow: Nauka; 1987. Karimov AKh, Klokov VV, Filatov YeI. Methods of Calculating Electrochemical Shaping. Kazan: Izd. Kazan Univ.; 1990. Klokov VV, Kosterin AV, Nuzhin MT. On The use of inverse boundary-value problems in the theory of dimensional electrochemical machining. Trudy Seminaar po Krayevym Zadacham Kazan 1972;9:132–40. Tatur TA. Foundations of Electromagnetic Field Theory: A Reference Textbook Electrotechnical Specialties in Universities. Moscow: Vyssh. Shkola; 1989. Gurevich MI. The Theory of Jets in an Ideal Fluid. Oxford: Pergamon; 1966. Birkhoff G, Zarantonello E. Jets, Wakes and Cavities. New York: Acad. Press; 1957.
Translated by E.L.S.
Contents lists available at SciVerse ScienceDirect
Journal of Applied Mathematics and Mechanics journal homepage: www.elsevier.com/locate/jappmathmech
The two-dimensional problem of the electrochemical machining of metals with a periodic cathode tool夽 N.M. Minazetdinov Naberezhnye Chelny, Russia
a r t i c l e
i n f o
Article history: Received 16 February 2011
a b s t r a c t The problem of the electrochemical machining of metals with a periodic cathode tool in the form of a lattice of plates is formulated and solved. The hydrodynamic analogue of the initial problem is the problem of a plane-parallel potential circulating flow of an ideal incompressible fluid around plate electrodes. Auxiliary schemes are considered in order to specify the initial data and conditions determining the parameters of the problem. The steady-state shapes of the anode boundaries are found. It is shown that different anode boundaries are obtained due to a change in the electric field characteristics provided that the properties of the metal and the electrolyte, the geometry of the electrode tool and its rate of advance are identical. © 2012 Elsevier Ltd. All rights reserved.
Two-dimensional electrochemical machining problems with a symmetrical triangular cathode and with a cathode with a curvilinear part of the boundary have been solved earlier1,2 using a hydrodynamic analogy. The solution of a problem, associated with determining the steady-state shape of the surface of a workpiece in to case of machining with a periodic cathode, is found below within the limits of a model of an ideal process.3 This electrode can be used to form the contoured surfaces of heat exchange equipment.
1. Model of the process A model of an ideal process is used as a first approximation in the theoretical analysis of the electrochemical machining of metals. The electric field in the interelectrode gap is assumed to be potential and the potential u of the field is assumed to be a harmonic function. The boundary of the anode (of the machined surface) and the boundary of the cathode tool (of the machining surface) are equipotential field lines. In the steady state, the shape of the machined surface does not change in a moving system of coordinates associated with the cathode. The distribution of the normal derivative of the potential on the steady-state anode boundary has the form2,3
where is the electrical conductivity of the medium, is the electrochemical equivalent of the metal, is the density of the anode material and is the angle between the velocity vector for the advancement of the cathode Vc and the vector na of the outward normal at a given point of the anode boundary. The constants a0 and a1 characterize the electrolyte properties; values found from experimental data5 obtained in machining 5HXM steel have been given for these constants.4 A two-dimensional model of the process is considered. A system of Cartesian coordinates x1 , y1 is introduced, and this system, which is associated with the cathode, moves in the direction of the ordinate axis. Using the prerequisites for a model of an ideal process,3 we will assume that a complex electric field potential exists in the interelectrode gap
夽 Prikl. Mat. Mekh., Vol. 76, No. 4, pp. 658–666, 2012. E-mail address: [email protected] 0021-8928/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jappmathmech.2012.09.015
476
N.M. Minazetdinov / Journal of Applied Mathematics and Mechanics 76 (2012) 475–481
Fig. 1.
where u(z1 ) is the field potential and (z1 ) is the stream function.6 We now introduce the current density i0 , that depends on the velocity at which the cathode advances, the properties of the machined metal and the length H,7 as well as the dimensionless variables
Here, ua and uc are the values of the field potential on the anode and cathode boundaries. We now change to the dimensionless complex potential
In the interelectrode gap region, the function is harmonic and, on the anode and cathode boundaries, it satisfies the conditions1
The electric field in the interelectrode gap is modelled by a fictitious plane-parallel potential flow of an ideal incompressible fluid.8 The velocity of the fictitious flow varies according to the law1 (1.1) where is the argument of the velocity vector. 2. Formulation and solution of the basic problem A cross-section of the interelectrode gap is shown schematically in Fig. 1. The cathode tool is formed by an infinite assembly of identical plates of length 2l lying in a plane, each of which is obtained from the adjacent plate by a parallel translation through the same distance 2 h along the abscissa. The thickness of the plates is negligibly small. We will confine ourselves to considering the region located between the lines of symmetry AF and BF. In this region, CDE is the cathode boundary and the line AB is the required anode boundary. On the line AF there is a branch point P of the equipotential electric field line at which the voltage potential is equal to zero9 and F is an infinitely distant point. The problem of determining the boundary of the flow AB with the given law of variation in the velocity (1.1) is the hydrodynamic analogue. The flow is created by a system of continuously distributed sources along the lines AP and EF and sinks on the lines BC and FP. The velocity of the fictitious fluid flow is equal to zero at the point P (Fig. 1). The flow domain in the physical plane z (Fig. 1) is denoted by Gz and the semicircle of unit radius in the parametric plane t = + i␦ (Fig. 2) is denoted by Gt . We shall seek a conformal mapping z(t) of the domain Gt onto Gz . The correspondence of the points is clear from Figs 1 and 2.
Fig. 2.
N.M. Minazetdinov / Journal of Applied Mathematics and Mechanics 76 (2012) 475–481
477
Fig. 3.
The complex potential W(t) = (t) + i(t) satisfies the conditions
(2.1) where is the polar angle in the parametric plane t. The function (t) takes constant values on the lines EF, BC, AP and PF. Without loss of generality, we shall assume that
The domain of the change in the complex potential is shown in Fig. 3. Using the method of conformal mapping,6 we find the derivative of the complex potential
(2.2) Integrating equality (2.2) along a semicircle of infinitesimal radius with centre at the point t = -f in the plane of the variable t, using the theory of residues6 we find
(2.3) Integrating equality (2.2) in the intervals [− , 0] and [− 1, − p] respectively, we obtain
(2.4) where
In the hydrodynamic interpretation of the problem, the parameter 0 is related to the circulation along a closed contour around the boundary of the cathode by the equality 0 = − /2.7 The parameter 0 determines the amount of fictitious fluid flowing between the anode (the workpiece) and the point P where the flow branches. The value of the parameter = 1 − 0 characterizes the electric current I flowing across the part of the anode boundary AB (s is a curved abscissa).
We now introduce the Zhukovskii function10
(2.5)
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N.M. Minazetdinov / Journal of Applied Mathematics and Mechanics 76 (2012) 475–481
where V0 = a + b is the value of the fictitious flow velocity at the point B(t = 1) and we represent it in the form of a sum10 (2.6) where * (t) = r* − i* , r* = ln(V* /V0 ) is the Zhukovskii function for the flow according to the given scheme with the condition V* /V0 on the ¯ t . On the boundary of anode boundary AB, while (t) is a function that is analytic in the domain Gt and continuous in the closed domain G the domain Gt , the functions (t) and * (t) satisfy the conditions
(2.7) Using Chaplygin’s method of singular
points,10
we find
(2.8) Taking account of equality (2.6) and boundary conditions (2.7), we find the non-linear boundary-value problem (2.9) (2.10) for the function (t). Here,
By virtue of condition (2.10), the function (t) is expanded in a power series with real coefficients (summation is carried out from k = 1 to =∞ everywhere) (2.11) We now introduce the function
From formulae (2.2), (2.5).(2.6), (2.8) and (2.11) we will have the equality
(2.12) Integrating equality (2.12) along a semicircle of infinitesimal radius with centre at the point t = -f using the theory of residues, we find the distance between the lines AF and BF
(2.13) Integrating equality (2.12) in the intervals [− d, 0] and [− , − d], we respectively find the lengths L1 and L2 of the parts CD and DE
(2.14) The coefficients of expansion (2.11) are determined by the collocation method in such a way that condition (2.9) is satisfied at a finite number of points on the required anode boundary. Since the position of the point P is unknown, there are only three geometric conditions, defined by Eqs (2.13) and (3.14), for determining the parameters d, , f and p for known values of the half-length l of the plates and the distance h between the lines AF and EF. We now introduce an additional condition, giving the value of the parameter 0 and using the second formula from relations (2.4). In the limiting case when the point P coincides with the infinitely distant point F, the amount of fictitious fluid flowing between the boundaries of the cathode tool and a stream line passing through the point P will be minimal, and the value of the function = 2 at the point P will correspond to the smallest value of the parameter 0 . Taking account of the fact that the value of the parameter 0 is less than unity, we obtain the conditions (2.15)
N.M. Minazetdinov / Journal of Applied Mathematics and Mechanics 76 (2012) 475–481
479
Fig. 4.
that are necessary for selecting the values of the parameter 0 . To determine the value of 2 , we will consider the problem for the above-mentioned limiting case. 3. Limiting case When account is taken of the equality f = p, the domain Gt of variation in the parametric variable t = + i␦ is identical to the domain shown in Fig. 2. On the boundary of the domain Gt , the imaginary part of the function W(t) satisfies conditions analogous to (2.1) and its real part takes the constant values:
on the lines AP, BC and EP. The domain of variation in the complex potential W is shown in Fig. 4, its derivative has the form (2.2) when f = p and, as a result of integration in the intervals [− p, − ] and [− , 0] respectively, we obtain
The formulae defining the Zhukovskii function (2.5) remain unchanged and are identical to (2.7) - (2.11). As in the general case, we find the geometric characteristics, including the distance between the lines AF and EF and the lengths of the segments CD and DE using formulae (2.12) - (2.14), where it is necessary to put f = p. In this case, the parameters d, and p are completely determined from the geometric conditions (2.13) and (2.14) for known values of l and h. 4. An auxiliary problem The quantity h, characterizing the distance between the cathode plates, occurs in the list of quantities that are specified for the numerical solution of the problem. When h increases, the mutual effect of the periodically arranged plates decreases, and then, starting from a certain critical value h = h* , each plate of the cathode carries out electrochemical machining independently of the adjacent electrodes. The condition (4.1) therefore has to be taken into account when specifying the value of h. In order to determine h* , we consider the auxiliary problem of electrochemical machining with a single plate electrode of length 2l. The left-hand symmetric part of the interelectrode gap and the scheme of the stream lines of the fictitious flow are shown in Fig. 5. The distance between the lines AF and EF is equal to h* . We separate the required anode boundary into two parts. In the part AB, the fictitious flow velocity varies according to the law (1.1). Along the part AF, the velocity decreases from a constant value V = a at point A to zero at the infinitely distant point F. A smooth separation condition11 is satisfied at point A. The method of solution in the case of electrochemical machining with a symmetric triangular shaped cathode has been presented earlier.1 The domain Gt of variation in the parametric variable t = + i␦, taking account of the fact that the branch point P is missing in this scheme, is identical to the domain shown in Fig. 2. The imaginary part of the function W(t) takes constant values on the anode and cathode boundaries and its real part is constant on the lines of symmetry BC and EF:
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N.M. Minazetdinov / Journal of Applied Mathematics and Mechanics 76 (2012) 475–481
Fig. 5.
The domain of variation of the complex potential is a rectangle with sides equal to unity and 3 . The derivative of the complex potential has the form (2.2), where now
The parameter 3 and the function (t) are determined using the formulae
The conditions and formulae determining the function (t) remain unchanged and are identical to relations (2.9) - (2.11). The condition for smooth separation at the point A has the form1 (4.2) The geometric characteristics of the flow are determined from the parametric relation
whence we find the distance h* and the lengths L1 and L2 of the segments CD and DE respectively,
(4.3)
(4.4) In this case, the parameters d, and p are determined from the smooth separation condition (4.1) and the geometric conditions (4.3) for a known value of the half-length of the plate. 5. Results of calculations In order to solve the auxiliary problem, the dimensionless geometric quantity l (the half-length of the plate), the coefficients a0 and a1 , characterizing the electrolyte properties, and the characteristic current density i0 are given. The auxiliary problem was solved for (5.1) As a result, we have
N.M. Minazetdinov / Journal of Applied Mathematics and Mechanics 76 (2012) 475–481
481
Fig. 6.
The anode boundary is the upper dashed curve in Fig. 6. Calculations for the limiting case were then carried out for the same values of the parameters (5.1) and a distance h = 0.9 that satisfies condition (4.1). The results are as follows:
The plot of the anode boundary is the lower dashed curve in Fig. 6. The basic problem is solved taking account of condition (2.15). Calculations were carried out for three values of the parameter 0 = 0.70, 0.80, 0.85 and with the same values of the specified parameters as in the preceding case. The results of the calculation of the parameters 0 , 1 and , the ordinates yA and yP of the points A and P, and of the distance L3 = |yA − yP | are given on the right-hand side of Fig. 6. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
Minazetdinov NM. A hydrodynamic interpretation of a problem in the theory of the dimensional electrochemical machining of metals. J Appl Math Mech 2009;73(1):41–7. Minazetdinov NM. A scheme for the electrochemical machining of metals by a cathode tool with a curvilinear part of the boundary. J Appl Math Mech 2009;73(5):592–8. Davydov AD, Kozak Ye. High-Speed Electrochemical Shaping. Moscow: Nauka; 1990. Kotlyar LM, Minazetdinov NM. Modelling of the Process of the Electrochemical Machining of a Metal for the Technological Preparation of the Process on Machine Tools with Computer Numerical Control. Moscow: Akademia; 2005. Sedykin FV, Orlov BP, Matasov VF. Investigation of the anode current efficiency in electrochemical machining using a constant and pulsed voltage potential. Tekhnologiya Mashinostroyeniya Tula 1975;39:3–10. Lavrent’ev MA, Shabat BV. Methods of the Theory of Functions of a Complex Variable. Moscow: Nauka; 1987. Karimov AKh, Klokov VV, Filatov YeI. Methods of Calculating Electrochemical Shaping. Kazan: Izd. Kazan Univ.; 1990. Klokov VV, Kosterin AV, Nuzhin MT. On The use of inverse boundary-value problems in the theory of dimensional electrochemical machining. Trudy Seminaar po Krayevym Zadacham Kazan 1972;9:132–40. Tatur TA. Foundations of Electromagnetic Field Theory: A Reference Textbook Electrotechnical Specialties in Universities. Moscow: Vyssh. Shkola; 1989. Gurevich MI. The Theory of Jets in an Ideal Fluid. Oxford: Pergamon; 1966. Birkhoff G, Zarantonello E. Jets, Wakes and Cavities. New York: Acad. Press; 1957.
Translated by E.L.S.